Fractional Inequalities In Banach Algebras 3031051475, 9783031051470

Citation preview

Studies in Systems, Decision and Control 441

George A. Anastassiou

Fractional Inequalities In Banach Algebras

Studies in Systems, Decision and Control Volume 441

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the worldwide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

More information about this series at https://link.springer.com/bookseries/13304

George A. Anastassiou

Fractional Inequalities In Banach Algebras

George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN, USA

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-031-05147-0 ISBN 978-3-031-05148-7 (eBook) https://doi.org/10.1007/978-3-031-05148-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to Covid-19 vaccine creators

Preface

Fractional Calculus and its long history started in 1695 by L’Hospital and Leibnitz. Over the years, many different kinds of it emerged according to the needs of science given by many famous mathematicians. In the last 50 years, fractional calculus due to its wide applications to many applied sciences has gained extreme popularity. By employing regular and iterated (sequential) generalized Caputo and Canavati fractional left and right vectorial Taylor formulae, we establish a great variety of mixed regular and iterated generalized fractional inequalities. In particular, we derive generalized Caputo Fractional Ostrowski and Grüss type inequalities involving several Banach algebra valued functions, the regular and iterated (sequential) versions. Also, we establish generalized Canavati fractional Ostrowski, Opial, Grüss and Hilbert-Pachpatte type inequalities for multiple Banach algebra valued functions. The same kind of the above types of inequalities are studied at univariate and multivariate settings for ordinary vectorial derivatives and vectorial partial derivatives, respectively. The counterpart studies continue, extensively, by using the p-Schatten norms over the von Neumann-Schatten classes (∗-ideals) B p (H ), p >_1, and we produce all the analogous inequalities. We provide a great variety of applications per chapter. So in this monograph, all that is presented is the original work by the author given at a very general level to cover a maximum number of different fractional calculus applications. As a result, this monograph is the natural and expected evolution of the author’s recent research work put in a book form for the first time. The presented approaches are original, and chapters are self-contained and can be read independently. This monograph is suitable to be used in related graduate classes and research projects. The motivation to write this monograph came by the following: various issues related to the modeling and analysis of ordinary and fractional order systems have gained increased popularity, as witnesses by many books and volumes in Springer’s program: http://www.springer.com/gp/search?query=fractional&submit=Prze%C5%9Blij

vii

viii

Preface

and the purpose of our book is to capture at a very general level a deeper formal analysis on some issues that are relevant to many areas, for instance: decision making, complex processes, systems modeling and control, and related areas. The above are deeply embedded in the fields of mathematics, engineering, computer science, physics, economics, social and life sciences. Next is listed the author’s recent monographs in Fractional Analysis and applications: • George Anastassiou, “Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations”, Springer, Studies in Computational Intelligence 734, Heidelberg, New York, 2018. • George Anastassiou, “Nonlinearity: Ordinary and Fractional Approximations by Sublinear and Max-product Operators”, Springer, Studies in Systems, Decision and Control 147, Heidelberg, New York, 2018. • George Anastassiou, “Ordinary and Fractional approximation by non-additive integrals: Choquet, Shilkret and Sugeno integral approximators”, Springer, Studies in Systems, Decision and Control 190, Heidelberg, New York, 2019. • George Anastassiou, “Intelligent Analysis: Fractional Inequalities and Approximations Expanded”, Springer, Studies in Computational Intelligence 886, Heidelberg, New York, 2020. • George Anastassiou, “Generalized Fractional Calculus: New Advancements and Applications”, Springer, Studies in Systems, Decision and Control 305, Heidelberg, New York, 2021. • George Anastassiou, “Constructive Fractional Analysis with Applications”, Springer, Studies in Systems, Decision and Control 362, Heidelberg, New York, 2021. • George Anastassiou, “Unification of Fractional Calculi with Applications”, Springer, Studies in Systems, Decision and Control 398, Heidelberg, New York, 2022. The complete list of presented topics here follows: • Generalized Fractional Ostrowski and Grüss type inequalities involving several Banach algebra valued functions, • Sequential Generalized Fractional Ostrowski and Grüss type inequalities for several Banach algebra valued functions, • Generalized Canavati Fractional Ostrowski, Opial and Grüss type inequalities for Banach algebra valued functions, • Generalized Canavati Fractional Hilbert-Pachpatte type inequalities for Banach algebra valued functions, • Generalized Ostrowski, Opial and Hilbert-Pachpatte type inequalities for Banach algebra valued functions involving integer vectorial derivatives, • Multivariate Ostrowski type inequalities for several Banach algebra valued functions, • p-Schatten norm generalized fractional Ostrowski and Grüss type inequalities involving several functions,

Preface

ix

• p-Schatten norm sequential generalized fractional Ostrowski and Grüss type inequalities for several functions, • p-Schatten norm generalized Canavati fractional Ostrowski, Opial and Grüss type inequalities involving several functions, • p-Schatten norm Generalized Canavati Fractional Hilbert-Pachpatte type inequalities for von Neumann-Schatten class B p (H ) valued functions, • p-Schatten norm Generalized Ostrowski, Opial and Hilbert Pachpatte type inequalities for von Neumann-Schatten class B p (H ) valued functions with integer vectorial derivatives, • p-Schatten norm Multivariate Ostrowski type inequalities for several NeumannSchatten class B p (H ) valued functions. The book’s results are expected to find applications in many areas of pure and applied mathematics, especially in fractional inequalities and fractional differential equations. Other possible applications can be in applied sciences like geophysics, physics, chemistry, economics, engineering, etc. All in all, what is presented here is a valuable tool for a large range of applications. Therefore, this monograph is suitable for researchers, graduate students, practitioners and seminars of the above disciplines, and also to be in all science and engineering libraries. The preparation of the book took place during 2021–22 at the University of Memphis, during the author’s home stay during the Covid-19 and Omicron outbreak keeping him alive, happy and in control! The author likes to thank Prof. Alina Lupas of University of Oradea, Romania, for checking and reading the manuscript. Memphis, TN, USA March 2022

George A. Anastassiou

Contents

1

2

3

Generalized Fractional Ostrowski and Grüss Inequalities with Multiple Banach Algebra Valued Functions . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Vectorial Background Fractional Calculus . . . . . . . . . . . . . . . . . . . 1.3 Banach Algebras Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iterated Generalized Fractional Ostrowski and Grüss Inequalities with Multiple Banach Algebra Valued Functions . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Vectorial Sequential Generalized Fractional Calculus Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Banach Algebras Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Canavati Fractional Ostrowski, Opial and Grüss Inequalities with Multiple Banach Algebra Valued Functions . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Background on Vectorial Generalized Canavati Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Banach Algebras Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 6 17 19 21 21 22 26 27 42 44 45 45 46 50 51 61 62 66

xi

xii

4

5

6

7

8

Contents

Generalized Canavati Fractional Hilbert–Pachpatte Inequalities for Banach Algebra Valued Functions . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Background on Vectorial Generalized Canavati Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Banach Algebras Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 70 73 75 79 79

Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities for Banach Algebra Valued Functions Involving Integer Vectorial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 About Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82 83 84 96 98

Multivariate Ostrowski Inequalities for Several Banach Algebra Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 About Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Vector Analysis Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 100 101 103 110

p-Schatten Norm Generalized Fractional Ostrowski and Grüss Inequalities for Multiple Functions . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Vectorial Background Fractional Calculus . . . . . . . . . . . . . . . . . . . 7.3 Banach Algebras Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 p-Schatten Norms Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 115 117 118 120 141 146

p-Schatten Norm Iterated Generalized Fractional Ostrowski and Grüss Inequalities for Multiple Functions . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Vectorial Sequential Generalized Fractional Calculus Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Banach Algebras Basic Background . . . . . . . . . . . . . . . . . . . . . . . . 8.4 p-Schatten Norms Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 151 155 156 158 190 193

Contents

9

p-Schatten Norm Generalized Canavati Fractional Ostrowski, Opial and Grüss Inequalities for Multiple Functions . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Background on Vectorial Generalized Canavati Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Basic Banach Algebras Background . . . . . . . . . . . . . . . . . . . . . . . . 9.4 p-Schatten Norms Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

195 195 197 201 202 204 229 235

10 γ -Schatten Norm Generalized Canavati Fractional Hilbert– Pachpatte Inequalities with von Neumann–Schatten Class Bγ (H) Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Background on Vectorial Generalized Canavati Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Basic Banach Algebras Background . . . . . . . . . . . . . . . . . . . . . . . . 10.4 p-Schatten Norms Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239 242 244 246 251 252

11 γ -Schatten Norm Generalized Ostrowski, Opial and Hilbert– Pachpatte Inequalities with von Neumann–Schatten Class Bγ (H) Valued Functions Using Ordinary Vectorial Derivatives . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 About Basic Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 p-Schatten Norms Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 258 258 260 261 282 286

12 γ -Schatten Norm Multivariate Ostrowski Inequalities for Multiple Neumann–Schatten Class Bγ (H) Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 About Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 p-Schatten Norms Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Vector Analysis Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 287 289 290 292 294 305

237 237

13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Chapter 1

Generalized Fractional Ostrowski and Grüss Inequalities with Multiple Banach Algebra Valued Functions

Using generalized Caputo fractional left and right vectorial Taylor formulae we establish mixed fractional Ostrowski and Grüss type inequalities involving several Banach algebra valued functions. The estimates are with respect to all norms · p , 1 ≤ p ≤ ∞. It follows [6].

1.1 Introduction The following results motivate our chapter. Theorem 1.1 (1938, Ostrowski [11]) Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, derivative f  : (a, b) → R is bounded on (a, b),    b) whose   sup    i.e., f ∞ := sup f (t) < +∞. Then t∈(a,b)

   1  b − a

a

b

 2       sup x − a+b 1 2 f  , + f (t) dt − f (x) ≤ − a) (b ∞ 4 (b − a)2

for any x ∈ [a, b]. The constant

1 4

(1.1)

is the best possible.

Ostrowski type inequalities have great applications to integral approximations in Numerical Analysis. ˇ Theorem 1.2 (1882, Cebyšev [7]) Let f, g : [a, b] → R be absolutely continuous   functions with f , g ∈ L ∞ ([a, b]). Then

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_1

1

2

1 Generalized Fractional Ostrowski …

   1  b − a

b

f (x) g (x) d x −

a

≤

1 b−a





b

f (x) d x a

1 b−a

    1 (b − a)2  f  ∞ g  ∞ . 12

 a

b

  g (x) d x  (1.2)

The above integrals are assumed to exist. The related Grüss type inequalities have many applications to Probability Theory. We presented also ([4], Chaps. 8, 9) mixed fractional Ostrowski and Grüss-Cebysev type inequalities for several functions, acting to all possible directions. The estimates involve the left and right Caputo fractional derivatives. See also the monographs written by the author [2], Chaps. 24–26 and [3], Chaps. 2–6. In this chapter we generalize [4], Chaps. 8, 9 for several Banach algebra valued functions. Now our left and right Caputo fractional derivatives are for Banach space valued functions and our integrals are of Bochner type. Several applications finish this chapter. Inspiration came also from [8, 9].

1.2 Vectorial Background Fractional Calculus Here all come from [5]. We need Definition 1.3 ([5], p. 106) Let α > 0, α = n, · the ceiling of the number. Let f ∈ C n ([a, b] , X ), where [a, b] ⊂ R, and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]). We define the left generalized g-fractional derivative X -valued of f of order α as follows:  x (n)   α  1 Da+;g f (x) := (g (x) − g (t))n−α−1 g  (t) f ◦ g −1 (g (t)) dt,  (n − α) a (1.3) ∀ x ∈ [a, b], where  is the gamma function. The last integral is of Bochner type ([1], pp. 422–428; [10]).  α f ∈ C ([a, b] , X ). If α ∈ / N, by Theorem 4.10 ([5], p. 98), we have that Da+;g We set  n n f (x) := f ◦ g −1 ◦ g (x) ∈ C ([a, b] , X ) , n ∈ N, (1.4) Da+;g 0 f (x) = f (x) , ∀ x ∈ [a, b] . Da+;g

When g = id, then

α α α f = Da+;id f = D∗a f, Da+;g

(1.5)

1.2 Vectorial Background Fractional Calculus

3

the usual left X -valued Caputo fractional derivative, see [5], Chap. 1. We also need Definition 1.4 ([5], p. 107) Let α > 0, α = n, · the ceiling of the number. Let f ∈ C n ([a, b] , X ), where [a, b] ⊂ R, and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]). We define the right generalized g-fractional derivative X -valued of f of order α as follows:  b (n)    α (−1)n Db−;g f (x) := (g (t) − g (x))n−α−1 g  (t) f ◦ g −1 (g (t)) dt,  (n − α) x (1.6) ∀ x ∈ [a, b]. The last integral is of Bochner type.  α f ∈ C ([a, b] , X ). If α ∈ / N, by Theorem 4.11 ([5], p. 101), we have that Db−;g We set  n n f (x) := (−1)n f ◦ g −1 ◦ g (x) ∈ C ([a, b] , X ) , n ∈ N, (1.7) Db−;g 0 f (x) := f (x) , ∀ x ∈ [a, b] . Db−;g

When g = id, then

α α α f (x) = Db−;id f (x) = Db− f, Db−;g

(1.8)

the usual right X -valued Caputo fractional derivative, see [5], Chap. 2. We mention the following generalized fractional Taylor formulae with integral remainders over Banach spaces. Theorem 1.5 ([5], p. 107) Let α > 0, n = α , and f ∈ C n ([a, b] , X ), where [a, b] ⊂ R and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]), a ≤ x ≤ b . Then f (x) = f (a) +

n−1

(g (x) − g (a))i 

i!

i=1

1  (α)



x

a

f (a) + 

(i)

(g (a)) +

  α f (t) dt = (g (x) − g (t))α−1 g  (t) Da+;g

n−1

(g (x) − g (a))i  i=1

1  (α)

f ◦ g −1

g(x) g(a)

i! (g (x) − z)α−1



f ◦ g −1

(i)

(g (a)) +

  α Da+;g f ◦ g −1 (z) dz.

(1.9)

4

1 Generalized Fractional Ostrowski …

We also mention Theorem 1.6 ([5], p. 108) Let α > 0, n = α , and f ∈ C n ([a, b] , X ), where [a, b] ⊂ R and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]), a ≤ x ≤ b. Then f (x) = f (b) +

n−1

(g (x) − g (b))i 

i!

i=1

1  (α)



b x

f (b) + 

(i)

(g (b)) +

  α f (t) dt = (g (t) − g (x))α−1 g  (t) Db−;g

n−1

(g (x) − g (b))i  i=1

1  (α)

f ◦ g −1

g(b) g(x)

i!

(z − g (x))α−1



f ◦ g −1

(i)

(g (b)) +

(1.10)

  α f ◦ g −1 (z) dz. Db−;g

If 0 < α ≤ 1, then the sums in (1.9), (1.10) disappear.     α , Also in (1.9), (1.10), we have that  Da+;g f ◦ g −1  ∞,[g(a),g(b)]     α  < ∞.  Db−;g f ◦ g −1  ∞,[g(a),g(b)]

1.3 Banach Algebras Background All here come from [12]. We need Definition 1.7 ([12], p. 245) A complex algebra is a vector space A over the complex filed C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(1.11)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(1.12)

α (x y) = (αx) y = x (αy) ,

(1.13)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality

1.3 Banach Algebras Background

5

x y ≤ x y (x ∈ A, y ∈ A)

(1.14)

and if A contains a unit element e such that xe = ex = x (x ∈ A)

(1.15)

e = 1,

(1.16)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 1.8 Commutativity of A will be explicited stated when needed. There exists at most one e ∈ A that satisfies (1.15). Inequality (1.14) makes multiplication to be continuous, more precisely left and right continuous, see [12], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [12], p. 247–248, Sect. 10.3. We also make Remark 1.9 Next we mention about integration of A-valued functions, see [12], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some compact  Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [12], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   f dμ = x f ( p) dμ ( p) (1.17) x Q

and



Q

 f dμ x = Q

f ( p) x dμ ( p) .

(1.18)

Q

The Bochner integrals we will involve in our chapter follow (1.17) and (1.18). Also, let f ∈ C ([a, b] , X ), where [a, b] ⊂ R, (X, ·) is a Banach space. By [5], p. 3, f is Bochner integrable.

6

1 Generalized Fractional Ostrowski …

1.4 Main Results We start with mixed generalized fractional Ostrowski type inequalities for several functions over a Banach algebra. A uniform estimate follows. Theorem 1.10 Let ( A, ·) be a Banach algebra, x0 ∈ [a, b] ⊂ R, α > 0 , n = α , f i ∈ C n ([a, b] , A), i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing, such (k)  that g −1 ∈ C n ([g (a) , g (b)]), with f i ◦ g −1 (g (x0 )) = 0, k = 1, ..., n − 1; i = 1, ..., r. Denote by θ ( f 1 , ..., fr ) (x0 ) := ⎡ r

i=1

⎢ ⎢ ⎣

 a

⎞

⎛ b

⎜ ⎜ ⎝

r  j=1 j=i

⎞ ⎞ ⎤ ⎛ ⎛  b  r ⎟ ⎟ ⎟ ⎥ ⎜ ⎜ ⎟ ⎥ ⎜ ⎜ f j (x)⎟ f j (x)⎟ ⎠ f i (x) d x − ⎝ a ⎝ ⎠ d x ⎠ f i (x0 )⎦ .

(1.19)

j=1 j=i

Then θ ( f 1 , ..., fr ) (x0 ) ≤

r 

 α  −1  1  D  x0 −;g f i ◦ g ∞,[g(a),g(x0 )]  (α + 1) i=1

⎛  ⎜ (g (x0 ) − g (a))α ⎜ ⎝

a

⎡   ⎢  α −1  ⎢ ⎣ Dx0 +;g f i ◦ g 

⎞

⎛ x0

⎞⎤

r ⎜  ⎟ ⎟⎥ ⎜  f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦ ⎝

(1.20)

j=1 j=i

⎛  ⎜ (g (b) − g (x0 ))α ⎜ ⎝ ∞,[g(x0 ),g(b)]

⎞

⎛ b x0

⎞⎤⎤

r ⎟ ⎟⎥⎥  ⎜  f j (x)⎟ d x ⎟⎥⎥ . ⎜ ⎠ ⎠⎦⎦ ⎝ j=1 j=i

(k)  Proof Since f i ◦ g −1 (g (x0 )) = 0, k = 1, ..., n − 1; i = 1, ..., r , we have by Theorem 1.5 that f i (x) − f i (x0 ) =

1  (α)



g(x)

g(x0 )

(g (x) − z)α−1



  Dxα0 +;g f i ◦ g −1 (z) dz, (1.21)

∀ x ∈ [x0 , b] , and by Theorem 1.6 that 1 f i (x) − f i (x0 ) =  (α) ∀ x ∈ [a, x0 ] ;



g(x0 )

g(x)

(z − g (x))α−1



  Dxα0 −;g f i ◦ g −1 (z) dz, (1.22)

1.4 Main Results

7

for all i = 1, ..., r.



Left multiplying (1.21) and (1.22) with ⎛

⎞

f j (x) we get, respectively,

r j=1 j=i

⎞

⎛

r r ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ f i (x) − ⎜ f f j (x)⎟ (x) j ⎝ ⎠ ⎠ f i (x0 ) = ⎝ j=1 j=i



f j (x) 

r j=1 j=i

 (α) ∀ x ∈ [x0 , b] , and

g(x) g(x0 )

(g (x) − z)α−1

⎞

⎛



  Dxα0 +;g f i ◦ g −1 (z) dz,

⎞

⎛

r r ⎟ ⎟ ⎜ ⎜ ⎟ f i (x) − ⎜ ⎜ f f j (x)⎟ (x) j ⎠ ⎠ f i (x0 ) = ⎝ ⎝ j=1 j=i

r j=1 j=i

(1.23)

j=1 j=i

(1.24)

j=1 j=i

f j (x) 

 (α)

g(x0 )

g(x)

(z − g (x))α−1



  Dxα0 −;g f i ◦ g −1 (z) dz,

∀ x ∈ [a, x0 ] ; for all i = 1, ..., r. Adding (1.23) and (1.24) as separate groups, we obtain ⎞

⎛ r

i=1

⎜ ⎜ ⎝

r  j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x) −

⎞

⎛ r

i=1

⎜ ⎜ ⎝

r  j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

⎞

⎛

 r r  ⎟ g(x)    1 ⎜ ⎟ ⎜ f j (x)⎠ (g (x) − z)α−1 Dxα0 +;g f i ◦ g −1 (z) dz, ⎝  (α) i=1 j=1 g(x0 ) j=i

∀ x ∈ [x0 , b] , and

⎞

⎛

⎞

⎛

r r r r

⎟ ⎟ ⎜ ⎜ ⎟ f i (x) − ⎜ ⎜ f f j (x)⎟ (x) j ⎠ ⎠ f i (x0 ) = ⎝ ⎝ i=1

j=1 j=i

i=1

j=1 j=i

(1.25)

8

1 Generalized Fractional Ostrowski …

⎞

⎛ r

1  (α)

⎜ ⎜ ⎝

i=1

r  j=1 j=i



⎟ f j (x)⎟ ⎠

g(x0 ) g(x)

(z − g (x))α−1



  Dxα0 −;g f i ◦ g −1 (z) dz, (1.26)

∀ x ∈ [a, x0 ] . Next we integrate (1.25) and (1.26) with respect to x ∈ [a, b]. We have r 

b x0

i=1

⎞

⎛

⎛

r r

⎟ ⎜ ⎜ ⎟ f i (x) d x − ⎜ ⎜ f (x) j ⎠ ⎝ ⎝ j=1 j=i



b x0

i=1

⎞

⎛

⎞

r ⎟ ⎟ ⎜ ⎟ ⎜ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) = ⎝ j=1 j=i

⎡ ⎛ ⎤ ⎞    b  r r  g(x)

⎢ ⎜ ⎥ ⎟ 1 ⎢ ⎜ f j (x)⎟ (g (x) − z)α−1 Dxα0 +;g f i ◦ g −1 (z) dz d x ⎥ ⎣ ⎝ ⎦, ⎠  (α) x0 g(x 0 ) i=1

j=1 j=i

(1.27) and r 

a

i=1

⎛

 r r x 0 ⎜

⎟ ⎜ ⎟ ⎜ ⎜ f j (x)⎠ f i (x) d x − ⎝ ⎝ j=1 j=i

⎡ 1  (α)

⎞

⎛

r

i=1

⎢ ⎢ ⎣

⎛



x0

a

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎟ f j (x)⎟ ⎠



g(x 0 ) g(x)

⎞

r ⎟ ⎟ ⎜ ⎟ ⎜ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) = ⎝

x0

a

i=1

⎞

⎛

j=1 j=i

(z − g (x))α−1



⎤



⎥ Dxα0 −;g f i ◦ g −1 (z) dz d x ⎥ ⎦.

(1.28) Finally, adding (1.27) and (1.28) we obtain the useful identity θ ( f 1 , ..., fr ) (x0 ) := ⎡

 r

⎢ ⎢ ⎣ i=1

a

⎞

⎛

⎛  r b ⎜ ⎟ ⎜ ⎟ f i (x) d x − ⎜ ⎜ f (x) j ⎠ ⎝ ⎝

a

j=1 j=i

⎞

⎛ b

⎞

⎤

r ⎟ ⎟ ⎥ ⎜ ⎟ ⎥ ⎜ f j (x)⎟ ⎠ d x ⎠ f i (x0 )⎦ = ⎝ j=1 j=i

⎡⎡ ⎛ ⎤ ⎞    x0  r r  g(x )

0 ⎢ ⎢ ⎜ ⎥ ⎟ 1 ⎢⎢ ⎜ f j (x)⎟ (z − g (x))α−1 Dxα0 −;g f i ◦ g −1 (z) dz d x ⎥ ⎣ ⎣ ⎝ ⎦ ⎠  (α) a g(x) i=1

j=1 j=i

1.4 Main Results

⎡ ⎢ +⎢ ⎣

9

⎞

⎛



b x0

⎜ ⎜ ⎝

r  j=1 j=i

⎟ f j (x)⎟ ⎠



g(x) g(x0 )

(g (x) − z)α−1





⎤⎤

⎥⎥   ⎥ Dxα0 +;g f i ◦ g −1 (z) dz d x ⎥ ⎦⎦ . (1.29)

Therefore we get that θ ( f 1 , ..., fr ) (x0 ) =  ⎞ ⎞ ⎞ ⎤ ⎡ ⎛ ⎛ ⎛       r r  r ⎢ b ⎜ ⎟ ⎟ ⎟ ⎥ ⎜ b ⎜ 1  ⎟ ⎟ ⎟ ⎥ ⎢ ⎜ ⎜ ⎜ f j (x)⎠ f i (x) d x − ⎝ f j (x)⎠ d x ⎠ f i (x0 )⎦   ≤  (α) ⎣ a ⎝ ⎝ a  i=1  j=1 j=1   j=i j=i (1.30) ⎤ ⎞ ⎡⎡ ⎛

      r ⎢⎢ x0 ⎜  r ⎥ ⎟  g(x0 ) 

⎟ ⎢⎢ ⎜ α−1 α −1 (z) dz d x ⎥ D ◦ g f f − g (x) (z (x)) ⎥ ⎟ ⎢⎢ ⎜ j x0 −;g i ⎦ ⎠ g(x) ⎣⎣ ⎝  i=1  a j=1   j=i ⎡ ⎤⎤ ⎞ ⎛        b r  g(x) ⎥⎥ ⎟ ⎜ ⎢ α−1 α −1 ⎥⎥ ≤ ⎟ ⎜ ⎢ D ◦ g dz d x − z) + f f (z) (x) (g (x) j i x 0 +;g ⎣ ⎦⎦ ⎠ ⎝ g(x 0 )   x0 j=1   j=i  ⎤ ⎛ ⎞       r  ⎥ ⎜ r ⎢ ⎟  g(x0 )  1 ⎢  ⎥ ⎢⎢ x0 ⎜  ⎟ α−1 α −1 f j (x)⎟ Dx −;g f i ◦ g (z − g (x)) (z) dz  d x ⎥ ⎜ ⎢⎢ 0  ⎦ ⎣⎣ a ⎝ ⎠ g(x)  (α)   j=1 i=1   j=i ⎡⎡

(1.31)  ⎤⎤ ⎛ ⎞       r  ⎥⎥ ⎟  g(x) ⎢ b ⎜    ⎥⎥ ⎜ ⎟ ⎢ α−1 α −1 Dx0 +;g f i ◦ g +⎢ f j (x)⎟ (z) dz  d x ⎥⎥ ≤ (g (x) − z) ⎜  ⎦⎦ ⎠ g(x0 ) ⎣ x0 ⎝   j=1   j=i ⎡

⎡⎡ ⎛ ⎞ ⎤    x0  r r    g(x )

0 ⎢ ⎢ ⎜   ⎟ ⎥ 1   ⎢⎢ ⎜  f j (x)⎟ (z − g (x))α−1  Dxα0 −;g f i ◦ g −1 (z) dz d x ⎥ ⎣ ⎣ ⎝ ⎠ ⎦  (α) a g(x) i=1

⎡ ⎢ +⎢ ⎣



j=1 j=i

⎛ b x0

⎞

r ⎜  ⎟ ⎜  f j (x)⎟ ⎝ ⎠ j=1 j=i



g(x) g(x 0 )

⎤⎤    ⎥⎥   ⎥ (g (x) − z)α−1  Dxα0 +;g f i ◦ g −1 (z) dz d x ⎥ ⎦⎦ =: (∗) .

(1.32) Hence it holds θ ( f 1 , ..., fr ) (x0 ) ≤ (∗) .

(1.33)

10

1 Generalized Fractional Ostrowski …

We have that (∗) ≤

1  (α + 1)

⎡⎡ r

i=1

  ⎢⎢  α −1  ⎢⎢ ◦ g D f    i x 0 −;g ⎣⎣

⎛



x0

∞,[g(a),g(x 0 )] a

⎡ ∞,[g(x 0 ),g(b)]

j=1 j=i

b x0

⎜ ⎜ ⎝

⎤

 ⎟ ⎥  f j (x)⎟ (g (x0 ) − g (x))α d x ⎥ ⎠ ⎦

⎛



  ⎢  α −1  ◦ g +⎢ D f    i x +;g ⎣ 0

⎜ ⎜ ⎝

⎞ r 

⎞ r  j=1 j=i

⎤⎤

 ⎟ ⎥⎥  f j (x)⎟ (g (x) − g (x0 ))α d x ⎥⎥ ≤ ⎠ ⎦⎦

(1.34)

  −1  1  D α  x0 −;g f i ◦ g ∞,[g(a),g(x0 )]  (α + 1) i=1 r

⎛  ⎜ (g (x0 ) − g (a))α ⎜ ⎝

⎞

⎛ x0

a

⎜ ⎜ ⎝

r  j=1 j=i

⎞⎤

 ⎟ ⎟⎥  f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦

⎡

⎛    ⎢ ⎜  α −1  α⎜ ⎢ D ◦ g − g f (g (b) (x ))     0 x 0 +;g i ⎣ ⎝ ∞,[g(x 0 ),g(b)]

⎛ b x0

(1.35) ⎞

⎞ ⎤⎤

r ⎜   ⎟ ⎟ ⎥⎥ ⎜  f j (x)⎟ d x ⎟⎥⎥ , ⎝ ⎠ ⎠ ⎦⎦ j=1 j=i



proving (1.20). Next comes an L 1 estimate. Theorem 1.11 All as in Theorem 1.10, with α ≥ 1. Then θ ( f 1 , ..., fr ) (x0 ) ≤ ⎡⎡ r

i=1

  ⎢⎢  α −1  ⎢⎢ ⎣⎣ Dx0 −;g f i ◦ g 

1  (α) ⎞

⎛



x0

L 1 ([g(a),g(x 0 )]) a

⎜ ⎜ ⎝

r  j=1 j=i

⎤

 ⎟ ⎥  f j (x)⎟ (g (x0 ) − g (x))α−1 d x ⎥ ⎠ ⎦

(1.36) ⎡   ⎢  α −1  +⎢ f ◦ g D    i x +;g ⎣ 0

 L 1 ([g(x 0 ),g(b)])

⎛ b x0

⎞

⎤⎤

r  ⎜ ⎟ ⎥⎥  f j (x)⎟ (g (x) − g (x0 ))α−1 d x ⎥⎥ . ⎜ ⎝ ⎠ ⎦⎦ j=1 j=i

1.4 Main Results

11

Proof Next we use that α ≥ 1. We observe that (1.33)

(∗) ≤ ⎡⎡ r  

⎢⎢  α −1  ⎢⎢ ⎣⎣ Dx0 −;g f i ◦ g  i=1

1  (α) ⎛



x0

L 1 ([g(a),g(x 0 )]) a

⎞

⎤

r ⎜  ⎟ ⎥ ⎜  f j (x)⎟ (g (x0 ) − g (x))α−1 d x ⎥ ⎝ ⎠ ⎦ j=1 j=i

(1.37) ⎡   ⎢  α −1  +⎢ ⎣ Dx0 +;g f i ◦ g 

⎛

 L 1 ([g(x 0 ),g(b)])

b x0

⎞

⎤⎤

r  ⎜ ⎟ ⎥⎥  f j (x)⎟ (g (x) − g (x0 ))α−1 d x ⎥⎥ , ⎜ ⎝ ⎠ ⎦⎦ j=1 j=i



proving Theorem 1.11. An L p estimate follows.

Theorem 1.12 All as in Theorem 1.10. Let now p, q > 1 : 1p + q1 = 1, with α > q1 . Then 1 θ ( f 1 , ..., fr ) (x0 ) ≤ 1 ( p (α − 1) + 1) p  (α) ⎡ r

i=1

⎛

  ⎢  α −1  ⎢ f ◦ g D    i x −;g ⎣ 0

⎜ ⎜



⎞

⎛ x0

q,[g(a),g(x 0 )] ⎝ a

α− q1

(g (x0 ) − g (x))

⎜ ⎜ ⎝

r  j=1 j=i

⎞

⎟ ⎟   f j (x)⎟ d x ⎟ ⎠ ⎠

(1.38)    +  Dxα0 +;g f i ◦ g −1 q,[g(x ⎛ ⎜ ⎜ ⎝



b x0

0 ),g(b)]

⎞

⎛

⎞⎤

r ⎟ ⎟⎥ 1 ⎜   f j (x)⎟ d x ⎟⎥ =:  (x0 ) . (g (x) − g (x0 ))α− q ⎜ ⎠ ⎠⎦ ⎝ j=1 j=i

Proof We have that ⎡⎡ (1.33)

(∗) ≤

1  (α)

r

i=1



⎢⎢ ⎢⎢ ⎣⎣



g(x0 )

g(x)

a

⎞

⎛ x0

⎜ ⎜ ⎝

 α  D

r  j=1 j=i

⎟   f j (x)⎟ ⎠

x0 −;g f i



◦g

−1





g(x0 )

g(x)

q (z) dz

(z − g (x)) p(α−1) dz

q1

 dx +

1p

12

1 Generalized Fractional Ostrowski …

⎡



⎢ ⎢ ⎣

⎞

⎛ ⎜ ⎜ ⎝

b x0



r  j=1 j=i

g(x)

 ⎟  f j (x)⎟ ⎠

 α  D



x0 +;g f i

g(x0 )

g(x)

g(x0 )



◦g

(g (x) − z)

 −1

q (z) dz

1p p(α−1)

dz



q1

≤

dx

(1.39)

⎤ ⎞ ⎡⎡ ⎛ 1  x0  r r α−1+     

p ⎥  ⎟ ⎢⎢ ⎜ 1    f j (x)⎟ (g (x0 ) − g (x)) ⎢⎢ ⎜ dx⎥  Dxα0 −;g f i ◦ g −1  1 ⎦ ⎠ ⎣⎣ ⎝ q,[g(a),g(x0 )]  (α) p a p − 1) + 1) ( (α i=1 j=1 j =i

⎡ ⎢ +⎢ ⎣

⎛



b x0

⎜ ⎜ ⎝

⎞

⎤⎤ α−1+ 1p

r 

 ⎟  f j (x)⎟ (g (x) − g (x0 )) 1 ⎠ ( p (α − 1) + 1) p j=1

     Dxα0 +;g f i ◦ g −1 (z)

q,[g(x 0 ),g(b)]

⎥⎥ ⎥ dx⎥ ⎦⎦

j=i

=

1 1

( p (α − 1) + 1) p  (α)

⎡ ⎛  r  

⎢ ⎜  α −1  ⎢ ⎜ D ◦ g f     i x 0 −;g ⎣ q,[g(a),g(x 0 )] ⎝

⎛ x0

α− q1

(g (x0 ) − g (x))

a

i=1

⎞

⎞

r ⎜  ⎟ ⎟ ⎜  f j (x)⎟ d x ⎟ ⎝ ⎠ ⎠ j=1 j=i

(1.40)   +  Dxα0 +;g

⎛   ⎜  ⎜ f i ◦ g −1  ⎝ q,[g(x0 ),g(b)]

⎛ b

α− q1

(g (x) − g (x0 ))

x0

⎞

⎞⎤

r  ⎜ ⎟ ⎟⎥  f j (x)⎟ d x ⎟⎥ , ⎜ ⎝ ⎠ ⎠⎦ j=1 j =i



proving Theorem 1.12.

We continue with generalized fractional Grüss-Cebysev type inequalities for several functions over a Banach algebra. A uniform estimate follows. Theorem 1.13 Let (A, ·) be a Banach algebra, 0 < α ≤ 1, f i ∈ C 1 ([a, b] , A), i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) ; x0 ∈ [a, b] ⊂ R and θ ( f 1 , ..., fr ) (x0 ) as in (1.19). Assume that M ( f 1 , ..., fr ) := max

i=1,..,r

   sup  Dxα0 −;g f i ◦ g −1 ∞,[g(a),g(x

x0 ∈[a,b]

   sup  Dxα0 +;g f i ◦ g −1 ∞,[g(x

x0 ∈[a,b]

! 0 ),g(b)]

< ∞.

0 )]

,

(1.41)

1.4 Main Results

13

Denote by



b

 ( f 1 , ..., fr ) :=

θ ( f 1 , ..., fr ) (x) d x =

a

⎛ ⎞ ⎛ ⎞  r r

⎜ b ⎜ ⎟ ⎢ ⎟ ⎢(b − a) ⎜ ⎜ ⎟ f j (x)⎟ ⎝ a ⎝ ⎠ f i (x) d x ⎠ − ⎣ ⎡

i=1

j=1 j=i

⎛ ⎜ ⎜ ⎝



⎞

⎛ b

a

⎜ ⎜ ⎝

r  j=1 j=i

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ dx⎠



b

a



⎤

⎥ f i (x) d x ⎥ ⎦.

(1.42)

Then it holds  ( f 1 , ..., fr ) =  ⎛ ⎞ ⎡ ⎛ ⎞    r  r ⎢ ⎜ b ⎜ ⎟ ⎟  ⎢(b − a) ⎜ ⎜ ⎟ f j (x)⎟  ⎝ ⎠ f i (x) d x ⎠ − ⎣ ⎝ a  i=1 j=1  j=i ⎛  ⎜ ⎜ ⎝

a

⎞

⎛ b

⎞

r ⎟ ⎟ ⎜ ⎟ ⎜ f j (x)⎟ ⎠ dx⎠ ⎝

 a

j=1 j=i

b

⎤ 

 ⎥  f i (x) d x ⎥ ⎦ ≤  

        M ( f 1 , ..., fr ) (b − a) (g (b) − g (a)) ⎜   f j  ⎜   ⎝  (α + 1)  i=1  j=1   j=i ⎛

α

2

⎞

r  r

⎟ ⎟. ⎠

(1.43)

∞,[a,b]

Proof We have that   ( f 1 , ..., fr ) ≤

b

(1.33)

θ ( f 1 , ..., fr ) (x0 ) d x0 ≤

a

1  (α + 1) i=1 r

⎧ ⎪ ⎪ ⎨



   sup  Dxα0 −;g f i ◦ g −1 ∞,[g(a),g(x

x0 ∈[a,b]

0 )]

⎫⎤ ⎞ ⎞⎤ ⎛ ⎛ ⎪ ⎪  r ⎬⎥ b⎢ x 0 ⎜   ⎟ ⎟ ⎥ ⎜ ⎢(g (x0 ) − g (a))α ⎜ ⎜  f j (x)⎟ d x ⎟⎥ d x0 ⎥ + ⎦ ⎠ ⎠⎦ ⎣ ⎝ ⎝ ⎪ ⎪ a ⎪ ⎪ j=1 ⎩ a ⎭ ⎡

j=i

(1.44)

14

1 Generalized Fractional Ostrowski …



   sup  Dxα0 +;g f i ◦ g −1 ∞,[g(x

x0 ∈[a,b]

⎧ ⎪ ⎪ ⎨

0 ),g(b)]

⎫⎤⎤ ⎞ ⎞⎤ ⎛ ⎛ ⎪ ⎪  r ⎬⎥⎥ b⎢  ⎟ ⎟⎥ ⎜ b ⎜ ⎢(g (b) − g (x0 ))α ⎜ ⎜  f j (x)⎟ d x ⎟⎥ d x0 ⎥⎥ ≤ ⎠ ⎠⎦ ⎣ ⎝ x ⎝ ⎪ ⎪⎦⎦ ⎪ ⎪ 0 j=1 ⎩ a ⎭ ⎡

j=i

r (g (b) − g (a))α  (α + 1) i=1

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 

b

a



   sup  Dxα0 −;g f i ◦ g −1 ∞,[g(a),g(x

x0 ∈[a,b]

⎫⎤ ⎞ ⎞ ⎛ ⎛ ⎪ ⎪  x0  r ⎬⎥ ⎜ ⎜  ⎟ ⎟ ⎜ ⎜  f j (x)⎟ d x ⎟ d x0 ⎥ + ⎠ ⎠ ⎝ a ⎝ ⎪⎦ ⎪ j=1 ⎭ j=i

   sup  Dxα0 +;g f i ◦ g −1 ∞,[g(x

x0 ∈[a,b]

0 ),g(b)]

⎧ ⎪ ⎪ ⎨

⎫⎤⎤ ⎞ ⎞ ⎛ ⎛ ⎪ ⎪ r ⎬⎥⎥ b ⎜ b ⎜  ⎟ ⎟ ⎜ ⎜  f j (x)⎟ d x ⎟ d x0 ⎥⎥ =: (∗∗) . ⎠ ⎠ ⎝ x ⎝ ⎪ ⎪⎦⎦ ⎪ ⎪ 0 j=1 ⎩ a ⎭ j=i

Hence it holds M ( f 1 , ..., fr ) (g (b) − g (a))α (b − a)2 2 (α + 1)   ⎡⎡ ⎤     r r

⎢⎢    ⎥ ⎢⎢ sup   f j (x) ⎥+  ⎣⎣x ∈[a,b]  ⎦   j=1 0 i=1   j=i

(∗∗) ≤

∞,[a,x0 ]

  ⎡     r   ⎢  ⎢ sup   f j (x)  ⎣x ∈[a,b]    j=1 0   j=i

∞,[x0 ,b]

⎤⎤ ⎥⎥ ⎥⎥ ⎦⎦

0 )]

(1.45)

1.4 Main Results

15

     2 α ⎜     M ( f 1 , ..., fr ) (b − a) (g (b) − g (a)) ⎜   f j  ≤   ⎝  (α + 1)  i=1  j=1   j=i ⎛

⎞

r  r

⎟ ⎟, ⎠

(1.46)

∞,[a,b]



proving (1.43). Next comes an L p estimate

Theorem 1.14 Let (A, ·) be a Banach algebra, 0 < α ≤ 1, f i ∈ C 1 ([a, b] , A), i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . Let p, q > 1 : 1p + q1 = 1, q1 < α ≤ 1, x0 ∈ [a, b] ⊂ R. Assume that    sup  Dxα0 −;g f i ◦ g −1  L

N ( f 1 , ..., fr ) := max

i=1,..,r

x0 ∈[a,b]

   sup  Dxα0 +;g f i ◦ g −1  L

x0 ∈[a,b]

and set

,

! q ([g(x 0 ),g(b)])

⎧⎛ ⎞   ⎪   ⎪ r ⎨⎜  ⎟ ⎜  ⎟   f j ⎠ Z ( f 1 , ..., fr ) := max ⎝  i=1,..,r ⎪   ⎪ ⎩ j=1  j=i

Then

q ([g(a),g(x 0 )])

< ∞, ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(1.47)

.

(1.48)

q,[a,b]

 ⎛ ⎞ ⎡ ⎛ ⎞    r  r ⎢ ⎜ b ⎜ ⎟ ⎟  ⎢(b − a) ⎜ ⎜ ⎟ f j (x)⎟  ⎝ ⎠ f i (x) d x ⎠ − ⎣ ⎝ a  i=1 j=1  j=i ⎛  ⎜ ⎜ ⎝

a

⎞

⎛ b

⎞

r ⎟ ⎟ ⎜ ⎟ ⎜ f j (x)⎟ ⎠ dx⎠ ⎝ j=1 j=i

2r pN ( f 1 , ..., fr ) Z ( f 1 , ..., fr ) 1 P

( p + 1) ( p (α − 1) + 1)  (α)



b

a

⎤ 

 ⎥  f i (x) d x ⎥ ⎦ ≤  

(b − a) p +1 (g (b) − g (a))α− q . 1

1

Proof Here  (x0 ) is as in (1.38). Clearly then it holds  (x0 ) ≤

N ( f 1 , ..., fr ) 1

( p (α − 1) + 1) p  (α)

(1.49)

16

1 Generalized Fractional Ostrowski … r

i=1

⎡⎛  ⎢⎜ x0 α− q1 ⎢⎜ ⎣⎝ a (g (x0 ) − g (x))

⎛  ⎜ ⎜ ⎝

⎞

⎛ ⎜ ⎜ ⎝

r  j=1 j=i

 ⎟ ⎟  f j (x)⎟ d x ⎟ + ⎠ ⎠ ⎞

⎛ 1 ⎜ (g (x) − g (x0 ))α− q ⎜ ⎝

b x0

r  j=1 j=i

⎞ (1.50)

⎞⎤

 ⎟ ⎟⎥  f j (x)⎟ d x ⎟⎥ . ⎠ ⎠⎦

Therefore we obtain   ( f 1 , ..., fr ) ≤

b

θ ( f 1 , ..., fr ) (x0 ) d x0 ≤

a



b

 (x0 ) d x0 ≤

a

⎡⎡ r

i=1

⎛



⎢⎢ ⎢⎢ ⎣⎣

b

a

⎜ ⎜ ⎝

N ( f 1 , ..., fr ) 1

( p (α − 1) + 1) p  (α) ⎞

⎛



x0



(g (x0 ) − g (x))

p(α−1)+1 p



a

⎜ ⎜ ⎝

r  j=1 j=i

⎞

⎤

⎥  ⎟ ⎟  f j (x)⎟ d x ⎟ d x0 ⎥ ⎠ ⎠ ⎦

⎡ ⎞ ⎞ ⎛ ⎛ ⎤⎤  b  b  r p(α−1)+1 ⎜  ⎢ ⎟ ⎟ ⎜ ⎥⎥ p ⎜ ⎜  f j (x)⎟ d x ⎟ d x0 ⎥⎥ ≤ +⎢ ⎣ a ⎝ x (g (x) − g (x0 )) ⎠ ⎠ ⎝ ⎦⎦ 0

j=1 j=i

⎞ ⎡⎛      r ⎜     ⎟ ⎢ N ( f 1 , ..., fr ) ⎜     ⎟ ⎢ f j ⎠ 1  ⎝ ⎣  ( p (α − 1) + 1) p  (α) i=1  j=1   j=i r

q,[a,b]



b



a

b



(g (x0 ) − g (x))

( p(α−1)+1)

1p dx

 d x0 +

a

a



x0

b

(g (x) − g (x0 ))

( p(α−1)+1)



1p dx

d x0

x0

Hence we get (η) ≤

r N ( f 1 , ..., fr ) Z ( f 1 , ..., fr ) 1

( p (α − 1) + 1) p  (α)

=: (η) .

(1.51)

1.5 Applications

17



b



a



x0

(g (x0 ) − g (x))

( p(α−1)+1)



1p

d x0 +

dx

a

b



a

b

(g (x) − g (x0 ))

( p(α−1)+1)



1p dx

d x0

≤

(1.52)

x0

2r pN ( f 1 , ..., fr ) Z ( f 1 , ..., fr ) 1 P

( p + 1) ( p (α − 1) + 1)  (α)

(g (b) − g (a))α− q (b − a) p +1 , 1

1



proving (1.49).

1.5 Applications We make Remark 1.15 Assume from now on that (A, ·) is a commutative Banach algebra. Then, we get that

(1.19)

θ ( f 1 , ..., fr ) (x0 ) = r

 b a

⎛ ⎝

r 

⎛

⎞ f j (x)⎠ d x −

j=1

⎞

⎞

⎛

r ⎜ b ⎜  r ⎟ ⎟

⎟ ⎟ ⎜ ⎜ f j (x)⎟ d x ⎟ f i (x0 ) , ⎜ ⎜ ⎠ ⎠ ⎝ a ⎝ i=1

j=1 j=i

(1.53) x0 ∈ [a, b] . Furthermore, it holds (0 < α ≤ 1 case) (1.42)



 ( f 1 , ..., fr ) = r (b − a)

⎛ b

⎝

a

r 

⎞ f j (x)⎠ d x−

j=1

⎡⎛ ⎛ ⎞ ⎞ ⎤

  r r b

⎢⎜ b ⎜ ⎟ ⎟ ⎥ ⎢⎜ ⎜ ⎟ dx⎟ ⎥ f (x) j ⎣⎝ a ⎝ ⎠ ⎠ a f i (x) d x ⎦ . i=1

(1.54)

j=1 j=i

When r = 2, we get that θ ( f 1 , f 2 ) (x0 ) = 2

x0 ∈ [a, b] , and

 b a

f 1 (x) f 2 (x) d x − f 1 (x0 )

 b a

f 2 (x) d x − f 2 (x0 )

 b a

f 1 (x) d x,

(1.55)

18

1 Generalized Fractional Ostrowski …

)   ( f 1 , f 2 ) = 2 (b − a)

b

 f 1 (x) f 2 (x) d x −

a



b

f 1 (x) d x

*

b

,

f 2 (x) d x

a

a

(1.56)

0 < α ≤ 1. We give Corollary 1.16 All as in Theorem 1.10, A is a commutative Banach algebra, r = 2. Then

  −1  1   D α x0 −;g f i ◦ g ∞,[g(a),g(x0 )]  (α + 1) i=1 2

θ ( f 1 , f 2 ) (x0 ) ≤

⎞ ⎞⎤ ⎛ ⎛  x0  2 ⎟ ⎟⎥ ⎜ ⎜  ⎜  f j (x)⎟ d x ⎟⎥ + (g (x0 ) − g (a))α ⎜ ⎠ ⎠⎦ ⎝ a ⎝ j=1 j=i

⎡

⎛  ⎜ (g (b) − g (x0 ))α ⎜ ⎝ ∞,[g(x 0 ),g(b)]

  ⎢  α −1  ⎢ ◦ g D f    i x +;g ⎣ 0

⎛ b x0

⎜ ⎜ ⎝

⎞ 2  j=1 j=i

⎞ ⎤⎤

  ⎟ ⎟ ⎥⎥  f j (x)⎟ d x ⎟⎥⎥ . ⎠ ⎠ ⎦⎦

(1.57) 

Proof By Theorem 1.10. We continue with

Corollary 1.17 All as in Theorem 1.13, A is a commutative Banach algebra, r = 2, 0 < α ≤ 1, M ( f 1 , f 2 ) as in (1.41). Then  ( f 1 , f 2 ) ≤

, M ( f 1 , f 2 ) (b − a)2 (g (b) − g (a))α +  f 1 ∞,[a,b] +  f 2 ∞,[a,b] .  (α + 1)

(1.58) 

Proof By Theorem 1.13. Finally we derive: Corollary 1.18 All as in Corollary 1.16, for g (t) = et . Then θ ( f 1 , f 2 ) (x0 ) ≤

2 

 α   1   D x0 −;et f i ◦ log ∞,[ea ,e x0 ]  (α + 1) i=1

⎛ 

e x0 − ea

α ⎜ ⎜ ⎝

 a

⎞

⎛ x0

⎜ ⎜ ⎝

2  j=1 j=i

⎞⎤

⎟ ⎟⎥   f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦

References

⎡ ⎢ α ⎢ D x0 +;et ⎣

19

⎞ ⎞ ⎤⎤ ⎛ ⎛  b  2 ⎟ ⎟⎥⎥ ⎜    α ⎜  ⎜  f j (x)⎟ d x ⎟⎥⎥ . f i ◦ log ∞,[ex0 ,eb ] eb − e x0 ⎜ ⎠ ⎠ ⎦⎦ ⎝ x ⎝ 0

j=1 j=i

(1.59) Proof By Corollary 1.16.



References 1. Aliprantis, C., Border, K.: Infinite Dimensional Analysis, 3rd edn. Springer, New York (2006) 2. Anastassiou, G.A.: Fractional Differentiation Inequalities. Research Monograph. Springer, New York (2009) 3. Anastassiou, G.A.: Advances on Fractional Inequalities. Research Monograph. Springer, New York (2011) 4. Anastassiou, G.A.: Intelligent Comparisons: Analytic Inequalities. Springer, Heidelberg, New York (2016) 5. Anastassiou, G.A.: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations. Springer, Heidelberg, New York (2018) 6. Anastassiou, G.A.: Generalized Fractional Ostrowski and Grüss type inequalities involving several Banach algebra valued functions. submitted for publication (2021) ˇ 7. Cebyšev: Sur les expressions approximatives des intégrales définies par les aures prises entre les mêmes limites 8. Dragomir, S.S.: Noncommutative Ostrowski type inequalities for functions in Banach algebras. RGMIA Res. Rep. Coll. 24, Art. 10, 24 pp (2021) 9. Dragomir, S.S.: New Noncommuttive Grüss type inequalities for functions in Banach algebras. RGMIA Res. Rep. Coll. 24, Art. 13, 19 pp (2021) 10. Mikusinski, J.: The Bochner Integral. Academic Press, New York (1978) 11. Ostrowski, A.: Über die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938) 12. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991)

Chapter 2

Iterated Generalized Fractional Ostrowski and Grüss Inequalities with Multiple Banach Algebra Valued Functions

Employing iterated generalized Caputo fractional left and right vectorial Taylor formulae we establish mixed sequential generalized fractional Ostrowski and Grüss type inequalities for several Banach algebra valued functions. The estimates are with respect to all norms · p , 1 ≤ p ≤ ∞. We finish with applications. It follows [5].

2.1 Introduction The following results motivate our chapter. Theorem 2.1 (1938, Ostrowski [10]) Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, derivative f  : (a, b) → R is bounded on (a, b),    b) whose   sup    i.e., f ∞ := sup f (t) < +∞. Then t∈(a,b)

   1  b − a

a

b

 2       sup x − a+b 1 2 f  , + f (t) dt − f (x) ≤ − a) (b ∞ 4 (b − a)2

for any x ∈ [a, b]. The constant

1 4

(2.1)

is the best possible.

Ostrowski type inequalities have great applications to integral approximations in Numerical Analysis. ˇ Theorem 2.2 (1882, Cebyšev [6]) Let f, g : [a, b] → R be absolutely continuous   functions with f , g ∈ L ∞ ([a, b]). Then

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_2

21

22

2 Iterated Generalized Fractional Ostrowski …

   1  b − a

b

f (x) g (x) d x −

a

≤

1 b−a





b

f (x) d x a

1 b−a

 a

b

  g (x) d x 

    1 (b − a)2  f  ∞ g  ∞ . 12

(2.2)

The above integrals are assumed to exist. The related Grüss type inequalities have many applications to Probability Theory. In this article we present sequential generalized fractional Ostrowski and CebysevGrüss type inequalities for several Banach algebra valued functions. Now our sequential generalized left and right Caputo fractional derivatives are for Banach space valued functions and our integrals are of Bochner type. The main motivation here is Theorem 2.13 at the end of Sect. 2.2. Several applications finish this article. Inspiration came also from [7, 8]. See also [1–3].

2.2 Vectorial Sequential Generalized Fractional Calculus Background We need Definition 2.3 ([4], p. 106) Let 0 < α ≤ 1, f ∈ C 1 ([a, b] , X ), where [a, b] ⊂ R, and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). We define the left generalized g-fractional derivative X -valued of f of order α as follows:  x    α  1 Da+;g f (x) := (g (x) − g (t))−α g  (t) f ◦ g −1 (g (t)) dt,  (1 − α) a (2.3) ∀ x ∈ [a, b], where  is the gamma function. The last integral is of Bochner type ([9]).  α If 0 < α < 1, by Theorem 4.10, p. 98, [4], we have that Da+;g f ∈ C ([a, b] , X ). We set   1 f (x) := f ◦ g −1 ◦ g (x) ∈ C ([a, b] , X ) , (2.4) Da+;g 0 f (x) = f (x) , ∀ x ∈ [a, b] . Da+;g

When g = id, then

α α α f = Da+;id f = D∗a f, Da+;g

the usual left X -valued Caputo fractional derivative, see [4], Chap. 1. We make

(2.5)

2.2 Vectorial Sequential Generalized Fractional Calculus Background

23

Remark 2.4 By (2.3) we have  x    1   (g (x) − g (t))−α g  (t)  f ◦ g −1 (g (t)) dt ≤  (1 − α) a       x  f ◦ g −1 ◦ g  ∞,[a,b] (g (x) − g (t))−α g  (t) dt =  (1 − α) a       f ◦ g −1 ◦ g  1−α ∞,[a,b] (g (x) − g (t)) = (2.6)  (1 − α) 1−α       f ◦ g −1 ◦ g  ∞,[a,b] (g (x) − g (a))1−α , ∀ x ∈ [a, b] .  (2 − α)

   a+;g f (x) ≤

 α  D

Hence



 α f (a) = 0. Da+;g

(2.7)

We need Definition 2.5 ([4], p. 107) Let 0 < α ≤ 1, f ∈ C 1 ([a, b] , X ), where [a, b] ⊂ R, and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . We define the right generalized g-fractional derivative X -valued of f of order α as follows:  b     α −1 Db−;g f (x) := (g (t) − g (x))−α g  (t) f ◦ g −1 (g (t)) dt,  (1 − α) x (2.8) ∀ x ∈ [a, b]. The last integral is of Bochner type.  α If 0 < α < 1, by Theorem 4.11, p. 101 ([4]), we have that Db−;g f ∈ C ([a, b] , X ). We set   1 Db−;g f (x) := − f ◦ g −1 ◦ g (x) ∈ C ([a, b] , X ) , (2.9) 0 f (x) := f (x) , ∀ x ∈ [a, b] . Db−;g

When g = id, then

α α α f (x) = Db−;id f (x) = Db− f, Db−;g

the usual right X -valued Caputo fractional derivative, see [4], Chap. 2. We make

(2.10)

24

2 Iterated Generalized Fractional Ostrowski …

Remark 2.6 By (2.8) we have  α  D

 b    1   (g (t) − g (x))−α g  (t)  f ◦ g −1 (g (t)) dt ≤  (1 − α) x       b  f ◦ g −1 ◦ g  ∞,[a,b] (2.11) (g (t) − g (x))−α g  (t) dt =  (1 − α) x       f ◦ g −1 ◦ g  1−α ∞,[a,b] (g (b) − g (x)) =  (1 − α) 1−α       f ◦ g −1 ◦ g  ∞,[a,b] (g (b) − g (x))1−α , ∀ x ∈ [a, b] .  (2 − α)

  b−;g f (x) ≤ 

Hence



 α f (b) = 0. Db−;g

(2.12)

We need Definition 2.7 ([4], p. 115) Denote by (0 < α ≤ 1) nα α α α Da+;g := Da+;g Da+;g ...Da+;g (n times), n ∈ N

(2.13)

0 = I (identity operator). and Da+;g

We also need Definition 2.8 ([4], p. 118) nα α α α Db−;g := Db−;g Db−;g ...Db−;g (n times), n ∈ N

(2.14)

0 = I (identity operator). and Db−;g

Based on (2.7) and Theorem 4.30, p. 117, ([4]), we have the following g-left generalized modified X -valued Taylor’s formula: Theorem 2.9 Let 0 < α ≤ 1, n ∈ N, f ∈ C 1 ([a, b] , X ), (X, ·) a Banach space, g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Let Fk := kα f , k = 1, ..., n, that fulfill Fk ∈ C 1 ([a, b] , X ), and Fn+1 ∈ C ([a, b] , X ) . Da+;g Then n

 (g (x) − g (a))iα  iα Da+;g f (a) + f (x) − f (a) =  (iα + 1) i=2

2.2 Vectorial Sequential Generalized Fractional Calculus Background

1  ((n + 1) α)



x

a

 (n+1)α f (t) dt, (g (x) − g (t))(n+1)α−1 g  (t) Da+;g

25

(2.15)

∀ x ∈ [a, b] . When n = 1 we obtain Corollary 2.10 Let 0 < α ≤ 1, f ∈ C 1 ([a, b] , X ), (X, ·) is a Banach space, g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Assume that α 2α f ∈ C 1 ([a, b] , X ), and Da+;g f ∈ C ([a, b] , X ) . Then Da+;g 1 f (x) − f (a) =  (2α)



  2α f (t) dt, (g (x) − g (t))2α−1 g  (t) Da+;g

x

a

(2.16)

∀ x ∈ [a, b] . Based on (2.12) and Theorem 4.33, p. 120, ([4]), we have the following g-right generalized modified X -valued Taylor’s formula: Theorem 2.11 Let f ∈ C 1 ([a, b] , X ), (X, ·) a Banach space, g ∈ C 1 ([a, b]), kα f,k = strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Suppose Fk := Db−;g 1 1, ..., n, fulfill Fk ∈ C ([a, b] , X ), and Fn+1 ∈ C ([a, b] , X ) , where 0 < α ≤ 1, n ∈ N. Then f (x) − f (b) =

n

(g (b) − g (x))iα  i=2

1  ((n + 1) α)



b x

 (iα + 1)

 iα Db−;g f (b) +

 (n+1)α f (t) dt, (g (t) − g (x))(n+1)α−1 g  (t) Db−;g

(2.17)

∀ x ∈ [a, b] . When n = 1 we obtain Corollary 2.12 Let 0 < α ≤ 1, f ∈ C 1 ([a, b] , X ), (X, ·) is a Banach space, g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Assume that α 2α f ∈ C 1 ([a, b] , X ), and Db−;g f ∈ C ([a, b] , X ) . Then Db−;g f (x) − f (b) =

1  (2α)



b x

 2α  f (t) dt, (g (t) − g (x))2α−1 g  (t) Db−;g

(2.18)

∀ x ∈ [a, b] . We are greatly motivated by the following sequential generalized fractional Ostrowski type inequality:

26

2 Iterated Generalized Fractional Ostrowski …

Theorem 2.13 (p. 140, [4]) Let g ∈ C 1 ([a, b]) and strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]), and 0 < α < 1, n ∈ N, f ∈ C 1 ([a, b] , X ), where (X, ·) is a Banach space. Let x0 ∈ [a, b] be fixed. Assume that Fkx0 := Dxkα0 −;g f , for  x0 k = 1, ..., n, fulfill Fkx0 ∈ C 1 ([a, b] , X ) and Fn+1 ∈ C ([a, x0 ] , X ) and Dxiα0 −;g f (x0 ) = 0, i = 1, ..., n. Similarly, we assume that G kx0 := Dxkα0 +;g f , for k = 1, ..., n, fulfill G kx0 ∈  x0 C 1 ([x0 , b] , X ) and G n+1 ∈ C ([x0 , b] , X ) and Dxiα0 +;g f (x0 ) = 0, i = 1, ..., n. Then    b   1 1  f (x) d x − f (x0 )  ≤ (b − a)  ((n + 1) α + 1) · b − a a      f (g (b) − g (x0 ))(n+1)α (b − x0 ) Dx(n+1)α  0 +;g     f (g (x0 ) − g (a))(n+1)α (x0 − a) Dx(n+1)α  0 −;g

∞,[x0 ,b]

+

(2.19)

 ∞,[a,x0 ]

.

2.3 Banach Algebras Background All here come from [11]. We need Definition 2.14 ([11], p. 245) A complex algebra is a vector space A over the complex filed C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(2.20)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(2.21)

α (x y) = (αx) y = x (αy) ,

(2.22)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A)

(2.23)

and if A contains a unit element e such that xe = ex = x (x ∈ A)

(2.24)

2.4 Main Results

27

and e = 1,

(2.25)

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 2.15 Commutativity of A will be explicited stated when needed. There exists at most one e ∈ A that satisfies (2.24). Inequality (2.23) makes multiplication to be continuous, more precisely left and right continuous, see [11], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [11], pp. 247–248, Sect. 10.3. We also make Remark 2.16 Next we mention about integration of A-valued functions, see [11], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some compact  Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [11], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   x f dμ = x f ( p) dμ ( p) (2.26) Q

and



Q

 f dμ x = Q

f ( p) x dμ ( p) .

(2.27)

Q

The Bochner integrals we will involve in our chapter follow (2.26) and (2.27). Also, let f ∈ C ([a, b] , X ), where [a, b] ⊂ R, (X, ·) is a Banach space. By [4], p. 3, f is Bochner integrable.

2.4 Main Results We start with mixed sequential generalized fractional Ostrowski type inequalities for several functions over a Banach algebra. A uniform estimate follows. Theorem 2.17 Let (A, ·) be a Banach algebra, x0 ∈ [a, b] ⊂ R, 0 < α < 1, n ∈ N, f i ∈ C 1 ([a, b] , A) , i = 1, ..., r . Let g ∈ C 1 ([a, b]) and strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Assume that Fkix0 := Dxkα0 −;g f i , for k = 1, ..., n, fulfill

28

2 Iterated Generalized Fractional Ostrowski …

 jα x0 Fkix0 ∈ C 1 ([a, x0 ] , A) and F(n+1)i ∈ C ([a, x0 ] , A) and Dx0 −;g f i (x0 ) = 0, j = x0 := Dxkα0 +;g f i , k = 1, ..., n, fulfill 2, ..., n; i = 1, ..., r . Similarly, we assume that G ki  jα x0 x0 G ki ∈ C 1 ([x0 , b] , A) and G (n+1)i ∈ C ([x0 , b] , A) and Dx0 +;g f i (x0 ) = 0, j = 2, ..., n; i = 1, ..., r. Denote by  ( f 1 , ..., fr ) (x0 ) :=

⎞ ⎞ ⎞ ⎤ ⎛ ⎛ ⎛   r r r

⎟ ⎟ ⎟ ⎥ ⎢ b ⎜ ⎜ b ⎜ ⎟ f i (x) d x − ⎜ ⎥ ⎟ ⎢ ⎜ ⎜ f f j (x)⎟ (x) j ⎠ ⎠ d x ⎠ f i (x0 )⎦ . ⎣ a ⎝ ⎝ a ⎝ ⎡

i=1

j=1 j =i

(2.28)

j=1 j =i

Then

1  ((n + 1) α + 1) i=1 r

 ( f 1 , ..., fr ) (x0 ) ≤

⎛ ⎜ (g (x0 ) − g (a))(n+1)α ⎜ ⎝ ⎡   ⎢  (n+1)α  ⎢ D f    i x +;g ⎣ 0



   (n+1)α   Dx0 −;g f i  ⎞

⎛ x0

a

⎜ ⎜ ⎝

r  j=1 j =i

∞,[a,x0 ]

⎞⎤

 ⎟ ⎟⎥  f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦

⎛  ⎜ (n+1)α ⎜ (g (b) − g (x0 )) ⎝ ∞,[x0 ,b]

⎞

⎛ b x0

(2.29) ⎞⎤⎤

r  ⎟ ⎟⎥⎥ ⎜ ⎜  f j (x)⎟ d x ⎟⎥⎥ . ⎠ ⎠⎦⎦ ⎝ j=1 j =i

Proof By Theorem 2.11 we obtain f i (x) − f i (x0 ) =

 x0  1 (n+1)α (g (t) − g (x))(n+1)α−1 g  (t) Dx −;g f i (t) dt, 0  ((n + 1) α) x

(2.30)

∀ x ∈ [a, x0 ] , i = 1, ..., r. Also, by Theorem 2.9, we get f i (x) − f i (x0 ) =

 x  1 (n+1)α (g (x) − g (t))(n+1)α−1 g  (t) Dx +;g f i (t) dt, 0  ((n + 1) α) x0

(2.31)

∀ x ∈ [x0 , b] , i = 1, ..., r. Left multiplying (2.30) and (2.31) with



r j=1 j =i

f j (x) we get, respectively,

2.4 Main Results

29

⎞

⎛ ⎜ ⎜ ⎝

r  j=1 j =i

⎟ ⎜ ⎜ f j (x)⎟ ⎠ f i (x) − ⎝

f j (x) 

j=1 j =i

 ((n + 1) α)

⎜ ⎜ ⎝

x

r  j=1 j =i

f j (x) 

j=1 j =i

j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

(2.32)

 f (t) dt, (g (t) − g (x))(n+1)α−1 g  (t) Dx(n+1)α i 0 −;g

x0

∀ x ∈ [a, x0 ] , i = 1, ..., r, and ⎛

r

r 



r



⎞

⎛

 ((n + 1) α)

x x0

⎞

⎞

⎛

⎟ ⎜ ⎜ f j (x)⎟ ⎠ f i (x) − ⎝

r  j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

 f (t) dt, (g (x) − g (t))(n+1)α−1 g  (t) Dx(n+1)α i 0 +;g

(2.33)

∀ x ∈ [x0 , b] , i = 1, ..., r. Adding (2.32) and (2.33) as separate groups, we obtain ⎞

⎛

⎞

⎛

r r r r

⎟ ⎟ ⎜ ⎜ ⎟ f i (x) − ⎜ ⎜ f f j (x)⎟ (x) j ⎠ ⎠ f i (x0 ) = ⎝ ⎝ i=1

r i=1

j=1 j =i

i=1



r j=1 j =i

f j (x) 

 ((n + 1) α) ∀ x ∈ [a, x0 ] , and

i=1

r j=1 j =i

⎜ ⎜ ⎝

 f i (t) dt, (g (t) − g (x))(n+1)α−1 g  (t) Dx(n+1)α 0 −;g (2.34)

⎞ r  j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x) −

⎞

⎛ r

i=1

⎜ ⎜ ⎝

r  j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

f j (x) 

 ((n + 1) α) ∀ x ∈ [x0 , b] .

x0 x

⎛ r

r i=1

j=1 j =i

x x0

 f (g (x) − g (t))(n+1)α−1 g  (t) Dx(n+1)α i (t) dt, (2.35) 0 +;g

30

2 Iterated Generalized Fractional Ostrowski …

Next, we integrate (2.34) and (2.35) with respect to x ∈ [a, b]. We have ⎜ ⎜ ⎝

a

i=1

⎞

⎛

r  x0

r  j=1 j =i

⎛

⎟ f j (x)⎟ ⎠ f i (x) d x −

r

i=1

⎜ ⎜ ⎝



x0 x

r 

x0

a

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) =

j=1 j =i

⎛

 r

⎢ 1 ⎢ ⎣  ((n + 1) α) i=1

⎜ ⎜ ⎝

x0

a

⎡

⎞

⎛



⎞

r ⎜ ⎟ ⎜ f j (x)⎟ ⎝ ⎠ j=1 j =i

!  dt dx , f (t) (g (t) − g (x))(n+1)α−1 g  (t) Dx(n+1)α i 0 −;g

(2.36)

and r  b

x0

i=1

⎞

⎛ ⎜ ⎜ ⎝

r  j=1 j =i

⎛

⎟ f j (x)⎟ ⎠ f i (x) d x −

r

i=1

⎜ ⎜ ⎝

1  ((n + 1) α) 

x

(g (x) − g (t))

(n+1)α−1

i=1



g (t)

b x0



x0

⎜ ⎜ ⎝

r  j=1 j =i

⎜ ⎜ ⎝

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) = ⎞

⎛



⎢ ⎢ ⎣

b x0

⎡ r

⎞

⎛



r  j=1 j =i

⎟ f j (x)⎟ ⎠

fi Dx(n+1)α 0 +;g





!

(t) dt d x .

(2.37)

Finally, adding (2.36) and (2.37) we obtain the useful identity  ( f 1 , ..., fr ) (x0 ) := ⎡ r

i=1

⎢ ⎢ ⎣

 a

⎞

⎛ b

⎜ ⎜ ⎝

r  j=1 j =i

⎞ ⎞ ⎤ ⎛ ⎛  b  r ⎟ ⎟ ⎟ ⎥ ⎜ ⎜ ⎥ ⎟ ⎜ ⎜ f j (x)⎟ f j (x)⎟ ⎠ f i (x) d x − ⎝ a ⎝ ⎠ d x ⎠ f i (x0 )⎦ = j=1 j =i

⎡⎡ 1  ((n + 1) α)

r

i=1

⎢⎢ ⎢⎢ ⎣⎣

 a

⎞

⎛ x0

⎜ ⎜ ⎝

r  j=1 j =i

⎟ f j (x)⎟ ⎠

2.4 Main Results

31



x0 x

⎡

!  dt dx f (t) (g (t) − g (x))(n+1)α−1 g  (t) Dx(n+1)α i 0 −;g ⎤⎤

⎞

⎛

⎥⎥ r ⎟  x ⎢ b ⎜   ⎟ ⎜ ⎥⎥ ⎢ (n+1)α  (n+1)α−1 +⎢ f j (x)⎟ g (t) Dx +;g f i (t) dt d x ⎥⎥ . (g (x) − g (t)) ⎜ 0 ⎠ x0 ⎦⎦ ⎣ x0 ⎝ j=1 j =i

(2.38) Therefore, we get that ⎞ ⎡⎡ ⎛    x0 r ⎟ ⎢⎢ ⎜ 1 ⎢⎢ ⎜  ( f 1 , ..., fr ) (x0 ) ≤ f j (x)⎟  ⎣ ⎠ ⎣ ⎝  ((n + 1) α) i=1  a j=1  j =i r



x0

(g (t) − g (x))

(n+1)α−1



g (t)



x

Dx(n+1)α fi 0 −;g





!  (t) dt d x  +

⎡ ⎞ ⎛ ⎤⎤    

⎥ r ⎥ ⎢ b ⎜  ⎟  x  ⎢ ⎟ ⎜ ⎥⎥ (n+1)α f j (x)⎟ (g (x) − g (t))(n+1)α−1 g  (t) Dx +;g f i (t) dt d x ⎥⎥ ⎢ ⎜ 0 ⎣ x0 ⎝ ⎠ x0 ⎦⎦   j=1   j =i

(2.39) ⎛ ⎞    r r

⎟ ⎢⎢ x0 ⎜ 1 ⎜ ⎢⎢ f j (x)⎟ ≤  ⎝ ⎠ ⎣ ⎣  ((n + 1) α) i=1 a  j=1  j =i ⎡⎡



x0 x

 !    dx + dt f (t) (g (t) − g (x))(n+1)α−1 g  (t) Dx(n+1)α i  0 −;g

⎛  ⎤⎤ ⎞    r  

  ⎥⎥ ⎟ ⎢ b ⎜   x ⎜  ⎥⎥ ⎟ ⎢ (n+1)α  (n+1)α−1 f j (x)⎟ g (t) Dx +;g f i (t) dt  d x ⎥⎥ (g (x) − g (t)) ⎜ ⎢ 0  ⎦⎦ ⎠ x0 ⎣ x0 ⎝  j=1   j =i  ⎡

⎡⎡



x0 x



⎢⎢ 1 ⎢⎢  ((n + 1) α) i=1 ⎣⎣ r

≤

a

⎞

⎛ x0

r ⎜  ⎟ ⎜  f j (x)⎟ ⎠ ⎝ j=1 j =i

  !   f (g (t) − g (x))(n+1)α−1 g  (t)  Dx(n+1)α i (t) dt d x + 0 −;g

32

2 Iterated Generalized Fractional Ostrowski …

⎡

⎞

⎛

⎤⎤

 x r  ⎢ b ⎜    ⎥⎥ ⎟ ⎢ ⎜   (n+1)α ⎥⎥  f j (x)⎟ (g (x) − g (t))(n+1)α−1 g  (t)  Dx +;g f i (t) dt d x ⎥⎥ ⎟ ⎢ ⎜ 0 ⎠ x0 ⎣ x0 ⎝ ⎦⎦ j=1 j =i

=: (∗) .

(2.40)

 ( f 1 , ..., fr ) (x0 ) ≤ (∗) .

(2.41)

Hence it holds

We have that

1  ((n + 1) α + 1)

(∗) ≤ ⎡⎡ r  

⎢⎢  (n+1)α  ⎢⎢ D f    i x −;g ⎣⎣ 0 ∞,[a,x0 ]



i=1

⎡   ⎢  (n+1)α  D +⎢ f    i x0 +;g ⎣

 ∞,[x0 ,b]

⎞

⎛ x0

a

r  ⎟ ⎜ ⎥ ⎜  f j (x)⎟ (g (x0 ) − g (x))(n+1)α d x ⎥ ⎠ ⎝ ⎦ j=1 j =i

x0

r ⎜ ⎥⎥  ⎟ ⎜  f j (x)⎟ (g (x) − g (x0 ))(n+1)α d x ⎥⎥ ≤ ⎠ ⎝ ⎦⎦ j=1 j =i

1  ((n + 1) α + 1) i=1 r

⎛ ⎜ (g (x0 ) − g (a))(n+1)α ⎜ ⎝

   (n+1)α   Dx0 −;g f i   a

x0

⎞⎤

r ⎟ ⎟⎥  ⎜ ⎜  f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦ ⎝

(2.42)

j=1 j =i

⎛ ∞,[x0 ,b]

∞,[a,x0 ]

⎞

⎛

⎡   ⎢  (n+1)α  ⎢ D f    i x0 +;g ⎣

⎤⎤

⎞

⎛ b

⎤

⎜ (g (b) − g (x0 ))(n+1)α ⎜ ⎝



⎞

⎛ b x0

⎜ ⎜ ⎝

r  j=1 j =i

⎞⎤⎤

⎟ ⎟⎥⎥   f j (x)⎟ d x ⎟⎥⎥ , ⎠ ⎠⎦⎦ 

proving (2.29). Next comes an L 1 estimate. Theorem 2.18 All as in Theorem 2.17, plus

1 (n+1)

 ( f 1 , ..., fr ) (x0 ) ≤

≤ α < 1, n ∈ N. Then

  g  ∞,[a,b]  ((n + 1) α)

2.4 Main Results

33

⎡⎡ r  

⎢⎢  (n+1)α  ⎢⎢ ⎣⎣ Dx0 −;g f i  i=1

⎛  ⎜ (g (x0 ) − g (a))(n+1)α−1 ⎜ ⎝ L 1 ([a,x0 ])

a

⎡   ⎢  (n+1)α  D +⎢ f    i +;g x ⎣ 0

L 1 ([x0 ,b])

(g (b) − g (x0 ))(n+1)α−1

⎞

⎛ x0

j=1 j =i

⎞

⎛



b x0

⎞⎤

r ⎟ ⎟⎥  ⎜  f j (x)⎟ d x ⎟⎥ ⎜ ⎠ ⎠⎦ ⎝

⎜ ⎜ ⎝

r  j=1 j =i

⎤⎤

 ⎟ ⎥⎥  f j (x)⎟ d x ⎥⎥ . ⎠ ⎦⎦ (2.43)

Proof We have that (by (2.40), (2.41)) (∗) ≤ ⎡⎡

  g 

∞,[a,b]

 ((n + 1) α) ⎞

⎛

⎤

 x0 ⎜  r  ⎥  ⎟  ⎢⎢ (n+1)α ⎜ ⎥  f j (x)⎟ ⎟ (g (x0 ) − g (x))(n+1)α−1 d x ⎥ ⎢⎢ Dx0 −;g f i  ⎜ L 1 ([a,x0 ]) a ⎝ ⎠ ⎣⎣ ⎦

r ⎢⎢

i=1

j=1 j =i

⎡   ⎢  (n+1)α  D +⎢ f    i x0 +;g ⎣

 L 1 ([x0 ,b])

b x0

≤

r

i=1

⎡⎡   ⎢⎢  (n+1)α ⎢⎢ ⎣⎣|| Dx0 −;g f i ||

⎞

⎛

⎤⎤

r  ⎟ ⎜ ⎥⎥ ⎜  f j (x)⎟ (g (x) − g (x0 ))(n+1)α−1 d x ⎥⎥ ⎠ ⎝ ⎦⎦ j=1 j =i

  g 

∞,[a,b]

 ((n + 1) α)

⎛  ⎜ (g (x0 ) − g (a))(n+1)α−1 ⎜ ⎝ L 1 ([a,x0 ])

⎞

⎛ x0

a

⎜ ⎜ ⎝

r  j=1 j =i

⎞⎤

⎟ ⎟⎥   f j (x)⎟ d x ⎟⎥ ⎠ ⎠⎦

(2.44) ⎡

⎛

⎞

⎛

⎞⎤⎤

r  ⎥⎥ ⎜ b ⎜  ⎢  ⎟ ⎟  ⎟⎥⎥ ⎜ ⎜ ⎢ (n+1)α  f j (x)⎟ d x ⎟⎥⎥ , + ⎢ Dx +;g f i  (g (b) − g (x0 ))(n+1)α−1 ⎜ ⎟ ⎜ 0 L 1 ([x0 ,b]) ⎠ ⎠⎦⎦ ⎝ x0 ⎝ ⎣ j=1 j =i



proving the claim. An L p estimate follows. Theorem 2.19 All as in Theorem 2.17, plus p, q > 1 : α < 1. Then

1 p

+

1 q

= 1, and

1 q(n+1)


0 in (0, h). Then  h  h     2 x (t) x  (t) dt ≤ h x (t) dt. (3.2) 4 0 0 In (3.2), the constant optimal function

h 4

is the best possible. Inequality (3.2) holds as equality for the x (t) =

ct, 0 ≤ t ≤ h2 , c (h − t) , h2 ≤ t ≤ h,

(3.3)

where c > 0 is an arbitrary constant. Opial-type inequalities are used a lot in proving uniqueness of solutions to differential equations and also to give upper bounds to their solutions. For an extensive study about fractional Opial inequalities see the author’s monograph [1]. In this chapter we also derive Opial type inequalities for Banach algebra valued functions with respect to their Canavati type generalized left and right fractional derivatives. We include applications for Ostrowski and Opial inequalities. We finish the chapter with related Grüss type inequalities.

3.2 Background on Vectorial Generalized Canavati Fractional Calculus All in this section come from [5], pp. 109–115 and [4]. Let g : [a, b] → R be a strictly increasing function. such that g ∈ C 1 ([a, b]), and −1 g ∈ C n ([g(a), g(b)]), n ∈ N, (X, ·) is a Banach space. Let f ∈ C n ([a, b] , X ), and call l := f ◦ g −1 : [g (a) , g (b)] → X . It is clear that l, l  , ..., l (n) are continuous functions from [g (a) , g (b)] into f ([a, b]) ⊆ X. Let ν ≥ 1 such that [ν] = n, n ∈ N as above, where [·] is the integral part of the number. Clearly when 0 < ν < 1, [ν] = 0. (I) Let h ∈ C ([g (a) , g (b)] , X ), we define the left Riemann-Liouville Bochner fractional integral as 

 Jνz0 h (z) :=

1  (ν)



z z0

(z − t)ν−1 h (t) dt,

(3.4)

3.2 Background on Vectorial Generalized Canavati Fractional Calculus

47

∞ for g (a) ≤ z 0 ≤ z ≤ g (b), where  is the gamma function;  (ν) = 0 e−t t ν−1 dt. We set J0z0 h = h. ν Let α := ν − [ν] (0 < α < 1). We define the subspace Cg(x ([g (a) , g (b)] , X ) 0) [ν] of C ([g (a) , g (b)] , X ), where x0 ∈ [a, b] as: ν Cg(x ([g (a) , g (b)] , X ) = 0)



g(x ) h ∈ C [ν] ([g (a) , g (b)] , X ) : J1−α0 h ([ν]) ∈ C 1 ([g (x0 ) , g (b)] , X ) .

(3.5)

ν So let h ∈ Cg(x ([g (a) , g (b)] , X ), we define the left g-generalized X -valued 0) fractional derivative of h of order ν, of Canavati type, over [g (x0 ) , g (b)] as



g(x0 ) ([ν]) ν h := J h . Dg(x 1−α 0)

(3.6)

ν Clearly, for h ∈ Cg(x ([g (a) , g (b)] , X ) , there exists 0)



 ν Dg(x h (z) = 0)

d 1  (1 − α) dz



z

g(x0 )

(z − t)−α h ([ν]) (t) dt,

(3.7)

for all g (x0 ) ≤ z ≤ g (b). ν In particular, when f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), we have that 0) 

ν Dg(x 0)



f ◦g

−1



d 1 (z) =  (1 − α) dz



z g(x0 )

 ([ν]) (z − t)−α f ◦ g −1 (t) dt, (3.8)

   (n) n f ◦ g −1 = f ◦ g −1 and for all g (x0 ) ≤ z ≤ g (b). We have that Dg(x ) 0   0 −1 −1 = f ◦ g , see [4]. Dg(x0 ) f ◦ g ν By [4], we have for f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ) , where x0 ∈ [a, b] the 0) following left generalized g-fractional, of Canavati type,X -valued Taylor’s formula: ν Theorem 3.3 Let f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), where x0 ∈ [a, b] is fixed. 0) (i) If ν ≥ 1, then

f (x) − f (x0 ) =

[ν]−1  k=1

1  (ν)



g(x)

g(x0 )

for all x0 ≤ x ≤ b. (ii) If 0 < ν < 1, we get



f ◦ g −1

(k) k!

(g (x0 ))

(g (x) − g (x0 ))k +

 ν   f ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(3.9)

48

3 Generalized Canavati Fractional Ostrowski …

1 f (x) =  (ν)



g(x) g(x0 )

 ν   f ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(3.10)

for all x0 ≤ x ≤ b. (II) Let h ∈ C ([g (a) , g (b)] , X ), we define the right Riemann-Liouville Bochner fractional integral as 

Jzν0 − h



1 (z) :=  (ν)



z0

(t − z)ν−1 h (t) dt,

(3.11)

z

for g (a) ≤ z ≤ z 0 ≤ g (b) . We set Jz00 − h = h. ν Let α := ν − [ν] (0 < α < 1). We define the subspace Cg(x ([g (a) , g (b)] , X ) 0 )− [ν] of C ([g (a) , g (b)] , X ), where x0 ∈ [a, b] as: ν Cg(x ([g (a) , g (b)] , X ) := 0 )−



1−α 1 ([ν]) h ∈ C [ν] ([g (a) , g (b)] , X ) : Jg(x h ∈ C , g , X . ([g (a) (x )] ) 0 )− 0

(3.12)

ν So let h ∈ Cg(x ([g (a) , g (b)] , X ), we define the right g-generalized X -valued 0 )− fractional derivative of h of order ν, of Canavati type, over [g (a) , g (x0 )] as

 1−α ν n−1 ([ν]) Dg(x J h := h . (−1) g(x0 )− 0 )−

(3.13)

ν Clearly, for h ∈ Cg(x ([g (a) , g (b)] , X ) , there exists 0 )−



 (−1)n−1 d ν Dg(x h = (z) 0 )−  (1 − α) dz



g(x0 )

(t − z)−α h ([ν]) (t) dt,

(3.14)

z

for all g (a) ≤ z ≤ g (x0 ) ≤ g (b) . ν In particular, when f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), we have that 0 )− 

  (−1)n−1 d ν Dg(x f ◦ g −1 (z) = 0 )−  (1 − α) dz



g(x0 )

 ([ν]) (t − z)−α f ◦ g −1 (t) dt,

z

(3.15)

for all g (a) ≤ z ≤ g (x0 ) ≤ g (b). We get that 

   (n) n f ◦ g −1 (z) = (−1)n f ◦ g −1 Dg(x (z) 0 )−

    0 f ◦ g −1 (z) = f ◦ g −1 (z), all z ∈ [g (a) , g (b)] , see [4]. and Dg(x 0 )−

(3.16)

3.2 Background on Vectorial Generalized Canavati Fractional Calculus

49

ν By [4], we have for f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ) , where x0 ∈ [a, b] is 0 )− fixed, the following right generalized g-fractional, of Canavati type, X -valued Taylor’s formula: ν Theorem 3.4 Let f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), where x0 ∈ [a, b] is fixed. 0 )− (i) If ν ≥ 1, then

f (x) − f (x0 ) =

[ν]−1  k=1

1  (ν)



g(x0 ) g(x)



f ◦ g −1

(k) k!

(g (x0 ))

(g (x) − g (x0 ))k +

 ν   f ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

(3.17)

for all a ≤ x ≤ x0 , (ii) If 0 < ν < 1, we get f (x) =

1  (ν)



g(x0 ) g(x)

 ν   f ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

(3.18)

all a ≤ x ≤ x0 . (III) Denote by mν ν ν ν Dg(x = Dg(x Dg(x ...Dg(x (m -times), m ∈ N. 0) 0) 0) 0)

(3.19)

We mention the following modified and generalized left X -valued fractional Taylor’s formula of Canavati type: 1 −1 Theorem 3.5 Let f ∈ C 1 ([a, b]  b]), strictly increasing: g ∈

, X ), g ∈ C ([a,  iν ν C 1 ([g (a) , g (b)]). Assume that Dg(x f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), 0 < 0) 0) ν < 1, x0 ∈ [a, b], for i = 0, 1, ..., m. Then

f (x) =

1  ((m + 1) ν)



g(x) g(x0 )

 (m+1)ν  −1 f ◦ g (z) dz, (g (x) − z)(m+1)ν−1 Dg(x 0) (3.20)

all x0 ≤ x ≤ b. (IV) Denote by mν ν ν ν Dg(x = Dg(x Dg(x ...Dg(x (m times), m ∈ N. 0 )− 0 )− 0 )− 0 )−

(3.21)

We mention the following modified and generalized right X -valued fractional Taylor’s formula of Canavati type:

50

3 Generalized Canavati Fractional Ostrowski …

−1 Theorem 3.6 Let f ∈ C 1 ([a, b] , X ), g ∈ C 1 ([a, b]),  strictly increasing: g ∈   iν ν C 1 ([g (a) , g (b)]). Assume that Dg(x f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), 0 )− 0 )− 0 < ν < 1, x0 ∈ [a, b], for all i = 0, 1, ..., m. Then

f (x) =

1  ((m + 1) ν)



g(x0 ) g(x)

 (m+1)ν  −1 f ◦ g (z − g (x))(m+1)ν−1 Dg(x (z) dz, 0 )− (3.22)

all a ≤ x ≤ x0 ≤ b.

3.3 Banach Algebras Background All here come from [11]. We need Definition 3.7 ([11], p. 245) A complex algebra is a vector space A over the complex filed C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(3.23)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(3.24)

α (x y) = (αx) y = x (αy) ,

(3.25)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A)

(3.26)

and if A contains a unit element e such that xe = ex = x (x ∈ A)

(3.27)

e = 1,

(3.28)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make

3.4 Main Results

51

Remark 3.8 Commutativity of A will be explicited stated when needed. There exists at most one e ∈ A that satisfies (3.27). Inequality (3.26) makes multiplication to be continuous, more precisely left and right continuous, see [11], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [11], pp. 247–248, Sect. 10.3. We also make Remark 3.9 Next we mention about integration of A-valued functions, see [11], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some compact

Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [11], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   f dμ = x f ( p) dμ ( p) (3.29) x Q



and

Q

  f dμ x = Q

f ( p) x dμ ( p) .

(3.30)

Q

The Bochner integrals we will involve in our chapter follow (3.29) and (3.30). Also, let f ∈ C ([a, b] , X ), where [a, b] ⊂ R, (X, ·) is a Banach space. By [5], p. 3, f is Bochner integrable.

3.4 Main Results We start with mixed generalized Canavati type fractional Ostrowski type inequalities for several functions over a Banach algebra. A uniform estimate follows. Theorem 3.10 Let (A, ·) be a Banach algebra, x0 ∈ [a, b] ⊂ R, ν ≥ 1, n = [ν], f i ∈ C n ([a, b] , A), i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]) strictly increasing, (k)  such that g −1 ∈ C n ([g (a) , g (b)]), with f i ◦ g −1 (g (x0 )) = 0, k = 1, ..., n − ν 1; i = 1, ..., r. Assume further that f i ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , A) ∩ 0 )− ν Cg(x0 ) ([g (a) , g (b)] , A), i=1,...,r. Denote by K ( f 1 , ..., fr ) (x0 ) := ⎡ r  i=1

⎢ ⎢ ⎣

 a

⎞

⎛ b

⎜ ⎜ ⎝

r  j=1 j =i

⎞ ⎞ ⎤ ⎛ ⎛  b  r ⎟ ⎟ ⎟ ⎥ ⎜ ⎜ ⎟ d x ⎟ f i (x0 )⎥ . ⎜ ⎜ f f j (x)⎟ f d x − (x) (x) i j ⎠ ⎠ ⎠ ⎦ ⎝ a ⎝ j=1 j =i

(3.31)

52

3 Generalized Canavati Fractional Ostrowski …

Then     1 −1   D ν K ( f 1 , ..., fr ) (x0 ) ≤ g(x0 )− f i ◦ g ∞,[g(a),g(x0 )]  (ν + 1) i=1 r

⎛ ⎜ (g (x0 ) − g (a))ν ⎜ ⎝

 a

⎞

⎛ x0

⎞⎤

r ⎟ ⎟ ⎥  ⎜ ⎜  f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦ ⎝

(3.32)

j=1 j =i

⎡

⎛

⎞

⎛

⎞⎤⎤

r  ⎥⎥ ⎢ ⎜ b ⎜ 

 ⎟ ⎟ ⎟⎥⎥  ⎢ ν ⎜ ⎜  f j (x)⎟ (g (b) − g (x0 ))ν ⎜ ⎟ d x ⎟⎥⎥ . ⎢Dg(x0 ) f i ◦ g −1  ⎜ ∞,[g(x0 ),g(b)] ⎠ ⎠⎦⎦ ⎣ ⎝ x0 ⎝ j=1 j =i

 (k) Proof Since f i ◦ g −1 (g (x0 )) = 0, k = 1, ..., [ν] − 1; i = 1, ..., r ; we have by Theorem 3.3 that f i (x) − f i (x0 ) =

1  (ν)



g(x)

g(x0 )

 ν   f i ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(3.33)

∀ x ∈ [x0 , b] , and by Theorem 3.4 that 1 f i (x) − f i (x0 ) =  (ν)



g(x0 ) g(x)

 ν   f i ◦ g −1 (t) dt, (3.34) (t − g (x))ν−1 Dg(x 0 )−

∀ x ∈ [a, x0 ] , for all i = 1, ..., r.

 r j=1 j =i

Left multiplying (3.33) and (3.34) with ⎛

⎞

 f j (x) we get, respectively, ⎞

⎛

r r ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ f i (x) − ⎜ f f j (x)⎟ (x) j ⎝ ⎠ ⎠ f i (x0 ) = ⎝ j=1 j =i

j=1 j =i



 r j=1 j =i

f j (x) 

 (ν) ∀ x ∈ [x0 , b] , and

g(x)

g(x0 )

 ν   f i ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(3.35)

3.4 Main Results

53

⎞

⎛ ⎜ ⎜ ⎝

r  j=1 j =i

⎞

⎛

⎟ ⎜ ⎜ f j (x)⎟ ⎠ f i (x) − ⎝

r  j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

(3.36)



 r j=1 j =i

f j (x) 

 (ν)

g(x0 ) g(x)

 ν   f i ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

∀ x ∈ [a, x0 ] , for all i = 1, ..., r. Adding (3.35) and (3.36) as separate groups, we obtain ⎞

⎛

⎞

⎛

r r r r   ⎟ ⎟ ⎜ ⎜ ⎟ f i (x) − ⎜ ⎜ f f j (x)⎟ (x) j ⎠ ⎠ f i (x0 ) = ⎝ ⎝ i=1

j=1 j =i

⎞

⎛ 1  (ν)

r  i=1

⎜ ⎜ ⎝

r  j=1 j =i

∀ x ∈ [x0 , b] , and

i=1

i=1



⎜ ⎜ ⎝

⎜ ⎜ ⎝

r  j=1 j =i

g(x)

g(x0 )

j=1 j =i



g(x0 ) g(x)

r  i=1

⎜ ⎜ ⎝

(3.37)

⎞

⎛

⎟ f j (x)⎟ ⎠ f i (x) −

⎟ f j (x)⎟ ⎠

j=1 j =i

 ν   f i ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

⎞ r 

⎞

⎛ 1  (ν)

⎟ f j (x)⎟ ⎠

⎛ r 

r 

i=1

r  j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

 ν   f i ◦ g −1 (t) dt, (3.38) (t − g (x))ν−1 Dg(x 0 )−

∀ x ∈ [a, x0 ] . Next, we integrate (3.37) and (3.38) with respect to x ∈ [a, b]. We have r  b  i=1

x0

⎞

⎛ ⎜ ⎜ ⎝

r  j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x) d x −

⎛ r  i=1

⎜ ⎜ ⎝



⎞

⎛ b x0

⎜ ⎜ ⎝

r  j=1 j =i

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) =

54

3 Generalized Canavati Fractional Ostrowski …

⎡ 1  (ν)

⎞

⎛

r ⎢ b ⎜  r 

⎟ ⎟ ⎢ ⎜ f j (x)⎟ ⎢ ⎜ ⎠ ⎣ x0 ⎝ i=1 j=1

!

⎤

"

⎥

 ⎥ ν f i ◦ g −1 (t) dt d x ⎥ , (g (x) − t)ν−1 Dg(x 0) ⎦ g(x0 ) g(x)

j =i

(3.39) and ⎜ ⎜ ⎝

a

i=1

⎞

⎛

r  x0 

r  j=1 j =i

⎡

⎛

⎟ f j (x)⎟ ⎠ f i (x) d x −

⎛

r  i=1

⎜ ⎜ ⎝

 a

⎞

⎛ x0

⎜ ⎜ ⎝

r  j=1 j =i

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) = ⎤

⎞

! "  r ⎥ ⎜ r ⎟  g(x0 )

 1 ⎢ ⎥ ⎢ x0 ⎜  ⎟ ν −1 f j (x)⎟ ◦ g f dt dx⎥ , (t − g (x))ν−1 Dg(x (t) ⎢ ⎜ i 0 )− ⎦ ⎣ a ⎝ ⎠ g(x)  (ν) i=1

j=1 j =i

(3.40) Finally, adding (3.39) and (3.40) we obtain the useful identity K ( f 1 , ..., fr ) (x0 ) := ⎡ r  i=1

⎛



⎢ ⎢ ⎣

b

a

⎜ ⎜ ⎝

⎛ ⎞ ⎞ ⎤ ⎛  b  r ⎜ ⎟ ⎟ ⎟ ⎥ ⎜ ⎜ ⎥ ⎜ ⎟ f j (x)⎟ f j (x)⎟ ⎠ f i (x) d x − ⎝ a ⎝ ⎠ d x ⎠ f i (x0 )⎦ = ⎞

r  j=1 j =i

j=1 j =i

⎤ ⎡⎡ ⎛ ⎞ " !  x0  r r 

g(x ) 0  ⎥ ⎢⎢ ⎜ ⎟   1 ν ⎢⎢ ⎜ f i ◦ g −1 (t) dt d x ⎥ f j (x)⎟ (t − g (x))ν−1 Dg(x ⎦ ⎣⎣ ⎝ ⎠ 0 )−  (ν) a g(x) i=1

⎡ ⎢ +⎢ ⎣



j=1 j =i

⎞

⎛ b x0

⎜ ⎜ ⎝

r  j=1 j =i

⎟ f j (x)⎟ ⎠



g(x) g(x0 )



⎤⎤

⎥⎥  ν   ⎥ f i ◦ g −1 (t) dt d x ⎥ (g (x) − t)ν−1 Dg(x 0) ⎦⎦ . (3.41)

Therefore, we get that K ( f 1 , ..., fr ) (x0 ) =  ⎞ ⎞ ⎞ ⎤ ⎡ ⎛ ⎛ ⎛       r r  r ⎢ b ⎜ ⎟ ⎟ ⎟ ⎥ ⎜ b ⎜ 1  ⎟ ⎟ ⎥ ⎟ ⎢ ⎜ ⎜ ⎜ f j (x)⎠ f i (x) d x − ⎝ f j (x)⎠ d x ⎠ f i (x0 )⎦   ≤  (ν) ⎣ ⎝ ⎝ a a  i=1  j=1 j=1   j =i j =i

3.4 Main Results

55

⎤ ⎞ ⎡⎡ ⎛   " !    x0  r r 

g(x )  0 ⎥ ⎟ ⎢⎢ ⎜    ν−1 ν −1 ⎥ ⎟ ⎢⎢ ⎜  f ◦ g f dt d x − g D (t) (x) (t (x)) j i g(x0 )− ⎦ ⎠ ⎣⎣ ⎝ g(x) a  i=1  j=1   j =i

⎡ ⎤ ⎤(3.42) ⎞ ⎛      ! " r ⎢ b ⎜  ⎥⎥ ⎟  g(x)

 ⎢ ⎥⎥ ⎟ ⎜ ν −1 f dt d x + ⎢ f j (x)⎟ ◦ g (t) (g (x) − t)ν−1 Dg(x ⎥⎥ ≤ ⎜ i ) 0 ⎣ x 0 ⎝ ⎦⎦ ⎠ g(x0 )   j=1   j =i  ⎤ ⎛ ⎞  " !   r

 g(x 0 ) ⎢⎢ x0 ⎜   ⎥ ⎟ 1 ν−1 ν −1 ⎢⎢  dx⎥ ⎟ ⎜ D f dt − g f ◦ g (t) (x) (t (x)) j i g(x 0 )−  ⎦ ⎝ ⎣⎣ ⎠  (ν) a g(x)   i=1   j=1 j =i ⎡⎡

r 



 ⎤(3.43) ⎛ ⎤ ⎞     ! " r  ⎥⎥ ⎟  g(x) ⎢ b ⎜ 

  ⎥⎥ ⎜ ⎟ ⎢ ν −1 f dt +⎢ f j (x)⎟ ◦ g (t) (g (x) − t)ν−1 Dg(x  d x ⎥⎥ ≤ ⎜ i 0)  ⎦⎦ ⎠ g(x0 ) ⎣ x0 ⎝   j=1   j =i ⎡

⎞ ⎤ ⎡⎡ ⎛ ! "  x0  r r  

 g(x ) 0   ⎟ ⎥  ⎢ ⎢ ⎜   1  ν  −1  f j (x)⎟ ⎢⎢ ⎜ (t − g (x))ν−1  Dg(x (t) dt d x ⎥ )− f i ◦ g ⎠ ⎦ ⎣ ⎣ ⎝ 0  (ν) a g(x) i=1

⎡  ⎢ +⎢ ⎣

j=1 j =i

⎞

⎛ b x0

⎜ ⎜ ⎝

r  j=1 j =i

⎟   f j (x)⎟ ⎠

!

⎤⎤ "  

g(x) ⎥⎥     ν ⎥ f i ◦ g −1 (t) dt d x ⎥ (g (x) − t)ν−1  Dg(x ⎦⎦ =: (∗) . 0)

g(x0 )

(3.44) Hence it holds K ( f 1 , ..., fr ) (x0 ) ≤ (∗) . We have that (∗) ≤

1  (ν + 1)

⎡⎡



r   ⎢⎢    ν −1  ⎢⎢  ⎣⎣ Dg(x0 )− f i ◦ g

∞,[g(a),g(x0 )] a

i=1

⎡  ⎢    ν −1  +⎢  ⎣ Dg(x0 ) f i ◦ g

∞,[g(x0 ),g(b)]

(3.45)



x0

⎤

r ⎥ ⎜ ⎟  ⎜  f j (x)⎟ (g (x0 ) − g (x))ν d x ⎥ ⎠ ⎦ ⎝ j=1 j =i

⎛ b

⎞

⎛ x0

⎞

⎤⎤

r ⎥⎥  ⎜ ⎟  f j (x)⎟ (g (x) − g (x0 ))ν d x ⎥⎥ ≤ ⎜ ⎦⎦ ⎝ ⎠ j=1 j =i

(3.46)

56

3 Generalized Canavati Fractional Ostrowski …

    1 −1   D ν g(x0 )− f i ◦ g ∞,[g(a),g(x0 )]  (ν + 1) i=1 r

⎛ ⎜ (g (x0 ) − g (a))ν ⎜ ⎝



⎞

⎛ ⎜ ⎜ ⎝

x0

a

r  j=1 j =i

⎞⎤

⎟ ⎟ ⎥   f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦

⎡

⎛

  ⎢ ⎜    ν −1  ν⎜ ⎢ − g ◦ g D f (g (b) (x ))     i 0 g(x0 ) ⎣ ⎝ ∞,[g(x0 ),g(b)]

⎛ b x0

(3.47) ⎞

⎞⎤⎤

r ⎟ ⎟⎥⎥  ⎜  f j (x)⎟ d x ⎟⎥⎥ , ⎜ ⎝ ⎠ ⎠⎦⎦ j=1 j =i



proving (3.32). Next comes an L 1 estimate. Theorem 3.11 All as in Theorem 3.10. Then |K ( f 1 , ..., fr ) (x0 ) ≤ ⎡⎡

 r   ⎢⎢    ν −1  ⎢⎢ D ◦ g f     i g(x0 )− ⎣⎣ L 1 ([g(a),g(x0 )])

⎡ 

 ⎢   ν −1 f +⎢ D ◦ g    i g(x 0 ) ⎣

b x0

⎤

r ⎟ ⎥  ⎜  f j (x)⎟ (g (x0 ) − g (x))ν−1 d x ⎥ ⎜ ⎠ ⎦ ⎝ j=1 j =i

⎛

 L 1 ([g(x 0 ),g(b)])

⎞

⎛ x0

a

i=1

1  (ν)

⎜ ⎜ ⎝

(3.48) ⎤ ⎤

⎞ r  j=1 j =i

 ⎟ ⎥⎥  f j (x)⎟ (g (x) − g (x0 ))ν−1 d x ⎥⎥ . ⎠ ⎦⎦

Proof We observe that (by (3.45) 1  (ν)

(∗) ≤ ⎡⎡ r  i=1



 ⎢⎢   ν −1 ⎢⎢ f D ◦ g    i g(x 0 )− ⎣⎣

⎡  ⎢    ν −1  +⎢  ⎣ Dg(x0 ) f i ◦ g

proving (3.48).

 L 1 ([g(a),g(x 0 )]) a

 L 1 ([g(x0 ),g(b)])

⎛ x0

⎜ ⎜ ⎝

⎛ b x0

⎜ ⎜ ⎝

⎞ r  j=1 j =i

 ⎟ ⎥  f j (x)⎟ (g (x0 ) − g (x))ν−1 d x ⎥ ⎠ ⎦

⎞ r  j=1 j =i

⎤

(3.49) ⎤⎤

 ⎥⎥ ⎟  f j (x)⎟ (g (x) − g (x0 ))ν−1 d x ⎥⎥ , ⎦⎦ ⎠



3.4 Main Results

57

An L p estimate follows. Theorem 3.12 All as in Theorem 3.10. Let now p, q > 1 : K ( f 1 , ..., fr ) (x0 ) ≤

1 p

⎛

 ⎢    ν −1  f +⎢ D ◦ g    i g(x ) ⎣ 0

⎜ ⎜



= 1. Then

1

( p (ν − 1) + 1) p  (ν) ⎞

⎛ x0

(g (x0 ) − g (x))

ν− q1

a

⎡

1 q

1

⎡⎡ ⎛  r    ⎢⎢ ⎜    ν −1  ⎢⎢ ⎜ D f ◦ g     i g(x0 )− ⎣⎣ q,[g(a),g(x0 )] ⎝ i=1

+

j=1 j =i

b

(g (x) − g (x0 ))

ν− q1

⎜ ⎜ ⎝

⎞(3.50) ⎤⎤

⎞

⎛

q,[g(x0 ),g(b)] ⎝ x0

⎞⎤

r ⎟ ⎟ ⎥ ⎜  ⎜  f j (x)⎟ d x ⎟⎥ ⎠ ⎠⎦ ⎝

r  j=1 j =i

 ⎟ ⎟⎥⎥  f j (x)⎟ d x ⎟⎥⎥ . ⎠ ⎠⎦ ⎦

Proof We have that ⎡⎡



⎢ 1 ⎢ ⎢⎢ (∗) ≤ ⎣  (ν) i=1 ⎣ r

(3.45)



g(x0 ) g(x)

⎡



⎢ ⎢ ⎣

a

⎞

⎛ x0

r  ⎟ ⎜ ⎜  f j (x)⎟ ⎠ ⎝ j=1 j =i

 ν  D

g(x0 )−

⎞

⎛ b x0



⎜ ⎜ ⎝



r  j=1 j =i

g(x0 ) g(x)

 ⎟  f j (x)⎟ ⎠

 ν  D

g(x0 )



fi ◦ g 

−1

g(x)

g(x0 )

fi ◦ g





g(x0 )

g(x)

q (t) dt

 q1

(g (x) − t)

 −1

q (t) dt

(t − g (x))

 1p p(ν−1)

dt

 dx +  1p

p(ν−1)

dt



 q1 dx

≤

(3.51)

⎤ ⎡⎡ ⎛ ⎞  x0  r r ν−1+ 1p      ⎥ ⎢ ⎢ ⎜  ⎟    1  ν  f j (x)⎟ (g (x0 ) − g (x)) ⎢⎢ ⎜ dx⎥  Dg(x0 )− f i ◦ g −1  1 ⎦ ⎣⎣ ⎝ ⎠ q,[g(a),g(x )]  (ν) 0 a ( p (ν − 1) + 1) p i=1 j=1 j =i

⎡ ⎢ +⎢ ⎣



⎤⎤ ⎞ 1 r    ⎥⎥ ⎟ (g (x) − g (x0 ))ν−1+ p   ⎜     ν ⎥  f j (x)⎟ ⎜ dx⎥  Dg(x0 ) f i ◦ g −1 (z) 1 ⎦⎦ ⎠ ⎝ q,[g(x ),g(b)] 0 ( p (ν − 1) + 1) p j=1 ⎛

b x0

j =i

58

3 Generalized Canavati Fractional Ostrowski …

1

= ⎡ r  i=1

1

( p (ν − 1) + 1) p  (ν)

⎛  ⎜ ⎜ q,[g(a),g(x 0 )] ⎝



 ⎢  ν  −1 ⎢ D f ◦ g    i g(x 0 )− ⎣

⎛ x0

(g (x0 ) − g (x))

ν− q1

a

⎛   ⎜    ν −1  ⎜ f +  Dg(x ◦ g   i 0) q,[g(x0 ),g(b)] ⎝

⎜ ⎜ ⎝

⎞ r  j=1 j =i

 ⎟ ⎟  f j (x)⎟ d x ⎟ ⎠ ⎠

⎛ b

ν− q1

(g (x) − g (x0 ))

x0

⎜ ⎜ ⎝

⎞ r  j=1 j =i

⎞

(3.52) ⎞ ⎤

 ⎟ ⎟⎥  f j (x)⎟ d x ⎟⎥ , ⎠ ⎠⎦



proving (3.50). Next we present a left generalized Canavati Opial type inequality:

Theorem 3.13 Let (A, ·) be a Banach algebra, x0 ∈ [a, b] ⊂ R, ν ≥ 1, n = [ν], f ∈ C n ([a, b] , A); g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ (k)  C n ([g (a) , g (b)]) , with f ◦ g −1 (g (x0 )) = 0, k = 0, 1, ..., n − 1. Assume furν ther that f ◦ g −1 ∈ Cg(x , g , A). Let also p, q > 1 : 1p + q1 = 1. Then ([g (a) (b)] 0) 

z g(x0 )

       −1  f ◦ g −1 (w) D ν (w)  dw ≤ g(x0 ) f ◦ g 

2− q (z − g (x0 ))ν+ p − q 1

1

1 1

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) + 2)] p

 ν  D

z



g(x0 )

g(x0 )

f ◦ g −1



(3.53) q (t) dt

 q2

,

for all g (x0 ) ≤ z ≤ g (b) . Proof By (3.9) and assumptions we get that 

f ◦g

−1



1 (z) =  (ν)



z g(x0 )

 ν   f ◦ g −1 (t) dt, (z − t)ν−1 Dg(x 0)

(3.54)

for all g (x0 ) ≤ z ≤ g (b) . By Hölder’s inequality we obtain     f ◦ g −1 (z) ≤ 1  (ν)



z g(x0 )

1  (ν)

(z − t)

p(ν−1)

p(ν−1)+1

1 (z − g (x0 )) p  (ν) ( p (ν − 1) + 1) 1p



z

g(x0 )

 ν    f ◦ g −1 (t) dt ≤ (z − t)ν−1  Dg(x 0)

 1p 

 ν  D

z

dt

g(x0 )

g(x0 )



z

g(x0 )

 ν  D

g(x0 )





f ◦g

f ◦g

−1



−1



q (t) dt

q (t) dt

 q1

.

 q1

=

(3.55)

3.4 Main Results

59



Call ϕ (z) := ϕ (g (x0 )) = 0. Thus

and

 ν  D

z

g(x0 )



g(x0 )

f ◦ g −1



q (t) dt,

(3.56)

 ν   q f ◦ g −1 (z) ≥ 0, ϕ  (z) =  Dg(x 0)

(3.57)

 ν       q1 f ◦ g −1 (z) ≥ 0, ϕ (z) =  Dg(x 0)

(3.58)

∀ z ∈ [g (x0 ) , g (b)] . Consequently, we get        −1  f ◦ g −1 (w) |||| D ν (w)  ≤ g(x0 ) f ◦ g p(ν−1)+1

1 1 (w − g (x0 )) p  ϕ (w) ϕ  (w) q , 1  (ν) ( p (ν − 1) + 1) p

(3.59)

∀ w ∈ [g (x0 ) , g (b)] . Then  z        (3.26) −1  f ◦ g −1 (w) D ν (w)  dw ≤ g(x0 ) f ◦ g g(x0 )



z g(x0 )

       −1  f ◦ g −1 (w)   D ν (w)  dw ≤ g(x0 ) f ◦ g 

1  (ν) ( p (ν − 1) + 1)

1 p

g(x0 )



1 1

 (ν) ( p (ν − 1) + 1) p

z

p(ν−1)+1 p

(w − g (x0 ))

 1 

z g(x0 )

(w − g (x0 ))

p(ν−1)+1

1 1

1



2− q (z − g (x0 ))ν+ p − q 1

1 1

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) + 2)] p

z

g(x0 )

z

p

dw

g(x0 )

(z − g (x0 ))

 (ν) ( p (ν − 1) + 1) p ( p (ν − 1) + 2) p 1

 1 ϕ (w) ϕ  (w) q dw ≤

 ν  D

g(x0 )

1 q ϕ (w) dϕ (w) =

p(ν−1)+2 p



(3.60)



f ◦g

ϕ 2 (z) 2

−1



 q1

=

q (t) dt

for all g (x0 ) ≤ z ≤ g (b), proving (3.53). It follows the corresponding right side fractional Opial type inequality:

 q2

,

(3.61) 

60

3 Generalized Canavati Fractional Ostrowski …

Theorem 3.14 All as in Theorem 3.13, however now it is ν Cg(x ([g (a) , g (b)] , A). Then 0 )− 

g(x0 ) z

2

− q1

f ◦ g −1 ∈

       −1  f ◦ g −1 (w) D ν (w)  dw ≤ g(x0 )− f ◦ g

(g (x0 ) − z)

!

ν+ 1p − q1 1

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) + 2)] p

g(x0 )  z

 q  ν   Dg(x0 )− f ◦ g −1 (t) dt

"2 q

,

(3.62)

for all g (a) ≤ z ≤ g (x0 ) . Proof Based on (3.17), and as similar to the proof of Theorem 3.13 is omitted.  It follows the modified generalized left A-valued fractional Opial inequality: Theorem 3.15 All as in Theorem 3.5 and let p, q > 1 : 1 < ν < 1. Then that (m+1)q 

z g(x0 )

1 p

+

1 q

= 1. Here we assume

   

(m+1)ν     Dg(x0 ) f ◦ g −1 (w)  dw ≤  f ◦ g −1 (w)

(3.63)

2− q (z − g (x0 ))(m+1)ν+ p − q 1

1

1 1

 ((m + 1) ν) [( p ((m + 1) ν − 1) + 1) ( p ((m + 1) ν − 1) + 2)] p 

z

g(x0 )

 q2    q  (m+1)ν  −1 , f ◦g (t) dt  Dg(x0 )

for all g (x0 ) ≤ z ≤ g (b) . 

Proof As in Theorem 3.13.

The corresponding modified generalized right A-valued fractional Opial inequality comes next: Theorem 3.16 All as in Theorem 3.6 and let p, q > 1 : 1 < ν < 1. Then that (m+1)q 

g(x0 ) z

1 p

+

1 q

= 1. Here we assume

   

(m+1)ν     Dg(x0 )− f ◦ g −1 (w)  dw ≤  f ◦ g −1 (w)

(3.64)

2− q (g (x0 ) − z)(m+1)ν+ p − q 1

1

1 1

 ((m + 1) ν) [( p ((m + 1) ν − 1) + 1) ( p ((m + 1) ν − 1) + 2)] p

3.5 Applications

61



g(x0 ) z

 q2     (m+1)ν  q −1 D f ◦ g dt , (t)  g(x0 )− 

for all g (a) ≤ z ≤ g (x0 ) . 

Proof As in Theorem 3.13.

3.5 Applications We make Remark 3.17 Assume in this section that ( A, ·) is a commutative Banach algebra. Then, we get that (3.31)



K ( f 1 , ..., fr ) (x0 ) = r a

⎛ b

⎝

⎛

⎞

r 

f j (x)⎠ d x −

j=1

r  i=1

⎜ ⎜ ⎝



⎞

⎛ b

a

⎜ ⎜ ⎝

r  j=1 j =i

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) ,

(3.65) x0 ∈ [a, b] . When r = 2, we find K ( f 1 , f 2 ) (x0 ) = 2

 b a

f 1 (x) f 2 (x) d x − f 1 (x0 )

 b a

f 2 (x) d x − f 2 (x0 )

 b a

f 1 (x) d x,

(3.66)

x0 ∈ [a, b] . We give Corollary 3.18 All as in Theorem 3.10, A is a commutative Banach algebra, r = 2. Then     1 −1   D ν g(x0 )− f i ◦ g ∞,[g(a),g(x0 )]  (ν + 1) i=1 2

K ( f 1 , f 2 ) (x0 ) ≤

⎛ ⎜ (g (x0 ) − g (a))ν ⎜ ⎝ ⎡  ⎢    ν −1  ⎢ ◦ g f D    i g(x0 ) ⎣

 a

⎞

⎛ x0

⎜ ⎜ ⎝

2  j=1 j =i

⎞⎤

 ⎟ ⎟⎥  f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦

⎛  ⎜ (g (b) − g (x0 ))ν ⎜ ⎝ ∞,[g(x0 ),g(b)]

⎛ b x0

⎜ ⎜ ⎝

⎞ 2  j=1 j =i

⎞⎤⎤

 ⎟ ⎟⎥⎥  f j (x)⎟ d x ⎟⎥⎥ . ⎠ ⎠⎦⎦

(3.67)

62

3 Generalized Canavati Fractional Ostrowski …



Proof By Theorem 3.10. We also give Corollary 3.19 All as in Corollary 3.18, for g (t) = et . Then K ( f 1 , f 2 ) (x0 ) ≤

2    ν  1  D x0 ( f i ◦ log) e − ∞,[ea ,e x0 ]  (ν + 1) i=1

⎛  x0 ν ⎜ e − ea ⎜ ⎝

 a

⎞

⎛ x0

⎜ ⎜ ⎝

2  j=1 j =i

⎡

⎞⎤

 ⎟ ⎟⎥  f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦ ⎛

⎢ ν   ⎜  b x0 ν ⎜ ⎢ D x0 ( f i ◦ log) − e e x b e 0 ∞,[e ,e ] ⎣ ⎝



⎞

⎛ b x0

⎜ ⎜ ⎝

(3.68)

2  j=1 j =i

⎞⎤⎤

 ⎟ ⎟⎥⎥  f j (x)⎟ d x ⎟⎥⎥ . ⎠ ⎠⎦⎦ 

Proof By Corollary 3.18. We add also the following: Corollary 3.20 (to Theorem 3.13) All as in Theorem 3.13 for g (t) = et . Then 

z e x0

    (( f ◦ log) (w)) D νx0 ( f ◦ log) (w)  dw ≤ e 

2− q (z − e x0 )ν+ p − q 1

1

1 1

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) + 2)] p

z e x0

 ν    D x0 ( f ◦ log) (t)q dt e

(3.69)  q2

,

for all e x0 ≤ z ≤ eb . Proof By Theorem 3.13.



3.6 Addendum We give the following generalized Canavati type fractional Grüss type inequalities for several functions over a Banach algebra. We start with a uniform estimate. Theorem 3.21 Let (A, ·) be a Banach algebra, x0 ∈ [a, b] ⊂ R, 1 ≤ ν < 2, f i ∈ C 1 ([a, b] , A), i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]) strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . Assume further that f i ◦ g −1 ∈ ν ν Cg(x ([g (a) , g (b)] , A) ∩ Cg(x ([g (a) , g (b)] , A), i=1,...,r. 0 )− 0)

3.6 Addendum

63

Assume that # M1 ( f 1 , ..., fr ) (x0 ) := max

i=1,...,r

 ν   f i ◦ g −1 ∞,[g(a),g(x sup  Dg(x 0 )−

x0 ∈[a,b]

 ν   f i ◦ g −1 ∞,[g(x sup  Dg(x 0)

x0 ∈[a,b]

Denote by



b

 ( f 1 , ..., fr ) :=

0 )]

,

$ < ∞.

0 ),g(b)]

(3.70)

K ( f 1 , ..., fr ) (x) d x =

a

⎡

i=1

⎛

⎛

 r  ⎢ ⎜ ⎢(b − a) ⎜ ⎣ ⎝

b

a

⎞

⎞

⎛

r ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ f j (x)⎟ ⎝ ⎠ f i (x) d x ⎠ − ⎝ j=1 j =i



⎛ b

a

⎞

⎞

r ⎜ ⎟ ⎟ ⎜ ⎟ f j (x)⎟ ⎝ ⎠ dx⎠

 a

j=1 j =i

b

⎤  ⎥ f i (x) d x ⎥ ⎦.

(3.71) Then  ( f 1 , ..., fr ) =  ⎞ ⎞ ⎛ ⎞ ⎞ ⎤ ⎡ ⎛ ⎛ ⎛    r  b     r r  ⎢ ⎟ ⎟ ⎜ b ⎜ ⎟ ⎟ ⎥ ⎜ b ⎜   ⎟ ⎜ ⎟ ⎢(b − a) ⎜ ⎜ ⎜ f j (x)⎟ f j (x)⎟ f i (x) d x ⎥  ⎠ f i (x) d x ⎠ − ⎝ ⎠ dx⎠ ⎦ ⎣ ⎝ ⎝ ⎝ a a a   i=1 j=1 j=1   j =i j =i

 ⎛     r r    M1 ( f 1 , ..., fr ) (b − a)2 (g (b) − g (a))ν  ⎜    ⎜ f j (x)  ≤  ⎝  (ν + 1)   i=1   j=1 j =i

⎞ ⎟ ⎟ . (3.72) ⎠

∞,[a,b]

Proof From (3.32) we get ⎛ R.H.S. (3.32) ≤

M1 ( f 1 , ..., fr ) (g (b) − g (a))ν  (ν + 1)

⎞

⎛

r  b ⎜ r  ⎜ ⎟ ⎟ ⎟ ⎜ ⎜  f j (x)⎟ ⎟ dx⎟ ≤ ⎜ ⎜ ⎠ ⎠ ⎝ a ⎝ i=1

j=1 j =i

     ν     ⎜ M1 ( f 1 , ..., fr ) (g (b) − g (a)) (b − a) ⎜   f j    ⎝  (ν + 1)  i=1  j=1   j =i ⎛

r  r 

∞,[a,b]

We have that

⎞

⎞ ⎟ ⎟ =: λ1 . (3.73) ⎠

64

3 Generalized Canavati Fractional Ostrowski …



b

 ( f 1 , ..., fr ) ≤

K ( f 1 , ..., fr ) (x0 ) d x0

((3.32),(3.73))



b

≤

a

λ1 d x =

a

 ⎛     r r    ⎜ M1 ( f 1 , ..., fr ) (g (b) − g (a))ν (b − a)2 ⎜ ⎜ ⎜  f j (x)  ⎝ ⎝  (ν + 1)   j=1 i=1   j =i ⎛

⎞⎞ ⎟⎟ ⎟⎟ , ⎠⎠

∞,[a,b]

proving (3.72).

(3.74) 

Next comes an L 1 -estimate. Theorem 3.22 All as in Theorem 3.21, however now we assume that #    f i ◦ g −1  sup  D ν M2 ( f 1 , ..., fr ) (x0 ) := max i=1,...,r

g(x0 )−

x0 ∈[a,b]

 ν   f i ◦ g −1  L sup  Dg(x 0)

x0 ∈[a,b]

L 1 ([g(a),g(x0 )])

,

$ < ∞.

1 ([g(x 0 ),g(b)])

(3.75)

Then  ( f 1 , ..., fr ) ≤

≤

M2 ( f 1 , ..., fr ) (b − a)2 (g (b) − g (a))ν−1  (ν)

 ⎛     r r    ⎜ ⎜  f j (x)  ⎝   j=1 i=1   j =i

⎞ ⎟ ⎟. ⎠

∞,[a,b]

(3.76)

Proof By (3.48) we get ⎛ R.H.S. (3.48) ≤

M2 ( f 1 , ..., fr ) (g (b) − g (a))ν−1  (ν)

⎞

⎛

⎞

r  b ⎜ r  ⎜ ⎟ ⎟ ⎟ ⎜ ⎜  f j (x)⎟ ⎟ dx⎟ ≤ ⎜ ⎜ ⎠ ⎠ ⎝ ⎝ a i=1

j=1 j =i

      r r     M2 ( f 1 , ..., fr ) (g (b) − g (a))ν−1 (b − a) ⎜   f j  ⎜   ⎝  (ν)  i=1  j=1   j =i ⎛

⎞ ⎟ ⎟ =: λ2 . (3.77) ⎠

∞,[a,b]

We have that   ( f 1 , ..., fr ) ≤ a

b

K ( f 1 , ..., fr ) (x0 ) d x0

((3.48),(3.76))



≤

a

b

λ2 d x =

3.6 Addendum

65

 ⎛     r ν−1 2 ⎜      ⎜ M2 ( f 1 , ..., fr ) (g (b) − g (a)) (b − a) ⎜ ⎜  f j (x)   ⎝ ⎝  (ν)   j=1 i=1   j =i ⎛

⎞⎞

r 

⎟⎟ ⎟⎟ , ⎠⎠

∞,[a,b]

(3.78) 

proving (3.76). An L p -estimate follows: Theorem 3.23 All as in Theorem 3.21, however here we assume that #    M3 ( f 1 , ..., fr ) (x0 ) := max f i ◦ g −1  sup  D ν i=1,...,r

x0 ∈[a,b]

g(x0 )−

 ν   f i ◦ g −1  L sup  Dg(x 0)

x0 ∈[a,b]

Above it is p, q > 1 : Then

1 p

+

1 q

L q ([g(a),g(x0 )])

,

$ q ([g(x 0 ),g(b)])

< ∞.

(3.79)

= 1.  ( f 1 , ..., fr ) ≤

≤

M3 ( f 1 , ..., fr ) (b − a)2 (g (b) − g (a))

ν− q1

1 p

( p (ν − 1) + 1)  (ν)

 ⎛     r r    ⎜  ⎜  f j (x)  ⎝   j=1 i=1   j =i

⎞ ⎟ ⎟. ⎠

∞,[a,b]

(3.80)

Proof By (3.50) we get ⎛ R.H.S. (3.50) ≤

M3 ( f 1 , ..., fr ) (g (b) − g

ν− 1 (a)) q

1

( p (ν − 1) + 1) p  (ν)

⎞

⎛

r  b ⎜ r  ⎜ ⎟ ⎟ ⎟ ⎜ ⎜  f j (x)⎟ ⎟ dx⎟ ≤ ⎜ ⎜ ⎠ ⎠ ⎝ a ⎝ i=1

j=1 j =i

        M3 ( f 1 , ..., fr ) (g (b) − g (a)) (b − a) ⎜   ⎜   f j  1  ⎝  ( p (ν − 1) + 1) p  (ν) i=1  j=1   j =i ⎛

ν− q1

⎞

r  r 

⎟ ⎟ =: λ3 . ⎠

∞,[a,b]

Hence it holds   ( f 1 , ..., fr ) ≤ a

b

⎞

K ( f 1 , ..., fr ) (x0 ) d x0

((3.50),(3.80))



≤

a

b

λ3 d x =

(3.81)

66

3 Generalized Canavati Fractional Ostrowski …

 ⎛     r 2 ⎜      ⎜ M3 ( f 1 , ..., fr ) (g (b) − g (a)) (b − a) ⎜     ⎜ f (x) j 1   ⎝ ⎝   j=1 ( p (ν − 1) + 1) p  (ν) i=1   j =i ⎛

ν− q1

r 

∞,[a,b]

proving (3.80).

⎞⎞ ⎟⎟ ⎟⎟ , ⎠⎠ (3.82) 

References 1. Anastassiou, G.A.: Fractional Differentiation Inequalities. Research Monograph. Springer, New York (2009) 2. Anastassiou, G.A.: Advances on Fractional Inequalities. Research Monograph. Springer, New York (2011) 3. Anastassiou, G.A.: Intelligent Comparisons: Analytic Inequalities. Springer, Heidelberg, New York (2016) 4. Anastassiou, G.A.: Strong mixed and generalized fractional calculus for Banach space valued functions. Mat. Vesnik 69(3), 176–191 (2017) 5. Anastassiou, G.A.: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations. Springer, Heidelberg, New York (2018) 6. Anastassiou, G.A.: Generalized Canavati Fractional Ostrowski, Opial and Grüss type inequalities for Banach algebra valued functions . Moroccan J. of Pure and Applied Analysis, accepted for publication (2022) 7. Dragomir, S.S.: Noncommutative Ostrowski type inequalities for functions in Banach algebras, RGMIA Res. Rep. Coll. 24, Art. 10, 24 pp (2021) 8. Mikusinski, J.: The Bochner Integral. Academic Press, New York (1978) 9. Opial, Z.: Sur une inegalite. Ann. Polon. Math. 8, 29–32 (1960) 10. Ostrowski, A.: Über die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938) 11. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991)

Chapter 4

Generalized Canavati Fractional Hilbert–Pachpatte Inequalities for Banach Algebra Valued Functions

Using generalized Canavati fractional left and right vectorial Taylor formulae we prove corresponding left and right fractional Hilbert–Pachpatte inequalities for Banach algebra valued functions. We cover also the sequential fractional case. We finish with applications. It follows [3].

4.1 Introduction Motivation follows: We need Definition 4.1 (see [6]) A definition of the Hausdorff measure h α goes as follows: if (T, d) is a metric space, A ⊆ T and δ > 0, let  ( A, δ) be the set of all arbitrary collections (C)i of subsets of T , such that A ⊆ ∪i Ci and diam (Ci ) ≤ δ (diam = diameter) for every i. Now, for every α > 0 define h δα (A) := inf



 (diamCi )α | (Ci )i ∈  ( A, δ) .

(4.1)

Then there exists lim h δα (A) = suph δα (A), and h α (A) := lim h δα (A) gives an outer δ→0

δ>0

δ→0

measure on the power set P (T ), which is countably additive on the σ -field of all Borel subsets of T . If T = Rn , then the Hausdorff measure h n , restricted to the σ field of the Borel subsets of Rn , equals the Lebesgue measure on Rn up to a constant multiple. In particular, h 1 (C) = μ (C) for every Borel set C ⊆ R, where μ is the Lebesgue measure.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_4

67

68

4 Generalized Canavati Fractional Hilbert–Pachpatte Inequalities …

We also need Definition 4.2 ([2], Chap. 1) Let [a, b] ⊂ R, X be a Banach space, ν > 0; n := ν ∈ N, · is the ceiling of the number, f : [a, b] → X . We assume that f (n) ∈ L 1 ([a, b] , X ). We call the Caputo-Bochner left fractional derivative of order ν: 

 ν D∗a f (x) :=

1  (n − ν)



x

(x − t)n−ν−1 f (n) (t) dt, ∀ x ∈ [a, b] .

(4.2)

a

ν 0 f := f (ν) the ordinary X -valued derivative, and also set D∗a f := If ν ∈ N, we set D∗a f. Here  is the gamma function and integrals are of Bochner type [4].  ν  ν By [2], Chap. 1, D∗a f (x) exists almost everywhere in x ∈ [a, b] and D∗a f ∈ L 1 ([a, b] , X ).   ν f ∈ C ([a, b] , X ) . If  f (n)  L ∞ ([a,b],X ) < ∞, then by [2], Chap. 1, D∗a We are motivated by a Hilbert–Pachpatte left fractional inequality:

Theorem 4.3 ([2], Chap. 1) Let p, q > 1 : 1p + q1 = 1, and ν1 > q1 , ν2 > 1p , n i := νi , i = 1, 2. Here [ai , bi ] ⊂ R, i = 1, 2; X is a Banach space. Let f i ∈ C ni −1 ([ai , bi ] , X ), i = 1, 2. Set Fxi (ti ) :=

n i −1 ji =0

(xi − ti ) ji ( ji ) f i (ti ) , ji !

(4.3)

∀ ti ∈ [ai , xi ], where xi ∈ [ai , bi ]; i = 1, 2. Assume that f i(ni ) exists outside a μ-null Borel set Bxi ⊆ [ai , xi ], such that    h 1 Fxi Bxi = 0, ∀ xi ∈ [ai , bi ] ; i = 1, 2.

(4.4)

We also assume that f i(ni ) ∈ L 1 ([ai , bi ] , X ), and

and



Then

f i(ki ) (ai ) = 0, ki = 0, 1, ..., n i − 1; i = 1, 2,

(4.5)

  ν2  ν1 f ∈ L q ([a1 , b1 ] , X ) , D∗a f ∈ L p ([a2 , b2 ] , X ) . D∗a 1 1 2 2

(4.6)



b1

a1



b2

a2



f 1 (x1 ) f 2 (x2 ) d x1 d x2

q (ν2 −1)+1 (x1 −a1 ) p(ν1 −1)+1 2 −a2 ) + (xq(q(ν p( p(ν1 −1)+1) 2 −1)+1)

≤

  ν  (b1 − a1 ) (b2 − a2 )   D ν1 f 1   D 2 f2  . ∗a1 ∗a 2 L q ([a1 ,b1 ],X ) L p ([a2 ,b2 ],X )  (ν1 )  (ν2 ) We need

(4.7)

4.1 Introduction

69

Definition 4.4 ([2], Chap. 2) Let [a, b] ⊂ R, X be a Banach space, α > 0, m := α . We assume that f (m) ∈ L 1 ([a, b] , X ), where f : [a, b] → X . We call the CaputoBochner right fractional derivative of order α: 

α Db−



(−1)m f (x) :=  (m − α)



b

(J − x)m−α−1 f (m) (J ) d J, ∀ x ∈ [a, b] .

(4.8)

x

m 0 f (x) = (−1)m f (m) (x) , for m ∈ N, and Db− f (x) = f (x) . We observe that Db−  α   α  By [2], Chap. 2, Db− f (x) exists almost everywhere on [a, b] and Db− f ∈ b] , X L 1 ([a, ).   α / N, then by [2], Chap. 2, Db− f ∈ C ([a, b] , X ) , If  f (m)  L ∞ ([a,b],X ) < ∞, and α ∈  α  hence  Db− f  ∈ C ([a, b]) .

We are motivated also by the following Hilbert–Pachpatte right fractional inequality: Theorem 4.5 ([2], Chap. 2) Let p, q > 1 : 1p + q1 = 1, and α1 > q1 , α2 > 1p , m i := αi , i = 1, 2. Here [ai , bi ] ⊂ R, i = 1, 2; X is a Banach space. Let f i ∈ C m i −1 ([ai , bi ] , X ), i = 1, 2. Set Fxi (ti ) :=

m i −1  ji =0

(xi − ti ) ji ( ji ) f i (ti ) , ji !

(4.9)

∀ ti ∈ [xi , bi ], where xi ∈ [ai , bi ]; i = 1, 2. Assume that f i(m i ) exists outside a μ-null Borel set Bxi ⊆ [xi , bi ], such that    h 1 Fxi Bxi = 0, ∀ xi ∈ [ai , bi ] ; i = 1, 2.

(4.10)

We also assume that f i(m i ) ∈ L 1 ([ai , bi ] , X ), and f i(ki ) (bi ) = 0, ki = 0, 1, ..., , m i − 1; i = 1, 2, and

Then



   Dba11− f 1 ∈ L q ([a1 , b1 ] , X ) , Dbα22− f 2 ∈ L p ([a2 , b2 ] , X ) . 

b1

a1



b2

a2



f 1 (x1 ) f 2 (x2 ) d x1 d x2

(b1 −x1 ) p(α1 −1)+1 (b2 −x2 )q (α2 −1)+1 + p( p(α1 −1)+1) q(q(α2 −1)+1)

(4.11)

(4.12)

≤

  α2  (b1 − a1 ) (b2 − a2 )   D f2   D α1 f 1  . b1 − b2 − L ,b L p ([a2 ,b2 ],X ) ],X ([a ) q 1 1  (α1 )  (α2 )

(4.13)

70

4 Generalized Canavati Fractional Hilbert–Pachpatte Inequalities …

In this chapter we derive Hilbert–Pachpatte inequalities for Banach algebra valued functions with respect to their Canavati type generalized left and right fractional derivatives. We cover also the sequential fractional case. We finish with applications.

4.2 Background on Vectorial Generalized Canavati Fractional Calculus All in this section come from [2], pp. 109–115 and [1]. Let g : [a, b] → R be a strictly increasing function. such that g ∈ C 1 ([a, b]), and −1 g ∈ C n ([g(a), g(b)]), n ∈ N, (X, · ) is a Banach space. Let f ∈ C n ([a, b] , X ), and call l := f ◦ g −1 : [g (a) , g (b)] → X . It is clear that l, l  , ..., l (n) are continuous functions from [g (a) , g (b)] into f ([a, b]) ⊆ X. Let ν ≥ 1 such that [ν] = n, n ∈ N as above, where [·] is the integral part of the number. Clearly when 0 < ν < 1, [ν] = 0. (I) Let h ∈ C ([g (a) , g (b)] , X ), we define the left Riemann–Liouville Bochner fractional integral as 

Jνz0 h



1 (z) :=  (ν)



z

(z − t)ν−1 h (t) dt,

(4.14)

z0

∞ for g (a) ≤ z 0 ≤ z ≤ g (b), where  is the gamma function;  (ν) = 0 e−t t ν−1 dt. z0 We set J0 h = h. ν Let α := ν − [ν] (0 < α < 1). We define the subspace C g(x ([g (a) , g (b)] , X ) 0) [ν] of C ([g (a) , g (b)] , X ), where x0 ∈ [a, b] as: ν C g(x ([g (a) , g (b)] , X ) = 0)



 g(x ) h ∈ C [ν] ([g (a) , g (b)] , X ) : J1−α0 h ([ν]) ∈ C 1 ([g (x0 ) , g (b)] , X ) .

(4.15)

ν So let h ∈ C g(x ([g (a) , g (b)] , X ), we define the left g-generalized X -valued 0) fractional derivative of h of order ν, of Canavati type, over [g (x0 ) , g (b)] as

 g(x ) ν h := J1−α0 h ([ν]) . Dg(x 0)

(4.16)

ν Clearly, for h ∈ C g(x ([g (a) , g (b)] , X ) , there exists 0)



 ν Dg(x h (z) = 0)

for all g (x0 ) ≤ z ≤ g (b) .

d 1  (1 − α) dz



z g(x0 )

(z − t)−α h ([ν]) (t) dt,

(4.17)

4.2 Background on Vectorial Generalized Canavati Fractional Calculus

71

ν In particular, when f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), we have that 0)



  ν Dg(x f ◦ g −1 (z) = 0)

d 1  (1 − α) dz



z g(x0 )

 ([ν]) (z − t)−α f ◦ g −1 (t) dt,

(4.18)     n 0 −1 −1 (n) f ◦ g = f ◦ g and Dg(x for all g (x0 ) ≤ z ≤ g (b). We have that Dg(x 0) 0)   −1 −1 f ◦g = f ◦ g , see [1]. ν By [1], we have for f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ) , where x0 ∈ [a, b] the 0) following left generalized g-fractional, of Canavati type, X -valued Taylor’s formula: ν Theorem 4.6 Let f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), where x0 ∈ [a, b] is fixed. 0) (i) If ν ≥ 1, then

f (x) − f (x0 ) =

[ν]−1 



f ◦ g −1

(k) k!

k=1

1  (ν)



g(x) g(x0 )

(g (x0 ))

(g (x) − g (x0 ))k +

 ν   f ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(4.19)

for all x0 ≤ x ≤ b. (ii) If 0 < ν < 1, we get f (x) =

1  (ν)



g(x) g(x0 )

 ν   f ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(4.20)

for all x0 ≤ x ≤ b. (II) Let h ∈ C ([g (a) , g (b)] , X ), we define the right Riemann–Liouville Bochner fractional integral as 

 Jzν0 − h (z) :=

1  (ν)



z0

(t − z)ν−1 h (t) dt,

(4.21)

z

for g (a) ≤ z ≤ z 0 ≤ g (b) . We set Jz00 − h = h. ν Let α := ν − [ν] (0 < α < 1). We define the subspace C g(x ([g (a) , g (b)] , X ) 0 )− of C [ν] ([g (a) , g (b)] , X ), where x0 ∈ [a, b] as: ν C g(x ([g (a) , g (b)] , X ) := 0 )−



 1−α 1 ([ν]) h ∈ C , g , X . h ∈ C [ν] ([g (a) , g (b)] , X ) : Jg(x ([g (a) (x )] ) 0 )− 0

(4.22)

ν So let h ∈ C g(x ([g (a) , g (b)] , X ), we define the right g-generalized X -valued 0 )− fractional derivative of h of order ν, of Canavati type, over [g (a) , g (x0 )] as

72

4 Generalized Canavati Fractional Hilbert–Pachpatte Inequalities …



 1−α ν n−1 ([ν]) J Dg(x h := h . (−1) g(x0 )− 0 )−

(4.23)

ν Clearly, for h ∈ C g(x ([g (a) , g (b)] , X ) , there exists 0 )−



ν h Dg(x 0 )−



(−1)n−1 d (z) =  (1 − α) dz



g(x0 )

(t − z)−α h ([ν]) (t) dt,

(4.24)

z

for all g (a) ≤ z ≤ g (x0 ) ≤ g (b) . ν In particular, when f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), we have that 0 )− 

  (−1)n−1 d ν −1 f ◦ g = Dg(x (z) 0 )−  (1 − α) dz



g(x0 )

 ([ν]) (t − z)−α f ◦ g −1 (t) dt,

z

(4.25)

for all g (a) ≤ z ≤ g (x0 ) ≤ g (b). We get that 

   (n) n f ◦ g −1 (z) = (−1)n f ◦ g −1 Dg(x (z) 0 )−

(4.26)

 

  0 −1 f ◦ g and Dg(x (z) = f ◦ g −1 (z), all z ∈ [g (a) , g (b)] , see [1]. 0 )−

ν By [1], we have for f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ) , where x0 ∈ [a, b] is 0 )− fixed, the following right generalized g -fractional, of Canavati type, X -valued Taylor’s formula: ν Theorem 4.7 Let f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), where x0 ∈ [a, b] is fixed. 0 )− (i) If ν ≥ 1, then

f (x) − f (x0 ) =

[ν]−1  k=1

1  (ν)



g(x0 ) g(x)



f ◦ g −1

(k) k!

(g (x0 ))

(g (x) − g (x0 ))k +

 ν   f ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

(4.27)

for all a ≤ x ≤ x0 , (ii) If 0 < ν < 1, we get f (x) =

1  (ν)



g(x0 ) g(x)

 ν   f ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

(4.28)

all a ≤ x ≤ x0 . (III) Denote by mν ν ν ν Dg(x = Dg(x Dg(x ...Dg(x (m-times), m ∈ N. 0) 0) 0) 0)

(4.29)

4.3 Banach Algebras Background

73

We mention the following modified and generalized left X -valued fractional Taylor’s formula of Canavati type: 1 Theorem 4.8 Let f ∈ C 1 ([a, b] b]), strictly increasing: g −1 ∈ , X ), g ∈ C ([a, 

iν ν 1 −1 C ([g (a) , g (b)]). Assume that Dg(x0 ) f ◦ g ∈ C g(x ([g (a) , g (b)] , X ), 0 < 0) ν < 1, x0 ∈ [a, b], for i = 0, 1, ..., m. Then

1 f (x) =  ((m + 1) ν)



g(x) g(x0 )



(m+1)ν  −1 f ◦ g (z) dz, (g (x) − z)(m+1)ν−1 Dg(x ) 0 (4.30)

all x0 ≤ x ≤ b. (IV) Denote by mν ν ν ν = Dg(x Dg(x ...Dg(x (m times), m ∈ N. Dg(x 0 )− 0 )− 0 )− 0 )−

(4.31)

We mention the following modified and generalized right X -valued fractional Taylor’s formula of Canavati type: Theorem 4.9 Let f ∈ C 1 ([a, b] , X ), g ∈ C 1 ([a, b]), strictly increasing: g −1 ∈   iν ν C 1 ([g (a) , g (b)]). Assume that Dg(x f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), 0 )− 0 )− 0 < ν < 1, x0 ∈ [a, b], for all i = 0, 1, ..., m. Then 1 f (x) =  ((m + 1) ν)



g(x0 ) g(x)



(m+1)ν  −1 f ◦ g (z) dz, (z − g (x))(m+1)ν−1 Dg(x 0 )− (4.32)

all a ≤ x ≤ x0 ≤ b.

4.3 Banach Algebras Background All here come from [5]. We need Definition 4.10 ([5], p. 245) A complex algebra is a vector space A over the complex filed C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(4.33)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(4.34)

α (x y) = (αx) y = x (αy) ,

(4.35)

and for all x, y and z in A and for all scalars α.

74

4 Generalized Canavati Fractional Hilbert–Pachpatte Inequalities …

Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality

x y ≤ x y (x ∈ A, y ∈ A)

(4.36)

and if A contains a unit element e such that xe = ex = x (x ∈ A)

(4.37)

e = 1,

(4.38)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 4.11 Commutativity of A will be explicated stated when needed. There exists at most one e ∈ A that satisfies (4.37). Inequality (4.36) makes multiplication to be continuous, more precisely left and right continuous, see [5], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [5], pp. 247–248, Sect. 10.3. We also make Remark 4.12 Next we mention about integration of A-valued functions, see [5], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some compact  Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [5], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   f dμ = x f ( p) dμ ( p) (4.39) x Q

and



Q

 f dμ x = Q

f ( p) x dμ ( p) .

(4.40)

Q

The Bochner integrals we will involve in our chapter follow (4.39) and (4.40). Also, let f ∈ C ([a, b] , X ), where [a, b] ⊂ R, (X, · ) is a Banach space. By [2], p. 3, f is Bochner integrable.

4.4 Main Results

75

4.4 Main Results We start with a left generalized Canavati fractional Hilbert–Pachpatte type inequality over a Banach algebra. Theorem 4.13 Let p, q > 1, such that 1p + q1 = 1, and (A, · ) is a Banach algebra; and i=1, 2. Let also x0i ∈ [ai , bi ] ⊂ R, νi ≥ 1, n i = [νi ], f i ∈ C ni ([ai , bi ] , A); gi ∈ C 1 ([ai , bi ]), strictly increasing, such that gi−1 ∈ C ni ([gi (ai ) , gi (bi )]), with (ki )  f i ◦ gi−1 (gi (x0i )) = 0, ki = 0, 1, ..., n i − 1. Assume further that f i ◦ gi−1 ∈ νi C gi (x0i ) ([gi (ai ) , gi (bi )] , A). Then 

g1 (b1 )



g1 (x01 )

g2 (b2 ) g2 (x02 )

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2

≤ (z 1 −g1 (x01 )) p(ν1 −1)+1 (z 2 −g2 (x02 ))q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

(g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 ))  (ν1 )  (ν2 )          ν 1  ν2   Dg1 (x01 ) f 1 ◦ g1−1  Dg2 (x02 ) f 2 ◦ g2−1  L q ([g1 (x01 ),g1 (b1 )],A)

(4.41)

L p ([g2 (x02 ),g2 (b2 )],A)

.

Proof By (4.19) and assumptions we get that 

 f i ◦ gi−1 (z i ) =

1  (νi )



zi gi (x0i )

 

(z i − ti )νi −1 Dgνii (x0i ) f i ◦ gi−1 (ti ) dti , (4.42)

for all gi (x0i ) ≤ z i ≤ gi (bi ); i = 1, 2. By Hölder’s inequality we obtain     f 1 ◦ g −1 (z 1 ) ≤ 1

1  (ν1 )



z1 g1 (x01 )

1  (ν1 )



z1 g1 (x01 )

(z 1 − t1 ) p(ν1 −1) dt1 p (ν1 −1)+1

1 (z 1 − g1 (x01 )) p  (ν1 ) ( p (ν1 − 1) + 1) 1p

   

  (z 1 − t1 )ν1 −1  Dgν11 (x01 ) f 1 ◦ g1−1 (t1 ) dt1 ≤

1  p

g1 (x01 )



That is     f 1 ◦ g −1 (z 1 ) ≤ 1

z1

z1 g1 (x01 )

1  q

q  ν1  =  Dg1 (x01 ) f 1 ◦ g1−1 (t1 ) dt1

q1  q  

 ν1   Dg1 (x01 ) f 1 ◦ g1−1 (t1 ) dt1 . (4.43) p (ν1 −1)+1

1 (z 1 − g1 (x01 )) p  (ν1 ) ( p (ν1 − 1) + 1) 1p

76

4 Generalized Canavati Fractional Hilbert–Pachpatte Inequalities …



z1 g1 (x01 )

q1  q  

 ν1   Dg1 (x01 ) f 1 ◦ g1−1 (t1 ) dt1 ,

(4.44)

for all g1 (x01 ) ≤ z 1 ≤ g1 (b1 ). Similarly, we prove that     f 2 ◦ g −1 (z 2 ) ≤ 2



z2 g2 (x02 )

q (ν2 −1)+1

1 (z 2 − g2 (x02 )) q  (ν2 ) (q (ν2 − 1) + 1) q1

  p 1p  

 ν2  −1 , (t2 ) dt2  Dg2 (x02 ) f 2 ◦ g2

(4.45)

for all g2 (x02 ) ≤ z 2 ≤ g2 (b2 ). Therefore we have     f 1 ◦ g −1 (z 1 ) ≤ 1

p (ν1 −1)+1

1 (z 1 − g1 (x01 )) p  (ν1 ) ( p (ν1 − 1) + 1) 1p

       ν1  Dg1 (x01 ) f 1 ◦ g1−1 

q,[g1 (x01 ),g1 (b1 )]

for all g1 (x01 ) ≤ z 1 ≤ g1 (b1 ); and     f 2 ◦ g −1 (z 2 ) ≤ 2

,

(4.46)

q (ν2 −1)+1

1 (z 2 − g2 (x02 )) q  (ν2 ) (q (ν2 − 1) + 1) q1

       ν2  Dg2 (x02 ) f 2 ◦ g2−1 

p,[g2 (x02 ),g2 (b2 )]

,

(4.47)

for all g2 (x02 ) ≤ z 2 ≤ g2 (b2 ). Hence we get that    



    f 1 ◦ g1−1 (z 1 )  f 2 ◦ g2−1 (z 2 ) ≤

(z 1 − g1 (x01 ))       ν 1   Dg1 (x01 ) f 1 ◦ g1−1 

1 1

1

 (ν1 )  (ν2 ) ( p (ν1 − 1) + 1) p (q (ν2 − 1) + 1) q

p (ν1 −1)+1 p

q,[g1 (x01 ),g1 (b1 )]

q (ν2 −1)+1

(z 2 − g2 (x02 )) q       ν2   Dg2 (x02 ) f 2 ◦ g2−1  1

1

(using Young’s inequality for a, b ≥ 0, a p b q ≤

a p

+ qb )

(4.48) p,[g2 (x02 ),g2 (b2 )]

≤

4.4 Main Results

77



1  (ν1 )  (ν2 )

(z 2 − g2 (x02 ))q(ν2 −1)+1 (z 1 − g1 (x01 )) p(ν1 −1)+1 + p( p (ν1 − 1) + 1) q(q (ν2 − 1) + 1)



   ν1  Dg1 (x01 ) f 1 ◦ g1−1 

L q ([g1 (x01 ),g1 (b1 )],A)



   ν2  Dg2 (x02 ) f 2 ◦ g2−1 

L p ([g2 (x02 ),g2 (b2 )],A)

∀ (z 1 , z 2 ) ∈ [g1 (x01 ) , g1 (b1 )] × [g2 (x02 ) , g2 (b2 )] . So far we have       f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) 1 2

≤ (z 1 −g1 (x01 )) p(ν1 −1)+1 (z 2 −g2 (x02 ))q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

       f 1 ◦ g −1 (z 1 )  f 2 ◦ g −1 (z 2 )

1 (z 1 −g1 (x01 )) p(ν1 −1)+1 p( p(ν1 −1)+1)

+

2

(z 2 −g2 (x02 ))q (ν2 −1)+1 q(q(ν2 −1)+1)



,

(4.50)

≤

(4.51)

     1  ν1   Dg1 (x01 ) f 1 ◦ g1−1  L q ([g1 (x01 ),g1 (b1 )],A)  (ν1 )  (ν2 )        ν2 ,  Dg2 (x02 ) f 2 ◦ g2−1  L p ([g2 (x02 ),g2 (b2 )],A)

∀ (z 1 , z 2 ) ∈ [g1 (x01 ) , g1 (b1 )] × [g2 (x02 ) , g2 (b2 )] . The denominators in (4.50), (4.51) can be zero only when both z 1 = g1 (x01 ) and z 2 = g2 (x02 ) . Therefore we obtain (4.41), by integrating (4.50), (4.51) over [g1 (x01 ) , g1 (b1 )] ×  [g2 (x02 ) , g2 (b2 )] . We continue with a right generalized Canavati fractional Hilbert–Pachpatte type inequality over a Banach algebra. Theorem 4.14 All as in Theorem 4.13, however now it is f i ◦ gi−1 ∈ C gνii (x0i )− ([gi (ai ) , gi (bi )] , A), for i = 1, 2. Then 

g1 (x01 ) g1 (a1 )



g2 (x02 ) g2 (a2 )

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2

≤ (g1 (x01 )−z 1 ) p(ν1 −1)+1 (g2 (x02 )−z 2 )q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

(g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 ))  (ν1 )  (ν2 ) 

   ν1 Dg1 (x01 )− f 1 ◦ g1−1 

L q ([g1 (a1 ),g1 (x01 )],A)



   ν2 Dg2 (x02 )− f 2 ◦ g2−1 

(4.52)

L p ([g2 (a2 ),g2 (x02 )],A)

.

78

4 Generalized Canavati Fractional Hilbert–Pachpatte Inequalities …



Proof Similar to Theorem 4.13, by using now (4.27).

Next comes a sequential left generalized Canavati fractional Hilbert–Pachpatte type inequality over a Banach algebra. Theorem 4.15 Let p, q > 1, such that 1p + q1 = 1, and (A, · ) is a Banach algebra; and i = 1, 2. Let also f i ∈ C 1 ([ai , bi ] , A); gi ∈ C 1 ([ai , bi ]), strictly increas1 < νi < 1, x0i ∈ ing, such that gi−1 ∈ C 1 ([gi (ai ) , gi (bi )]). Assume that (m i +1)q   ji νi νi −1 [ai , bi ], and Dgi (x0i ) f i ◦ gi ∈ C gi (x0i ) ([gi (ai ) , gi (bi )] , A) , for ji = 0, 1, ..., m i ∈ N. Then       g1 (b1 )  g2 (b2 )  f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2

≤ (z 1 −g1 (x01 )) p((m 1 +1)ν1 −1)+1 (z 2 −g2 (x02 ))q ((m 2 +1)ν2 −1)+1 g1 (x01 ) g2 (x02 ) + p( p((m 1 +1)ν1 −1)+1) q(q((m 2 +1)ν2 −1)+1) (g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 ))  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 ) 

   (m 1 +1)ν1 f 1 ◦ g1−1  Dg1 (x01 )

L q ([g1 (x01 ),g1 (b1 )],A)



   (m 2 +1)ν2 f 2 ◦ g2−1  Dg2 (x02 )

(4.53)

L p ([g2 (x02 ),g2 (b2 )],A)

Proof Using (4.30), as similar to Theorem 4.13 the proof is omitted.

.



The right side analog of Theorem 4.15 follows: Theorem 4.16 Let p, q > 1, such that 1p + q1 = 1, and (A, · ) is a Banach algebra; and i = 1, 2. Let also f i ∈ C 1 ([ai , bi ] , A); gi ∈ C 1 ([ai , bi ]), strictly increas1 < νi < 1, x0i ∈ ing, such that gi−1 ∈ C 1 ([gi (ai ) , gi (bi )]). Assume that (m i +1)q   ji νi νi −1 [ai , bi ], and Dgi (x0i )− f i ◦ gi ∈ C gi (x0i )− ([gi (ai ) , gi (bi )] , A) , for ji = 0, 1, ..., m i ∈ N. Then       g1 (x01 )  g2 (x02 )  f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2

≤ (g1 (x01 )−z 1 ) p((m 1 +1)ν1 −1)+1 (g2 (x02 )−z 2 )q ((m 2 +1)ν2 −1)+1 g1 (a1 ) g2 (a2 ) + p( p((m 1 +1)ν1 −1)+1) q(q((m 2 +1)ν2 −1)+1) (g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 ))  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 ) 

   (m 1 +1)ν1 Dg1 (x01 )− f 1 ◦ g1−1 

L q ([g1 (a1 ),g1 (x01 )],A)



   (m 2 +1)ν2 Dg2 (x02 )− f 2 ◦ g2−1 

Proof Using (4.32), as similar to Theorem 4.13 is omitted.

(4.54)

L p ([g2 (a2 ),g2 (x02 )],A)

.



References

79

4.5 Applications We give Corollary 4.17 (to Theorem 4.13) All as in Theorem 4.13 for gi (t) = et , i = 1, 2. Then  e b1  e b2

( f 1 ◦ log) (z 1 ) ( f 2 ◦ log) (z 2 ) dz 1 dz 2

≤ −e x02 )q (ν2 −1)+1 (z 1 −e x01 ) p(ν1 −1)+1 x x e 01 e 02 + (z2q(q(ν p( p(ν1 −1)+1) −1)+1) 2

 ν1  D x01 e

 b   e 1 − e x01 eb2 − e x02 (4.55)  (ν1 )  (ν2 )    ( f 1 ◦ log) L q ([ex01 ,eb1 ],A)  Deνx202 ( f 2 ◦ log) L p ([ex02 ,eb2 ],A) .

We finish with Corollary 4.18 (to Theorem 4.15) All as in Theorem 4.15 for [a1 , b1 ] ⊂ R, [a2 , b2 ] ⊂ (0, ∞), and g1 (t) = et and g2 (t) = log t. Then 

e b1

e x01



log(b2 )

log(x02 )



    ( f 1 ◦ log) (z 1 ) f 2 ◦ et (z 2 ) dz 1 dz 2 (z 1 −e x01 ) p((m 1 +1)ν1 −1)+1 p( p((m 1 +1)ν1 −1)+1)

+

(z 2 −log(x02 ))q ((m 2 +1)ν2 −1)+1 q(q((m 2 +1)ν2 −1)+1)

≤

 b  e 1 − e x01 log (b2 /x02 ) (4.56)  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 )      (m 2 +1)ν2   (m 1 +1)ν1  t  D . f ◦ e ( f 1 ◦ log)    Dex01  2 log(x ) 02 L q ([e x01 ,eb1 ],A) L p ([log(x02 ),log(b2 )],A)

References 1. Anastassiou, G.A.: Strong mixed and generalized fractional calculus for Banach space valued functions. Mat. Vesnik 69(3), 176–191 (2017) 2. Anastassiou, G.A.: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations. Springer, Heidelberg (2018) 3. Anastassiou, G.A.: Generalized Canavati Fractional Hilbert-Pachpatte type inequalities for Banach algebra valued functions. J. Comput. Anal. Appl. 30(1), 66–77 (2022) 4. Mikusinski, J.: The Bochner Integral. Academic, New York (1978) 5. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991) 6. Volintiru, C.: A proof of the fundamental theorem of Calculus using Hausdorff measures. Real Anal. Exch. 26(1), 381–390 (2000/2001)

Chapter 5

Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities for Banach Algebra Valued Functions Involving Integer Vectorial Derivatives

Using a generalized vectorial Taylor formula involving ordinary vector derivatives we establish mixed Ostrowski, Opial and Hilbert-Pachpatte inequalities for several Banach algebra valued functions. The estimates are with respect to all norms · p , 1 ≤ p ≤ ∞. We finish with applications. It follows [3].

5.1 Introduction The following result motivates this chapter. Theorem 5.1 (1938, Ostrowski [7]) Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, derivative f  : (a, b) → R is bounded on (a, b), i.e.,   sup  b) whose   f  := sup  f  (t) < +∞. Then ∞ t∈(a,b)

   1  b − a

a

b

 2       sup x − a+b 1 2 f  , + f (t) dt − f (x) ≤ − a) (b ∞ 4 (b − a)2

for any x ∈ [a, b]. The constant

1 4

(5.1)

is the best possible.

Ostrowski type inequalities have great applications to integral approximations in Numerical Analysis. We present ([1], Chap. 8,9) mixed fractional Ostrowski inequalities for several functions for various norms. In this chapter we generalize [1], Chap. 8,9 for several Banach algebra valued functions by using ordinary vector valued derivatives and our integrals here are of Bochner type [5]. Motivation comes also from [4]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_5

81

82

5 Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities …

We are also inspired by Opial [6], 1960, famous inequality. Theorem 5.2 Let x (t) ∈ C 1 ([0, h]) be such that x (0) = x (h) = 0, and x (t) > 0 in (0, h). Then  h  h     2 x (t) x  (t) dt ≤ h x (t) dt. (5.2) 4 0 0 In (5.2), the constant optimal function

h 4

is the best possible. Inequality (5.2) holds as equality for the x (t) =

ct, 0 ≤ t ≤ h2 , c (h − t) , h2 ≤ t ≤ h,

(5.3)

where c > 0 is an arbitrary constant. Opial-type inequalities are used a lot in proving uniqueness of solutions to differential equations and also to give upper bounds to their solutions. In this chapter we also derive Opial type inequalities for Banach algebra valued functions with respect to ordinary vector valued derivatives. Additionally we include in this chapter related Hilbert-Pachpatte type inequalities, [8]. We finish with selective applications to Ostrowski, Opial and Hilbert-Pachpatte inequalities.

5.2 About Banach Algebras All here come from [9]. We need Definition 5.3 ([9], p. 245) A complex algebra is a vector space A over the complex field C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(5.4)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(5.5)

α (x y) = (αx) y = x (αy) ,

(5.6)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A) and if A contains a unit element e such that

(5.7)

5.3 Background

83

xe = ex = x (x ∈ A)

(5.8)

e = 1,

(5.9)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 5.4 Commutativity of A will be explicited stated when needed. There exists at most one e ∈ A that satisfies (5.8). Inequality (5.7) makes multiplication to be continuous, more precisely left and right continuous, see [9], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [9], p. 247-248, § 10.3. We also make Remark 5.5 Next we mention about integration of A-valued functions, see [9], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some compact

Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chapter 3 of [9], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then 

 f dμ =

x Q

and



x f ( p) dμ ( p)

 f dμ x = Q

(5.10)

Q

f ( p) x dμ ( p) .

(5.11)

Q

The Bochner integrals we will involve in our chapter follow (5.10) and (5.11).

5.3 Background We use the following generalized vector Taylor’s formula: Theorem 5.6 ([2], p. 97) Let n ∈ N and f ∈ C n ([a, b] , X ), where [a, b] ⊂ R and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]). Let any x, y ∈ [a, b]. Then

84

5 Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities …

f (x) = f (y) +

n−1

(g (x) − g (y))i  i=1



1 + (n − 1)!

g(x)

i!

f ◦ g −1

(i)

(g (y))

(5.12)

 (n) (g (x) − z)n−1 f ◦ g −1 (z) dz.

g(y)

The derivatives here are defined similarly to the numerical ones, see [10], pp. 83-86. The above integral is of Bochner type [5], and so are the integrals in this work. By [2], p. 3, if f ∈ C ([a, b] , X ) then f is Bochner integrable.

5.4 Main Results We start with mixed generalized Ostrowski type inequalities for several functions that are Banach algebra valued. A uniform estimate follows. Theorem 5.7 Let n ∈ N and f i ∈ C n ([a, b] , A), i = 1, ..., r ∈ N − {1}; where [a, b] ⊂ R and (A, ·) is a Banach algebra. Let g ∈ C 1 ([a, b]) , strictly increas( j)  ing, such that g −1 ∈ C n ([g (a) , g (b)]). We assume that f i ◦ g −1 (g (x0 )) = 0, j = 1, ..., n − 1; i = 1, ..., r ; where x0 ∈ [a, b] be fixed. Denote by E ( f 1 , ..., fr ) (x0 ) := ⎡ r

i=1



⎢ ⎢ ⎣

a

⎞

⎛ b

⎜ ⎜ ⎝

r  j=1 j =i

⎞ ⎞ ⎤ ⎛ ⎛  b  r ⎟ ⎟ ⎟ ⎥ ⎜ ⎜ ⎟ ⎥ ⎜ ⎜ f j (x)⎟ f j (x)⎟ ⎠ f i (x) d x − ⎝ a ⎝ ⎠ d x ⎠ f i (x0 )⎦ .

(5.13)

j=1 j =i

Then (1) E ( f 1 , ..., fr ) (x0 ) = ⎡

⎡

1 (n − 1)!

⎞

⎛

⎤

  r ⎟  g(x0 ) ⎢ x0 ⎜  ⎥  (n) ⎟ ⎢ ⎢ ⎜ ⎥ n n−1 −1 fi ◦ g f j (x)⎟ (z − g (x)) (z) dz d x ⎥ + ⎢(−1) ⎢ ⎜ ⎠ g(x) ⎣ ⎣ a ⎝ ⎦

r ⎢

i=1

j=1 j =i

(5.14)

⎢ ⎢ ⎣



⎞

⎛

⎡

b x0

⎜ ⎜ ⎝

r  j=1 j =i

⎟ f j (x)⎟ ⎠



g(x) g(x0 )



⎤⎤

⎥⎥  (n) ⎥ (g (x) − z)n−1 f i ◦ g −1 (z) dz d x ⎥ ⎦⎦ ,

5.4 Main Results

85

and (2) E ( f 1 , ..., fr ) (x0 ) ≤

1 n!

⎧ ⎞ ⎞⎤ ⎡⎡ ⎛ ⎛ ⎪ ⎪     ⎪  r ⎢⎢ r  ⎥ ⎜ x0 ⎜  ⎨ (n)  ⎟ ⎟ ⎟⎥ ⎢⎢ ⎜ ⎜  f j (x)⎟  (g (x0 ) − g (a))n ⎜ ⎟ d x ⎟⎥ + ⎢⎢ f i ◦ g −1 ⎜   ⎪ ⎠ ⎠⎦ ⎣⎣ ⎝ a ⎝ ⎪ ∞,[g(a),g(x0 )] ⎪ j=1 ⎩i=1 j =i

⎞⎤⎤⎫ (5.15) ⎪ ⎪ ⎪   r  ⎟⎥⎥⎬ ⎢ ⎜ b ⎜  (n)   ⎟  ⎟ ⎟⎥⎥ ⎢ ⎜ ⎜ −1 n       f j (x) ⎟ d x ⎟⎥⎥ . (g (b) − g (x0 )) ⎜ ⎢ f i ◦ g ⎜   ⎠ ⎠⎦⎦⎪ ⎣ ⎝ x0 ⎝ ⎪ ∞,[g(x0 ),g(b)] ⎪ j=1 ⎭ ⎡

⎛

⎞

⎛

j =i

( j)  Proof Let x0 ∈ [a, b] such that f i ◦ g −1 (g (x0 )) = 0, j = 1, ..., n − 1; i = 1, ..., r. Let x ∈ [a, x0 ], then by Theorem 5.6 we have f i (x) − f i (x0 ) =

1 (n − 1)!

(−1)n = (n − 1)!





g(x) g(x0 )

g(x0 )

 (n) (g (x) − z)n−1 f i ◦ g −1 (z) dz

(5.16)

 (n) (z − g (x))n−1 f i ◦ g −1 (z) dz,

g(x)

for i = 1, ..., r. And for x ∈ [x0 , b], then again by Theorem 5.6 we get f i (x) − f i (x0 ) = for i = 1, ..., r. We multiply (5.16) by

 &r j=1 j =i

g(x) g(x0 )

j=1 j =i

j=1 j =i

(n − 1)! ∀ x ∈ [a, x0 ] ; for i = 1, ..., r.

g(x0 ) g(x)

⎞

⎛

⎟ ⎜ ⎜ f j (x)⎟ ⎠ f i (x) − ⎝

f j (x) (−1)n 

(5.17)

f j (x) to get:

⎞ r 

 (n) (g (x) − z)n−1 f i ◦ g −1 (z) dz,

 &r

⎛ ⎜ ⎜ ⎝



1 (n − 1)!

r  j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

 (n) (z − g (x))n−1 f i ◦ g −1 (z) dz,

(5.18)

86

5 Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities …

Similarly, we get (by (5.17)) ⎛ ⎜ ⎜ ⎝

⎞ r  j=1 j =i

⎞

⎛

⎟ ⎜ ⎜ f f j (x)⎟ − (x) i ⎠ ⎝

r 

⎟ f j (x)⎟ ⎠ f i (x0 ) =

j=1 j =i



 &r j=1 j =i

f j (x) 

(n − 1)!

g(x) g(x0 )

 (n) (g (x) − z)n−1 f i ◦ g −1 (z) dz,

(5.19)

∀ x ∈ [x0 , b] ; for i = 1, ..., r. Adding (5.18) and (5.19) as separate groups, we obtain ⎛ r

i=1

⎜ ⎜ ⎝

j=1 j =i

(−1)n (n − 1)!

i=1

⎜ ⎜ ⎝

∀ x ∈ [a, x0 ] , and

r  j=1 j =i

⎟ f j (x)⎟ ⎠

⎛ r

i=1

⎟ f j (x)⎟ ⎠ f i (x) − ⎞

⎛ r

⎜ ⎜ ⎝

⎛

⎞ r 



g(x0 )

r

i=1

j=1 j =i

j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

 (n) (z − g (x))n−1 f i ◦ g −1 (z) dz,

(5.20)

g(x)

⎛

⎞ r 

⎜ ⎜ ⎝

⎞ r 

⎟ f j (x)⎟ ⎠ f i (x) −

r

i=1

⎜ ⎜ ⎝

⎞ r  j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

⎞

⎛

 r r

⎟ g(x) ⎜  (n) 1 ⎟ ⎜ f j (x)⎠ (g (x) − z)n−1 f i ◦ g −1 (z) dz, ⎝ (n − 1)! i=1 j=1 g(x0 )

(5.21)

j =i

∀ x ∈ [x0 , b] . Next, we integrate (5.20) and (5.21) with respect to x ∈ [a, b]. We have r  x0

i=1

a

⎞

⎛ ⎜ ⎜ ⎝

r  j=1 j =i

⎟ f j (x)⎟ ⎠ f i (x) d x −

⎛ r

i=1

⎜ ⎜ ⎝

 a

⎞

⎛ x0

⎜ ⎜ ⎝

r  j=1 j =i

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) = (5.22)

5.4 Main Results

87

⎡

⎞

⎛

 ⎜ r r (−1)n ⎢ ⎢ x0 ⎜ 

(n − 1)!

i=1

⎢ ⎣ a

⎜ ⎝

j=1 j =i

⎤   ⎟ ⎥  (n) g(x0 ) ⎟ ⎥ f j (x)⎟ (z − g (x))n−1 f i ◦ g −1 (z) dz d x ⎥ , ⎠ g(x) ⎦

and ⎛ ⎞ ⎞  r r r b ⎜

⎜ b ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ f i (x) d x − f f j (x)⎟ (x) j ⎝ ⎝ ⎝ ⎠ ⎠ d x ⎠ f i (x0 ) = (5.23) x x ⎛

r 

0

i=1

j=1 j =i

r

i=1



⎢ ⎢ ⎣

0

i=1

⎡ 1 (n − 1)!

⎛

⎞

⎞

⎛ b x0

⎜ ⎜ ⎝

r  j=1 j =i

⎟ f j (x)⎟ ⎠



g(x) g(x0 )

j=1 j =i

⎤



⎥  (n) (g (x) − z)n−1 f i ◦ g −1 (z) dz d x ⎥ ⎦.

Finally, adding (5.22) and (5.23) we obtain the useful identity E ( f 1 , ..., fr ) (x0 ) := ⎡

 r

⎢ ⎢ ⎣

a

i=1

⎞

⎛

⎛  r b ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ f j (x)⎠ f i (x) d x − ⎝ ⎝

a

j=1 j =i

⎡

⎡

⎞

⎛ b

⎤

⎞

r ⎟ ⎟ ⎥ ⎜ 1 ⎥ ⎟ ⎜ f j (x)⎟ ⎠ d x ⎠ f i (x0 )⎦ = (n − 1)! ⎝ j=1 j =i

⎞

⎛

⎤

  r ⎟  g(x0 ) ⎥ ⎢ x0 ⎜   (n) ⎟ ⎥ ⎢ ⎢ ⎜ n n−1 −1 fi ◦ g f j (x)⎟ (z − g (x)) (z) dz d x ⎥ + ⎢(−1) ⎢ ⎜ ⎠ g(x) ⎦ ⎣ ⎣ a ⎝

r ⎢

i=1

j=1 j =i

⎡



⎢ ⎢ ⎣

⎛ b x0

⎞

r ⎜ ⎟ ⎜ f j (x)⎟ ⎝ ⎠



j=1 j =i

g(x) g(x0 )



⎤⎤

⎥⎥  (n) ⎥ (g (x) − z)n−1 f i ◦ g −1 (z) dz d x ⎥ ⎦⎦ , (5.24)

proving (5.14). Therefore, we get that E ( f 1 , ..., fr ) (x0 ) =  ⎡ ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ ⎤    r   r r  ⎢ b ⎜  ⎟ ⎜ b ⎜  ⎟ ⎟ ⎥ 1  ⎢ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎥ f j (x)⎟ f i (x) d x − ⎜ f j (x)⎟ d x ⎟ f i (x0 )⎥ ≤  ⎢ ⎜ ⎜  ⎣ a ⎝ ⎠ ⎝ a ⎝ ⎠ ⎠ ⎦ (n − 1)! i=1  j=1 j=1   j =i j =i

88

5 Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities … ⎧ ⎡⎡ ⎛ ⎞ ⎤   ⎪   ⎪   ⎪ r ⎢⎢ x0 ⎜  r ⎨ ⎟  g(x0 ) ⎥  (n)  ⎢⎢ ⎜ ⎟ ⎥ n−1 −1 f j (x)⎟ fi ◦ g (z − g (x)) (z) dz d x ⎥ ⎢⎢ ⎜ ⎪ ⎣⎣ a ⎝ ⎠ g(x) ⎦ ⎪  ⎪ j=1 ⎩i=1    j =i ⎡ ⎛ ⎞ ⎤⎤⎫   ⎪    ⎪   r ⎢ b ⎜  ⎬ ⎟  g(x) ⎥⎥⎪  (n) ⎢ ⎥ ⎜ ⎟ ⎥ n−1 −1 + ⎢ f j (x)⎟ fi ◦ g (g (x) − z) (z) dz d x ⎥⎥ ≤ ⎜ ⎣ x 0 ⎝ ⎠ g(x0 ) ⎦⎦⎪   ⎪ ⎪ j=1   ⎭ j =i  ⎤ ⎛ ⎧ ⎡⎡ ⎞   ⎪  r ⎪   ⎪  r ⎜  ⎥ ⎢  ⎢ ⎨ ⎟   g(x0 )

⎢⎢ x0 ⎜  (n) 1  ⎥ ⎟ f j (x)⎟ (z − g (x))n−1 f i ◦ g −1 (z) dz  d x ⎥ ⎜ ⎢⎢  ⎦ ⎣⎣ a ⎝ ⎠ g(x) (n − 1)! ⎪ ⎪   j=1 ⎪ ⎩i=1   j =i

⎛  ⎤⎤⎫ ⎞   ⎪ ⎪  r   ⎢ b ⎜  ⎬  ⎥⎥⎪ ⎟  g(x)  (n) ⎢ ⎥ ⎥ ⎜  ⎟ n−1 −1 + ⎢ f j (x)⎟ fi ◦ g (g (x) − z) (z) dz  d x ⎥⎥ ≤ ⎜  ⎦⎦⎪ ⎣ x0 ⎝ ⎠ g(x0 ) ⎪ ⎪  j=1  ⎭  j =i  ⎡

(5.25)

⎧ ⎤ ⎞ ⎡⎡ ⎛ ⎪ ⎪   ⎪    r r ⎥ ⎟ ⎨ ⎢ ⎢ ⎜   g(x0 )

⎢⎢ x 0 ⎜      (n) 1 ⎥  f j (x)⎟ dz d x ⎥ f i ◦ g −1 (z − g (x))n−1  (z) ⎟ ⎢⎢ ⎜   ⎦ ⎠ g(x) ⎣⎣ a ⎝ (n − 1)! ⎪ ⎪ ⎪ j=1 ⎩i=1 j =i

⎞ ⎤⎤⎫ ⎛ ⎪ ⎪     r  ⎬ ⎢ b ⎜  ⎟ ⎥⎥⎪   g(x) ⎟   (n) ⎥ ⎢ ⎥ ⎜ n−1 −1     + ⎢ fi ◦ g f j (x) ⎟ (g (x) − z) (z) dz d x ⎥⎥ =: (ξ ) . ⎜  ⎣ x0 ⎝ ⎠ g(x0 ) ⎦⎦⎪ ⎪ ⎪ j=1 ⎭ ⎡

j =i

(5.26) Hence it holds E ( f 1 , ..., fr ) (x0 ) ≤ (ξ ) .

(5.27)

We have that ⎧ ⎞ ⎤ ⎡⎡ ⎛ ⎪ ⎪     ⎪  r r ⎢  ⎥   x0 ⎜   ⎟ 1 ⎨ ⎢ ⎥ ⎢⎢ ⎜ −1 (n)    f j (x)⎟ (ξ ) ≤ ⎟ (g (x0 ) − g (x))n d x ⎥ ⎢⎢ ⎜  f i ◦ g  ⎣ ⎠ ⎦ ⎣ ⎝ n! ⎪ a ⎪ ∞, g(a),g(x ) [ ] 0 ⎪ j=1 ⎩i=1 j =i

⎡ ⎞ ⎤⎤⎫ ⎛ ⎪ ⎪      r ⎬ ⎢ ⎥⎥⎪ (n)  b ⎜  ⎟ ⎢ ⎟ ⎥⎥ ⎜ −1 n     + ⎢ f i ◦ g f j (x) ⎟ (g (x) − g (x0 )) d x ⎥⎥ ≤ ⎜  ⎪ ⎦ ⎣ ⎠ ⎦ ⎝ ⎪ ∞,[g(x0 ),g(b)] x0 j=1 ⎪ ⎭ j =i

(5.28)

5.4 Main Results

89

⎧ ⎞ ⎞⎤ ⎡⎡ ⎛ ⎛ ⎪ ⎪     ⎪  r r ⎥ ⎢ ⎢ ⎜ ⎜ ⎨    ⎟ ⎟ (n)  1 ⎢⎢ ⎟⎥ ⎜ x0 ⎜    f i ◦ g −1  f j (x)⎟  (g (x0 ) − g (a))n ⎜ ⎟ d x ⎟⎥ ⎢⎢ ⎜     ⎪ ⎠ ⎠⎦ ⎣⎣ ⎝ a ⎝ n! ⎪ ∞,[g(a),g(x0 )] ⎪ j=1 ⎩i=1 j =i

⎞ ⎞⎤⎤⎫ ⎛ ⎛ ⎪ ⎪    b ⎜ r ⎢ ⎟ ⎟⎥⎥⎪ ⎬ ⎜       (n) ⎢ ⎟ ⎟ ⎥ ⎥ ⎜ ⎜ −1 n       + ⎢  f i ◦ g f j (x) ⎟ d x ⎟⎥⎥ , (g (b) − g (x0 )) ⎜ ⎜   ⎣ ⎠ ⎠⎦⎦⎪ ⎝ x0 ⎝ ⎪ ∞,[g(x0 ),g(b)] ⎪ j=1 ⎭ ⎡

(5.29)

j =i



proving (5.15). Next comes an L 1 estimate. Theorem 5.8 All as in Theorem 5.7. Then E ( f 1 , ..., fr ) (x0 ) ≤

1 (n − 1)!

⎧ ⎡⎡ ⎛ ⎞ ⎤ ⎪   ⎪  x0  r r ⎨ (n)  ⎢⎢ ⎜     ⎟ ⎥ ⎢⎢ f i ◦ g −1 ⎜   f j (x)⎟ (g (x0 ) − g (x))n−1 d x ⎥     ⎣ ⎣ ⎝ ⎠ ⎦ ⎪ ⎪ L 1 ([g(a),g(x0 )]) a j=1 ⎩ i=1 j =i

⎡

  (n)   ⎢   f i ◦ g −1 +⎢  ⎣

 L 1 ([g(x0 ),g(b)])

⎤ ⎤⎫ ⎪ ⎪ b⎜  ⎟ ⎥ ⎥⎬ ⎜  f j (x)⎟ (g (x) − g (x0 ))n−1 d x ⎥⎥ . ⎝ ⎠ ⎦ ⎦⎪ x0 ⎪ j=1 ⎭ ⎛

⎞

r  j =i

(5.30) Proof By (5.26), (5.27), we get that E ( f 1 , ..., fr ) (x0 ) ≤ (ξ ) ≤

1 (n − 1)!

⎧ ⎡⎡ ⎛ ⎞ ⎤ ⎪  ⎪   x0  r r ⎨  (n)  ⎢⎢  ⎜  ⎟ ⎥ ⎢⎢ f i ◦ g −1 ⎜   f j (x)⎟ (g (x0 ) − g (x))n−1 d x ⎥  ⎣⎣  ⎝ ⎠ ⎦ ⎪ a ⎪ L 1 ([g(a),g(x0 )]) j=1 ⎩ i=1 j =i

⎡   (n)   ⎢   f i ◦ g −1 +⎢  ⎣

 L 1 ([g(x0 ),g(b)])

⎞ ⎤ ⎤⎫ ⎪ ⎪ r ⎜  ⎟ ⎥ ⎥⎬ n−1 ⎜  f j (x)⎟ (g (x) − g (x0 )) ⎥⎥ , d x ⎝ ⎠ ⎦ ⎦⎪ ⎪ j=1 ⎭ ⎛

b x0

j =i

(5.31) proving (5.30). An L p estimate follows.



90

5 Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities …

Theorem 5.9 All as in Theorem 5.7, and let p, q > 1 : E ( f 1 , ..., fr ) (x0 ) ≤

1 p

+

1 1

(n − 1)! ( p (n − 1) + 1) p

⎡

⎛   ⎜ x0 (n)  ⎢ ⎜ n− 1  (g (x0 ) − g (x)) q ⎢ f i ◦ g −1 ⎜   ⎣ L q ([g(a),g(x0 )]) ⎝ a i=1 ⎛

⎞

⎛

 r ⎢



  (n)     f i ◦ g −1  +  

= 1. Then

1 q

⎜ ⎝

⎟ ⎟ ⎟  f j (x)⎟ ⎟ dx⎟ ⎠ ⎠

j=1 j =i

⎞

⎛

⎜ b n− 1 ⎜ (g (x) − g (x0 )) q ⎜ L q ([g(x0 ),g(b)]) ⎝ x0

⎞

r  ⎜ ⎜

⎞⎤

r  ⎥ ⎜ ⎟ ⎟ ⎟⎥ ⎜  f j (x)⎟ ⎟ d x ⎟⎥ . ⎜ ⎠ ⎠⎦ ⎝ j=1 j =i

(5.32) Proof By (5.26), (5.27), we get that E ( f 1 , ..., fr ) (x0 ) ≤ (ξ ) ≤

1 (n − 1)!

⎧ ⎞ ⎡⎡ ⎛ ⎪ ⎪  g(x0 ) 1p  r r ⎨ ⎢⎢ x0 ⎜  ⎟ p(n−1) ⎢⎢ ⎜  f j (x)⎟ dz (z − g (x)) ⎠ ⎣⎣ ⎝ ⎪ g(x) a ⎪ j=1 ⎩ i=1 j =i



g(x0 )

  q q1  (n)   (z) dz d x +  f i ◦ g −1

g(x)

⎡  ⎢ ⎢ ⎣ 

⎞

⎛ b x0

r  ⎟ ⎜ ⎜  f j (x)⎟ ⎠ ⎝

g(x) g(x0 )

j=1 j =i



g(x) g(x0 )

(g (x) − z)

 q q1    −1 (n) (z) dz d x  fi ◦ g

⎧ ⎞ ⎡⎡ ⎛ ⎪ ⎪ p(n−1)|+1 ⎪ r ⎢⎢ x0 ⎜  r  ⎨ p ⎟ ⎟ (g (x0 ) − g (x)) ⎢⎢ ⎜   f j (x) ⎟ ⎢⎢ ⎜ 1 ⎪ ⎠ ⎣⎣ a ⎝ ⎪ ⎪ j=1 ( p (n − 1) + 1) p ⎩i=1 j =i

1p p(n−1)

' =

dz

(5.33)

1 (n − 1)! ⎤

  (n)     f i ◦ g −1   

⎥ ⎥ dx⎥ ⎦ L q ([g(a),g(x0 )])

5.4 Main Results ⎡

91 ⎞

⎛

p(n−1)|+1 r  ⎢ b ⎜  p ⎟ ⎟ (g (x) − g (x0 )) ⎜ ⎢   f j (x) ⎟ +⎢ ⎜ 1 ⎠ ⎣ x0 ⎝ j=1 ( p (n − 1) + 1) p

  (n)     f i ◦ g −1   

j =i

⎤⎤⎫ ⎪ ⎪ ⎪ ⎥⎥⎬ ⎥⎥ d x ⎥⎥ ⎦⎦⎪ ⎪ L q ([g(x0 ),g(b)]) ⎪ ⎭

1

=

1

(n − 1)! ( p (n − 1) + 1) p ⎧ ⎛ ⎡ ⎪ ⎪   ⎪ r ⎢ ⎜ x 0 ⎨   (n)  n− 1 ⎜ ⎢  f i ◦ g −1  (g (x0 ) − g (x)) q ⎜ ⎢     ⎪ ⎝ ⎣ a ⎪ L q ([g(a),g(x0 )]) ⎪ ⎩i=1 ⎛       −1 (n)   +  f i ◦ g 

⎞

⎞

⎛

r  ⎜ ⎟ ⎟ ⎟ ⎜  f j (x)⎟ ⎟ dx⎟ ⎜ ⎠ ⎠ ⎝ j=1 j =i

⎞ ⎞⎤⎫ ⎪ ⎪ r  ⎬ ⎟⎥⎪ ⎜ ⎟ ⎥ ⎟ ⎟ ⎜   f j (x) ⎟ d x ⎟⎥ , ⎜ ⎠ ⎠⎦⎪ ⎝ ⎪ ⎪ j=1 ⎭ ⎛

⎜ b n− 1 ⎜ (g (x) − g (x0 )) q ⎜ ⎝ x L q ([g(x0 ),g(b)]) 0

j =i

(5.34) 

proving (5.32).

Next we present a left generalized Opial type inequality for ordinary derivatives: Theorem 5.10 Let p, q > 1 : 1p + q1 = 1, and n ∈ N, f ∈ C n ([a, b] , A); where [a, b] ⊂ R and (A, ·) is a Banach algebra. Let g ∈ C 1 ([a, b]), strictly increasing, ( j)  such that g −1 ∈ C n ([g (a) , g (b)]) . We assume that f ◦ g −1 (g (x0 )) = 0, j = 0, 1, ..., n − 1; where x0 ∈ [a, b] be fixed. Then 

g(x) g(x0 )

   (n)    (z) dz ≤  f ◦ g −1 (z) f ◦ g −1 

(g (x) − g (x0 ))n+ p − q 1

1

1

1

2 q (n − 1)! [( p (n − 1) + 1) ( p (n − 1) + 2)] p

g(x) g(x0 )

 q q2    −1 (n) (z) dz ,  f ◦g (5.35)

for all x0 ≤ x ≤ b.

 ( j) Proof Let x0 ∈ [a, b] such that f ◦ g −1 (g (x0 )) = 0, j = 0, 1, ..., n − 1. For x ∈ [x0 , b] by Theorem 5.6 we have 

 f ◦ g −1 (g (z)) =

1 (n − 1)!



g(x) g(x0 )

 (n) (g (x) − z)n−1 f ◦ g −1 (z) dz.

(5.36)

By Hölder’s inequality we obtain     f ◦ g −1 (g (x)) ≤

1 (n − 1)!



g(x) g(x0 )

 (n)    (g (x) − z)n−1  f ◦ g −1 (z) dz ≤ (5.37)

92

5 Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities …

1 (n − 1)!



g(x) g(x0 )

(g (x) − z) p(n−1) dt p(n−1)+1

1 (g (x) − g (x0 )) p (n − 1)! ( p (n − 1) + 1) 1p Call

 ϕ (g (x)) :=

ϕ (g (x0 )) = 0. Thus

g(x) g(x0 )

1p 

g(x) g(x0 )



g(x) g(x0 )

 q q1    −1 (n) f ◦ g = (z)   dz

 q q1    −1 (n) (z) dz .  f ◦g

 (n)   q (z) dz,  f ◦ g −1

q (n) dϕ (g (x))    =  f ◦ g −1 (g (x)) ≥ 0, dg (x)

and



dϕ (g (x)) dg (x)

q1

  (n)   =  f ◦ g −1 (g (x)) ≥ 0,

(5.38)

(5.39)

(5.40)

∀ g (x) ∈ [g (x0 ) , g (b)] . Consequently, we get      (n)   f ◦ g −1 (g (w))  (g (w)) ≤  f ◦ g −1 1  dϕ (g (w)) q ϕ , (g (w)) 1 dg (w) (n − 1)! ( p (n − 1) + 1) p (g (w) − g (x0 ))

p(n−1)+1 p

(5.41)

∀ g (w) ∈ [g (x0 ) , g (b)] . Then we observe that 

g(x) g(x0 )



g(x) g(x0 )

    (n) (5.7)   (g (w)) dg (w) ≤  f ◦ g −1 (g (w)) f ◦ g −1

     (n)   f ◦ g −1 (g (w))  (g (w)) dg (w) ≤  f ◦ g −1 1 1

(n − 1)! ( p (n − 1) + 1) p 

g(x) g(x0 )

(g (w) − g (x0 ))

p(n−1)+1 p

 1 dϕ (g (w)) q ϕ (g (w)) dg (w) ≤ dg (w)

(5.42)

5.4 Main Results

93

1 1

(n − 1)! ( p (n − 1) + 1) p 

g(x) g(x0 )

 1 

g(x)

p

(g (w) − g (x0 )) p(n−1)+1 dg (w)

g(x0 )

ϕ (g (w))

dϕ (g (w)) dg (w) dg (w)

1 q

=

1 1

1

(n − 1)! ( p (n − 1) + 1) p ( p (n − 1) + 2) p (g (x) − g (x0 ))

p(n−1)+2 p



g(x) g(x0 )

ϕ (g (w)) dϕ (g (w)) 

(g (x) − g (x0 ))n+ p − q 1

1

1

1

(n − 1)! ( p (n − 1) + 1) p ( p (n − 1) + 2) p 

(g (x) − g (x0 ))n+ p − q 1

1

1

1

2 q (n − 1)! (( p (n − 1) + 1) ( p (n − 1) + 2)) p

g(x) g(x0 )

q1

ϕ 2 (g (x)) 2

=

q1

(5.43)

=

 q q2    −1 (n) (z) dz ,  f ◦g (5.44) 

for all g (x0 ) ≤ g (x) ≤ g (b), proving (5.35). The corresponding right generalized Opial type inequality follows: Theorem 5.11 All as in Theorem 5.10. Then 

g(x0 )

   (n)    (z) dz ≤  f ◦ g −1 (z) f ◦ g −1

g(x)



(g (x0 ) − g (x))n+ p − q 1

1

1

g(x0 )

1

2 q (n − 1)! (( p (n − 1) + 1) ( p (n − 1) + 2)) p

 q q2    −1 (n) (z) dz ,  f ◦g

g(x)

(5.45)

for all a ≤ x ≤ x0 . Proof As similar to Theorem 5.10 is omitted.



Next we present a left generalized Hilbert-Pachpatte inequality for ordinary derivatives. Theorem 5.12 Let i = 1, 2; p, q > 1 : 1p + q1 = 1, and n i ∈ N, f i ∈ C ni ([ai , bi ] , A); where [ai , bi ] ⊂ R and (A, ·) is a Banach algebra. Let gi ∈ C 1 ([ai , bi ]), strictly increasing, such that gi−1 ∈ C ni ([gi (ai ) , gi (bi )]). We assume that ( ji )  f i ◦ gi−1 (gi (x0i )) = 0, ji = 0, 1, ..., n i − 1; where x0i ∈ [ai , bi ] be fixed. Then

94

5 Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities …



g1 (b1 )



g1 (x01 )

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2   ≤ q (n 2 −1)+1 (z 1 −g1 (x01 )) p(n 1 −1)+1 2 (x 02 )) + (z2 −g p( p(n 1 −1)+1) q(q(n 2 −1)+1)

g2 (b2 ) g2 (x02 )

(g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 )) (n 1 − 1)! (n 2 − 1)!      (n 1 )  (n 2 )       f 2 ◦ g2−1   f 1 ◦ g1−1 L q ([g1 (x01 ),g1 (b1 )],A)

L p ([g2 (x02 ),g2 (b2 )],A)

Proof Let i = 1, 2; x0 ∈ [ai , bi ], such ji = 0, 1, ..., n i − 1. For xi ∈ [x0i , bi ] by Theorem 5.6 we have 

 f i ◦ gi−1 (gi (xi )) =

1 (n i − 1)!



gi (xi ) gi (x0i )

(5.46)

that



f i ◦ gi−1

( ji )

.

(gi (x0i )) = 0,

 (ni ) (gi (xi ) − z i )ni −1 f i ◦ gi−1 (z i ) dz i . (5.47)

As in (5.37) we have     f 1 ◦ g −1 (g1 (x1 )) ≤ 1



g1 (x1 ) g1 (x01 )

p (n 1 −1)+1

1 (g1 (x1 ) − g1 (x01 )) p 1 (n 1 − 1)! ( p (n 1 − 1) + 1) p

 q q1    −1 (n 1 ) ≤ (z) dz  f 1 ◦ g1 p (n 1 −1)+1

1 (g1 (x1 ) − g1 (x01 )) p 1 (n 1 − 1)! ( p (n 1 − 1) + 1) p

  (n 1 )     f 1 ◦ g1−1 

L q ([g1 (x01 ),g1 (b1 )])

, (5.48)

for all x1 ∈ [x01 , b1 ] . Similarly, we obtain that     f 2 ◦ g −1 (g2 (x2 )) ≤ 2

q (n 2 −1)+1 q

1 (g2 (x2 ) − g2 (x02 )) 1 (n 2 − 1)! (q (n 2 − 1) + 1) q

  (n 2 )     f 2 ◦ g2−1 

L p ([g2 (x02 ),g2 (b2 )])

,

for all x2 ∈ [x02 , b2 ] . By (5.48) and (5.49) we get       f 1 ◦ g −1 (g1 (x1 )) f 2 ◦ g −1 (g2 (x2 )) ≤ 1

2

(5.49)

5.4 Main Results

95

       f 1 ◦ g −1 (g1 (x1 ))  f 2 ◦ g −1 (g2 (x2 )) ≤ 1

2

(g1 (x1 ) − g1 (x01 ))

p (n 1 −1)+1 p 1

( p (n 1 − 1) + 1) p   (n 1 )     f 1 ◦ g1−1 

L q ([g1 (x01 ),g1 (b1 )])

(g2 (x2 ) − g2 (x02 ))

(5.50)

1

L p ([g2 (x02 ),g2 (b2 )])

1

1 (n 1 − 1)! (n 2 − 1)!

q (n 2 −1)+1 q

(q (n 2 − 1) + 1) q   (n 2 )     f 2 ◦ g2−1  1

(using Young’s inequality for a, b ≥ 0, a p b q ≤ 

1 (n 1 − 1)! (n 2 − 1)!

a p

≤

+ qb )

(g2 (x2 ) − g2 (x02 ))q(n 2 −1)+1 (g1 (x1 ) − g1 (x01 )) p(n 1 −1)+1 + p ( p (n 1 − 1) + 1) q (q (n 2 − 1) + 1)



(5.51)   (n 1 )     f 1 ◦ g1−1 

L q ([g1 (x01 ),g1 (b1 )])

  (n 2 )     f 2 ◦ g2−1 

L p ([g2 (x02 ),g2 (b2 )])

∀ (x1 , x2 ) ∈ [x01 , b1 ] × [x02 , b2 ] . So far we have       f 1 ◦ g −1 (g1 (x1 )) f 2 ◦ g −1 (g2 (x2 )) 1 2  ≤ (g1 (x1 )−g1 (x01 )) p(n 1 −1)+1 )−g2 (x02 ))q (n 2 −1)+1 + (g2 (x2q(q(n p( p(n 1 −1)+1) −1)+1) 2

,

(5.52)

  (n 1 )  1    f 1 ◦ g1−1  L q ([g1 (x01 ),g1 (b1 )],A) (n 1 − 1)! (n 2 − 1)!   (n 2 )    ,  f 2 ◦ g2−1  L p ([g2 (x02 ),g2 (b2 )],A)

∀ (x1 , x2 ) ∈ [x01 , b1 ] × [x02 , b2 ] . The denominator in (5.52) can be zero, only when both g1 (x1 ) = g1 (x01 ) and g2 (x2 ) = g2 (x02 ) . Therefore we obtain (5.46), by integrating (5.52) over [g1 (x01 ) , g1 (b1 )] ×  [g2 (x02 ) , g2 (b2 )] . It follows the right generalized Hilbert-Pachpate inequality for ordinary derivatives. Theorem 5.13 All as in Theorem 5.12. Then       g1 (x01 )  g2 (x02 )   f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2   ≤ (g1 (x01 )−z 1 ) p(n 1 −1)+1 (g2 (x02 )−z 2 )q (n 2 −1)+1 g1 (a1 ) g2 (a2 ) + p( p(n 1 −1)+1) q(q(n 2 −1)+1)

96

5 Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities …

(g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 )) (n 1 − 1)! (n 2 − 1)!     (n 1 )  (n 2 )        f 2 ◦ g2−1   f 1 ◦ g1−1 L q ([g1 (a1 ),g1 (x01 )],A)

(5.53)

L p ([g2 (a2 ),g2 (x02 )],A)

. 

Proof As similar to Theorem 5.12 is omitted.

5.5 Applications We make Remark 5.14 Assume next that (A, ·) is a commutative Banach algebra. Then, we get that ⎞ ⎞ ⎛ ⎛ ⎞ ⎛   b  r r ⎜ b ⎜ r ⎟ ⎟

(5.13) ⎟ ⎟ ⎜ ⎜ ⎝ E ( f 1 , ..., fr ) (x0 ) = r f j (x)⎠ d x − f j (x)⎟ d x ⎟ f i (x0 ) , ⎜ ⎜ ⎠ ⎠ ⎝ a ⎝ a j=1

i=1

j=1 j =i

(5.54) x0 ∈ [a, b] . When r = 2, we have that E ( f 1 , f 2 ) (x0 ) = 2

 b a

f 1 (x) f 2 (x) d x − f 1 (x0 )

 b a

f 2 (x) d x − f 2 (x0 )

 b a

f 1 (x) d x,

(5.55)

x0 ∈ [a, b] . We give Corollary 5.15 (to Theorem 5.7) All as in Theorem 5.7, (A, ·) is a commutative Banach algebra, r = 2. Then 1 n! i=1 2

E ( f 1 , f 2 ) (x0 ) ≤

⎛ ⎜ (g (x0 ) − g (a))n ⎜ ⎝ ⎡

 a

((  (n)     f i ◦ g −1 

∞,[g(a),g(x0 )]

⎞

⎛ x0

⎜ ⎜ ⎝

2  j=1 j =i

⎞⎤

⎟ ⎟⎥   f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦ ⎛

 ⎢ ⎜    −1 (n)  ⎢ (g (b) − g (x0 ))n ⎜  ⎣ f i ◦ g ⎝ ∞,[g(x0 ),g(b)]



⎞

⎛ b x0

⎜ ⎜ ⎝

2  j=1 j =i

⎞⎤⎤

 ⎟ ⎟⎥⎥  f j (x)⎟ d x ⎟⎥⎥ . ⎠ ⎠⎦⎦ (5.56)

5.5 Applications

97

It follows Corollary 5.16 (to Corollary 5.15) All as in Corollary 5.15, with g (t) = et . Then E ( f 1 , f 2 ) (x0 ) ≤

2   1 )) ( f i ◦ log)(n)  ∞,[ea ,e x0 ] n! i=1

⎛   x0 n ⎜ e − ea ⎜ ⎝

a

⎞

⎛ x0

⎞⎤

2  ⎟ ⎟⎥ ⎜ ⎜  f j (x)⎟ d x ⎟⎥ + ⎠ ⎠⎦ ⎝ j=1 j =i

⎡

⎛  ⎢   ⎜  b x0 n ⎜ ⎢( f i ◦ log)(n)  − e e x ∞,[e 0 ,eb ] ⎣ ⎝

⎞

⎛ b x0

⎜ ⎜ ⎝

2  j=1 j =i

⎞⎤⎤

 ⎟ ⎟⎥⎥  f j (x)⎟ d x ⎟⎥⎥ . (5.57) ⎠ ⎠⎦⎦

We continue with Corollary 5.17 (to Theorem 5.10) All as in Theorem 5.10 for g (t) = et . Then 

ex e x0

  (( f ◦ log) (z)) ( f ◦ log)(n) (z) dz ≤ 

(e x − e x0 )n+ p − q 1

1

1

1

2 q (n − 1)! [( p (n − 1) + 1) ( p (n − 1) + 2)] p

z

e x0

  ( f ◦ log)(n) (z)q dz

q2

,

(5.58)

for all x0 ≤ x ≤ b. We finish with Corollary 5.18 (to Theorem 5.12) All as in Theorem 5.12 for gi (t) = et , i = 1, 2. Then  e b1  e b2 ( f 1 ◦ log) (z 1 ) ( f 2 ◦ log) (z 2 ) dz 1 dz 2   ≤ (z 1 −e x01 ) p(n 1 −1)+1 (z 2 −e x02 )q (n 2 −1)+1 e x01 e x02 + p( p(n 1 −1)+1) q(q(n 2 −1)+1)  b   e 1 − e x01 eb2 − e x02 (n 1 − 1)! (n 2 − 1)!   ( f 1 ◦ log)(n 1 ) 

L q ([e x01 ,eb1 ],A)

  ( f 2 ◦ log)(n 2 ) 

(5.59) L p ([e x02 ,eb2 ],A)

.

The simplest applications derive when g (t) = t and A = R, leading to basic known results.

98

5 Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities …

References 1. Anastassiou, G.A.: Intelligent Comparisons: Analytic Inequalities. Springer, Heidelberg, New York (2016) 2. Anastassiou, G.A.: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations. Springer, Heidelberg, New York (2018) 3. Anastassiou, G.A.: Generalized Ostrowski, Opial and Hilbert-Pachpatte type inequalities for Banach algebra valued functions involving integer vectorial derivatives. J. Comput. Anal. Appli. 30(1), 78–94 (2022) 4. Dragomir, S.S.: Noncommutative Ostrowski type inequalities for functions in Banach algebras. RGMIA Res. Rep. Coll. 24, Art. 10, 24 pp (2021) 5. Mikusinski, J.: The Bochner Integral. Academic Press, New York (1978) 6. Opial, Z.: Sur une inegalite. Ann. Polon. Math. 8, 29–32 (1960) 7. Ostrowski, A.: Über die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938) 8. Pachpatte, B.G.: Inequalities similar to the integral analogue of Hilbert’s inequalities. Tamkang J. Math. 30(1), 139–146 (1999) 9. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991) 10. Shilov, G.E.: Elementary Functional Analysis. Dover Publications Inc., New York (1996)

Chapter 6

Multivariate Ostrowski Inequalities for Several Banach Algebra Valued Functions

Here we are dealing with several smooth functions from a compact convex set of Rk , k ≥ 2 to a Banach algebra. For these we prove general multivariate Ostrowski inequalities with estimates. It follows [2].

6.1 Introduction In 1938, A Ostrowski [6] proved the following famous inequality: Theorem 6.1 (1938, Ostrowski [7]) Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, derivative f  : (a, b) → R is bounded on (a, b), i.e.,   sup   b) whose  f  := sup  f  (t) < +∞. Then ∞ t∈(a,b)

   1  b − a

a

b

 2       sup x − a+b 1 2 f  , + f (t) dt − f (x) ≤ − a) (b ∞ 4 (b − a)2

for any x ∈ [a, b]. The constant

1 4

(6.1)

is the best possible.

Since then there has been a lot of activity around these inequalities with important applications to Numerical Analysis and Probability. This chapter is also greatly motivated by the following result:  k

Theorem 6.2 (see [1]) Let f ∈ C 1 [ai , bi ] , where ai < bi ; ai , bi ∈ R, i = i=1

→ 1, . . . , k, ans let − x0 := (x01 , . . . , x0k ) ∈

k

[ai , bi ] be fixed. Then

i=1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_6

99

100

6 Multivariate Ostrowski Inequalities for Several Banach Algebra Valued Functions

       bi  bk  b1   − 1  → ··· ··· f (z 1 , . . . , z k ) dz 1 · · · dz k − f x0  ≤ (6.2)  k 

 ai ak  (bi − ai ) a1   i=1    k  (x0i − ai )2 + (bi − x0i )2   ∂f  .  2 (bi − ai ) ∂z i ∞ i=1 Inequality (6.2) is sharp, here the optimal function is ∗

f (z 1 , . . . , z k ) :=

k

|z i − x0i |αi , αi > 1.

i=1

Clearly inequality (6.2) generalizes inequality (6.1) to multidimension. We are inspired also by [3]. In this chapter we establish multivariate Ostrowski type inequalities for several smooth functions from a compact convex subset of Rk , k ≥ 2, to a Banach algebra. These involve the norms · p , 1 ≤ p ≤ ∞.

6.2 About Banach Algebras All here come from [7]. We need Definition 6.3 ([7], p. 245) A complex algebra is a vector space A over the complex field C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(6.3)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(6.4)

α (x y) = (αx) y = x (αy) ,

(6.5)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A) and if A contains a unit element e such that

(6.6)

6.3 Vector Analysis Background

101

xe = ex = x (x ∈ A)

(6.7)

e = 1,

(6.8)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 6.4 Commutativity of A will be explicited stated when needed. There exists at most one e ∈ A that satisfies (6.7). Inequality (6.6) makes multiplication to be continuous, more precisely left and right continuous, see [7], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [7], pp. 247–248, Sect. 10.3. We also make Remark 6.5 Next we mention about integration of A-valued functions, see [7], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some compact

Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [7], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   f dμ = x f ( p) dμ ( p) (6.9) x Q

and



Q

  f dμ x = Q

f ( p) x dμ ( p) .

(6.10)

Q

The vector integrals we will involve in our chapter follow (6.9) and (6.10).

6.3 Vector Analysis Background (see [9], pp. 83–94) Let f (t) be a function defined on [a, b] ⊆ R taking values in a real or complex normed linear space (X, · ), Then f (t) is said to be differentiable at a point t0 ∈ [a, b] if the limit f (t0 + h) − f (t0 ) (6.11) f  (t0 ) = lim h→0 h

102

6 Multivariate Ostrowski Inequalities for Several Banach Algebra Valued Functions

exists in X , the convergence is in · . This is called the derivative of f (t) at t = t0 . We call f (t) differentiable on [a, b], iff there exists f  (t) ∈ X for all t ∈ [a, b]. Similarly and inductively are defined higher order derivatives of f , denoted f  , f (3) , . . . , f (k) , k ∈ N, just as for numerical functions. For all the properties of derivatives see [9], pp. 83–86. Let now (X, · ) be a Banach space, and f : [a, b] → X.

b We define the vector valued Riemann integral a f (t) dt ∈ X as the limit of the vector valued Riemann sums in X , convergence is in · . The definition is as for the numerical

b valued functions. If a f (t) dt ∈ X we call f integrable on [a, b]. If f ∈ C ([a, b] , X ), then f is integrable, [9], p. 87. For all the properties of vector valued Riemann integrals see [9], pp. 86–91. We define the space C n ([a, b] , X ), n ∈ N, of n-times continuousky differentiable functions from [a, b] into X ; here continuity is with respect to · and defined in the usual way as for numerical functions·. Let (X, · ) be a Banach space and f ∈ C n ([a, b] , X ), then we have the vector valued Taylor’s formula, see [9], pp. 93–94, and also [8], (IV, 9; 47). It holds f (y) − f (x) − f  (x) (y − x) −

1  1 f (n−1) (x) (y − x)n−1 f (x) (y − x)2 − · · · − 2 (n − 1)!

(6.12)

=

1 (n − 1)!



y

(y − t)n−1 f (n) (t) dt, ∀ x, y ∈ [a, b] .

x

In particular (6.12) is true when X = Rm , Cm , m ∈ N, etc. A function f (t) with values in a normed linear space X is said to be piecewise continuous (see [9], p. 85) on the interval a ≤ t ≤ b if there exists a partition a = t0 < t1 < t2 < · · · < tn = b such that f (t) is continuous on every open interval tk < t < tk+1 and has finite limits f (t0 + 0) , f (t1 − 0) , f (t1 + 0) , f (t2 − 0) , f (t2 + 0) , . . . , f (tn − 0) . Here f (tk − 0) = lim f (t) , f (tk + 0) = lim f (t) . t↑tk

t↓tk

The values of f (t) at the points tk can be arbitrary or even undefined. A function f (t) with values in normed linear space X is said to be piecewise smooth on [a, b], if it is continuous on [a, b] and has a derivative f  (t) at all but a finite number of points of [a, b] , and if f  (t) is piecewise continuous on [a, b] (see [9], p. 85). Let u (t) and v (t) be two piecewise smooth functions on [a, b], one a numerical function and the other a vector function with values in Banach space X . Then we have the following integration by parts formula  a

see [9], p. 93.

b

 u (t) dv (t) = u

(t) v (t) |ab

− a

b

v (t) du (t) ,

(6.13)

6.4 Main Results

103

We mention also the mean value theorem for Banach space valued functions. Theorem 6.6 (see [5], p. 3) Let  f ∈C ([a, b] , X ), where X is a Banach space. Assume f  exists on [a, b] and  f  (t) ≤ K , a < t < b, then f (b) − f (a) ≤ K (b − a) .

(6.14)

Here the multiple Riemann integral of a function from a real box or a real compact and convex subset to a Banach space is defined similarly to numerical one however convergence is with respect to · . Similarly are defined the vector valued partial derivatives as in the numerical case. We mention the equality of vector valued mixed partiasl derivatives. Proposition 6.7 (see Proposition 4.11 of [4], p. 90) Let Q = (a, b) × (c, d) ⊆ R2 and f ∈ C (Q, X ), where (X, · ) is a Banach space. Assume that ∂t∂ f (s, t), ∂ ∂2 ∂2 f t) and ∂t∂s f (s, t) exist and are continuous for (s, t) ∈ Q, then ∂s∂t f (s, t) ∂s (s, exists for (s, t) ∈ Q and ∂2 ∂2 f (s, t) = f (s, t) , for (s, t) ∈ Q. ∂s∂t ∂t∂s

(6.15)

6.4 Main Results We present general Ostrowski type inequalities results regarding several Banach algebra valued functions. Theorem 6.8 Let p, q > 1 : 1p + q1 = 1; (A, · ) a Banach algebra and f i ∈ C n+1 → x0 ∈ Q ⊂ Rk , k ≥ 2, whereα Q (Q, A), i = 1, . . . , r ; r ∈ N, n ∈ Z+ , and fixed − is a compact and convex subset. Here all vector partial derivatives f iα := ∂∂z αfi , k  αλ = j, j = 1, . . . , n, where α = (α1 , . . . , αk ), αλ ∈ Z+ , λ = 1, . . . , k, |α| = λ=1 −  fulfill f iα → x0 = 0, i = 1, . . . , r. Denote (6.16) Dn+1 ( f i ) := max f iα ∞,Q , α:|α|=n+1

i = 1, . . . , r, and

k −  → → |z λ − x0λ | . z −− x0 l1 := λ=1

Then

(6.17)

104

6 Multivariate Ostrowski Inequalities for Several Banach Algebra Valued Functions

  ⎛ ⎛ ⎛ ⎞ ⎞ ⎞      r  r r r    ⎜ − ⎜ ⎜ ⎟ ⎟ ⎟        → − → − → − → − → − →   ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ z f z d z − z d z f x f f ρ i ρ i 0 ≤  ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ Q  i=1 Q ρ=1  ρ=1 i=1   ρ=i ρ=i ⎞ ⎛ ⎛ ⎞ (6.18)  max Dn+1 ( f i ) r r  →⎟ − n+1 →⎟ ⎜ ⎜ → → ⎜ ⎜  fρ − z −− x0 l1 d − z ⎟ z⎟ ⎠ ⎝ ⎝ ⎠≤ (n + 1)! Q ρ=1 i=1

i∈{1,...,r }

ρ=i

⎧ ⎪ ⎪ ⎨ 

max Dn+1 ( f i )

i∈{1,...,r }

(n + 1)!

min

⎪ ⎪ ⎩

Q

⎛ ⎞⎤ ⎡  r r − n+1 → ⎢ ⎜   ⎟⎥ → → ⎜  f ρ  ⎟⎥ , z −− x0 l1 d − z ⎢ ∞,Q ⎠⎦ ⎝ ⎣ i=1

⎛ ⎞    r ⎜  ⎟ ⎢ ⎜  f ρ ⎟ ⎢ ⎝ ⎠ ∞,Q ⎣  i=1  ρ=1   ρ=i ⎡

   − →n+1   · − x0 l1 

ρ=1 ρ=i

⎤

r 

⎞ ⎡⎛     r r    ⎜      ⎟ ⎢ ⎢ n+1   − → ⎜     ⎟ ⎢ ⎢ f ρ ⎠  · − x 0 l1    ⎝ ⎣ ⎣ L p (Q,A)  i=1  ρ=1   ρ=i

⎥ ⎥, ⎦ L 1 (Q,A)

⎡

L q (Q,A)

⎤⎤⎫ ⎪ ⎪ ⎥⎥⎬ ⎥⎥ . ⎦⎦⎪ ⎪ ⎭

(6.19)

→  → − → Proof Take gi − x0 + t − z −→ x0 , 0 ≤ t ≤ 1; i = 1, . . . , r . Notice that := f i − z (t) −    → − → → → → gi − z (0) = f i x 0 and gi − z (1) = f i z . The jth derivative of gi − z (t), based on Proposition 6.7, is given by ⎡# $j ⎤ k ∂ ( j) gi − f i ⎦ (x01 + t (z 1 − x01 ) , . . . , x0k + t (z k − x0k )) (t) = ⎣ (z λ − x0λ ) → z ∂z λ λ=1 (6.20) and

⎡# $j ⎤ k  → ∂ ( j) gi − fi ⎦ − x0 , (0) = ⎣ (z λ − x0λ ) → z ∂z λ λ=1

(6.21)

for j = 1, . . . , n + 1; i = 1, . . . , r. Let f iα be a partialderivative of f i ∈ C n+1 (Q, A). Because by assumption of the  − → theorem we have f iα x0 = 0 for all α : |α| = j, j = 1, . . . , n, we find that ( j)

gi − (0) = 0, j = 1, . . . , n; i = 1, . . . , r. → z

6.4 Main Results

105

Hence by vector Taylor’s theorem (6.12) we see that ( j)

n gi − (0) −  −  →  →  → z → → z , 0 = Rin − z ,0 , fi z − fi x0 = + Rin − j! j=1

(6.22)

where →  Rin − z , 0 :=





1 0

t1



0

%

tn−1

··· 0

 &  (n) · · · dt1 , (6.23) gi(n) − g dt (t ) (0) − → − → n n z i z

i = 1, . . . , r. Therefore,  →   Rin − z ,0  ≤



1



0

t1



tn−1

···

0

0

     (n+1)  (ξ (tn )) tn dtn · · · dt1 , gi − → z ∞

(6.24) over t by the vector mean value Theorem 6.6 applied on gi(n) Moreover, we get (0, ). − → n z 

   →      Rin − z , 0  ≤ g (n+1) − →  i z

∞,[0,1]

1



t1

 ···

0

0

tn−1

   (n+1)   gi − → z

∞,[0,1]

(6.25)

∞,[0,1]

That is    (n+1)  gi −  → z

∞,[0,1]

    = gi(n+1) (t ) − → i0  . z

 ⎡# $n+1 ⎤   k  −  − ∂ → − → →  ⎣ ⎦ x0 + ti0 z − z 0i  = fi (z λ − x0λ )  ∂z λ   λ=1

⎡#  k  ∂ ⎣ |z λ − x0λ |  ≤  ∂z λ=1

I.e.,

tn dtn · · · dt1

0

. (n + 1)!     However, there exists a ti0 ∈ [0, 1] such that gi(n+1)  − → z =



   (n+1)  gi −  → z

∞,[0,1]

i = 1, . . . , r. Hence by (6.26) we get

λ

⎤ $n+1  → −  →  z −→ z 0i . x0 + ti0 − fi ⎦ − 

⎡#  k  ∂  |z λ − x0λ |  ≤⎣  ∂z λ=1

λ

⎤  $n+1   fi ⎦ ,  ∞

(6.26)

106

6 Multivariate Ostrowski Inequalities for Several Banach Algebra Valued Functions

  →   Rin − z ,0  ≤

k 

λ=1

   n+1  ∂  |z λ − x0λ |  ∂zλ  fi ∞

≤

(n + 1)!

$n+1 # k n+1 Dn+1 ( f i )  Dn+1 ( f i ) → → − |z λ − x0λ | z −− x0 l1 , = + 1)! + 1)! (n (n λ=1

(6.27)

i = 1, . . . , r. Therefore it holds max Dn+1 ( f i )   n+1 →  i∈{1,...,r } → →  Rin − − z −− x0 l1 , z ,0  ≤ (n + 1)!

(6.28)

for i = 1, . . . , r. By (6.22) we get that ⎛ ⎜ ⎜ ⎝

⎞ r 

ρ=1 ρ=i

⎞

⎛

→⎟ −  ⎜ → f z ⎟ z −⎜ fρ − i ⎠ ⎝

r 

ρ=1 ρ=i

⎞

⎛

→⎟ −  ⎜ → f z ⎟ x =⎜ fρ − i 0 ⎠ ⎝

r  ρ=1 ρ=i

→⎟ − →  z ⎟ fρ − ⎠ Rin z , 0 , (6.29)

for all i = 1, . . . , r. Hence ⎛ r i=1

⎜ ⎜ ⎝

r 

ρ=1 ρ=i

⎛

⎞ →⎟ − → z ⎟ fρ − ⎠ fi z −

r

⎛ =

i=1

⎜ ⎜ ⎝

⎞ r  ρ=1 ρ=i

→⎟ − → z ⎟ fρ − ⎠ fi x0

⎞

r r ⎜ →⎟ − →  ⎜ z ⎟ fρ − ⎝ ⎠ Rin z , 0 . i=1

(6.30)

ρ=1 ρ=i

Therefore we find E ( f 1 , . . . , fr ) (x0 ) := r  i=1

⎛ ⎜ ⎜ ⎝ Q

⎞ r 

ρ=1 ρ=i

→⎟ − → − → z ⎟ fρ − ⎠ fi z d z −

r i=1

⎛ ⎛ ⎞ ⎞  r ⎜ ⎜ − ⎟ − − →⎟ → ⎜ ⎜ ⎟ z ⎟ fρ → ⎝ Q⎝ ⎠ d z ⎠ fi x0 = ρ=1 ρ=i

6.4 Main Results

107 r  i=1

⎛ ⎜ ⎜ ⎝ Q

⎞ r  ρ=1 ρ=i

→⎟ − →  − → z ⎟ fρ − ⎠ Rin z , 0 d z .

(6.31)

Consequently, we have that E ( f 1 , . . . , fr ) (x0 ) =   ⎞ ⎛ ⎛ ⎛ ⎞ ⎞    r    r r r    ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ −        − → → − → − → − → − →  ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ fρ z ⎠ fi z d z − f ρ z ⎠ d z ⎠ fi x0   = ⎝ ⎝ ⎝ Q  i=1 Q ρ=1  ρ=1 i=1   ρ=i ρ=i   ⎛ ⎞      r  r  ⎜ ⎟ −    → − → − →  ⎜ ⎟ f ρ z ⎠ Rin z , 0 d z   ≤ ⎝  i=1 Q ρ=1    ρ=i      i=1  

r 

⎛ ⎜ ⎜ ⎝ Q

(6.32)

    ⎟ −    − → − → ≤ z ⎟ R z , 0 d z fρ →  ⎠ in   ⎞

r  ρ=1 ρ=i

 ⎞ ⎛ ⎛ ⎞       r ⎜ ⎜ →⎟ − (6.6) →⎟ →  ⎟  − ⎜ ⎜ fρ − z ⎟ ⎝ Q ⎝ ⎠ Rin z , 0  d z ⎠ ≤  ρ=1  i=1  ρ=i 

r

r i=1

⎛  ⎜ ⎜ ⎝

⎞

⎛

Q

⎜ ⎜ ⎝

r 

ρ=1 ρ=i

⎞

 − ⎟  − (6.28) →⎟ →    − ⎟  fρ → z ⎟ ⎠ Rin z , 0 d z ⎠ ≤

⎞ ⎛ ⎛ ⎞  max Dn+1 ( f i ) r r  →⎟ − n+1 →⎟ ⎜ ⎜ i∈{1,...,r } → → ⎜ ⎜  fρ − z −− x0 l1 d − z ⎟ z⎟ ⎠ ⎝ ⎝ ⎠. (n + 1)! Q i=1

ρ=1 ρ=i

So far we have proved E ( f 1 , . . . , fr ) (x0 ) ≤

(6.33)

108

6 Multivariate Ostrowski Inequalities for Several Banach Algebra Valued Functions

⎞ ⎛ ⎛ ⎞  max Dn+1 ( f i ) r r n+1 →⎟ ⎜ ⎜  − ⎟ − i∈{1,...,r } → → ⎜ ⎜  fρ → z −− x0 l1 d − z ⎟ z⎟ ⎠ ⎝ ⎝ ⎠ =: (ξ ) . (n + 1)! Q ρ=1 i=1 ρ=i

(6.34) Furthermore it holds ⎛ ⎞⎤ ⎡  max Dn+1 ( f i )   r r n+1 → ⎢ ⎜   ⎟⎥ i∈{1,...,r } → → − ⎜  f ρ  ⎟⎥ , z −− x0 l1 d − z ⎢ (ξ ) ≤ ∞,Q ⎝ ⎠⎦ ⎣ (n + 1)! Q ρ=1 i=1 ρ=i

(6.35) and ⎛ ⎞ ⎡     max Dn+1 ( f i )  r r   ⎜      ⎟ ⎢ i∈{1,...,r } n+1   − → ⎜     ⎟ ⎢ f ρ ⎠ (ξ ) ≤  · − x 0 l1    ⎝ ⎣ ∞,Q (n + 1)!  i=1  ρ=1   ρ=i

⎤ ⎥ ⎥, ⎦ L 1 (Q,A)

(6.36)

and finally ⎞ ⎡⎛ ⎡      max Dn+1 ( f i ) ⎢ r r ⎜     ⎟ ⎢ i∈{1,...,r } ⎜     ⎟ ⎢ ⎢ f (ξ ) ≤ ρ  ⎠ ⎣ ⎝ ⎣ (n + 1)!   i=1   ρ=1 ρ=i

⎤⎤  n+1  ⎥⎥  → · − −   ⎥⎥  x ,  0 l1  L (Q,A) ⎦⎦ p L q (Q,A)

(6.37) 

proving (6.18), (6.19). We give

Corollary 6.9 (to Theorem 6.8) All as in Theorem 6.8, with f 1 = · · · = fr = f , r ∈ N. Then       → −  → − −  → → → r − r −1 −  z d z f x0  f z d z − f  ≤ Q

Q

Dn+1 ( f ) (n + 1)! Dn+1 ( f ) min (n + 1)!    − →n+1   · − x0 l1 

∞,Q

 Q

  − n+1 → r −1 → − f → z  − z −→ x0 l1 d − z ≤

'  Q

   f r −1 

(6.38)

 − n+1 − r −1  − → → →  f ∞,Q z − x0 l d z , 1

 n+1   − →   · − x ,  0 L 1 (Q,A) l1 

L p (Q,A)

   f r −1 

( L q (Q,A)

.

(6.39) We also give

6.4 Main Results

109

Corollary 6.10 (to Theorem 6.8) All as in Theorem 6.8, with (A, · ) being a commutative Banach algebra. Then   ⎞ ⎛ ⎛ ⎞   ⎞ ⎛     r r r  −  ⎟ ⎜ ⎜  ⎟  −  −  − → → − → → → ⎟ ⎜ ⎜ r ⎟ ⎠ ⎝ d z − fρ z f ρ z ⎠ d z ⎠ fi x0  ≤  ⎝ ⎝ Q   Q ρ=1 ρ=1 i=1   ρ=i Right hand side of (6.18) ≤ Right hand side of (6.19).

(6.40)

We make Remark 6.11 Of great interest are applications of Theorem 6.8 when Q = [aλ , bλ ], where [aλ , bλ ] ⊂ R, λ = 1, . . . , k. We observe that by the multinomial theorem we get: # k

 k

[aλ ,bλ ]

|z λ − x0λ |

 [aλ ,bλ ]



dz 1 · · · dz k =

λ=1

λ=1

k

$n+1

ρ1 +ρ2 +···ρk

k

λ=1

(n + 1)! ρ !ρ2 ! · · · ρk ! 1 =n+1

|z 1 − x01 |ρ1 |z 2 − x02 |ρ2 · · · |z k − x0k |ρk dz 1 · · · dz k =

(6.41)

λ=1

ρ1 +ρ2 +···ρk

 k  bλ (n + 1)!  ρλ |z λ − x0λ | dz λ = ρ !ρ ! · · · ρk ! λ=1 aλ =n+1 1 2



(n + 1)!  k

k  ρλ ! λ=1 ρλ =n+1 k



x0λ

(x0λ − z λ )ρλ dz λ +

aλ



bλ

(z λ − x0λ )ρλ dz λ

 =

x0λ

λ=1

λ=1

 k (n + 1)!  (x0λ − aλ )ρλ +1 + (bλ − x0λ )ρλ +1 . k

ρλ + 1 k λ=1  ρλ ! ρλ =n+1

(6.42)

λ=1

λ=1



We have found that

k

[aλ ,bλ ]

− n+1 → → → z −− x0  d − z = l1

λ=1

 k (n + 1)!  (bλ − x0λ )ρλ +1 + (x0λ − aλ )ρλ +1 . k

ρλ + 1 k λ=1  ρλ ! ρλ =n+1

λ=1

λ=1

(6.43)

110

6 Multivariate Ostrowski Inequalities for Several Banach Algebra Valued Functions

Based on (6.18), (6.19) and (6.43) we conclude: Theorem 6.12 Let (A, · ) a Banach algebra and f i ∈ C → x0 ∈ i = 1, . . . , r ; r ∈ N, n ∈ Z+ , and fixed − α

k

λ=1

n+1

k

λ=1

 [aλ , bλ ] , A ,

[aλ , bλ ] ⊂ Rk , k ≥ 2. Here all vec-

tor partial derivatives f iα := ∂∂z αfi , where α = (α1 , . . . , αk ), αλ ∈ Z+ , λ = 1, . . . , k, k →  |α| = x0 = 0, i = 1, . . . , r. αλ = j, j = 1, . . . , n, fulfill f iα − λ=1

Denote Dn+1 ( f i ) := i = 1, . . . , r. Then

max f iα

α:|α|=n+1

∞,

k

λ=1

[aλ ,bλ ]

,

(6.44)

 ⎛ ⎞   r  r  ⎜ − ⎟ −  →  ⎜ fρ → z ⎟ fi → z d− z− k 

⎝ ⎠ [aλ ,bλ ] ρ=1  i=1 λ=1  ρ=i ⎛ r i=1

⎜ ⎜ ⎝

⎛

 k

[aλ ,bλ ]

λ=1

⎜ ⎜ ⎝

   ⎟ ⎟ −  −  → → − → ⎟ ⎟ f ρ z ⎠ d z ⎠ fi x0  ≤   ⎞

r 

ρ=1 ρ=i

⎞

(6.45)

⎡ ⎛ ⎞⎤  r ⎢ r ⎜   ⎟⎥ ⎜  f ρ  k ⎟⎥ max Dn+1 ( f i ) ⎢ ⎣ ⎝ i∈{1,...,r } ∞, [aλ ,bλ ] ⎠⎦



i=1

⎡ ⎢ ⎢ ⎢ ⎣

ρ=1 ρ=i

λ=1

⎤ k 

λ=1

k

ρλ =n+1

λ=1

ρλ !

1 k

λ=1

(ρλ + 1)

k   ⎥ ⎥ (bλ − x0λ )ρλ +1 + (x0λ − aλ )ρλ +1 ⎥ . ⎦ λ=1

References 1. Anastassiou, G.A.: Multivariate Ostrowski type inequalities. Acta Math. Hung. 76(4), 267–278 (1997) 2. Anastassiou, G.A.: Multivariate Ostrowski type inequalities for several Banach algebra valued functions. J. Comput. Anal. Appl. 30(1), 95–106 (2022)

References

111

3. Dragomir, S.S.: Noncommutative Ostrowski type inequalities for functions in Banach algebras. RGMIA Res. Rep. Coll. 24(10), (2021) 4. Driver, B.: Analysis Tools with Applications. Springer, N.Y., Heidelberg (2003) 5. Ladas, G., Laksmikantham, V.: Differential Equations in Abstract Spaces. Academic Press, New York, London (1972) 6. Ostrowski, A.: Über die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938) 7. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991) 8. Schwartz, L.: Analyse Mathematique. Hermann, Paris (1967) 9. Shilov, G.E.: Elementary Functional Analysis. The MIT Press Cambridge, Massachusetts (1974)

Chapter 7

p-Schatten Norm Generalized Fractional Ostrowski and Grüss Inequalities for Multiple Functions

Employing generalized Caputo fractional left and right vectorial Taylor formulae we establish generalized fractional Ostrowski and Grüss inequalities involving several functions that take values in the von Neumann-Schatten class B p (H ), 1 ≤ p < ∞. The estimates are with respect to all p-Schatten norms, 1 ≤ p < ∞. We finish with applications. It follows [6].

7.1 Introduction The following results motivate this chapter. Theorem 7.1 (1938, Ostrowski [15]) Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, derivative f  : (a, b) → R is bounded on (a, b),     b) whose  sup    i.e., f ∞ := sup f (t) < +∞. Then t∈(a,b)

   1  b − a

a

b

 2       sup x − a+b 1 2 f  , + f (t) dt − f (x) ≤ − a) (b ∞ 4 (b − a)2

for any x ∈ [a, b]. The constant

1 4

(7.1)

is the best possible.

Ostrowski type inequalities have great applications to integral approximations in Numerical Analysis. ˇ Theorem 7.2 (1882, Cebyšev [8]) Let f, g : [a, b] → R be absolutely continuous   functions with f , g ∈ L ∞ ([a, b]). Then © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_7

113

114

7

   1  b − a

b

f (x) g (x) d x −

a

p-Schatten Norm Generalized Fractional Ostrowski …

1 b−a





b

f (x) d x a

1 b−a



b

a

  g (x) d x 

    1 (b − a)2  f  ∞ g  ∞ . 12

≤

(7.2)

The above integrals are assumed to exist. The related Grüss type inequalities have many applications to Probability Theory. We presented also ([4], Chaps. 8, 9) mixed fractional Ostrowski and Grüss-Cebysev type inequalities for several functions, acting to all possible directions. The estimates involve the left and right Caputo fractional derivatives. See also the monographs written by the author [2], Chaps. 24–26 and [3], Chaps. 2–6. We are motivated also by S. Dragomir [11] recent work: An operator A ∈ B (H ) is said to belong to the von Neumann-Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite  1   A p := tr |A| p p < ∞. Assume that A : [a, b] → B p (H ), B : [a, b] → Bq (H ), p, q > 1 with are continuous and B is strongly differentiable on (a, b), then    

b

 A (t) B (t) dt −

a

  sup  B  (t)q ×

t∈[a,b]

a

b

1 p

+

1 q

= 1,

  A (s) ds B (u)  ≤

1

⎧   b  1  ⎪ A (t) p dt, − a) + u − a+b ⎪ a 2 (b 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  β1   α1  ⎪ β+1 β+1 b ⎪ ⎨ (u−a) +(b−u) A (t)αp , β+1

a

for α, β > 1 with α1 + β1 = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  2  ⎪ ⎪ ⎪ 14 (b − a)2 + u − a+b sup A (t) p , ⎩ 2

(7.3)

t∈[a,b]

for all u ∈ [a, b], an Ostrowski type inequality. Further inspiration comes from S. Dragomir [12] recent work on Grüss inequalities: For two continuous functions A, B : [a, b] → B (H ) we define the noncommutative Cebysev fractional  D (A, B) := (b − a) a

b

 A (t) B (t) dt −



b

A (t) dt a

b

B (t) dt.

(7.4)

a

If p, q > 1 with 1p + q1 = 1, let A : [a, b] → B p (H ), B : [a, b] → Bq (H ) be strongly differentiable functions on the interval (a, b), then

7.2 Vectorial Background Fractional Calculus



b

D (A, B)1 ≤ D a

1 (b − a)2 4

 a

b

115

    A (u) du, p

    A (u) du p

 a



b

a

b

    B (u) du q

≤

    B (u) du. q

In this chapter we generalize [4], Chaps. 8, 9 for several Banach algebra B p (H ) valued functions, in the sense of developing fractional Ostrowski and Grüss type inequalities. Now our left and right generalized Caputo fractional derivatives are for Banach space valued functions and our integrals are of Bochner type [13]. Applications finish the chapter.

7.2 Vectorial Background Fractional Calculus Here all come from [5]. We need Definition 7.3 ([5], p. 106) Let α > 0, α = n, · the ceiling of the number. Let f ∈ C n ([a, b] , X ), where [a, b] ⊂ R, and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]) . We define the left generalized g-fractional derivative X -valued of f of order α as follows:  x  (n)  α  1 Da+;g f (x) := (g (x) − g (t))n−α−1 g  (t) f ◦ g −1 (g (t)) dt,  (n − α) a (7.5) ∀ x ∈ [a, b], where  is the gamma function. The last integral is of Bochner type ([1], pp. 422–428; [13]).   α f ∈ C ([a, b] , X ). If α ∈ / N, by Theorem 4.10 ([5], p. 98), we have that Da+;g We set   n n f (x) := f ◦ g −1 ◦ g (x) ∈ C ([a, b] , X ) , n ∈ N, (7.6) Da+;g 0 f (x) = f (x) , ∀x ∈ [a, b] . Da+;g

When g = id, then

α α α f = Da+;id f = D∗a f, Da+;g

the usual left X -valued Caputo fractional derivative, see [5], Chap. 1. We also need

(7.7)

116

7

p-Schatten Norm Generalized Fractional Ostrowski …

Definition 7.4 ([5], p. 107) Let α > 0, α = n, · the ceiling of the number. Let f ∈ C n ([a, b] , X ), where [a, b] ⊂ R, and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]) . We define the right generalized g-fractional derivative X -valued of f of order α as follows:  b  (n)   α (−1)n Db−;g f (x) := (g (t) − g (x))n−α−1 g  (t) f ◦ g −1 (g (t)) dt,  (n − α) x (7.8) ∀ x ∈ [a, b]. The last integral is of Bochner type.   α f ∈ C ([a, b] , X ). If α ∈ / N, by Theorem 4.11 ([5], p. 101), we have that Db−;g We set   n n f (x) := (−1)n f ◦ g −1 ◦ g (x) ∈ C ([a, b] , X ) , n ∈ N, (7.9) Db−;g 0 f (x) := f (x) , ∀x ∈ [a, b] . Db−;g

When g = id, then

α α α f (x) = Db−;id f (x) = Db− f, Db−;g

(7.10)

the usual right X -valued Caputo fractional derivative, see [5], Chap. 2. We mention the following generalized fractional Taylor formulae with integral remainders over Banach spaces. Theorem 7.5 ([5], p. 107) Let α > 0, n = α , and f ∈ C n ([a, b] , X ), where [a, b] ⊂ R and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]), a ≤ x ≤ b. Then f (x) = f (a) +

n−1  (g (x) − g (a))i 

i!

i=1

1  (α)



x

a

f (a) +

We also mention



(i)

(g (a)) +

  α f (t) dt = (g (x) − g (t))α−1 g  (t) Da+;g

n−1  (g (x) − g (a))i  i=1

1  (α)

f ◦ g −1

g(x) g(a)

i! (g (x) − z)α−1



f ◦ g −1

(i)

(g (a)) +

  α Da+;g f ◦ g −1 (z) dz.

(7.11)

7.3 Banach Algebras Background

117

Theorem 7.6 ([5], p. 108) Let α > 0, n = α , and f ∈ C n ([a, b] , X ), where [a, b] ⊂ R and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]), a ≤ x ≤ b. Then f (x) = f (b) +

n−1  (g (x) − g (b))i 

i!

i=1

1  (α)



b x

f (b) + 

(i)

(g (b)) +

  α f (t) dt = (g (t) − g (x))α−1 g  (t) Db−;g

n−1  (g (x) − g (b))i 

i!

i=1

1  (α)

f ◦ g −1

g(b)

(z − g (x))α−1



g(x)

f ◦ g −1

(i)

(g (b)) +

(7.12)

  α Db−;g f ◦ g −1 (z) dz.

If 0 < α ≤ 1, then the sums in (7.11), (7.12) disappear.    α  Also in (7.11), (7.12), we have that  Da+;g f ◦ g −1  , ∞,[g(a),g(b)]       α < ∞.  Db−;g f ◦ g −1  ∞,[g(a),g(b)]

7.3 Banach Algebras Background All here come from [16]. We need Definition 7.7 ([16], p. 245) A complex algebra is a vector space A over the complex filed C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(7.13)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(7.14)

α (x y) = (αx) y = x (αy) ,

(7.15)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A)

(7.16)

118

7

p-Schatten Norm Generalized Fractional Ostrowski …

and if A contains a unit element e such that xe = ex = x (x ∈ A)

(7.17)

e = 1,

(7.18)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 7.8 Commutativity of A will be explicited stated when needed. There exists at most one e ∈ A that satisfies (7.17). Inequality (7.16) makes multiplication to be continuous, more precisely left and right continuous, see [16], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [16], pp. 247–248, Sect. 10.3. We also make Remark 7.9 Next we mention about integration of A-valued functions, see [16], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some  compact Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [16], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   f dμ = x f ( p) dμ ( p) (7.19) x Q

and



Q

 f dμ x = Q

f ( p) x dμ ( p) .

(7.20)

Q

The Bochner integrals we will involve in our chapter follow (7.19) and (7.20). Also, let f ∈ C ([a, b] , X ), where [a, b] ⊂ R, (X, ·) is a Banach space. By [5], p. 3, f is Bochner integrable.

7.4

p-Schatten Norms Background

In this basic section all come from [11].

7.4 p-Schatten Norms Background

119

Let (H, ·, ·) be a complex Hilbert space and B (H ) the Banach algebra of all bounded linear operators on H . If {ei }i∈I an orthonormal basis of H , we say that A ∈ B (H ) is of trace class if A1 :=



|A| ei , ei  < ∞.

(7.21)

i∈I

The definition of A1 does not depend on the choice of the orthornormal basis {ei }i∈I . We denote by B1 (H ) the set of trace class operators in B (H ). We define the trace of a trace class operator A ∈ B1 (H ) to be tr (A) :=



Aei , ei  ,

(7.22)

i∈I

where {ei }i∈I an orthonormal basis of H . Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series (7.22) converges absolutely and it is independent from the choice of basis. The following result collects some properties of the trace: Theorem 7.10 We have: (i) If A ∈ B1 (H ) then A∗ ∈ B1 (H ) and   tr A∗ = tr (A);

(7.23)

(ii) If A ∈ B1 (H ) and T ∈ B (H ), then AT, T A ∈ B1 (H ) and tr (AT ) = tr (T A) and |tr (AT )| ≤ A1 T  ;

(7.24)

(iii) tr (·) is a bounded linear functional on B1 (H ) with tr  = 1; (iv) If A, B ∈ B2 (H ) then AB, B A ∈ B1 (H ) and tr (AB) = tr (B A) ; (v) B f in (H ) (finite rank operators) is a dense subspace of B1 (H ) . An operator A ∈ B (H ) is said to belong to the von Neumann-Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite [18, pp. 60–64]  1   A p := tr |A| p p < ∞, |A| p is an operator notation and not a power. For 1 < p < q < ∞ we have that B1 (H ) ⊂ B p (H ) ⊂ Bq (H ) ⊂ B (H )

(7.25)

A1 ≥  A p ≥  Aq ≥ A .

(7.26)

and

120

7

p-Schatten Norm Generalized Fractional Ostrowski …

For p ≥ 1 the functional · p is a norm on the ∗-ideal B p (H ), which is a Banach   algebra, and B p (H ) , · p is a Banach space. Also, see for instance [18, pp. 60–64], for p ≥ 1,   A p =  A∗  p , A ∈ B p (H )

(7.27)

AB p ≤  A p B p , A, B ∈ B p (H )

(7.28)

and AB p ≤ A p B , B A p ≤ B A p , A ∈ B p (H ) , B ∈ B (H ) . (7.29) This implies that C AB p ≤ C A p B , A ∈ B p (H ) , B, C ∈ B (H ) .

(7.30)

In terms of p-Schatten norm we have the Hölder inequality for p, q > 1 with 1 =1: q (|tr (AB)| ≤) AB1 ≤ A p Bq , A ∈ B p (H ) , B ∈ Bq (H ) .

1 p

+

(7.31)

For the theory of trace functionals and their applications the interested reader is referred to [17, 18]. For some classical trace inequalities see [9, 10, 14], which are continuations of the work of Bellman [7].

7.5 Main Results We start with 1-Schatten norm weighted mixed generalized fractional Ostrowski type inequalities involving several functions taking values in the Banach algebra B2 (H ) ⊂ B (H ): Theorem 7.11 Let the ∗-ideal B2 (H ), which (B2 (H ) , ·2 ) is a Banach algebra; x0 ∈ [a, b] ⊂ R, α > 0, n = α ; Ai ∈ C n ([a, b] , B2 (H )), i = 1, . . . , r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing such that g −1 ∈ C n ([g (a) , g (b)]), with (k)  Ai ◦ g −1 (g (x0 )) = 0, k = 1, . . . , n − 1; i = 1, . . . , r. Then (1) it holds,  (A1 , . . . , Ar ) (x0 ) :=

7.5 Main Results

⎡ ⎢ ⎢ ⎣

i=1

b

a

⎜ ⎜ ⎝

⎡⎡ 1  (α)

⎡ ⎢ +⎢ ⎣

r  i=1



⎞

⎛



r 

121

⎢⎢ ⎢⎢ ⎣⎣

r  j=1 j=i

 a

⎞ ⎞ ⎤ ⎛ ⎛  b  r ⎟ ⎟ ⎟ ⎥ ⎜ ⎜ ⎟ ⎥ ⎜ ⎜ A j (x)⎟ A j (x)⎟ ⎠ Ai (x) d x − ⎝ a ⎝ ⎠ d x ⎠ Ai (x0 )⎦ = (7.32) j=1 j=i

⎛ x0

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎞

⎛ b x0

⎟ A j (x)⎟ ⎠

r ⎟ ⎜ ⎜ A j (x)⎟ ⎠ ⎝



1 γ

g(x) g(x0 )

j=1 j=i

(2) for γ, δ > 1 :

$

+

1 δ

g(x0 )

(z − g (x))α−1

g(x)

(g (x) − z)α−1

= 1, with α >

1 δ



     α   Dx0 +;g Ai ◦ g −1 2 

⎥ Dxα0 −;g Ai ◦ g −1 (z) dz d x ⎥ ⎦



⎤⎤

⎥⎥   ⎥ Dxα0 +;g Ai ◦ g −1 (z) dz d x ⎥ ⎦⎦ ,

1

 (α) (γ (α − 1) + 1) γ x0

(7.33)

r     A j (x) (g (x0 ) − g (x))α− 1δ d x+ 2 j=1 j=i

⎤



δ,[g(x0 ),g(b)]

⎤

%



we have that

2 δ,[g(a),g(x0 )] a

i=1



1

 ( A1 , . . . , Ar ) (x0 )1 ≤ ⎡ r      ⎢  α −1   ⎢ ⎣ Dx0 −;g Ai ◦ g  





r b  x0 j=1 j=i

 ⎥  A j (x) (g (x) − g (x0 ))α− 1δ d x ⎥ , 2 ⎦

(3) if α ≥ 1, we obtain  (A1 , . . . , Ar ) (x0 )1 ≤ r  i=1

⎡      ⎢  α −1   ⎢ D A ◦ g     x0 −;g i ⎣ 2 L 1 ([g(a),g(x0 )])

     α   Dx0 +;g Ai ◦ g −1 2 

x0

a

 L 1 ([g(x0 ),g(b)])

1  (α)

(7.34)

r     A j (x) (g (x0 ) − g (x))α−1 d x+ 2 j=1 j=i

⎤ r b  x0 j=1 j=i

 ⎥  A j (x) (g (x) − g (x0 ))α−1 d x ⎥ , 2 ⎦

122

7

p-Schatten Norm Generalized Fractional Ostrowski …

and (4)  ( A1 , . . . , Ar ) (x0 )1 ≤ ⎡ r  i=1

 α   ⎢  −1    D ⎢ A ◦ g   x0 −;g i ⎣ 2 ∞,[g(a),g(x0 )]

     α   Dx0 +;g Ai ◦ g −1 2 

 ∞,[g(x0 ),g(b)]



1  (α + 1)

r     A j (x) (g (x0 ) − g (x))α d x+ 2

x0

a

j=1 j=i

⎤ r b  x0 j=1 j=i

 ⎥  A j (x) (g (x) − g (x0 ))α d x ⎥ . 2 ⎦ (7.35)



Proof Since Ai ◦ g Theorem 7.5 that

 −1 (k)

(g (x0 )) = 0, k = 1, . . . , n − 1; i = 1, . . . , r , we have by

1 Ai (x) − Ai (x0 ) =  (α)



g(x) g(x0 )

(g (x) − z)α−1



  Dxα0 +;g Ai ◦ g −1 (z) dz, (7.36)

∀ x ∈ [x0 , b] , and by Theorem 7.6 that 1 Ai (x) − Ai (x0 ) =  (α)



g(x0 )

(z − g (x))α−1

g(x)



  Dxα0 −;g Ai ◦ g −1 (z) dz, (7.37)

∀ x ∈ [a, x0 ] ; for all i = 1, . . . , r.

⎛

⎜ Let multiplying (7.36) and (7.37) with ⎝ ⎞

⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎝

r  j=1 j=i

⎞ r & j=1 j=i

⎞

⎛

⎟ ⎜ ⎜ A j (x)⎟ ⎠ Ai (x) − ⎝

⎟ A j (x)⎠ we get, respectively,

r  j=1 j=i

⎟ A j (x)⎟ ⎠ Ai (x0 ) =

⎞ r & j=1 j=i

⎟ A j (x)⎠

 (α) ∀ x ∈ [x0 , b] ,



g(x) g(x0 )

(g (x) − z)α−1



  Dxα0 +;g Ai ◦ g −1 (z) dz,

(7.38)

7.5 Main Results

123

⎛

and

⎜ ⎜ ⎝ ⎛ ⎜ ⎝

⎞ r  j=1 j=i

⎞

⎛

⎟ ⎜ ⎜ A j (x)⎟ ⎠ Ai (x) − ⎝

r  j=1 j=i

⎟ A j (x)⎟ ⎠ Ai (x0 ) =

⎞ r & j=1 j=i

⎟ A j (x)⎠



 (α)

g(x0 )

(z − g (x))α−1



g(x)

  Dxα0 −;g Ai ◦ g −1 (z) dz,

(7.39)

∀ x ∈ [a, x0 ] ; for all i = 1, . . . , r. Adding (7.38) and (7.39) as separate groups, we obtain ⎞

⎛ r  i=1

⎜ ⎜ ⎝

r  j=1 j=i

1  (α)

i=1

⎜ ⎜ ⎝

r  j=1 j=i

∀ x ∈ [x0 , b] , and

i=1

i=1



⎜ ⎜ ⎝

⎜ ⎜ ⎝

r  j=1 j=i

g(x) g(x0 )

i=1

j=1 j=i



g(x0 )

⎟ A j (x)⎟ ⎠ Ai (x0 ) =

j=1 j=i



  Dxα0 +;g Ai ◦ g −1 (z) dz, (7.40)

⎛

⎟ A j (x)⎟ ⎠ Ai (x) −

⎟ A j (x)⎟ ⎠

⎜ ⎜ ⎝

r 

(g (x) − z)α−1

⎞ r 

⎞

⎛ 1  (α)

⎟ A j (x)⎟ ⎠

⎛

r 

r 

⎟ A j (x)⎟ ⎠ Ai (x) − ⎞

⎛ r 

⎞

⎛ r 

r  i=1

⎜ ⎜ ⎝

⎞ r  j=1 j=i

(z − g (x))α−1

g(x)



⎟ A j (x)⎟ ⎠ Ai (x0 ) =

  Dxα0 −;g Ai ◦ g −1 (z) dz, (7.41)

∀ x ∈ [a, x0 ] . Next we integrate (7.40) and (7.41) with respect to x ∈ [a, b]. We have r  b  i=1

x0

⎞

⎛ ⎜ ⎜ ⎝

r  j=1 j=i

⎟ A j (x)⎟ ⎠ Ai (x) d x −

⎛ r  i=1

⎜ ⎜ ⎝



⎞

⎛ b x0

⎜ ⎜ ⎝

r  j=1 j=i

⎞

⎟ ⎟ ⎟ A j (x)⎟ ⎠ d x ⎠ Ai (x0 ) =

124

7

p-Schatten Norm Generalized Fractional Ostrowski …

⎤ ⎡ ⎛ ⎞ % $  b  r r    g(x)  ⎥ ⎢ ⎜ ⎟ 1 ⎢ ⎜ A j (x)⎟ (g (x) − z)α−1 Dxα0 +;g Ai ◦ g −1 (z) dz d x ⎥ ⎦, ⎣ ⎝ ⎠  (α) g(x0 ) x0 i=1

j=1 j=i

(7.42) and r  x0  a

i=1

r 

1  (α)

i=1

⎞

⎛ ⎜ ⎜ ⎝

⎡  ⎢ ⎢ ⎣

r  j=1 j=i

⎟ A j (x)⎟ ⎠ Ai (x) d x −

⎛ x0

a

⎜ ⎜ ⎝

⎛

⎞ r  j=1 j=i

⎟ A j (x)⎟ ⎠

$

g(x0 )

r  i=1

⎜ ⎜ ⎝

⎞

⎛



x0

a

⎜ ⎜ ⎝

(z − g (x))α−1

r  j=1 j=i



g(x)

⎞

⎟ ⎟ ⎟ A j (x)⎟ ⎠ d x ⎠ Ai (x0 ) =



⎤

%



⎥ Dxα0 −;g Ai ◦ g −1 (z) dz d x ⎥ ⎦.

(7.43) Finally, adding (7.42) and (7.43) we obtain the useful identity  (A1 , . . . , Ar ) (x0 ) := ⎞ ⎞ ⎞ ⎤ ⎛ ⎛ ⎛   r r r  ⎟ ⎟ ⎟ ⎥ ⎢ b ⎜ ⎜ b ⎜ ⎟ Ai (x) d x − ⎜ ⎥ ⎟ ⎢ ⎜ ⎜ A A j (x)⎟ (x) j ⎠ ⎠ d x ⎠ Ai (x0 )⎦ = (7.44) ⎣ a ⎝ ⎝ a ⎝ ⎡

i=1

j=1 j=i

⎡⎡ 1  (α)

⎡ ⎢ +⎢ ⎣

r  i=1



⎢⎢ ⎢⎢ ⎣⎣

 a

x0

⎜ ⎜ ⎝

⎛ x0

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎟ A j (x)⎟ ⎠

⎞

⎛ b

j=1 j=i

r  j=1 j=i

⎟ A j (x)⎟ ⎠



$

g(x) g(x0 )

g(x0 )

(z − g (x))α−1

g(x)

(g (x) − z)α−1





%

⎤

  ⎥ Dxα0 −;g Ai ◦ g −1 (z) dz d x ⎥ ⎦



⎤⎤

⎥⎥   ⎥ Dxα0 +;g Ai ◦ g −1 (z) dz d x ⎥ ⎦⎦ .

Next we take the 1-Schatten norm in (7.44) and we get:  (A1 , . . . , Ar ) (x0 )1 ≤

1  (α)

 ⎞ ⎡ ⎛     

 r r g(x0 )   ⎟ ⎢ x0 ⎜    α−1 α −1 ⎟ ⎢ ⎜ Dx0 −;g Ai ◦ g A j (x)⎠ (z) dz d x  (z − g (x))  ⎣ ⎝ g(x)  i=1  a j=1   j=i

1

7.5 Main Results

125

  ⎞ ⎛    b   g(x)

 r   ⎟ ⎜    ⎜ + A j (x)⎟ (g (x) − z)α−1 Dxα0 +;g Ai ◦ g −1 (z) dz d x    ⎠ ⎝ g(x0 )   x0 j=1   j=i

⎤ ⎥ ⎥≤ ⎦ 1

 ⎛ ⎞  % $    r r    x g(x ) 0 ⎜  0  ⎟ 1 ⎢ α−1 α −1  dx ⎜ ⎢ ⎟ dz − g A A D ◦ g (z) (x) (z (x)) j i x −;g  ⎝ ⎣ ⎠ 0  (α) g(x) a   i=1   j=1 j=i ⎡

1

(7.45) ⎤

⎛  ⎞     

r b ⎜ g(x)  ⎟ ⎥    α−1 α −1 ⎜  ⎟ ⎥ Dx0 +;g Ai ◦ g A j (x)⎠ + (z) dz  (g (x) − z) ⎝  dx⎦ ≤ x0  j=1 g(x0 )   j=i  

1

(by using the p-Schatten norm and Hölder’s type inequality (7.31) for p = q = 2) 1  (α)

r  i=1

⎡  ⎢ ⎢ ⎣

a

x0

      r    A j (x)    j=1   j=i 

    g(x0 )      α−1 α −1 Dx0 −;g Ai ◦ g (z) dz  d x (z − g (x))    g(x) 2

2

      r b    + A j (x)   x0  j=1   j=i 

   



⎤  g(x)  ⎥    ⎥ (g (x) − z)α−1 Dxα0 +;g Ai ◦ g −1 (z) dz   dx⎦ ≤ g(x 0)

2

2

(7.46) ⎡ % $  g(x0 )  x0  r r       1 ⎢  α−1  α −1 ⎢  A j (x) (z) dz d x (z − g (x))  Dx0 −;g Ai ◦ g 2 ⎣ 2  (α) g(x) a i=1

 +

j=1 j=i

r b 

  A j (x)

x0 j=1 j=i

2



g(x) g(x0 )



⎤

 ⎥    (g (x) − z)α−1  Dxα0 +;g Ai ◦ g −1 (z)2 dz d x ⎥ ⎦.

We have proved, so far, that  (A1 , . . . , Ar ) (x0 )1 ≤

1  (α)

126

7

⎡  r  ⎢ ⎢ ⎣

r     A j (x)

a

i=1



x0

$ 2

  A j (x)

$

2

x0 j=1 j=i

%      α −1 (z) dz d x+  Dx0 −;g Ai ◦ g

α−1 

(z − g (x))

2

g(x)

j=1 j=i

r b 

g(x0 )

p-Schatten Norm Generalized Fractional Ostrowski …

⎤ %     ⎥   (g (x) − z)α−1  Dxα0 +;g Ai ◦ g −1 (z) dz d x ⎥ ⎦ =: (ξ) . 2

g(x) g(x0 )

Let now γ, δ > 1 such that γ1 + in (7.47); α > 1δ . Then we have

1 δ

(7.47) = 1, and we apply the usual Hölder’s inequality

 (A1 , . . . , Ar ) (x0 )1 ≤ (ξ) ≤ ⎡

 r  ⎢ ⎢ ⎣

a

i=1



x0

r  γ(α−1)+1    A j (x) (g (x0 ) − g (x)) γ 2

γ(α−1)+1   A j (x) (g (x) − g (x0 )) γ 2

1

 (α) (γ (α − 1) + 1) γ

g(x0 )  g(x)

j=1 j =i

r b  x0 j=1 j=i

$

1

$

 

g(x)  g(x0 )

 

  δ  Dxα0 −;g Ai ◦ g −1 (z) dz

%1 δ

d x+

2



Dxα0 +;g Ai ◦ g −1



δ  (z) dz 2

%1 δ

⎤ ⎥ dx⎥ ⎦

(7.48) ≤

r  i=1

1 1

 (α) (γ (α − 1) + 1) γ

⎡     ⎢  α −1   ⎢ D A ◦ g     i x0 −;g ⎣



x0

2 δ,[g(a),g(x0 )] a

     α   Dx0 +;g Ai ◦ g −1 2 

δ,[g(x0 ),g(b)]



r     A j (x) (g (x0 ) − g (x))α− 1δ d x+ 2 j=1 j=i

⎤ r b  x0 j=1 j=i

 ⎥  A j (x) (g (x) − g (x0 ))α− 1δ d x ⎥ , 2 ⎦ (7.49)

proving (7.33). If α ≥ 1 we obtain  (A1 , . . . , Ar ) (x0 )1 ≤ (ξ) ≤

1  (α)

7.5 Main Results

127

⎡ r      ⎢  α −1   ⎢ ⎣ Dx0 −;g Ai ◦ g  



r     A j (x) (g (x0 ) − g (x))α−1 d x+ 2

x0

2 L 1 ([g(a),g(x0 )]) a

i=1

     α   Dx0 +;g Ai ◦ g −1 2 

⎤

 L 1 ([g(x0 ),g(b)])

j=1 j=i

r b  x0 j=1 j=i

⎥   A j (x) (g (x) − g (x0 ))α−1 d x ⎥ , 2 ⎦ (7.50)

proving (7.34). At last we derive  (A1 , . . . , Ar ) (x0 )1 ≤ (ξ) ≤ ⎡ r  i=1

 α   ⎢  −1    D ⎢ ◦ g A   i x −;g 0 ⎣ 2 ∞,[g(a),g(x0 )]

∞,[g(x0 ),g(b)]

r     A j (x) (g (x0 ) − g (x))α d x+ 2

x0

a

j=1 j=i

⎤



     α   Dx0 +;g Ai ◦ g −1 2 



1  (α + 1)

r b  x0 j=1 j=i

 ⎥  A j (x) (g (x) − g (x0 ))α d x ⎥ , 2 ⎦ (7.51)

proving (7.35). The theorem is proved.



When r = 2 we obtain the following operator related Ostrowski type fractional inequalities. Theorem 7.12 Let p, q > 1 : 1p + q1 = 1, and let the ∗-ideals B p (H ), Bq (H ), for     which B p (H ) , · p , Bq (H ) , ·q are Banach algebras; x0 ∈ [a, b] ⊂ R, α >     0, n = α ; A1 ∈ C n [a, b] , B p (H ) , A2 ∈ C n [a, b] , Bq (H ) ; g ∈ C 1 ([a, b]), (k)  strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]), with Ai ◦ g −1 (g (x0 )) = 0, k = 1, . . . , n − 1; i = 1, 2. Then (1) it holds   (A1 , A2 ) (x0 ) :=

b

 A2 (x) A1 (x) d x +

a



A2 (x) d x a

A1 (x) A2 (x) d x− a



b

b

 A1 (x0 ) −



b

A1 (x) d x a

A2 (x0 ) =

128

7

1  (α)

'(

x0

 A2 (x) 

b

A2 (x) x0

g(x) g(x0 )

x0

(



A1 (x) x0



(z − g (x))α−1

g(x) g(x0 )

(2) for γ, δ > 1 :

1 γ

(g (x) − z)

+



α−1



Dxα0 −;g A1



◦g

−1





)

(z) dz d x +

)   Dxα0 +;g A1 ◦ g −1 (z) dz d x +

g(x)



b



(g (x) − z)α−1

g(x0 )

A1 (x)

a

(

(z − g (x))

α−1

g(x)

a

(

g(x0 )

p-Schatten Norm Generalized Fractional Ostrowski …

)   Dxα0 −;g A2 ◦ g −1 (z) dz d x +

Dxα0 +;g A2



◦g

−1





)* ,

(z) dz d x

(7.52)

= 1, with α > 1δ , we have that

1 δ

1

 (A1 , A2 ) (x0 )1 ≤

1

 (α) (γ (α − 1) + 1) γ

'(        Dxα0 −;g A1 ◦ g −1  p 



(        Dxα0 +;g A1 ◦ g −1  p 



(        Dxα0 −;g A2 ◦ g −1 q 



(        Dxα0 +;g A2 ◦ g −1 q 



δ,[g(a),g(x0 )]

x0

) 1 A2 (x)q (g (x0 ) − g (x))α− δ d x +

a

δ,[g(x0 ),g(b)]

b

A2 (x)q (g (x) − g (x0 ))

dx +

α− 1δ

(7.53) ) dx +

x0

δ,[g(a),g(x0 )]

x0

)

α− 1δ

A1 (x) p (g (x0 ) − g (x))

a

δ,[g(x0 ),g(b)]

b

A1 (x) p (g (x) − g (x0 ))α− δ d x

)*

1

,

x0

(3) if α ≥ 1, we obtain  (A1 , A2 ) (x0 )1 ≤ '(        Dxα0 −;g A1 ◦ g −1  p  (        Dxα0 +;g A1 ◦ g −1  p  (        Dxα0 −;g A2 ◦ g −1 q 

 L 1 ([g(a),g(x0 )])

A2 (x)q (g (x0 ) − g (x))

α−1

) dx +

a

 L 1 ([g(x0 ),g(b)])

b

) A2 (x)q (g (x) − g (x0 ))α−1 d x +

x0

 L 1 ([g(a),g(x0 )])

x0

1  (α)

a

x0

A1 (x) p (g (x0 ) − g (x))

α−1

) dx + (7.54)

7.5 Main Results

129

(        Dxα0 +;g A2 ◦ g −1 q 

 L 1 ([g(x0 ),g(b)])

b

A1 (x) p (g (x) − g (x0 ))

α−1

)* ,

dx

x0

and (4)  ( A1 , A2 ) (x0 )1 ≤ '(        Dxα0 −;g A1 ◦ g −1  p 



(        Dxα0 +;g A1 ◦ g −1  p 



(        Dxα0 −;g A2 ◦ g −1 q 



(        Dxα0 +;g A2 ◦ g −1 q 



∞,[g(a),g(x0 )]

x0

1  (α + 1)

)  A2 (x)q (g (x0 ) − g (x))α d x +

a

∞,[g(x0 ),g(b)]

b

)

α

A2 (x)q (g (x) − g (x0 )) d x +

x0

∞,[g(a),g(x0 )]

x0

) A1 (x) p (g (x0 ) − g (x))α d x +

a

∞,[g(x0 ),g(b)]

b

A1 (x) p (g (x) − g (x0 ))α d x

)*

x0

. (7.55)

Proof Here we have that (acting as in the proof of Theorem 7.11 for r = 2) 

b

 (A1 , A2 ) (x0 ) :=

 A2 (x) A1 (x) d x +

A1 (x) A2 (x) d x−

a



a



b

A2 (x) d x



'(

x0







A1 (x) x0

g(x) g(x0 )



(z − g (x))α−1

g(x)



b

(g (x) − z)α−1

g(x0 )

A1 (x)

a

(

g(x) g(x0 )

x0 x0

(z − g (x))

α−1

g(x)

A2 (x) (

g(x0 )

A2 (x)

b

(7.32)

A1 (x) d x

A2 (x0 ) =





a

a

(



b

A1 (x0 ) −

a

1  (α)

b

(g (x) − z)

α−1



Dxα0 −;g A1

◦g

−1





)

(z) dz d x +

)   Dxα0 +;g A1 ◦ g −1 (z) dz d x +



)   Dxα0 −;g A2 ◦ g −1 (z) dz d x +

Dxα0 +;g A2



◦g

Therefore it holds by taking the 1-Schatten norm that

−1



(z) dz d x

)* .

(7.56)

130

7

   (A1 , A2 ) (x0 )1 =  

b

p-Schatten Norm Generalized Fractional Ostrowski …

 A2 (x) A1 (x) +

a





A1 (x) d x

a

'(  1   (α) 

x0

a



A2 (x)  A1 (x)

1  (α)

'( a

x0

( a

(



◦g

−1



 )  (z) dz d x   +

1

(g (x) − z)α−1



 )    Dxα0 +;g A1 ◦ g −1 (z) dz d x   + 1

g(x0 )

  x0   A2 (x) 

    A2 (x) 

  x0   A1 (x)    b  A1 (x) 

x0

Dxα0 −;g A1

(z − g (x))α−1



 )    Dxα0 −;g A2 ◦ g −1 (z) dz d x   + 1

g(x0 )

x0

b



 g(x)

 )*   α   α−1 −1 ≤ A1 (x) Dx0 +;g A2 ◦ g (z) dz d x  (g (x) − z) 

b

(

α−1

g(x)

a

(   

g(x) g(x0 )

x0 x0

(z − g (x))

  A2 (x0 )  ≤ 1

g(x)



b

(   

g(x0 )

A2 (x)

a

(   



b

A1 (x0 ) −

A2 (x) d x

A1 (x) A2 (x) d x− a



b

b

1

g(x)

g(x) g(x0 )

 (7.57) ) g(x0 )   α   α−1 −1 Dx0 −;g A1 ◦ g d x + (z) dz  (z − g (x))  1

(g (x) − z)α−1



 )    Dxα0 +;g A1 ◦ g −1 (z) dz  d x +  1

 ) g(x0 )   α   α−1 −1  Dx0 −;g A2 ◦ g (z) dz  d x + (z − g (x))

g(x)

1

 )* g(x)   α   α−1 −1  Dx0 +;g A2 ◦ g ≤ (z) dz  d x (g (x) − z)

g(x0 )

1

(7.58) (by using the p-Schatten norm and Hölder’s type inequality (7.31) for p, q > 1 : 1 + q1 = 1) p ⎧⎡ ⎤ $ %       g(x0 ) 1 ⎨⎣ x0   A2 (x)q  (z − g (x))α−1 Dxα0 −;g A1 ◦ g −1 (z) dz  d x ⎦ +   g(x)  (α) ⎩ a p



b x0

 a

x0

   A2 (x)q  

g(x) g(x0 )

   A1 (x) p  

(g (x) − z)

g(x0 ) g(x)

α−1



Dxα0 +;g A1



◦g

−1





   (z) dz  d x + p

(z − g (x))

 α−1



Dxα0 −;g A2 ◦ g

 −1



  (z) dz   dx + q

7.5 Main Results



   A1 (x) p  

b x0

,

1  (α)

(

131

x0

a

A2 (x)q

A2 (x)q

g(x)



x0

(g (x) − z)



g(x0 )

(

b

 A1 (x) p

◦g

−1



+

   ≤ (z) dz  d x

p

(g (x) − z)

α−1

(z − g (x))

 α  D

α−1

x0 +;g A1

 α  D



x0 −;g A2

◦g



−1

◦g



−1

)   (z) p dz d x +



)   (z) q dz d x +

)*     (g (x) − z)α−1  Dxα0 +;g A2 ◦ g −1 (z)q dz d x .

g(x) g(x0 )

x0



% (7.59)        α−1 α −1 (z) dz d x + (z − g (x))  Dx0 −;g A1 ◦ g

g(x)

a

Dxα0 +;g A2

q

g(x0 )

A1 (x) p

α−1

g(x)

g(x0 )

x0

(

g(x0 )

$



b

g(x)

(7.60)

We have proved, so far, that  (A1 , A2 ) (x0 )1 ≤ ,

1  (α)

(

a

b

x0

$ A2 (x)q

 A2 (x)q

(

x0

g(x)

 A1 (x) p

b x0

$ A1 (x) p

g(x) g(x0 )

%        (z − g (x))α−1  Dxα0 −;g A1 ◦ g −1 (z) dz d x + p

)     (g (x) − z)α−1  Dxα0 +;g A1 ◦ g −1 (z) p dz d x +

g(x0 ) g(x)

a



g(x)

g(x0 )

x0

g(x0 )

)     (z − g (x))α−1  Dxα0 −;g A2 ◦ g −1 (z)q dz d x +

(g (x) − z)

% +      α −1 =: (λ) . (z) dz d x  Dx0 +;g A2 ◦ g

α−1 

q

(7.61) Let now γ, δ > 1 such that γ1 + 1δ = 1, and we apply the usual Hölder’s inequality in (7.61); α > 1δ . Then we have that  ( A1 , A2 ) (x0 )1 ≤ (λ) ≤

1 1

 (α) (γ (α − 1) + 1) γ

⎧⎡ ⎤ $ %1 δ ⎨  x0   δ g(x 0 )  γ(α−1)+1   α −1 ⎣ A2 (x)q (g (x0 ) − g (x)) γ dx⎦ + (z) dz  Dx0 −;g A1 ◦ g ⎩ a p g(x)

132

7

⎡  ⎣

b x0

⎡  ⎣

x0

a

A2 (x)q (g (x) − g (x0 ))

γ(α−1)+1 γ

A1 (x) p (g (x0 ) − g (x))

p-Schatten Norm Generalized Fractional Ostrowski …

$

g(x) 

 

g(x0 )

γ(α−1)+1 γ

$



Dxα0 +;g A1 ◦ g −1

g(x0 ) 

 

g(x)

Dxα0 −;g A2



◦g



δ  (z) dz

⎤

%1 δ

dx⎦ +

p

−1



δ  (z) dz

⎤

%1 δ

dx⎦ +

q

(7.62) ⎡  ⎣

b x0

A1 (x) p (g (x) − g (x0 ))

≤

γ(α−1)+1 γ

$

g(x) 

 

g(x0 )

Dxα0 +;g A2



◦g

−1



δ  (z) dz

%1 δ

q

⎤⎫ ⎬ dx⎦ ⎭

1 1

 (α) (γ (α − 1) + 1) γ

'(        Dxα0 −;g A1 ◦ g −1  p 



(        Dxα0 +;g A1 ◦ g −1  p 



(        Dxα0 −;g A2 ◦ g −1 q 



(        Dxα0 +;g A2 ◦ g −1 q 



δ,[g(a),g(x0 )]

x0

) 1 A2 (x)q (g (x0 ) − g (x))α− δ d x +

a

δ,[g(x0 ),g(b)]

b

A2 (x)q (g (x) − g (x0 ))

)

α− 1δ

dx +

α− 1δ

(7.63) ) dx +

x0

δ,[g(a),g(x0 )]

x0

A1 (x) p (g (x0 ) − g (x))

a

δ,[g(x0 ),g(b)]

b

A1 (x) p (g (x) − g (x0 ))

α− 1δ

)* ,

dx

x0

proving (7.53). If α ≥ 1, we obtain  (A1 , A2 ) (x0 )1 ≤ (λ) ≤ '(        Dxα0 −;g A1 ◦ g −1  p  (        Dxα0 +;g A1 ◦ g −1  p  (        Dxα0 −;g A2 ◦ g −1 q 

 L 1 ([g(a),g(x0 )])

 L 1 ([g(x0 ),g(b)])

L 1 ([g(a),g(x0 )])

b

A2 (x)q (g (x) − g (x0 ))

α−1

) dx +

x0 x0

) A1 (x) p (g (x0 ) − g (x))α−1 d x +

b

(7.64) )* α−1 A1 (x) p (g (x) − g (x0 )) dx ,

a

 L 1 ([g(x0 ),g(b)])

) A2 (x)q (g (x0 ) − g (x))α−1 d x +

a



(        Dxα0 +;g A2 ◦ g −1 q 

x0

1  (α)

x0

7.5 Main Results

133

proving (7.54). At last we derive  (A1 , A2 ) (x0 )1 ≤ (λ) ≤ '(        Dxα0 −;g A1 ◦ g −1  p 



(        Dxα0 +;g A1 ◦ g −1  p 



(        Dxα0 −;g A2 ◦ g −1 q 



(        Dxα0 +;g A2 ◦ g −1 q 



∞,[g(a),g(x0 )]

x0

1  (α + 1) α

)

 A2 (x)q (g (x0 ) − g (x)) d x +

a

∞,[g(x0 ),g(b)]

b

) A2 (x)q (g (x) − g (x0 ))α d x +

x0

∞,[g(a),g(x0 )]

x0

α

)

A1 (x) p (g (x0 ) − g (x)) d x +

a

∞,[g(x0 ),g(b)]

b

A1 (x) p (g (x) − g (x0 ))α d x

x0

)* , (7.65)

proving (7.55). The theorem is proved.



We continue with p-Schatten norm weighted mixed generalized fractional Ostrowski type inequalities involving several functions taking values in the Banach algebra B p (H ) ⊂ B (H ), p > 1.   Theorem 7.13 Let the ∗-ideal B p (H ), which B p (H ) , · p , p > 1, is a Banach   algebra; x0 ∈ [a, b] ⊂ R, α > 0, n= α ; Ai ∈ C n [a, b] , B p (H ) , i = 1, . . . , r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing such that g −1 ∈ C n ([g (a) , g (b)]), with (k)  Ai ◦ g −1 (g (x0 )) = 0, k = 1, . . . , n − 1; i = 1, . . . , r. Let  (A1 , . . . , Ar ) (x0 ) be as in (7.32). Then (1) for γ, δ > 1 : γ1 + 1δ = 1, with α > 1δ we have that  (A1 , . . . , Ar ) (x0 ) p ≤ ⎡   r    ⎢ α −1  ⎢ ⎣ Dx0 −;g Ai ◦ g  i=1

   p

 δ,[g(a),g(x0 )] a

1 1

 (α) (γ (α − 1) + 1) γ x0

r     A j (x) (g (x0 ) − g (x))α− 1δ d x+ p j=1 j=i

134

7

p-Schatten Norm Generalized Fractional Ostrowski …

⎤



     α   Dx0 +;g Ai ◦ g −1  p 

δ,[g(x0 ),g(b)]

r b  x0 j=1 j=i

 ⎥  A j (x) (g (x) − g (x0 ))α− 1δ d x ⎥ , p ⎦ (7.66)

(2) if α ≥ 1, we obtain  ( A1 , . . . , Ar ) (x0 ) p ≤ r  i=1

⎡     ⎢ α −1  ⎢ D A ◦ g   i x0 −;g ⎣

     α   Dx0 +;g Ai ◦ g −1  p 

   p



x0

L 1 ([g(a),g(x0 )]) a

r     A j (x) (g (x0 ) − g (x))α−1 d x+ p j=1 j=i

⎤

 L 1 ([g(x0 ),g(b)])

1  (α)

r b  x0 j=1 j=i

⎥   A j (x) (g (x) − g (x0 ))α−1 d x ⎥ , p ⎦ (7.67)

and (3)  (A1 , . . . , Ar ) (x0 ) p ≤ ⎡ r  i=1

 α   ⎢  −1    ⎢  ⎣ Dx0 −;g Ai ◦ g p ∞,[g(a),g(x0 )]

     α   Dx0 +;g Ai ◦ g −1  p 

 ∞,[g(x0 ),g(b)]



x0

a

1  (α + 1)

r     A j (x) (g (x0 ) − g (x))α d x+ p j=1 j=i

⎤ r b  x0 j=1 j=i

 ⎥  A j (x) (g (x) − g (x0 ))α d x ⎥ . p ⎦ (7.68)

Proof As similar to the proof of Theorem 7.11 is omitted. Use of (7.28).



We make Remark 7.14 (to Theorem 7.11) (i) for γ, δ > 1 : γ1 + 1δ = 1, with α > 1δ ; case of inequality (7.33): Call and assume M1 ( f 1 , . . . , fr ) := , max

i=1,...,r

      sup  Dxα0 −;g Ai ◦ g −1 2 

x0 ∈[a,b]

δ,[g(a),g(x0 )]

,

(7.69)

7.5 Main Results

135

+

      sup  Dxα0 +;g Ai ◦ g −1 2 

< +∞.

δ,[g(x0 ),g(b)]

x0 ∈[a,b]

Then  (A1 , . . . , Ar ) (x0 )1 ≤ Right hand side (7.33) ≤ ⎛ α− 1δ

M1 ( f 1 , . . . , fr ) (g (b) − g (a))  (α) (γ (α − 1) + 1)

1 γ

r  i=1

⎜ ⎜ ⎝



⎞ r b 

a

j=1 j=i

 ⎟  A j (x) d x ⎟ . 2 ⎠

(ii) Case of inequality (7.35): Call and assume M2 ( f 1 , . . . , fr ) := , max

i=1,...,r

(7.71)

      sup  Dxα0 −;g Ai ◦ g −1 2 

∞,[g(a),g(x0 )]

x0 ∈[a,b]

,

+

      sup  Dxα0 +;g Ai ◦ g −1 2 

∞,[g(x0 ),g(b)]

x0 ∈[a,b]

(7.70)

< +∞.

Then  (A1 , . . . , Ar ) (x0 )1 ≤ Right hand side (7.35) ≤ ⎛

⎞

 b r r   ⎟ M2 ( f 1 , . . . , fr ) (g (b) − g (a))α  ⎜  A j (x) d x ⎟ . ⎜ 2 ⎝ ⎠  (α + 1) a j=1 i=1 j=i

We make Remark 7.15 (to Theorem 7.12) (i) for γ, δ > 1 : γ1 + 1δ = 1, with α > 1δ ; case of inequality (7.53): Call and assume N1 ( f 1 , f 2 ) := , max

      sup  Dxα0 −;g A1 ◦ g −1  p 

δ,[g(a),g(x0 )]

x0 ∈[a,b]

      sup  Dxα0 +;g A1 ◦ g −1  p 

,

      sup  Dxα0 −;g A2 ◦ g −1 q 

,

δ,[g(x0 ),g(b)]

x0 ∈[a,b]

x0 ∈[a,b]

δ,[g(a),g(x0 )]

,

(7.72)

136

7

p-Schatten Norm Generalized Fractional Ostrowski …

+

      sup  Dxα0 +;g A2 ◦ g −1 q 

< +∞.

δ,[g(x0 ),g(b)]

x0 ∈[a,b]

(7.73)

Then  ( A1 , A2 ) (x0 )1 ≤ right hand side (7.53) ≤ N1 ( f 1 , f 2 ) (g (b) − g (a))α− δ

1

(

1

 (α) (γ (α − 1) + 1) γ

b

 A1 (x) p d x +

a

b

) A2 (x)q d x . (7.74)

a

(ii) Case of inequality (7.55): Call and assume N2 ( f 1 , f 2 ) := , max

      sup  Dxα0 −;g A1 ◦ g −1  p 

∞,[g(a),g(x0 )]

x0 ∈[a,b]

      sup  Dxα0 +;g A1 ◦ g −1  p 

,

      sup  Dxα0 −;g A2 ◦ g −1 q 

,

∞,[g(x0 ),g(b)]

x0 ∈[a,b]

∞,[g(a),g(x0 )]

x0 ∈[a,b]

+

      sup  Dxα0 +;g A2 ◦ g −1 q 

< +∞.

∞,[g(x0 ),g(b)]

x0 ∈[a,b]

,

(7.75)

Then  ( A1 , A2 ) (x0 )1 ≤ right hand side (7.55) ≤ N2 ( f 1 , f 2 ) (g (b) − g (a))α  (α + 1)

(

b

 A1 (x) p d x +

a

b

) A2 (x)q d x .

(7.76)

a

We also make Remark 7.16 (to Theorem 7.13) (i) for γ, δ > 1 : γ1 + 1δ = 1, with α > 1δ ; case of inequality (7.66): Call and assume R1 ( f 1 , . . . , fr ) := , max

i=1,...,r

      sup  Dxα0 −;g Ai ◦ g −1  p 

δ,[g(a),g(x0 )]

x0 ∈[a,b]

      sup  Dxα0 +;g Ai ◦ g −1  p 

x0 ∈[a,b]

δ,[g(x0 ),g(b)]

,

+ < +∞.

(7.77)

7.5 Main Results

137

Then ( p > 1)  (A1 , . . . , Ar ) (x0 ) p ≤ right hand side (7.66) ≤ ⎛

R1 ( f 1 , . . . , fr ) (g (b) − g (a))  (α) (γ (α − 1) + 1)

α− 1δ

1 γ

⎞

 r r  ⎜ b ⎟   ⎜  A j (x) d x ⎟ . p ⎝ a ⎠ i=1

(7.78)

j=1 j=i

(ii) Case of inequality (7.68): Call and assume R2 ( f 1 , . . . , fr ) := , max

i=1,...,r

(7.79)

      sup  Dxα0 −;g Ai ◦ g −1  p 

∞,[g(a),g(x0 )]

x0 ∈[a,b]

+

      sup  Dxα0 +;g Ai ◦ g −1  p 

< +∞.

∞,[g(x0 ),g(b)]

x0 ∈[a,b]

,

Then ( p > 1)  (A1 , . . . , Ar ) (x0 ) p ≤ Right hand side (7.68) ≤ ⎛

⎞  r r b   ⎟ R2 ( f 1 , . . . , fr ) (g (b) − g (a))α  ⎜  A j (x) d x ⎟ . ⎜ p ⎝ a ⎠  (α + 1) i=1

(7.80)

j=1 j=i

Next come 1-Schatten norm generalized fractional Grüss type inequalities involving several functions taking values in the Banach algebra B2 (H ) ⊂ B (H ): Theorem 7.17 Let the ∗-ideal B2 (H ), which (B2 (H ) , ·2 ) is a Banach algebra; 0 < α < 1, and Ai ∈ C 1 ([a, b] , B2 (H )), i = 1, . . . , r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing such that g −1 ∈ C 1 ([g (a) , g (b)]). Here M1 ( f 1 , . . . , fr ) is as in (7.69), and M2 ( f 1 , . . . , fr ) is as in (7.71). Denote by  b  (A1 , . . . , Ar ) (x0 ) d x0 =  (A1 , . . . , Ar ) := a

⎡ r  i=1

⎢ ⎢(b − a) ⎣

 a

⎛ b

⎜ ⎜ ⎝

⎛  ⎜ ⎟ ⎜ d x − A j (x)⎟ A (x) i ⎝ ⎠

⎛

⎞

r  j=1 j=i

a

b

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎞

⎟ ⎟ ⎟ A j (x)⎟ ⎠ dx⎠

 a

b



⎤

⎥ Ai (x) d x ⎥ ⎦,

(7.81) where  (A1 , . . . , Ar ) (x0 ) as in (7.32).

138

7

Then (i) for γ, δ > 1 :

1 γ

+

1 δ

p-Schatten Norm Generalized Fractional Ostrowski …

= 1, with α >

1 δ

we have that

M1 ( f 1 , . . . , fr ) (g (b) − g (a))α− δ (b − a) 1

 (A1 , . . . , Ar )1 ≤

1

 (α) (γ (α − 1) + 1) γ ⎛

⎞  r r    ⎜ b ⎟  A j (x) d x ⎟ , ⎜ 2 ⎝ a ⎠ i=1

and (ii)  ( A1 , . . . , Ar )1 ≤

M2 ( f 1 , . . . , fr ) (g (b) − g (a))α (b − a)  (α + 1)

⎛ r  i=1

(7.82)

j=1 j=i

⎜ ⎜ ⎝



⎞ r b 

a

j=1 j=i

 ⎟  A j (x) d x ⎟ . 2 ⎠

(7.83)

Proof (i) By (7.81) we have that 

b

 (A1 , . . . , Ar )1 ≤

(7.70)

 ( A1 , . . . , Ar ) (x)1 d x ≤

a

M1 ( f 1 , . . . , fr ) (g (b) − g (a))

α− 1δ

 (α) (γ (α − 1) + 1)

(b − a)

1 γ

r  i=1

⎛ ⎞  b r   ⎜ ⎟  A j (x) d x ⎟ , (7.84) ⎜ 2 ⎝ a ⎠ j=1 j=i

proving (7.82). (ii) Also it holds   (A1 , . . . , Ar )1 ≤

b

(7.72)

 ( A1 , . . . , Ar ) (x)1 d x ≤

a

⎛ M2 ( f 1 , . . . , fr ) (g (b) − g (a))α (b − a)  (α + 1) proving (7.83). The theorem is proved.

r  i=1

⎜ ⎜ ⎝

 a

⎞ r b  j=1 j=i

 ⎟  A j (x) d x ⎟ , 2 ⎠

(7.85)



7.5 Main Results

139

When r = 2 we obtain the following operator related Grüss type fractional inequalities. Theorem 7.18 Let p, q > 1 : 1p + q1 = 1, and let the ∗-ideals B p (H ) , Bq (H ),     for which B p (H ) , · p , Bq (H ) , ·q are Banach algebras; 0 < α < 1, A1 ∈     C 1 [a, b] , B p (H ) , A2 ∈ C 1 [a, b] , Bq (H ) ; g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Here N1 ( f 1 , f 2 ) is as in (7.73), and N2 ( f 1 , f 2 ) is as in (7.75). Denote by  b  (A1 , A2 ) (x0 ) d x0 =  ( A1 , A2 ) := a

 (b − a)



b

A2 (x) A1 (x) d x +

a



b

A2 (x) d x a

A1 (x) A2 (x) d x −

(7.86)

a



b

b





A1 (x) d x −



b

b

A1 (x) d x

a

a

A2 (x) d x ,

a

where  (A1 , A2 ) (x0 ) as in (7.52) Then (i) for γ, δ > 1 : γ1 + 1δ = 1, with α > 1δ , we have that N1 ( f 1 , f 2 ) (g (b) − g (a))α− δ (b − a) 1

 (A1 , A2 )1 ≤ (

b

1

 (α) (γ (α − 1) + 1) γ 

A1 (x) p d x +

a

) A2 (x)q d x ,

(7.87)

a

and (ii)  ( A1 , A2 )1 ≤ (

b

b

N2 ( f 1 , f 2 ) (g (b) − g (a))α (b − a)  (α + 1) 

A1 (x) p d x +

a

b

) A2 (x)q d x .

a

Proof (i) By (7.86) we have that   ( A1 , A2 )1 ≤ a

b

(7.74)

 ( A1 , A2 ) (x)1 d x ≤

(7.88)

140

7

p-Schatten Norm Generalized Fractional Ostrowski …

N1 ( f 1 , f 2 ) (g (b) − g (a))α− δ (b − a) 1

(

1

 (α) (γ (α − 1) + 1) γ

b

 A1 (x) p d x +

a

b

) A2 (x)q d x ,

a

(7.89)

proving (7.87). (ii) Also we get 

b

 ( A1 , A2 )1 ≤

(7.76)

 ( A1 , A2 ) (x)1 d x ≤

a

N2 ( f 1 , f 2 ) (g (b) − g (a))α (b − a)  (α + 1)

(

b

  A1 (x) p d x +

a

b

) A2 (x)q d x ,

a

(7.90) 

proving (7.88). The theorem is proved.

We continue with p-Schatten norm generalized fractional Grüss type inequalities involving several functions takin values in the Banach algebra B p (H ) ⊂ B (H ), p > 1.   Theorem 7.19 Let the ∗-ideal B p (H ), which B p (H ) , · p , p > 1, is a Banach   algebra, 0 < α < 1. Here Ai ∈ C 1 [a, b] , B p (H ) , i = 1, . . . , r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing such that g −1 ∈ C 1 ([g (a) , g (b)]). Here R1 ( f 1 , . . . , fr ) is as in (7.77), and R2 ( f 1 , . . . , fr ) is as in (7.79). Furthermore  (A1 , . . . , Ar ) is as in (7.81) with  (A1 , . . . , Ar ) (x0 ) as in (7.32). Then (i) for γ, δ > 1 : γ1 + 1δ = 1, with α > 1δ we have that R1 ( f 1 , . . . , fr ) (g (b) − g (a))α− δ (b − a) 1

 ( A1 , . . . , Ar ) p ≤

r  i=1

1

 (α) (γ (α − 1) + 1) γ

⎛ ⎞  b r   ⎜ ⎟  A j (x) d x ⎟ , ⎜ p ⎝ a ⎠

and (ii)  (A1 , . . . , Ar ) p ≤

R2 ( f 1 , . . . , fr ) (g (b) − g (a))α (b − a)  (α + 1)

⎛  r  ⎜ ⎜ ⎝ i=1

Proof (i) By (7.81) we have

(7.91)

j=1 j=i

a

⎞ r b  j=1 j=i

 ⎟  A j (x) d x ⎟ . p ⎠

(7.92)

7.6 Applications

141



b

 (A1 , . . . , Ar ) p ≤

(7.78)

 ( A1 , . . . , Ar ) (x) p d x ≤

a

⎛

R1 ( f 1 , . . . , fr ) (g (b) − g (a))

 r (b − a)  ⎜ ⎜ ⎝

α− 1δ

 (α) (γ (α − 1) + 1)

1 γ

i=1

a

⎞ r b  j=1 j=i

 ⎟  A j (x) d x ⎟ , (7.93) p ⎠

proving (7.91). (ii) Also we have 

b

 (A1 , . . . , Ar ) p ≤

(7.80)

 ( A1 , . . . , Ar ) (x) p d x ≤

a

R2 ( f 1 , . . . , fr ) (g (b) − g (a))α (b − a)  (α + 1)

r  i=1

⎛ ⎞  b r   ⎜ ⎟  A j (x) d x ⎟ , ⎜ p ⎝ a ⎠

(7.94)

j=1 j=i



proving (7.92). The theorem is proved.

7.6 Applications We start with special Ostrowski type inequalities. We give Corollary 7.20 (to Theorem 7.11) All as in Theorem 7.11, with g (t) = t, ∀ t ∈ [a, b]. Then (1) for γ, δ > 1 : γ1 + 1δ = 1, α > 1δ we have that  (A1 , . . . , Ar ) (x0 )1 ≤ ⎡ r  i=1

⎢ α   ⎢ D Ai   x0 − ⎣ 2 δ,[a,x0 ]

 α    D Ai   ∗x0 2 δ,[x

(2) if α ≥ 1, we obtain

 a

 0 ,b]

x0

1 1

 (α) (γ (α − 1) + 1) γ

r     A j (x) (x0 − x)α− 1δ d x+ 2 j=1 j=i

⎤ r b  x0 j=1 j=i

 ⎥  A j (x) (x − x0 )α− 1δ d x ⎥ , 2 ⎦

(7.95)

142

7

p-Schatten Norm Generalized Fractional Ostrowski …

 (A1 , . . . , Ar ) (x0 )1 ≤ ⎡ r  i=1

⎢ α   ⎢ D Ai   x0 − ⎣ 2 L 1 ([a,x0 ])

 α    D Ai   ∗x0 2 L

 1 ([x 0 ,b])



1  (α)

r     A j (x) (x0 − x)α−1 d x+ 2

x0

a

j=1 j=i

⎤ r b  x0 j=1 j=i

 ⎥  A j (x) (x − x0 )α−1 d x ⎥ , 2 ⎦

(7.96)

and (3)  ( A1 , . . . , Ar ) (x0 )1 ≤

1  (α + 1)

⎡

 r  ⎢ α   ⎢ D Ai   x0 − ⎣ 2 ∞,[a,x0 ]

r     A j (x) (x0 − x)α d x+ 2

x0

a

i=1

 α    D A i   ∗x0 2 ∞,[x

 0 ,b]

j=1 j=i

⎤ r b  x0 j=1 j=i

 ⎥  A j (x) (x − x0 )α d x ⎥ . 2 ⎦

(7.97)

We present Corollary 7.21 (to Theorem 7.12) All as in Theorem 7.12, with g (t) = et , ∀ t ∈ [a, b]. Then (1) for γ, δ > 1 : γ1 + 1δ = 1, with α > 1δ , we have that 1

 (A1 , A2 ) (x0 )1 ≤

1

 (α) (γ (α − 1) + 1) γ

'(        Dxα0 −;et A1 ◦ log p 



δ,[ea ,e x0 ]

(        Dxα0 +;et A1 ◦ log p 



δ,[e x0 ,eb ]

(        Dxα0 −;et A2 ◦ logq 

x0

)  x0  1 x α− δ A2 (x)q e − e dx +

a

)  α− 1δ A2 (x)q e x − e x0 dx +

b x0

δ,[ea ,e x0 ]

 a

x0

)  x0  1 x α− δ A1 (x) p e − e dx +

(7.98)

7.6 Applications

143

(        Dxα0 +;et A2 ◦ logq 



δ,[e x0 ,eb ]

b

 α− 1δ A1 (x) p e x − e x0 dx

)* ,

x0

(2) if α ≥ 1, we obtain  (A1 , A2 ) (x0 )1 ≤ '(        Dxα0 −;et A1 ◦ log p  (        Dxα0 +;et A1 ◦ log p  (        Dxα0 −;et A2 ◦ logq 

 L 1 ([ea ,e x0 ])

L 1 ([e x0 ,eb ])

L 1 ([ea ,e x0 ])

(        Dxα0 +;et A2 ◦ logq 

x0

)  α−1  A2 (x)q e x0 − e x dx +

b

)  x  x0 α−1  A2 (x)q e − e dx +

a

 

1  (α)

x0

x0

)  α−1 A1 (x) p e x0 − e x dx +

(7.99)

a

 L 1 ([e x0 ,eb ])

b

 α−1 A1 (x) p e x − e x0 dx

)* ,

x0

and (3)  ( A1 , A2 ) (x0 )1 ≤ '(        Dxα0 −;et A1 ◦ log p 



(        Dxα0 +;et A1 ◦ log p 



(        Dxα0 −;et A2 ◦ logq 



∞,[ea ,e x0 ]

(        Dxα0 +;et A2 ◦ logq 

∞,[e x0 ,eb ]



)  α A2 (x)q e x0 − e x d x +

a

∞,[e x0 ,eb ]

∞,[ea ,e x0 ]

x0

1  (α + 1)

b

)  x  x0 α A2 (x)q e − e dx +

x0 x0

)  x0  x α A1 (x) p e − e dx +

a b

 α A1 (x) p e x − e x0 d x

)* .

(7.100)

x0

We continue with Corollary 7.22 (to Theorem 7.13) All as in Theorem 7.13, eith g (t) = log t, ∀ t ∈ [a, b] ⊂ R+ − {0}. Then (1) for γ, δ > 1 : γ1 + 1δ = 1, with α > 1δ we have that

144

7

p-Schatten Norm Generalized Fractional Ostrowski …

1

 (A1 , . . . , Ar ) (x0 ) p ≤

1

 (α) (γ (α − 1) + 1) γ

⎡

  r      ⎢ α t  ⎢ ⎣ Dx0 −;log Ai ◦ e  p 



δ,[log a,log x0 ]

i=1

     α  t   D x0 +;log Ai ◦ e    p



δ,[log x0 ,log b]

x0

a

r α− 1δ      A j (x) log x0 d x+ p x j=1 j=i

⎤

α− 1δ  ⎥  A j (x) log x dx⎥ p ⎦, x0 j=1

r b  x0

j=i

(7.101) (2) if α ≥ 1, we obtain  ( A1 , . . . , Ar ) (x0 ) p ≤ ⎡ r  i=1

     ⎢  α t  ⎢ D ◦ e A   i x −;log  0 ⎣ p

     α  t   D x0 +;log Ai ◦ e    p

 L 1 ([log a,log x0 ])

x0

a

1  (α)

r α−1      A j (x) log x0 d x+ p x j=1 j=i

⎤

r α−1 b   ⎥  A j (x) log x dx⎥ p ⎦, x0 L 1 ([log x0 ,log b]) x0 j=1 

j=i

(7.102) and (3)  (A1 , . . . , Ar ) (x0 ) p ≤ ⎡ r  i=1

      ⎢ α t  ⎢ D ◦ e A    x0 −;log i ⎣ p



∞,[log a,log x0 ]

     α  t   D ◦ e A   x0 +;log i  p

∞,[log x0 ,log b]



a

x0

1  (α + 1) r α      A j (x) log x0 d x+ p x j=1 j=i

⎤

α  ⎥  A j (x) log x dx⎥ p ⎦ . (7.103) x0 j=1

r b  x0

j=i

We continue with special Grüss type inequalities.

7.6 Applications

145

Corollary 7.23 (to Theorem 7.17) All as in Theorem 7.17, with g (t) = t, ∀ t ∈ [a, b]. Then (i) for γ, δ > 1 : γ1 + 1δ = 1, α > 1δ we have that M1 ( f 1 , . . . , fr ) (b − a)

 ( A1 , . . . , Ar )1 ≤

α+ γ1

 (α) (γ (α − 1) + 1)

1 γ

r  i=1

⎛  ⎜ ⎜ ⎝

⎞ r b 

a

 ⎟  A j (x) d x ⎟ , 2 ⎠

j=1 j=i

(7.104) and (ii) ⎛

M2 ( f 1 , . . . , fr ) (b − a)α+1  (α + 1)

 ( A1 , . . . , Ar )1 ≤

 r  ⎜ ⎜ ⎝ i=1

⎞ r b 

a

j=1 j=i

 ⎟  A j (x) d x ⎟ . 2 ⎠ (7.105)

Next comes Corollary 7.24 (to Theorem 7.18) All as in Theorem 7.18, with g (t) = et , ∀ t ∈ [a, b]. Then (i) for γ, δ > 1 : γ1 + 1δ = 1, α > 1δ , we have that  (A1 , A2 )1 ≤ (

b

α− 1δ  N1 ( f 1 , f 2 ) eb − ea (b − a) 

b

A1 (x) p d x +

a

and (ii)

1

 (α) (γ (α − 1) + 1) γ )

A2 (x)q d x ,

(7.106)

a

α  N2 ( f 1 , f 2 ) eb − ea (b − a)  (A1 , A2 )1 ≤  (α + 1) ( a

b

 A1 (x) p d x +

b

) A2 (x)q d x .

(7.107)

a

We finish with Corollary 7.25 (to Theorem 7.19) All as in Theorem 7.19, with g (t) = log t, ∀ t ∈ [a, b] ⊂ R+ − {0}. Then (i) for γ, δ > 1 : γ1 + 1δ = 1, α > 1δ we have that

146

7

 (A1 , . . . , Ar ) p ≤

r  i=1

and (ii)

⎛  ⎜ ⎜ ⎝

p-Schatten Norm Generalized Fractional Ostrowski …

α− 1δ  R1 ( f 1 , . . . , fr ) log ab (b − a) 1

 (α) (γ (α − 1) + 1) γ ⎞ r b 

a

j=1 j=i

 ⎟  A j (x) d x ⎟ , p ⎠

(7.108)

α  R2 ( f 1 , . . . , fr ) log ab (b − a)  ( A1 , . . . , Ar ) p ≤  (α + 1) ⎛  r  ⎜ ⎜ ⎝ i=1

a

⎞ r b  j=1 j=i

⎟   A j (x) d x ⎟ . p ⎠

(7.109)

References 1. Aliprantis, C., Border, K.: Infinite Dimensional Analysis, 3rd edn. Springer, New York (2006) 2. Anastassiou, G.A.: Fractional Differentiation Inequalities. Research Monograph, Springer, New York (2009) 3. Anastassiou, G.A.: Advances on Fractional Inequalities. Research Monograph, Springer, New York (2011) 4. Anastassiou, G.A.: Intelligent Comparisons: Analytic Inequalities. Springer, Heidelberg, New York (2016) 5. Anastassiou, G.A.: Intelligent Computations: Abstract Fractional Calculus. Inequalities, Approximations, Springer, Heidelberg, New York (2018) 6. Anastassiou, G.A.: p-Schatten norm generalized fractional Ostrowski and Grüss type inequalities involving several functions. In: Progress in Fractional Differentiation and Applications, accepted for publication (2021) 7. Bellman, R.: Some inequalities for positive definite matrices. In: Beckenbach, E.F. (ed.) General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, pp. 89–90. Birkhauser, Basel (1980) ˇ 8. Cebyšev, Sur les expressions approximatives des intégrales définies par les aures prises entre les mêmes limites. Proc. Math. Soc. Charkov 2(1882), 93–98 9. Chang, D.: A matrix trace inequality for products of Hermitian matrices. J. Math. Anal. Appl. 237, 721–725 (1999) 10. Coop, I.D.: On matrix trace inequalities and related topics for products of Hermitian matrix. J. Math. Anal. Appl. 188, 999–1001 (1994) 11. Dragomir, S.S.: p-Schatten norm inequalities of Ostrowski’s type. RGMIA Res. Rep. Coll. 24(108), 19 (2021) 12. Dragomir, S.S.: p-Schatten norm inequalities of Grüss type. RGMIA Res. Rep. Coll. 24(115), 16 (2021) 13. Mikusinski, J.: The Bochner Integral. Academic Press, New York (1978)

References

147

14. Neudecker, H.: A matrix trace inequality. J. Math. Anal. Appl. 166, 302–303 (1992) 15. Ostrowski, A.: Über die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938) 16. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991) 17. Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979) 18. Zagrebvov, V.A.: Gibbs Semigroups. Operator Theory: Advances and Applications, vol. 273. Birkhauser (2019)

Chapter 8

p-Schatten Norm Iterated Generalized Fractional Ostrowski and Grüss Inequalities for Multiple Functions

Using iterated generalized Caputo fractional left and right vectorial Taylor formulae we establish sequential generalized fractional Ostrowski and Grüss inequalities for several functions that take values in the von Neumann–Schatten class B p (H ), 1 ≤ p < ∞. The estimates are given for all p-Schatten norms, 1 ≤ p < ∞. We finish with applications. It follows [6].

8.1 Introduction The following results inspire this chapter. Theorem 8.1 (1938, Ostrowski [15]) Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, derivative f  : (a, b) → R is bounded on (a, b),     b) whose  sup    i.e., f ∞ := sup f (t) < +∞. Then t∈(a,b)

   1  b − a

a

b

 2       sup x − a+b 1 2 f  , + f (t) dt − f (x) ≤ − a) (b ∞ 4 (b − a)2

for any x ∈ [a, b]. The constant

1 4

(8.1)

is the best possible.

Ostrowski type inequalities have great applications to integral approximations in Numerical Analysis. ˇ Theorem 8.2 (1882, Cebyšev [8]) Let f, g : [a, b] → R be absolutely continuous   functions with f , g ∈ L ∞ ([a, b]). Then © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_8

149

150

8

   1  b − a

b

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

f (x) g (x) d x −

a

1 b−a





b

f (x) d x a

1 b−a



b

a

  g (x) d x 

    1 (b − a)2  f  ∞ g  ∞ . 12

≤

(8.2)

The above integrals are assumed to exist. The related Grüss type inequalities have many applications to Probability Theory. We presented also ([4], Chaps. 8, 9) mixed fractional Ostrowski and Grüss–Cebysev type inequalities for several functions, acting to all possible directions. The estimates involve the left and right Caputo fractional derivatives. See also the monographs written by the author [2], Chaps. 24–26 and [3], Chaps. 2–6. We are motivated also by Dragomir [11] recent work: An operator A ∈ B (H ) is said to belong to the von Neumann–Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite    1 A p := tr |A| p p < ∞. Assume that A : [a, b] → B p (H ), B : [a, b] → Bq (H ), p, q > 1 with are continuous and B is strongly differentiable on (a, b), then    

b

 A (t) B (t) dt −

a

  sup  B  (t)q ×

t∈[a,b]

a

b

1 p

+

1 q

= 1,



 A (s) ds B (u)  ≤ 1

⎧    b 1 a+b   ⎪ A (t) p dt, ⎪ a ⎪ 2 (b − a) + u − 2 ⎪ ⎪ ⎪ ⎪ ⎪  β1   α1 ⎪ β+1 β+1 b ⎪ ⎨ (u−a) +(b−u) A (t)αp , β+1

a

for α, β > 1 with α1 + β1 = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪    ⎪ 2 1 a+b 2 ⎪ ⎪ sup A (t) p , + u − − a) (b ⎩ 4 2

(8.3)

t∈[a,b]

for all u ∈ [a, b], an Ostrowski type inequality. Further inspiration comes from Dragomir [12] recent work on Grüss inequalities: For two continuous functions A, B : [a, b] → B (H ) we define the noncommutative Cebysev fractional  D (A, B) := (b − a) a

b

 A (t) B (t) dt −



b

A (t) dt a

b

B (t) dt.

(8.4)

a

If p, q > 1 with 1p + q1 = 1, let A : [a, b] → B p (H ), B : [a, b] → Bq (H ) be strongly differentiable functions on the interval (a, b), then

8.2 Vectorial Sequential Generalized Fractional Calculus Background



b

D (A, B)1 ≤ D a

1 (b − a)2 4

 a

b

    A (u) du, p

    A (u) du p

 a



b

a

b

    B (u) du q

151

≤

    B (u) du. q

In this chapter we generalize [4], Chaps. 8, 9 for several Banach algebra B p (H ) valued functions, in the sense of developing sequential fractional Ostrowski and Grüss type inequalities. Now our left and right sequential generalized Caputo fractional derivatives are for Banach space valued functions and our integrals are of Bochner type [1, 13]. Applications finish the chapter.

8.2 Vectorial Sequential Generalized Fractional Calculus Background We need Definition 8.3 ([5], p. 106) Let 0 < α ≤ 1, f ∈ C 1 ([a, b] , X ), where [a, b] ⊂ R, and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . We define the left generalized g-fractional derivative X -valued of f of order α as follows:  x    α  1 Da+;g f (x) := (g (x) − g (t))−α g  (t) f ◦ g −1 (g (t)) dt,  (1 − α) a (8.5) ∀ x ∈ [a, b], where  is the gamma function. The last integral is of Bochner type [13].   α f ∈ C ([a, b] , X ). If 0 < α < 1, by Theorem 4.10, p. 98, [5], we have that Da+;g We set    1 f (x) := f ◦ g −1 ◦ g (x) ∈ C ([a, b] , X ) , (8.6) Da+;g 0 f (x) = f (x) , ∀ x ∈ [a, b] . Da+;g

When g = id, then

α α α f = Da+;id f = D∗a f, Da+;g

the usual left X -valued Caputo fractional derivative, see [5], Chap. 1.

(8.7)

152

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

We make Remark 8.4 By (8.5) we have  x    1   (g (x) − g (t))−α g  (t)  f ◦ g −1 (g (t)) dt ≤  (1 − α) a       x  f ◦ g −1 ◦ g  ∞,[a,b] (g (x) − g (t))−α g  (t) dt =  (1 − α) a       f ◦ g −1 ◦ g  1−α ∞,[a,b] (g (x) − g (t)) = (8.8)  (1 − α) 1−α       f ◦ g −1 ◦ g  ∞,[a,b] (g (x) − g (a))1−α , ∀ x ∈ [a, b] .  (2 − α)

   a+;g f (x) ≤

 α  D

Hence



 α f (a) = 0. Da+;g

(8.9)

We need Definition 8.5 ([5], p. 107) Let 0 < α ≤ 1, f ∈ C 1 ([a, b] , X ), where [a, b] ⊂ R, and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . We define the right generalized g-fractional derivative X -valued of f of order α as follows:  b     α −1 Db−;g f (x) := (g (t) − g (x))−α g  (t) f ◦ g −1 (g (t)) dt,  (1 − α) x (8.10) ∀ x ∈ [a, b]. The last integral is of Bochner type.   α f ∈ If 0 < α < 1, by Theorem 4.11, p. 101 [5], we have that Db−;g C ([a, b] , X ). We set    1 Db−;g f (x) := − f ◦ g −1 ◦ g (x) ∈ C ([a, b] , X ) , (8.11) 0 f (x) := f (x) , ∀ x ∈ [a, b] . Db−;g

When g = id, then

α α α f (x) = Db−;id f (x) = Db− f, Db−;g

the usual right X -valued Caputo fractional derivative, see [5], Chap. 2.

(8.12)

8.2 Vectorial Sequential Generalized Fractional Calculus Background

153

We make Remark 8.6 By (8.10) we have  α  D

 b    1   (g (t) − g (x))−α g  (t)  f ◦ g −1 (g (t)) dt ≤  (1 − α) x       b  f ◦ g −1 ◦ g  ∞,[a,b] (8.13) (g (t) − g (x))−α g  (t) dt =  (1 − α) x       f ◦ g −1 ◦ g  1−α ∞,[a,b] (g (b) − g (x)) =  (1 − α) 1−α       f ◦ g −1 ◦ g  ∞,[a,b] (g (b) − g (x))1−α , ∀ x ∈ [a, b] .  (2 − α)

   b−;g f (x) ≤

Hence



 α f (b) = 0. Db−;g

(8.14)

We need Definition 8.7 ([5], p. 115) Denote by (0 < α ≤ 1) nα α α α Da+;g := Da+;g Da+;g ...Da+;g (n times), n ∈ N

(8.15)

0 = I (identity operator). and Da+;g

We also need Definition 8.8 ([5], p. 118) nα α α α Db−;g := Db−;g Db−;g ...Db−;g (n times), n ∈ N

(8.16)

0 = I (identity operator). and Db−;g

Based on (8.9) and Theorem 4.30, p. 117, [5], we have the following g-left generalized modified X -valued Taylor’s formula: Theorem 8.9 Let 0 < α ≤ 1, n ∈ N, f ∈ C 1 ([a, b] , X ), (X, ·) a Banach space, g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Let Fk := kα f , k = 1, ..., n, that fulfill Fk ∈ C 1 ([a, b] , X ), and Fn+1 ∈ C ([a, b] , X ) . Da+;g Then

154

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

f (x) − f (a) =

n  (g (x) − g (a))iα  i=2

1  ((n + 1) α)



x

a

 (iα + 1)

 iα Da+;g f (a) +

  (n+1)α f (t) dt, (g (x) − g (t))(n+1)α−1 g  (t) Da+;g

(8.17)

∀ x ∈ [a, b] . When n = 1 we obtain Corollary 8.10 Let 0 < α ≤ 1, f ∈ C 1 ([a, b] , X ), (X, ·) is a Banach space, g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Assume that α 2α f ∈ C 1 ([a, b] , X ), and Da+;g f ∈ C ([a, b] , X ) . Then Da+;g f (x) − f (a) =

1  (2α)



 2α  f (t) dt, (g (x) − g (t))2α−1 g  (t) Da+;g

x

a

(8.18)

∀ x ∈ [a, b] . Based on (8.14) and Theorem 4.33, p. 120, [5], we have the following g-right generalized modified X -valued Taylor’s formula: Theorem 8.11 Let f ∈ C 1 ([a, b] , X ), (X, ·) a Banach space, g ∈ C 1 ([a, b]), kα f,k = strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Suppose Fk := Db−;g 1 1, ..., n, fulfill Fk ∈ C ([a, b] , X ), and Fn+1 ∈ C ([a, b] , X ) , where 0 < α ≤ 1, n ∈ N. Then f (x) − f (b) =

n  (g (b) − g (x))iα  i=2

1  ((n + 1) α)



b x

 (iα + 1)

 iα Db−;g f (b) +

  (n+1)α f (t) dt, (g (t) − g (x))(n+1)α−1 g  (t) Db−;g

(8.19)

∀ x ∈ [a, b] . When n = 1 we obtain Corollary 8.12 Let 0 < α ≤ 1, f ∈ C 1 ([a, b] , X ), (X, ·) is a Banach space, g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]). Assume that α 2α f ∈ C 1 ([a, b] , X ), and Db−;g f ∈ C ([a, b] , X ) . Then Db−;g f (x) − f (b) = ∀ x ∈ [a, b] .

1  (2α)



b x

 2α  f (t) dt, (g (t) − g (x))2α−1 g  (t) Db−;g

(8.20)

8.3 Banach Algebras Basic Background

155

We are greatly motivated by the following sequential generalized fractional Ostrowski type inequality: Theorem 8.13 (p. 140, [5]) Let g ∈ C 1 ([a, b]) and strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]), and 0 < α < 1, n ∈ N, f ∈ C 1 ([a, b] , X ), where (X, ·) is a Banach space. Let x0 ∈ [a, b] be fixed. Assume that Fkx0 := Dxkα0 −;g f , for   x0 k = 1, ..., n, fulfill Fkx0 ∈ C 1 ([a, b] , X ) and Fn+1 ∈ C ([a, x0 ] , X ) and Dxiα0 −;g f (x0 ) = 0, i = 1, ..., n. Similarly, we assume that G kx0 := Dxkα0 +;g f , for k = 1, ..., n, fulfill G kx0 ∈ C 1   x0 ∈ C ([x0 , b] , X ) and Dxiα0 +;g f (x0 ) = 0, i = 1, ..., n. ([x0 , b] , X ) and G n+1 Then    b   1 1  f (x) d x − f (x0 )  ≤ (b − a)  ((n + 1) α + 1) · b − a a      f (g (b) − g (x0 ))(n+1)α (b − x0 )  Dx(n+1)α  0 +;g (g (x0 ) − g (a))

(n+1)α

    f (x0 − a) Dx(n+1)α  −;g 0

∞,[x0 ,b]

+

(8.21)

 ∞,[a,x0 ]

.

8.3 Banach Algebras Basic Background All here come from [16]. We need Definition 8.14 ([16], p. 245) A complex algebra is a vector space A over the complex filed C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(8.22)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(8.23)

α (x y) = (αx) y = x (αy) ,

(8.24)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A) and if A contains a unit element e such that

(8.25)

156

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

xe = ex = x (x ∈ A)

(8.26)

e = 1,

(8.27)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 8.15 There exists at most one e ∈ A that satisfies (8.26). Inequality (8.25) makes multiplication to be continuous, more precisely left and right continuous, see [16], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [16], pp. 247–248, Sect. 10.3. We also make Remark 8.16 Next we mention about integration of A-valued functions, see [16], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some  compact Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [16], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   x f dμ = x f ( p) dμ ( p) (8.28) Q

and



Q

 f dμ x = Q

f ( p) x dμ ( p) .

(8.29)

Q

The Bochner integrals we will involve in our chapter follow (8.28) and (8.29). Also, let f ∈ C ([a, b] , X ), where [a, b] ⊂ R, (X, ·) is a Banach space. By [5], p. 3, f is Bochner integrable.

8.4

p-Schatten Norms Background

In this advanced section all come from [11]. Let (H, ·, · ) be a complex Hilbert space and B (H ) the Banach algebra of all bounded linear operators on H . If {ei }i∈I an orthonormal basis of H , we say that A ∈ B (H ) is of trace class if

8.4 p-Schatten Norms Background

157

A1 :=



|A| ei , ei < ∞.

(8.30)

i∈I

The definition of A1 does not depend on the choice of the orthornormal basis {ei }i∈I . We denote by B1 (H ) the set of trace class operators in B (H ). We define the trace of a trace class operator A ∈ B1 (H ) to be tr (A) :=



Aei , ei ,

(8.31)

i∈I

where {ei }i∈I an orthonormal basis of H . Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series (8.31) converges absolutely and it is independent from the choice of basis. The following result collects some properties of the trace: Theorem 8.17 We have: (i) If A ∈ B1 (H ) then A∗ ∈ B1 (H ) and   tr A∗ = tr (A);

(8.32)

(ii) If A ∈ B1 (H ) and T ∈ B (H ), then AT, T A ∈ B1 (H ) and tr (AT ) = tr (T A) and |tr (AT )| ≤ A1 T  ;

(8.33)

(iii) tr (·) is a bounded linear functional on B1 (H ) with tr  = 1; (iv) If A, B ∈ B2 (H ) then AB, B A ∈ B1 (H ) and tr (AB) = tr (B A) ; (v) B f in (H ) (finite rank operators) is a dense subspace of B1 (H ) . An operator A ∈ B (H ) is said to belong to the von Neumann–Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite [18, pp. 60–64]    1 A p := tr |A| p p < ∞, |A| p is an operator notation and not a power. For 1 < p < q < ∞ we have that B1 (H ) ⊂ B p (H ) ⊂ Bq (H ) ⊂ B (H )

(8.34)

A1 ≥  A p ≥  Aq ≥ A .

(8.35)

and For p ≥ 1 the functional · p is a norm on the ∗-ideal B p (H ), which is a Banach   algebra, and B p (H ) , · p is a Banach space.

158

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

Also, see for instance [18, pp. 60–64], for p ≥ 1,   A p =  A∗  p , A ∈ B p (H )

(8.36)

AB p ≤  A p B p , A, B ∈ B p (H )

(8.37)

and AB p ≤ A p B , B A p ≤ B A p , A ∈ B p (H ) , B ∈ B (H ) . (8.38) This implies that C AB p ≤ C  A p B , A ∈ B p (H ) , B, C ∈ B (H ) .

(8.39)

In terms of p-Schatten norm we have the Hölder inequality for p, q > 1 with 1 = 1: q (|tr (AB)| ≤) AB1 ≤ A p Bq , A ∈ B p (H ) , B ∈ Bq (H ) .

1 p

+

(8.40)

For the theory of trace functionals and their applications the interested reader is referred to [17, 18]. For some classical trace inequalities see [9, 10, 14], which are continuations of the work of Bellman [7].

8.5 Main Results We start with 1-Schatten norm weighted mixed sequential generalized fractional Ostrowski type inequalities for several functions taking values in the Banach algebra B2 (H ) ⊂ B (H ): Theorem 8.18 Let the ∗-ideal B2 (H ), which (B2 (H ) , ·2 ) is a Banach algebra; x0 ∈ [a, b] ⊂ R, 0 < α < 1; Ai ∈ C 1 ([a, b] , B2 (H )), i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . Assume that Fkix0 := Dxkα0 −;g Ai , for k = 1, ..., n ∈ N, fulfill Fkix0 ∈ C 1 ([a, x0 ] , B2 (H )) and   jα x0 ∈ C ([a, x0 ] , B2 (H )), and Dx0 −;g Ai (x0 ) = 0, j = 2, ..., n; i = 1, ..., r. F(n+1)i x0 x0 := Dxkα0 +;g Ai , k = 1, ..., n, fulfill G ki ∈ C1 Similarly, we assume that G ki   jα x0 ([x0 , b] , B2 (H )) and G (n+1)i ∈ C ([x0 , b] , B2 (H )), and Dx0 +;g Ai (x0 ) = 0, j = 2, ..., n; i = 1, ..., r.

8.5 Main Results

159

Denote by  (A1 , ..., Ar ) (x0 ) := ⎡ r  i=1



⎢ ⎢ ⎣

⎞

⎛ b

a

⎜ ⎜ ⎝

r  j=1 j=i

⎞

⎛

⎛  ⎟ ⎜ ⎜ A A j (x)⎟ d x − (x) ⎠ i ⎝

b

a

⎜ ⎜ ⎝

r 

⎞

⎤

⎟ ⎟ ⎥ ⎟ ⎥ A j (x)⎟ ⎠ d x ⎠ Ai (x0 )⎦ . (8.41)

j=1 j=i

Then  1  ((n + 1) α + 1) i=1 r

 (A1 , ..., Ar ) (x0 )1 ≤

⎛  ⎜ (g (x0 ) − g (a))(n+1)α ⎜ ⎝

&&      (n+1)α  Dx0 −;g Ai  

2 ∞,[a,x0 ]

⎛ x0

a

⎜ ⎜ ⎝

⎡

⎞ r  j=1 j=i

⎞⎤

  ⎟ ⎟⎥  A j (x) ⎟ d x ⎟⎥ + 2⎠ ⎠⎦ ⎛

⎛

(8.42) ⎞

⎞⎤⎤

r  ⎢ ⎜ b ⎜  ⎥⎥    ⎟ ⎟   ⎢ (n+1)α ⎜ ⎜ ⎟⎥⎥  A j (x) ⎟ (g (b) − g (x0 ))(n+1)α ⎜ ⎢ Dx0 +;g Ai   ⎜ ⎟ d x ⎟⎥⎥ . 2 2 ∞,[x0 ,b] ⎣ ⎝ x0 ⎝ ⎠ ⎠⎦⎦ j=1 j=i

Proof By Theorem 8.11 we obtain Ai (x) − Ai (x0 ) =

 x0   1 (n+1)α (g (t) − g (x))(n+1)α−1 g  (t) Dx −;g Ai (t) dt, 0  ((n + 1) α) x

(8.43)

∀ x ∈ [a, x0 ] , i = 1, ..., r. Also, by Theorem 8.9, we get Ai (x) − Ai (x0 ) =

 x   1 (n+1)α (g (x) − g (t))(n+1)α−1 g  (t) Dx +;g Ai (t) dt, 0  ((n + 1) α) x0

(8.44)

∀ x ∈ [x0 , b] , i = 1, ..., r.



Left multiplying (8.43) and (8.44) with ⎛ ⎜ ⎜ ⎝ 'r j=1 j=i

⎞ r  j=1 j=i

A j (x) 

 ((n + 1) α)

x0 x

'r j=1 j=i

A j (x) we get, respectively, ⎞

⎛

⎟ ⎜ ⎜ A j (x)⎟ ⎠ Ai (x) − ⎝



r  j=1 j=i

⎟ A j (x)⎟ ⎠ Ai (x0 ) =

  A (t) dt, (g (t) − g (x))(n+1)α−1 g  (t) Dx(n+1)α i 0 −;g

(8.45)

160

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

∀ x ∈ [a, x0 ] , i = 1, ..., r, and ⎛ ⎜ ⎜ ⎝ 'r j=1 j=i

r  j=1 j=i

A j (x) 

 ((n + 1) α)

⎞

⎟ ⎜ ⎜ A j (x)⎟ ⎠ Ai (x) − ⎝

x x0

⎞

⎛ r 

⎟ A j (x)⎟ ⎠ Ai (x0 ) =

j=1 j=i

  A (t) dt, (g (x) − g (t))(n+1)α−1 g  (t) Dx(n+1)α i 0 +;g

(8.46)

∀ x ∈ [x0 , b] , i = 1, ..., r. Adding (8.45) and (8.46) as separate groups, we obtain ⎞

⎛ r  i=1

⎜ ⎜ ⎝

⎟ A j (x)⎟ ⎠ Ai (x) −

A j (x) 

j=1 j=i

i=1

j=1 j=i

 ((n + 1) α) ∀ x ∈ [a, x0 ] , and

i=1

⎜ ⎜ ⎝

'r

(r

j=1 j=i

i=1

x0 x

⎛

r 

⎞

⎛ r  i=1

⎜ ⎜ ⎝

r  j=1 j=i

⎟ A j (x)⎟ ⎠ Ai (x0 ) =



'r

(r

r 

  Ai (t) dt, (g (t) − g (x))(n+1)α−1 g  (t) Dx(n+1)α 0 −;g (8.47) ⎛

⎞ r  j=1 j=i

⎟ A j (x)⎟ ⎠ Ai (x) −

r  i=1

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎟ A j (x)⎟ ⎠ Ai (x0 ) =

A j (x) 

 ((n + 1) α)

x x0

  A (g (x) − g (t))(n+1)α−1 g  (t) Dx(n+1)α i (t) dt, 0 +;g (8.48)

∀ x ∈ [x0 , b] . Next, we integrate (8.47) and (8.48) with respect to x ∈ [a, b]. We have r   i=1

a

⎞

⎛ x0

⎛

r r  ⎟ ⎜ ⎜ ⎟ Ai (x) d x − ⎜ ⎜ A (x) j ⎠ ⎝ ⎝ j=1 j=i

i=1

 a

⎞

⎛ x0

⎞

r ⎟ ⎟ ⎜ ⎟ ⎜ A j (x)⎟ ⎠ d x ⎠ Ai (x0 ) = ⎝ j=1 j=i

8.5 Main Results

161

⎡ 1  ((n + 1) α) 

x0

(g (t) − g (x))

r  i=1

⎢ ⎢ ⎣



(n+1)α−1

g (t)

⎞

⎛



x0

a



⎜ ⎜ ⎝

r  j=1 j=i

⎟ A j (x)⎟ ⎠

Dx(n+1)α Ai 0 −;g

x





)

(t) dt d x ,

(8.49)

and r  b  i=1

x0

⎞

⎛ ⎜ ⎜ ⎝

r  j=1 j=i

⎛

⎟ A j (x)⎟ ⎠ Ai (x) d x −

r  i=1

⎜ ⎜ ⎝

⎞

⎛



b x0

⎜ ⎜ ⎝

r  j=1 j=i

⎞

⎟ ⎟ ⎟ A j (x)⎟ ⎠ d x ⎠ Ai (x0 ) =

⎞ ⎛  r r  ⎟ ⎢ b ⎜ 1 ⎢ ⎜ A j (x)⎟ ⎠ ⎣ ⎝  ((n + 1) α) x ⎡

0

i=1



x x0

j=1 j=i

)   dt dx . A (t) (g (x) − g (t))(n+1)α−1 g  (t) Dx(n+1)α i 0 +;g

(8.50)

Finally, adding (8.49) and (8.50) we obtain the useful identity  (A1 , ..., Ar ) (x0 ) := ⎞ ⎞ ⎞ ⎤ ⎛ ⎛ ⎛   r r r  ⎟ ⎟ ⎟ ⎥ ⎢ b ⎜ ⎜ b ⎜ ⎟ Ai (x) d x − ⎜ ⎟ ⎥ ⎢ ⎜ ⎜ A A j (x)⎟ (x) j ⎠ ⎠ d x ⎠ Ai (x0 )⎦ = ⎣ a ⎝ ⎝ a ⎝ ⎡

i=1

j=1 j=i

j=1 j=i

⎡⎡



 ⎢⎢ 1 ⎢⎢  ((n + 1) α) i=1 ⎣⎣ r

 ⎡

x0 x

⎛

a

⎞

⎛ x0

r ⎟ ⎜ ⎜ A j (x)⎟ ⎠ ⎝ j=1 j=i

)   dt dx A (t) (g (t) − g (x))(n+1)α−1 g  (t) Dx(n+1)α i 0 −;g ⎞

⎤⎤

⎥⎥ r ⎟  x ⎢ b ⎜    ⎟ ⎜ ⎥⎥ ⎢ (n+1)α +⎢ A j (x)⎟ (g (x) − g (t))(n+1)α−1 g  (t) Dx +;g Ai (t) dt d x ⎥⎥ . ⎜ 0 ⎠ x0 ⎦⎦ ⎣ x0 ⎝ j=1 j=i

(8.51)

162

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

Therefore, we get (by using the p-Schatten norm and Hölder’s type inequality (8.40) for p = q = 2) ⎞ ⎡⎡ ⎛    x0  r ⎟ ⎢⎢ ⎜ 1 ⎟ ⎢⎢ ⎜  ( A1 , ..., Ar ) (x0 )1 ≤ A (x) j  ⎠  ((n + 1) α) i=1 ⎣⎣ a ⎝ j=1  j=i r 



x0

(g (t) − g (x))

(n+1)α−1





g (t)

x

Ai Dx(n+1)α 0 −;g





)  (t) dt d x   + 1

⎡ ⎞ ⎛ ⎤    

⎥ r  ⎢ b ⎜  ⎟  x   ⎢ ⎟ ⎜ ⎥ (n+1)α A j (x)⎟ (g (x) − g (t))(n+1)α−1 g  (t) Dx +;g Ai (t) dt d x ⎥ ⎢ ⎜ 0 ⎣ x0 ⎝ ⎠ x0 ⎦   j=1   j=i

x0 x

⎥ ⎥ ⎥ ⎦ 1

(8.52)

⎛ ⎞    r r  ⎟ ⎢⎢ x0 ⎜ 1 ⎜ ⎢⎢ ≤ A j (x)⎟  ⎝ ⎠ ⎣ ⎣  ((n + 1) α) i=1 a  j=1  j=i ⎡⎡



⎤

 )     dx + dt A (t) (g (t) − g (x))(n+1)α−1 g  (t) Dx(n+1)α i  0 −;g 1

 ⎛ ⎞ ⎤⎤     r 

  ⎟ ⎢ b ⎜  ⎥⎥   x  ⎜ ⎟ ⎢ ⎥⎥ (n+1)α A j (x)⎟ (g (x) − g (t))(n+1)α−1 g  (t) Dx +;g Ai (t) dt  d x ⎥⎥ ⎜ ⎢ 0  ⎠ x0 ⎣ x0 ⎝ ⎦⎦   j=1   j=i ⎡

1

(by (8.37), (8.40)) ⎡⎡ ≤ 

x0 x

⎡ ⎢ ⎢ ⎣



⎛ b x0

⎜ ⎜ ⎝

1  ((n + 1) α)

j=1 j=i

i=1

⎢⎢ ⎢⎢ ⎣⎣



⎛ x0

a

⎜ ⎜ ⎝

⎞ r  j=1 j=i

  ⎟  A j (x) ⎟ 2⎠

)      A dt dx + (t) (g (t) − g (x))(n+1)α−1 g  (t)  Dx(n+1)α  i 0 −;g 2

⎞ r 

r 

  ⎟  A j (x) ⎟ 2⎠



x x0

⎤⎤

   ⎥⎥   (n+1)α ⎥ (g (x) − g (t))(n+1)α−1 g  (t)  Dx0 +;g Ai (t) dt d x ⎥ ⎦⎦ 2

(8.53) =: (∗) .

8.5 Main Results

163

Hence it holds  (A1 , ..., Ar ) (x0 )1 ≤ (∗) . We have that (∗) ≤

(8.54)

1  ((n + 1) α + 1)

⎡⎡

⎛

⎞

⎤

 x0 ⎜  r ⎢⎢ r  ⎥     ⎟   ⎢⎢ (n+1)α ⎜ (n+1)α d x ⎥  A j (x) ⎟ − g (g (x ) (x)) ⎢⎢ Dx0 −;g Ai   ⎜ ⎟ ⎥ 0 2⎠ 2 ∞,[a,x0 ] a ⎝ ⎣⎣ ⎦ i=1

j=1 j=i

⎡

⎛

⎤⎤

⎞

 b ⎜ r  ⎥⎥ ⎢    ⎟   ⎥⎥ ⎢ (n+1)α ⎜  A j (x) ⎟ + ⎢ Dx +;g Ai   ⎜ ⎟ (g (x) − g (x0 ))(n+1)α d x ⎥⎥ ≤ 2 0 2 ∞,[x0 ,b] x0 ⎝ ⎦⎦ ⎣ ⎠ j=1 j=i

 1  ((n + 1) α + 1) i=1 r

&&    (n+1)α    Dx0 −;g Ai  

2 ∞,[a,x0 ]

⎛  ⎜ (g (x0 ) − g (a))(n+1)α ⎜ ⎝

a

⎛ x0

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎞⎤

  ⎟ ⎟⎥  A j (x) ⎟ d x ⎟⎥ + 2⎠ ⎠⎦

⎡

⎛

⎛

(8.55) ⎞⎤⎤

⎞

r  ⎥⎥ ⎢ ⎜ b ⎜     ⎟ ⎟   ⎢ (n+1)α ⎜ ⎜ ⎟⎥⎥ (n+1)α  A j (x) ⎟ D A − g (g (b) (x0 )) ⎢ x0 +;g i   ⎜ ⎜ ⎟ d x ⎟⎥⎥ , 2 2 ∞,[x0 ,b] ⎣ ⎝ x0 ⎝ ⎠ ⎠⎦⎦ j=1 j=i



proving (8.42). Next comes an L 1 estimate. Theorem 8.19 All as in Theorem 8.18, plus

1 (n+1)

 ( A1 , ..., Ar ) (x0 )1 ≤ ⎡⎡

≤ α < 1, n ∈ N. Then

  g 

∞,[a,b]

 ((n + 1) α) ⎛

⎛

⎞

i=1

⎞⎤

r  ⎥ ⎜ x 0 ⎜     ⎟ ⎟ ⎟⎥   ⎢⎢ (n+1)α ⎜ ⎜  A j (x) ⎟ (g (x0 ) − g (a))(n+1)α−1 ⎜ ⎢⎢ Dx −;g Ai   ⎜ ⎟ d x ⎟⎥ 2 0 2 L 1 ([a,x0 ]) ⎣⎣ ⎝ a ⎝ ⎠ ⎠⎦

r ⎢⎢ 

j=1 j =i

164

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

⎛

⎡

⎞

⎤⎤

 b ⎜ r  ⎢ ⎥    ⎟ ⎥ ⎜   ⎢ (n+1)α ⎥⎥  A j (x) ⎟ + ⎢ Dx +;g Ai   d x ⎥⎥ . (g (b) − g (x0 ))(n+1)α−1 ⎜ ⎟ 2 0 2 L 1 ([x0 ,b]) ⎠ ⎦⎦ ⎣ x0 ⎝ j=1 j=i

(8.56) Proof We have that (by (8.53), (8.54))   g 

∞,[a,b]

(∗) ≤

 ((n + 1) α)

⎡⎡

⎛

⎤

⎞

 x0 ⎜  r  ⎥    ⎟   ⎥ ⎢⎢ (n+1)α ⎜  A j (x) ⎟ ⎢⎢ Dx0 −;g Ai   ⎜ ⎟ (g (x0 ) − g (x))(n+1)α−1 d x ⎥ 2 2 L 1 ([a,x0 ]) a ⎝ ⎦ ⎣⎣ ⎠

r ⎢⎢  i=1

j=1 j=i

⎡

⎛

⎤⎤

⎞

 b ⎜ r  ⎢ ⎥⎥    ⎟ ⎢ (n+1)α   ⎥⎥ ⎜  A j (x) ⎟ + ⎢ Dx +;g Ai   (g (x) − g (x0 ))(n+1)α−1 d x ⎥⎥ ⎜ ⎟ 2 0 2 L 1 ([x0 ,b]) x0 ⎝ ⎣ ⎦⎦ ⎠ j=1 j=i

≤

  g 

∞,[a,b]

 ((n + 1) α)

⎡⎡ r  i=1

⎛

   ⎢⎢  (n+1)α   ⎢⎢ ⎣⎣ Dx0 −;g Ai  

2 L 1 ([a,x 0 ])

⎡    ⎢    (n+1)α +⎢ A D     i x +;g ⎣ 0 2

⎜ (g (x0 ) − g (a))(n+1)α−1 ⎜ ⎝

⎛  ⎜ (g (b) − g (x0 ))(n+1)α−1 ⎜ ⎝ L 1 ([x 0 ,b])



⎛ x0

a

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎛ b x0

⎞⎤

  ⎟ ⎟⎥  A j (x) ⎟ d x ⎟⎥ 2⎠ ⎠⎦

(8.57) ⎞⎤ ⎤

⎞

r  ⎜  ⎟ ⎟ ⎥⎥  A j (x) ⎟ d x ⎟⎥⎥ , ⎜ 2⎠ ⎝ ⎠ ⎦⎦ j=1 j=i



proving the claim. An L γ estimate follows. Theorem 8.20 All as in Theorem 8.18, plus γ, δ > 1 : 1. Then  ( A1 , ..., Ar ) (x0 )1 ≤

+

1 δ

1 = 1, and δ(n+1) 1 : 1, and 2δ1 < α < 1. Then  (A1 , ..., Ar ) (x0 )1 ≤

1 1

 (2α) (γ (2α − 1) + 1) γ

r &&         Dx2α0 −;g Ai ◦ g −1 2 

δ,[g(a),g(x0 )]

i=1

1

(g (x0 ) − g (a))2α− δ

⎛ ⎛ ⎞ ⎞⎤  x0   ⎟ ⎟⎥ ⎜ ⎜ r  ⎜ ⎜  A j (x) ⎟ d x ⎟⎥ + 2⎠ ⎝ a ⎝ ⎠⎦ j=1 j=i

&        Dx2α0 +;g Ai ◦ g −1 2 

δ,[g(x0 ),g(b)]

1

(g (b) − g (x0 ))2α− δ

1 γ

+

1 δ

=

8.5 Main Results

169

⎛ ⎛ ⎞ ⎞⎤⎤  b  r  ⎟ ⎟⎥⎥ ⎜ ⎜  ⎜ ⎜  A j (x) ⎟ d x ⎟⎥⎥ = 3 (x0 ) . 2⎠ ⎝ x ⎝ ⎠⎦⎦ 0

(8.65)

j=1 j=i



Proof By Theorem 8.20.

Next come p-Schatten norm, p > 1, fractional Ostrowski type inequalities for B p (H ) valued functions, B p (H ) ⊂ B (H ):   Theorem 8.24 Let p > 1, the ∗-ideal B p (H ), which B p (H ) , · p is a Banach   algebra; x0 ∈ [a, b] ⊂ R, 0 < α < 1; Ai ∈ C 1 [a, b] , B p (H ) , i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . Assume  that Fkix0 := Dxkα0 −;g Ai , for k = 1, ..., n ∈ N, fulfill Fkix0 ∈ C 1 [a, x0 ] , B p (H ) and     jα x0 ∈ C [a, x0 ] , B p (H ) , and Dx0 −;g Ai (x0 ) = 0, j = 2, ..., n; i = 1, ..., r. F(n+1)i x0 x0 := Dxkα0 +;g Ai , k = 1, ..., n, fulfill G ki ∈ C1 Similarly, we assume that G ki       jα x0 ∈ C [x0 , b] , B p (H ) , and Dx0 +;g Ai (x0 ) = 0, j = [x0 , b] , B p (H ) and G (n+1)i 2, ..., n; i = 1, ..., r. Denote by  (A1 , ..., Ar ) (x0 ) :=

⎡ r  i=1

⎢ ⎢ ⎣

 a

⎞

⎛ b

⎜ ⎜ ⎝

r  j=1 j=i

⎞ ⎞ ⎤ ⎛ ⎛  b  r ⎟ ⎟ ⎟ ⎥ ⎜ ⎜ ⎟ d x ⎟ Ai (x0 )⎥ . (8.66) ⎜ ⎜ A A j (x)⎟ A d x − (x) (x) i j ⎠ ⎠ ⎠ ⎦ ⎝ a ⎝ j=1 j=i

Then  1  ( A1 , ..., Ar ) (x0 ) p ≤  ((n + 1) α + 1) i=1 r

⎛ ⎜ (g (x0 ) − g (a))(n+1)α ⎜ ⎝ ⎡

 a

       (n+1)α   D A  x0 −;g i  p 

∞,[a,x0 ]

⎛ x0

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎞⎤

  ⎟ ⎟⎥  A j (x) ⎟ d x ⎟⎥ + p⎠ ⎠⎦ ⎛

⎛

(8.67) ⎞

⎞⎤⎤

r   ⎥⎥ ⎢ ⎜ b ⎜     ⎟ ⎟  (n+1)α   ⎢ ⎜ ⎟⎥⎥ (n+1)α ⎜   A j (x) ⎟ D d x A − g (g (b) (x ))  ⎥⎥ .  ⎢ ⎜ ⎜ ⎟ ⎟ 0 i x0 +;g p⎠ p ∞,[x ,b] ⎣ ⎝ x0 ⎝ ⎠⎦⎦ 0 j=1 j=i

Proof As similar to Theorem 8.18 is omitted. Use of (8.37). Next comes an L 1 estimate.



170

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

Theorem 8.25 All as in Theorem 8.24, plus

1 (n+1)

 (A1 , ..., Ar ) (x0 ) p ≤ ⎡⎡ r ⎢⎢      (n+1)α   ⎢⎢  D A  ⎢⎢ i   x0 −;g p ⎣⎣ i=1

≤ α < 1, n ∈ N. Then

  g 

∞,[a,b]

 ((n + 1) α) ⎛

⎛

⎞

⎞⎤

r  ⎥ ⎜ x0 ⎜   ⎟ ⎟ ⎟⎥ ⎜ ⎜  A j (x) ⎟ (g (x0 ) − g (a))(n+1)α−1 ⎜ ⎜ ⎟ d x ⎟⎥ p ⎝ a ⎝ ⎠ ⎠⎦ L 1 ([a,x0 ]) j=1 j =i

⎡    ⎢   (n+1)α ⎢ + ⎣  Dx0 +;g Ai 

   p

L 1 ([x0 ,b])

(g (b) − g (x0 ))(n+1)α−1

⎛



b x0

⎞

⎤(8.68) ⎤

r  ⎜  ⎟ ⎥⎥  A j (x) ⎟ d x ⎥⎥ . ⎜ p⎠ ⎝ ⎦⎦ j=1 j =i



Proof As similar to the proof of Theorem 8.19 is omitted. An L γ estimate follows. Theorem 8.26 All as in Theorem 8.24, plus γ, δ > 1 : 1. Then  (A1 , ..., Ar ) (x0 ) p ≤ r  i=1

1 γ

+

1 δ

1 = 1, and δ(n+1) 1, the ∗-ideal B p (H ); x0 ∈ [a, b] ⊂  R, 0 < α < 1; Ai ∈ C 1 [a, b] , B p (H ) , i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . Assume that Dxα0 −;g Ai ∈     C 1 [a, x0 ] , B p (H ) and Dx2α0 −;g Ai ∈ C [a, x0 ] , B p (H ) , i = 1, ..., r. Similarly,     assume that Dxα0 +;g Ai ∈ C 1 [x0 , b] , B p (H ) and Dx2α0 +;g Ai ∈ C [x0 , b] , B p (H ) ; i = 1, ..., r ;  ( A1 , ..., Ar ) (x0 ) is as in (8.66). Then  1  (2α + 1) i=1 r

 ( A1 , ..., Ar ) (x0 ) p ≤

&&       Dx2α0 −;g Ai  p 

∞,[a,x0 ]

⎛ ⎛ ⎞ ⎞⎤  x0  r  ⎟ ⎟⎥ ⎜ ⎜  ⎜  A j (x) ⎟ d x ⎟⎥ + (g (x0 ) − g (a))2α ⎜ p⎠ ⎝ a ⎝ ⎠⎦

(8.70)

j=1 j=i

⎡

⎛

 2α ⎢ ⎜    D   ⎢ A (g (b) − g (x0 ))2α ⎜   i x0 +;g ⎣ ⎝ p ∞,[x0 ,b]

⎛



b x0

⎞

⎞⎤⎤

r   ⎟ ⎟⎥⎥ ⎜ ⎜  A j (x) ⎟ d x ⎟⎥⎥ p⎠ ⎝ ⎠⎦⎦ j=1 j=i

=: 4 (x0 ) . 

Proof By Theorem 8.24 and Corollaries 8.10, 8.12. We continue with Corollary 8.28 (to Theorem 8.25) All as in Corollary 8.27, plus  ( A1 , ..., Ar ) (x0 ) p ≤

  g 

∞,[a,b]

 (2α)

1 2

≤ α < 1. Then

r &&        Dx2α0 −;g Ai  p  i=1

L 1 ([a,x0 ])

⎛ ⎛ ⎞ ⎞⎤  x0   ⎟ ⎟⎥ ⎜ ⎜ r  ⎜  A j (x) ⎟ d x ⎟⎥ + (g (x0 ) − g (a))2α−1 ⎜ p⎠ ⎝ a ⎝ ⎠⎦

(8.71)

j=1 j=i

⎡

⎛ ⎛ ⎞ ⎞⎤⎤  r  2α ⎢  ⎟ ⎟⎥⎥  ⎜ b ⎜   2α−1 ⎜  D   ⎢ ⎜  A j (x) ⎟ d x ⎟⎥⎥ A − g (g (b) (x ))   i 0 x0 +;g p⎠ ⎣ ⎝ x ⎝ ⎠⎦⎦ p L 1 ([x0 ,b]) 0

= 5 (x0 ) .

j=1 j=i

172

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …



Proof By Theorem 8.25. We also give Corollary 8.29 (to Theorem 8.26) All as in Corollary 8.27, plus γ, δ > 1 : 1, and 2δ1 < α < 1. Then  ( A1 , ..., Ar ) (x0 ) p ≤

1 γ

+

1 δ

=

1 1

 (2α) (γ (2α − 1) + 1) γ

r &&         Dx2α0 −;g Ai ◦ g −1  p 

δ,[g(a),g(x0 )]

i=1

⎛ 1 ⎜ (g (x0 ) − g (a))2α− δ ⎜ ⎝

 a

⎛ x0

⎜ ⎜ ⎝

&        Dx2α0 +;g Ai ◦ g −1  p 

⎞ r  j=1 j=i

 ⎟ ⎟⎥   A j (x) ⎟ d x ⎟⎥ + p⎠ ⎠⎦

⎛ b x0

Proof By Theorem 8.26.

⎜ ⎜ ⎝

⎞ r  j=1 j=i

(8.72)

1

δ,[g(x0 ),g(b)]

⎛  ⎜ ⎜ ⎝

⎞⎤

(g (b) − g (x0 ))2α− δ

⎞⎤⎤

 ⎟ ⎟⎥⎥   A j (x) ⎟ d x ⎟⎥⎥ = 6 (x0 ) . p⎠ ⎠⎦⎦ 

When r = 2 we obtain the following operator related sequential fractional Ostrowski type inequalities. Theorem 8.30 Let p, q > 1 : 1p + q1 = 1, and let the ∗-ideals B p (H ), Bq (H ),     for which B p (H ) , · p , Bq (H ) , ·q are Banach algebras; x0 ∈ [a, b] ⊂ R,     0 < α < 1, A1 ∈ C 1 [a, b] , B p (H ) , A2 ∈ C 1 [a, b] , Bq (H ) ; g ∈ C 1 ([a, b]), x0 strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)])  . Assume that  xF0ki := x0 kα 1 Dx0 −;g Ai , for k = 1, ..., n ∈ N, i = 1, 2, fulfill Fk1 ∈ C [a, x0 ] , B p (H ) , F(n+1)1 ∈       x0 x0 ∈ C 1 [a, x0 ] , Bq (H ) , F(n+1)2 ∈ C [a, x0 ] , Bq (H ) , C [a, x0 ] , B p (H ) ; Fk2   jα x0 and Dx0 −;g Ai (x0 ) = 0, j = 2, ..., n; i = 1, 2. Similarly, we assume that G ki :=   x0 x0 kα 1 Dx0 +;g Ai , k = 1, ..., n, i = 1, 2, fulfill G k1 ∈ C [x0 , b] , B p (H ) , G (n+1)1 ∈       x0 x0 ∈ C 1 [x0 , b] , Bq (H ) , G (n+1)2 ∈ C [x0 , b] , Bq (H ) , C [x0 , b] , B p (H ) , G k2   jα and Dx0 +;g Ai (x0 ) = 0, j = 2, ..., n; i = 1, 2. Then (1) it holds

8.5 Main Results

173

  ( A1 , A2 ) (x0 ) :=

b

 A2 (x) A1 (x) d x +

a





x0

a

 A2 (x) 

b

A2 (x) x0

 A1 (x) 

b

A1 (x) x0

g(x)

g(x0 )

(2) for γ, δ > 1 :

1 γ

(z − g (x))(n+1)α−1



(n+1)α−1

(g (x) − z)

= 1, with

1 δ

 (A1 , A2 ) (x0 )1 ≤

,       (n+1)α −1    D A ◦ g    1 x0 −;g  p

1 δ(n+1)



◦ g −1



)(8.73) (z) dz d x +



)   −1 Dx(n+1)α ◦ g dz dx + A (z) 2 0 −;g

Dx(n+1)α A2 0 +;g



◦g

−1





) ,

(z) dz d x

< α < 1, we have that 1 1

 ((n + 1) α) (γ ((n + 1) α − 1) + 1) γ



x0

δ,[g(a),g(x0 )] a

       (n+1)α −1    D ◦ g A    1  x0 +;g p

1  ((n + 1) α)

)   −1 Dx(n+1)α ◦ g dz dx + A (z) 1 0 −;g

Dx(n+1)α A1 0 +;g



(z − g (x))(n+1)α−1

+





(g (x) − z)(n+1)α−1

g(x0 )

g(x)

a

&

g(x)

g(x0 )

x0

&

g(x0 ) g(x)

a

&

A2 (x0 ) =

A1 (x) d x

a

&



b

A1 (x0 ) −

A2 (x) d x

A1 (x) A2 (x) d x− a



b

b

 b

δ,[g(x0 ),g(b)] x0

        (n+1)α −1    Dx0 −;g A2 ◦ g q 

 x0

        (n+1)α −1    Dx0 +;g A2 ◦ g q 

 b

δ,[g(a),g(x0 )] a

δ,[g(x0 ),g(b)] x0

 1

A2 (x)q (g (x0 ) − g (x))(n+1)α− δ d x +

 1 A2 (x)q (g (x) − g (x0 ))(n+1)α− δ d x

(8.74)  A1 (x) p (g (x0 ) − g

1 (x))(n+1)α− δ

dx + -

1 A1 (x) p (g (x) − g (x0 ))(n+1)α− δ d x

(3) if (n + 1) α ≥ 1, we obtain  ( A1 , A2 ) (x0 )1 ≤ ,       (n+1)α −1    D ◦ g A    1 x0 −;g  p

 L 1 ([g(a),g(x 0 )]) a

+

x0

1  ((n + 1) α) 

A2 (x)q (g (x0 ) − g (x))

(n+1)α−1

dx +

,

174

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

       (n+1)α −1    ◦ g A D    x0 +;g 1 p        (n+1)α −1    D ◦ g A    2 x0 −;g  q

 L 1 ([g(x0 ),g(b)])



b

A2 (x)q (g (x) − g (x0 ))(n+1)α−1 d x +

x0





x0

A1 (x) p (g (x0 ) − g (x))(n+1)α−1 d x +

L 1 ([g(a),g(x 0 )]) a

       (n+1)α −1    A D ◦ g    2 x0 +;g  q

 L 1 ([g(x 0 ),g(b)])

b x0

(8.75) A1 (x) p (g (x) − g (x0 ))

(n+1)α−1

,

dx

and (4) 1  ((n + 1) α + 1)

 ( A1 , A2 ) (x0 )1 ≤ ,       (n+1)α −1    D ◦ g A    1  x0 −;g p

 x0

∞,[g(a),g(x0 )] a

       (n+1)α −1     Dx0 +;g A1 ◦ g  p 



       (n+1)α −1     Dx0 −;g A2 ◦ g q 



       (n+1)α −1    D ◦ g A    2 x0 +;g  q



∞,[g(x0 ),g(b)]

∞,[g(a),g(x0 )]

∞,[g(x0 ),g(b)]

 A2 (x)q (g (x0 ) − g (x))(n+1)α d x

+



b

A2 (x)q (g (x) − g (x0 ))(n+1)α d x +

x0

(8.76) 

x0

A1 (x) p (g (x0 ) − g (x))(n+1)α d x +

a b

 A1 (x) p (g (x) − g (x0 ))

(n+1)α

dx

.

x0

Proof Here we have that (acting as in the proof of Theorem 8.18 for r = 2)   ( A1 , A2 ) (x0 ) :=

b

 A2 (x) A1 (x) d x +

a





b

A2 (x) d x x0

 A1 (x0 ) −

 A2 (x) 

b

A2 (x)

a

x0

g(x) g(x0 )

 A1 (x)

g(x0 ) g(x)

x0

&



b

A1 (x) d x

(8.51)

A2 (x0 ) =

a

a

&

A1 (x) A2 (x) d x− a

a

&

b

g(x0 )

g(x)

(z − g (x))(n+1)α−1

(g (x) − z)

(n+1)α−1

(z − g (x))(n+1)α−1





)   −1 Dx(n+1)α ◦ g dz dx + A (z) 1 0 −;g

Dx(n+1)α A1 0 +;g



1  ((n + 1) α)

 

◦g

−1



Dx(n+1)α A2 ◦ g −1 0 −;g



)

(z) dz d x +



)(8.77) (z) dz d x +

8.5 Main Results

&

175



b

A1 (x)

g(x)

g(x0 )

x0

(n+1)α−1

(g (x) − z)



Dx(n+1)α A2 0 +;g



◦g

−1





) .

(z) dz d x

Therefore it holds by taking the 1-Schatten norm that    ( A1 , A2 ) (x0 )1 =  

b

 A2 (x) A1 (x) d x +

a





b

a

A1 (x) A2 (x) d x− a

 A1 (x0 ) −

A2 (x) d x

b



b

A1 (x) d x a

  A2 (x0 )  ≤ 1

1  ((n + 1) α)

+   * ,  x0     g(x0 )   (n+1)α −1 (n+1)α−1 Dx −;g A1 ◦ g A2 (x) (z) dz d x  + (z − g (x))  0  a  g(x) 1

&   



b

A2 (x)

g(x0 )

x0

&   

x0

 A1 (x) 

b

A1 (x) x0

1

 )    g(x0 )  (n+1)α −1 (n+1)α−1 Dx0 −;g A2 ◦ g (z − g (x)) (z) dz d x   +

g(x)

a

&   

 )    g(x)  (n+1)α −1 (n+1)α−1 Dx0 +;g A1 ◦ g (z) dz d x  (g (x) − z)  +

g(x)

g(x0 )

1

(g (x) − z)(n+1)α−1



 )    −1 ≤ Dx(n+1)α ◦ g dz dx A (z) 2  0 +;g 1

1  ((n + 1) α) ,

 +  *     g(x0 )   (n+1)α −1 (n+1)α−1 Dx −;g A1 ◦ g (z) dz  d x + (z − g (x))  A2 (x) 0   g(x)

x0 

a

1

  b  A2 (x) 

&

g(x0 )

x0

&

x0

a

&

b x0

 (8.78) )    g(x)  (n+1)α −1 (n+1)α−1  Dx0 +;g A1 ◦ g (z) dz  d x + (g (x) − z)

    A1 (x)      A1 (x) 

1

g(x0 ) g(x) g(x)

g(x0 )

(z − g (x))(n+1)α−1



)

    −1  Dx(n+1)α ◦ g A d x + dz (z) 2  0 −;g 1

(g (x) − z)(n+1)α−1



 )    −1  dx Dx(n+1)α ◦ g dz A ≤ (z) 2  0 +;g 1

(8.79) (by using the p-Schatten norm and Hölder’s type inequality (8.40) for p, q > 1 : 1 + q1 = 1) p

176

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

1  ((n + 1) α) ⎧⎡ ⎤ * +   ⎨  x0    g(x 0 )   (n+1)α −1 (n+1)α−1 ⎣ A2 (x)q  Dx0 −;g A1 ◦ g (z) dz  d x ⎦ + (z − g (x))   g(x) ⎩ a p

⎡ ⎣



b x0

⎡ ⎣



⎤ * +      g(x)   (n+1)α −1 (n+1)α−1 A2 (x)q  Dx0 +;g A1 ◦ g (z) dz  d x ⎦ + (g (x) − z)   g(x0 ) p

⎤ * +      g(x 0 )   (n+1)α A1 (x) p  (z − g (x))(n+1)α−1 Dx0 −;g A2 ◦ g −1 (z) dz  d x ⎦ +   g(x)

x0

a

⎡ ⎣



q

b x0

⎤⎫ * +   ⎬    g(x)   (n+1)α A1 (x) p  (g (x) − z)(n+1)α−1 Dx0 +;g A2 ◦ g −1 (z) dz  d x ⎦ ≤   g(x0 ) ⎭ q

(8.80) 1  ((n + 1) α) ,

x0

a

&

b

* A2 (x)q



*

x0

A1 (x) p

a b x0

g(x) g(x0 )

x0



g(x)

 A2 (x)q

g(x 0 )

A1 (x) p

(g (x) − z)

g(x 0 )

g(x) g(x 0 )

p

(n+1)α−1



Dx(n+1)α A1 0 +;g



◦g

−1





)

(z) || p dz d x +

+        (n+1)α (z − g (x))(n+1)α−1  Dx0 −;g A2 ◦ g −1 (z) dz d x + q

g(x)

*

(z − g (x))

+       (n+1)α −1 (z) dz d x +  Dx0 −;g A1 ◦ g

(n+1)α−1 

+      (n+1)α −1 . (8.81) (z) dz d x  Dx0 +;g A2 ◦ g

(n+1)α−1 

(g (x) − z)

q

We have proved, so far, that  ( A1 , A2 ) (x0 )1 ≤ ,

x0

a



b x0

* A2 (x)q *

A2 (x)q

g(x 0 ) g(x) g(x)

g(x 0 )

(z − g (x))

1  ((n + 1)α + 1) +       (n+1)α −1 (z) dz d x +  Dx0 −;g A1 ◦ g

(n+1)α−1 

p

+        (n+1)α (g (x) − z)(n+1)α−1  Dx0 +;g A1 ◦ g −1 (z) dz d x + p

8.5 Main Results 

x0

a



b x0

177 *

A1 (x) p

g(x)

* A1 (x) p

g(x 0 )

g(x)

g(x0 )

+        n+1)α (z − g (x))(n+1)α−1  Dx0 −;g A2 ◦ g −1 (z) dz d x + q

+      (n+1)α −1 =: (λ) . (z) dz d x  Dx +;g A2 ◦ g

 (g (x) − z)(n+1)α−1

0

q

(8.82) Let now γ, δ > 1 such that γ1 + 1δ = 1, and we apply the usual Hölder’s inequality 1 in (8.82); δ(n+1) < α < 1. Then we have that  (A1 , A2 ) (x0 )1 ≤ (λ) ≤

1 1

 ((n + 1) α) (γ ((n + 1) α − 1) + 1) γ

⎧⎡ ⎤ +1 * δ ⎨  x0   δ g(x0 )  γ((n+1)α−1)+1   (n+1)α −1 γ ⎣ A2 (x)q (g (x0 ) − g (x)) dx⎦ + (z) dz  Dx0 −;g A1 ◦ g ⎩ a p g(x) ⎡ ⎣



b x0

⎡  ⎣

x0

a

A2 (x)q (g (x) − g (x0 ))

A1 (x) p (g (x0 ) − g (x))

*

γ((n+1)α−1)+1 γ

γ((n+1)α−1)+1 γ

g(x)

g(x0 )

*

   δ   (n+1)α  Dx0 +;g A1 ◦ g −1 (z) dz

dx⎦ + ⎤

+1 δ

q

g(x)

δ

p

   δ   (n+1)α  Dx0 −;g A2 ◦ g −1 (z) dz

g(x0 )

⎤

+1

dx⎦ +

(8.83)

⎤⎫ +1 * ⎪  b ⎬   δ δ g(x)  γ((n+1)α−1)+1   ⎢ ⎥ (n+1)α γ A1 (x) p (g (x) − g (x0 )) dx⎦  Dx +;g A2 ◦ g −1 (z) dz ⎣ 0 ⎪ q g(x0 ) x0 ⎭ ⎡

≤

1 1

 ((n + 1) α) (γ ((n + 1) α − 1) + 1) γ

,       (n+1)α −1    ◦ g D A    1 x0 −;g  p        (n+1)α −1    A D ◦ g    1 x0 +;g  p

 δ,[g(a),g(x 0 )] a

 δ,[g(x 0 ),g(b)]

       (n+1)α −1    D ◦ g A    2 x0 −;g  q

x0

b x0

 A2 (x)q (g (x) − g (x0 ))



x0

δ,[g(a),g(x 0 )] a

       (n+1)α −1    D ◦ g A    2 x0 +;g  q

 δ,[g(x 0 ),g(b)]

proving (8.74). If (n + 1) α ≥ 1, we obtain

 1

A2 (x)q (g (x0 ) − g (x))(n+1)α− δ d x +

b x0

(n+1)α− 1δ

d x + (8.84)  1

A1 (x) p (g (x0 ) − g (x))(n+1)α− δ d x + A1 (x) p (g (x) − g (x0 ))

(n+1)α− 1δ

dx

,

178

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

 (A1 , A2 ) (x0 )1 ≤ (λ) ≤ ,       (n+1)α −1    A ◦ g D    1 x0 −;g  p        (n+1)α −1    ◦ g D A    1 x0 +;g  p        (n+1)α −1    D ◦ g A    x0 −;g 2 q



A2 (x)q (g (x0 ) − g (x))(n+1)α−1 d x +

L 1 ([g(a),g(x 0 )]) a

L 1 ([g(x 0 ),g(b)])



b

A2 (x)q (g (x) − g (x0 ))

x0



x0

L 1 ([g(a),g(x 0 )]) a

       (n+1)α −1    D ◦ g A    x0 +;g 2 q



x0



 L 1 ([g(x 0 ),g(b)])

1  ((n + 1) α)

(n+1)α−1

dx + 

A1 (x) p (g (x0 ) − g (x))(n+1)α−1 d x +

(8.85) -

b

A1 (x) p (g (x) − g (x0 ))(n+1)α−1 d x

x0

,

proving (8.75). At last we derive  ( A1 , A2 ) (x0 )1 ≤ (λ) ≤ ,       (n+1)α −1    D ◦ g A    1 x0 −;g  p

 ∞,[g(a),g(x 0 )] a

       (n+1)α −1     Dx0 +;g A1 ◦ g  p 



       (n+1)α −1     Dx0 −;g A2 ◦ g q 



∞,[g(x0 ),g(b)]

∞,[g(x 0 ),g(b)]

proving (8.76). The theorem is proved.



b

 A2 (x)q (g (x0 ) − g (x))

(n+1)α

dx +

  A2 (x)q (g (x) − g (x0 ))

(n+1)α

dx +

x0

∞,[g(a),g(x0 )]

       (n+1)α −1    ◦ g D A    2 x0 +;g  q

x0

1  ((n + 1) α + 1)

x0

 A1 (x) p (g (x0 ) − g (x))

(n+1)α

dx +

a b

x0

A1 (x) p (g (x) − g (x0 ))

(n+1)α

dx

, (8.86)



When r = 2 and n = 1 we obtain the following special operator related sequential fractional Ostrowski type inequalities. Corollary 8.31 (to Theorem 8.30) Let p, q > 1 : 1p + q1 = 1, and let the ∗-ideals     B p (H ), Bq (H ), for which B p (H ) , · p , Bq (H ) , ·q are Banach algebras;     x0 ∈ [a, b] ⊂ R, 0 < α < 1, A1 ∈ C 1 [a, b] , B p (H ) , A2 ∈ C 1 [a, b] , Bq (H ) ; g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . Assume  xthat x0 0 ∈ C 1 [a, x0 ] , B p (H ) , F2,1 ∈ Fkix0 := Dxkα0 −;g Ai , for k = 1, 2, i = 1, 2, fulfill F1,1

8.5 Main Results

179

    x0  x0  C [a, x0 ] , B p (H ) ; F1,2 ∈ C 1 [a, x0 ] , Bq (H ) , F2,2 ∈ C [a, x0 ] , Bq (H ) . Simx0 x0 := Dxkα0 +;g Ai , k = 1, 2, i = 1, 2, fulfill G 1,1 ∈ C1 ilarly, we assume that G ki     x0  x0  x0 1 ∈ [x0 , b] , B p (H ) , G 2,1 ∈ C [x0 , b] , B p (H ) , G 1,2 ∈ C [x0 , b] , Bq (H ) , G 2,2   C [x0 , b] , Bq (H ) . Then (1) it holds   ( A1 , A2 ) (x0 ) :=

b



a



A2 (x) d x



&

x0

 A2 (x) 

b

A2 (x) x0

g(x)

 A1 (x)

A1 (x)

(z − g (x))

2α−1

(g (x) − z)

g(x0 ) g(x)



b x0

g(x)

g(x0 )

a

&

g(x0 )

2α−1

x0

&

A1 (x) d x

A2 (x0 ) =





a

a

&



b

A1 (x0 ) −

a

1  (2α)

A1 (x) A2 (x) d x− a



b

b

A2 (x) A1 (x) d x +

g(x)

g(x0 )

(2) for γ, δ > 1 :

1 γ

(z − g (x))2α−1 

(g (x) − z)2α−1

+

1 δ



= 1, with

&        Dx2α0 −;g A1 ◦ g −1  p 

δ,[g(a),g(x0 )]



)

(z) dz d x +

)

(z) dz d x +

)   Dx2α0 −;g A2 ◦ g −1 (z) dz d x +

)   Dx2α0 +;g A2 ◦ g −1 (z) dz d x ,

(8.87)

1

a

&        Dx2α0 −;g A2 ◦ g −1 q 



δ,[g(x0 ),g(b)]

b

) A2 (x)q (g (x) − g (x0 ))

2α− 1δ

x0 x0

A1 (x) p (g (x0 ) − g (x))

2α− 1δ

dx + (8.88) ) dx +

a



δ,[g(x0 ),g(b)]

◦g

−1



 (2α) (γ (2α − 1) + 1) γ )  x0 1 A2 (x)q (g (x0 ) − g (x))2α− δ d x +



&        Dx2α0 +;g A2 ◦ g −1 q 







1

&        Dx2α0 +;g A1 ◦ g −1  p 

δ,[g(a),g(x0 )]

Dx2α0 +;g A1

◦g

−1

< α < 1, we have that

1 2δ

 (A1 , A2 ) (x0 )1 ≤

Dx2α0 −;g A1

b

) 1

A1 (x) p (g (x) − g (x0 ))2α− δ d x

x0

=: 7 (x0 ) ,

180

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

(3) if α ≥ 21 , we obtain  ( A1 , A2 ) (x0 )1 ≤ &        Dx2α0 −;g A1 ◦ g −1  p  &        Dx2α0 +;g A1 ◦ g −1  p  &        Dx2α0 −;g A2 ◦ g −1 q 

 L 1 ([g(a),g(x0 )])

)

x0

A2 (x)q (g (x0 ) − g (x))

2α−1

dx +

a

 L 1 ([g(x0 ),g(b)])

)  A2 (x)q (g (x) − g (x0 ))2α−1 d x +

b x0

 L 1 ([g(a),g(x0 )])

&        Dx2α0 +;g A2 ◦ g −1 q 

1  (2α)

)

x0

 A1 (x) p (g (x0 ) − g (x))

2α−1

a

 L 1 ([g(x0 ),g(b)])

b

A1 (x) p (g (x) − g (x0 ))

2α−1

dx + (8.89) ) dx

x0

=: 8 (x0 ) , and (4)  (A1 , A2 ) (x0 )1 ≤ &        Dx2α0 −;g A1 ◦ g −1  p 



&        Dx2α0 +;g A1 ◦ g −1  p 



&        Dx2α0 −;g A2 ◦ g −1 q 



∞,[g(a),g(x0 )]

) A2 (x)q (g (x0 ) − g (x))2α d x +

a

∞,[g(x0 ),g(b)]

b

) A2 (x)q (g (x) − g (x0 ))2α d x +

x0

∞,[g(a),g(x0 )]

&        Dx2α0 +;g A2 ◦ g −1 q 

x0

1  (2α + 1)

)  A1 (x) p (g (x0 ) − g (x))

2α

dx +

a



∞,[g(x0 ),g(b)]

x0

b

)  A1 (x) p (g (x) − g (x0 ))2α d x

x0

(8.90)

=: 9 (x0 ) . Proof By Theorem 8.30. We make Remark 8.32 Let  ( A1 , ..., Ar ) (x0 ) as in (8.41). Denote by



8.5 Main Results

181



b

 (A1 , ..., Ar ) :=

 (A1 , ..., Ar ) (x0 ) d x0 =

a

⎡

⎛

⎛

⎞

⎛

⎞

⎞

⎤

* +  b ⎜ r r ⎜ b ⎜  ⎥ ⎟ ⎟ ⎟  b ⎜ ⎥ ⎢ ⎜ ⎜ ⎟ ⎟ ⎟ A j (x)⎟ Ai (x) d x − ⎜ A j (x)⎟ d x ⎟ Ai (x) d x ⎥ , ⎢(b − a) ⎜ ⎜ ⎝ ⎠ ⎦ ⎣ ⎝ ⎝ ⎠ ⎠ a a a i=1 j=1 j=1

r ⎢ 

j =i

(8.91)

j =i

r ∈ N − {1}. In particular, we have that 

b

 (A1 , A2 ) :=

 (A1 , A2 ) (x0 ) d x0 =

a





b

(b − a)

A2 (x) A1 (x) d x +

a



b

A2 (x) d x a

A1 (x) A2 (x) d x −

(8.92)

a



b

b

 A1 (x) d x −

a



b

b

A1 (x) d x

a

A2 (x) d x ,

a

Clearly, it holds that 

b

 (A1 , ..., Ar ) p ≤

 ( A1 , ..., Ar ) (x0 ) p d x0 ,

(8.93)

a

∀ p ≥ 1. We need Remark 8.33 (i) Call and assume , W1 (A1 , ..., Ar ) := max

i=1,...,r

     sup  Dx2α0 −;g Ai 2 

∞,[a,x0 ]

x0 ∈[a,b]

-

     sup  Dx2α0 +;g Ai 2 

∞,[x0 ,b]

x0 ∈[a,b]

,

< ∞.

(8.94)

Hence by (8.63) we obtain  (A1 , ..., Ar ) (x0 )1 ≤ 1 (x0 ) ≤ ⎡

W1 (A1 , ..., Ar ) (g (b) − g (a))  (2α + 1)

 r 2α  ⎢ b i=1

⎢ ⎣

a

⎛ ⎜ ⎜ ⎝

(8.95) ⎞

r  j=1 j=i

⎤

 ⎟ ⎥   A j (x) ⎟ d x ⎥ , 2⎠ ⎦

182

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

and  (A1 , ..., Ar )1 ≤ ⎡ W1 (A1 , ..., Ar ) (g (b) − g (a))2α (b − a)  (2α + 1)

r  i=1

⎢ ⎢ ⎣

⎛



⎜ ⎜ ⎝

b

a

⎤

⎞ r  j=1 j=i

  ⎟ ⎥  A j (x) ⎟ d x ⎥ . 2⎠ ⎦ (8.96)

(ii) Here

1 2

≤ α < 1. Call and assume ,

W2 (A1 , ..., Ar ) := max

i=1,...,r

     sup  Dx2α0 −;g Ai 2 

x0 ∈[a,b]

     sup  Dx2α0 +;g Ai 2 

x0 ∈[a,b]

L 1 ([a,x0 ])

,

< ∞.

L 1 ([x0 ,b])

(8.97)

Hence by (8.64) we obtain  ( A1 , ..., Ar ) (x0 )1 ≤ 2 (x0 ) ≤ ⎡

  g 

∞,[a,b]

 (2α)

r 

W2 (A1 , ..., Ar ) (g (b) − g (a))2α−1

i=1



⎢ ⎢ ⎣

⎛ b

a

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎤

  ⎟ ⎥  A j (x) ⎟ d x ⎥ , 2⎠ ⎦ (8.98)

and  (A1 , ..., Ar )1 ≤

  g 

∞,[a,b]

 (2α) ⎡

W2 (A1 , ..., Ar ) (g (b) − g (a))2α−1 (b − a)

r  i=1

⎢ ⎢ ⎣

 a

⎛ b

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎤

 ⎟ ⎥   A j (x) ⎟ d x ⎥ . 2⎠ ⎦ (8.99)

(iii) Here γ, δ > 1 :

1 γ

+

1 δ

= 1, and ,

W3 (A1 , ..., Ar ) := max

i=1,...,r

1 2δ

< α < 1. Call and assume

      sup  Dx2α0 −;g Ai ◦ g −1 2 

δ,[g(a),g(x0 )]

x0 ∈[a,b]

      sup  Dx2α0 +;g Ai ◦ g −1 2 

x0 ∈[a,b]

δ,[g(x0 ),g(b)]

,

< ∞.

(8.100)

8.5 Main Results

183

Therefore by (8.65) we get  ( A1 , ..., Ar ) (x0 )1 ≤ 3 (x0 ) ≤ ⎡ W3 (A1 , ..., Ar ) (g (b) − g (a))  (2α) (γ (2α − 1) + 1)

r 

2α− 1δ

1 γ

i=1

⎛



⎢ ⎢ ⎣

⎜ ⎜ ⎝

b

a

⎞ r  j=1 j=i

⎤

 ⎟ ⎥   A j (x) ⎟ d x ⎥ , 2⎠ ⎦

(8.101)

and  (A1 , ..., Ar )1 ≤ ⎡ W3 (A1 , ..., Ar ) (g (b) − g (a))

2α− 1δ

 (2α) (γ (2α − 1) + 1)

r 

(b − a)

1 γ

i=1



⎢ ⎢ ⎣

⎛ b

a

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎤

  ⎟ ⎥  A j (x) ⎟ d x ⎥ . 2⎠ ⎦ (8.102)

We need Remark 8.34 (i) Call and assume , W4 (A1 , ..., Ar ) := max

i=1,...,r

     sup  Dx2α0 −;g Ai  p 

∞,[a,x0 ]

x0 ∈[a,b]

-

     sup  Dx2α0 +;g Ai  p 

< ∞.

∞,[x0 ,b]

x0 ∈[a,b]

,

(8.103)

Hence by (8.70) we obtain  ( A1 , ..., Ar ) (x0 ) p ≤ 4 (x0 ) ≤ ⎡

W4 (A1 , ..., Ar ) (g (b) − g (a))  (2α + 1)

 r 2α  ⎢ b i=1

⎢ ⎣

a

⎛ ⎜ ⎜ ⎝

(8.104) ⎞

r  j=1 j=i

⎤

  ⎟ ⎥  A j (x) ⎟ d x ⎥ , p⎠ ⎦

and  ( A1 , ..., Ar ) p ≤ ⎡ W4 (A1 , ..., Ar ) (g (b) − g (a))2α (b − a)  (2α + 1)

r  i=1

⎢ ⎢ ⎣

 a

⎛ b

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎤

  ⎟ ⎥  A j (x) ⎟ d x ⎥ , p⎠ ⎦ (8.105)

184

8

(ii) Here

1 2

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

≤ α < 1. Call and assume ,

W5 (A1 , ..., Ar ) := max

i=1,...,r

     sup  Dx2α0 −;g Ai  p 

x0 ∈[a,b]

     sup  Dx2α0 +;g Ai  p 

x0 ∈[a,b]

L 1 ([a,x0 ])

,

< ∞.

L 1 ([x0 ,b])

(8.106)

Hence by (8.71) we obtain  (A1 , ..., Ar ) (x0 ) p ≤ 5 (x0 ) ≤ ⎡

  g 

∞,[a,b]

 (2α)

W5 (A1 , ..., Ar ) (g (b) − g (a))2α−1



r 

⎢ ⎢ ⎣

i=1

⎛ b

a

⎜ ⎜ ⎝

⎤

⎞ r  j=1 j=i

  ⎟ ⎥  A j (x) ⎟ d x ⎥ , p⎠ ⎦ (8.107)

  g 

and

∞,[a,b]

 ( A1 , ..., Ar ) p ≤

 (2α) ⎡ r 

W5 (A1 , ..., Ar ) (g (b) − g (a))2α−1 (b − a)

i=1

⎢ ⎢ ⎣

 a

⎛ b

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎤

 ⎟ ⎥   A j (x) ⎟ d x ⎥ . p⎠ ⎦ (8.108)

(iii) Here γ, δ > 1 :

1 γ

+

1 δ

= 1, and ,

W6 (A1 , ..., Ar ) := max

i=1,...,r

1 2δ

< α < 1. Call and assume

      sup  Dx2α0 −;g Ai ◦ g −1  p 

δ,[g(a),g(x0 )]

x0 ∈[a,b]

-

      sup  Dx2α0 +;g Ai ◦ g −1  p 

δ,[g(x0 ),g(b)]

x0 ∈[a,b]

,

< ∞.

(8.109)

Therefore by (8.72) we get  (A1 , ..., Ar ) (x0 ) p ≤ 6 (x0 ) ≤ ⎡ W6 (A1 , ..., Ar ) (g (b) − g (a))  (2α) (γ (2α − 1) + 1)

1 γ

2α− 1δ

r  i=1

⎢ ⎢ ⎣

 a

⎛ b

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎤

 ⎟ ⎥   A j (x) ⎟ d x ⎥ , p⎠ ⎦

(8.110)

8.5 Main Results

185

and  ( A1 , ..., Ar ) p ≤ ⎡ W6 (A1 , ..., Ar ) (g (b) − g (a))

2α− 1δ 1

 (2α) (γ (2α − 1) + 1) γ

⎛



(b − a)  ⎢ ⎢ ⎣ r

i=1

b

a

⎞

⎤

r   ⎟ ⎥ ⎜ ⎜  A j (x) ⎟ d x ⎥ . p⎠ ⎝ ⎦ j=1 j=i

(8.111) We also need Remark 8.35 (i) Call and assume W7 (A1 , A2 ) := , max

      sup  Dx2α0 −;g A1 ◦ g −1  p 

δ,[g(a),g(x0 )]

x0 ∈[a,b]

      sup  Dx2α0 +;g A1 ◦ g −1  p 

,

      sup  Dx2α0 −;g A2 ◦ g −1 q 

,

δ,[g(x0 ),g(b)]

x0 ∈[a,b]

δ,[g(a),g(x0 )]

x0 ∈[a,b]

-

      sup  Dx2α0 +;g A2 ◦ g −1 q 

δ,[g(x0 ),g(b)]

x0 ∈[a,b]

Let γ, δ > 1 : γ1 + 1δ = 1, with Then, by (8.88), we get

1 2δ

,

< ∞.

(8.112)

< α < 1.

 (A1 , A2 ) (x0 )1 ≤ 7 (x0 ) ≤ 1

W7 (A1 , A2 ) (g (b) − g (a))2α− δ

&

1

 (2α) (γ (2α − 1) + 1) γ

b

)   A1 (x) p + A2 (x)q d x ,

(8.113)

a

and  (A1 , A2 )1 ≤ 1

W7 (A1 , A2 ) (g (b) − g (a))2α− δ (b − a)

&

1

 (2α) (γ (2α − 1) + 1) γ

b

)   A1 (x) p + A2 (x)q d x .

a

(8.114)

186

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

(ii) Call and assume W8 (A1 , A2 ) := , max

      sup  Dx2α0 −;g A1 ◦ g −1  p 

x0 ∈[a,b]

      sup  Dx2α0 +;g A1 ◦ g −1  p 

x0 ∈[a,b]

      sup  Dx2α0 −;g A2 ◦ g −1 q 

x0 ∈[a,b]

      sup  Dx2α0 +;g A2 ◦ g −1 q 

x0 ∈[a,b]

L 1 ([g(a),g(x0 )])

L 1 ([g(x0 ),g(b)])

L 1 ([g(a),g(x0 )])

,

, ,

L 1 ([g(x0 ),g(b)])

< ∞.

(8.115)

If α ≥ 21 , by (8.89), we get  (A1 , A2 ) (x0 )1 ≤ 8 (x0 ) ≤ W8 (A1 , A2 ) (g (b) − g (a))2α−1  (2α)

&

b

)   A1 (x) p + A2 (x)q d x ,

(8.116)

a

and  (A1 , A2 )1 ≤ W8 (A1 , A2 ) (g (b) − g (a))2α−1 (b − a)  (2α)

&

b

)   A1 (x) p + A2 (x)q d x .

a

(8.117)

(iii) Call and assume W9 (A1 , A2 ) := , max

      sup  Dx2α0 −;g A1 ◦ g −1  p 

∞,[g(a),g(x0 )]

x0 ∈[a,b]

      sup  Dx2α0 +;g A1 ◦ g −1  p 

,

      sup  Dx2α0 −;g A2 ◦ g −1 q 

,

∞,[g(x0 ),g(b)]

x0 ∈[a,b]

∞,[g(a),g(x0 )]

x0 ∈[a,b]

      sup  Dx2α0 +;g A2 ◦ g −1 q 

x0 ∈[a,b]

∞,[g(x0 ),g(b)]

,

< ∞.

(8.118)

8.5 Main Results

187

Therefore, by (8.90), we get  (A1 , A2 ) (x0 )1 ≤ 9 (x0 ) ≤ W9 (A1 , A2 ) (g (b) − g (a))2α  (2α + 1)

&

b

)   A1 (x) p + A2 (x)q d x ,

(8.119)

a

and  (A1 , A2 )1 ≤ W9 (A1 , A2 ) (g (b) − g (a))2α (b − a)  (2α + 1)

&

b

)   A1 (x) p + A2 (x)q d x .

a

(8.120)

Based on the Remarks 8.32–8.35 we formulate the following generalized sequential fractional Grüss type inequalities results. First come 1-Schatten norm generalized sequential fractional Grüss type inequalities involving several functions taking values in the Banach algebra B2 (H ) ⊂ B (H ): Theorem 8.36 Let the ∗-ideal B2 (H ), 0 < α < 1; Ai ∈ C 1 ([a, b] , B2 (H )), i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing such that g −1 ∈ C 1 ([g (a) , g (b)]). Assume ∀ x0 ∈ [a, b] that Dxα0 −;g Ai ∈ C 1 ([a, x0 ] , B2 (H )) and Dx2α0 −;g Ai ∈ C ([a, x0 ] , B2 (H )); i = 1, ..., r . Similarly, assume ∀ x0 ∈ [a, b] that Dxα0 +;g Ai ∈ C 1 ([x0 , b] , B2 (H )) and Dx2α0 +;g Ai ∈ C ([x0 , b] , B2 (H )); i = 1, ..., r . Here  (A1 , ..., Ar ) is as in (8.91), W1 (A1 , ..., Ar ) as in (8.94), W2 (A1 , ..., Ar ) is as in (8.97), and W3 (A1 , ..., Ar ) is as in (8.100). Then (i) W1 (A1 , ..., Ar ) (g (b) − g (a))2α (b − a)  (A1 , ..., Ar )1 ≤  (2α + 1) ⎡ r  i=1

(ii) if

1 2

⎢ ⎢ ⎣

 a

⎛ b

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎤

 ⎟ ⎥   A j (x) ⎟ d x ⎥ , 2⎠ ⎦

(8.121)

≤ α < 1, we have  (A1 , ..., Ar )1 ≤

  g 

∞,[a,b]

 (2α) ⎡ ⎛ ⎞ ⎤  r r  ⎢ b ⎜   ⎟ ⎥ ⎢ ⎜  A j (x) ⎟ d x ⎥ , W2 (A1 , ..., Ar ) (g (b) − g (a))2α−1 (b − a) 2⎠ ⎣ a ⎝ ⎦ i=1

j=1 j=i

(8.122)

188

8

(iii) if γ, δ > 1 :

1 γ

+

1 δ

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

= 1;

1 2δ

< α < 1, we have

 (A1 , ..., Ar )1 ≤ ⎡ W3 (A1 , ..., Ar ) (g (b) − g (a))

2α− 1δ

 (2α) (γ (2α − 1) + 1)

(b − a)

1 γ

r  i=1

⎛



⎢ ⎢ ⎣

b

a

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎤

 ⎟ ⎥   A j (x) ⎟ d x ⎥ . 2⎠ ⎦ (8.123)

We continue with p-Schatten norm generalized sequential fractional Grüss type inequalities involving several functions taking values in the Banach algebra B p (H ) ⊂ B (H ), p > 1.   Theorem 8.37 Let p > 1, the ∗-ideal B p (H ), 0 < α < 1; Ai ∈ C 1 [a, b] , B p (H ) , i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing such that g −1∈ C 1 ([g (a) , g (b)]). Assume ∀ x0 ∈ [a, b] that Dxα0 −;g Ai ∈ C 1 [a, x0 ] , B p (H ) and   Dx2α0 −;g Ai ∈ C [a, x0 ] , B p (H ) ; i = 1, ..., r . Similarly, assume ∀ x0 ∈ [a, b] that     Dxα0 +;g Ai ∈ C 1 [x0 , b] , B p (H ) and Dx2α0 +;g Ai ∈ C [x0 , b] , B p (H ) ; i = 1, ..., r . Here  (A1 , ..., Ar ) is as in (8.91), W4 (A1 , ..., Ar ) as in (8.103), W5 (A1 , ..., Ar ) is as in (8.106), and W6 (A1 , ..., Ar ) is as in (8.109). Then (i) W4 (A1 , ..., Ar ) (g (b) − g (a))2α (b − a)  ( A1 , ..., Ar ) p ≤  (2α + 1) ⎡

 r  ⎢ ⎢ ⎣ i=1

(ii) if

1 2

a

⎛ b

⎞

⎤

r   ⎟ ⎥ ⎜ ⎜  A j (x) ⎟ d x ⎥ , p⎠ ⎝ ⎦

(8.124)

j=1 j=i

≤ α < 1, we have  ( A1 , ..., Ar ) p ≤

  g 

∞,[a,b]

 (2α) ⎡

W5 (A1 , ..., Ar ) (g (b) − g (a))2α−1 (b − a)

r  i=1

⎢ ⎢ ⎣

 a

⎛ b

⎜ ⎜ ⎝

⎞ r  j=1 j=i

⎤

 ⎟ ⎥   A j (x) ⎟ d x ⎥ , p⎠ ⎦ (8.125)

8.5 Main Results

(iii) if γ, δ > 1 :

189 1 γ

+

1 δ

= 1;

< α < 1, we have

1 2δ

 ( A1 , ..., Ar ) p ≤ ⎡ W6 (A1 , ..., Ar ) (g (b) − g (a))

2α− 1δ

 (2α) (γ (2α − 1) + 1)

1 γ



(b − a)  ⎢ ⎢ ⎣ r

i=1

a

⎛ b

⎞

⎤

r ⎜   ⎟ ⎥ ⎜  A j (x) ⎟ d x ⎥ . p⎠ ⎦ ⎝ j=1 j=i

(8.126) When r = 2 we obtain the following operator related Grüss type sequential fractional inequalities. Theorem 8.38 Let p, q > 1 : 1p + q1 = 1, and let the ∗-ideals B p (H ), Bq (H ),     for which B p (H ) , · p , Bq (H ) , ·q are Banach algebras; 0 < α < 1, A1 ∈     C 1 [a, b] , B p (H ) , A2 ∈ C 1 [a, b] , Bq (H ) ; g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C 1 ([g (a) , g (b)]) . Assume ∀ x0 ∈ [a, b] that Fkix0 := Dxkα0 −;g Ai , for    x0  x0 ∈ C 1 [a, x0 ] , B p (H ) , F2,1 ∈ C [a, x0 ] , B p (H ) ; k = 1, 2, i = 1, 2, fulfill F1,1     x0 x0 ∈ C 1 [a, x0 ] , Bq (H ) , F2,2 ∈ C [a, x0 ] , Bq (H ) . Similarly, we assume ∀ F1,2 x0 x0 kα ∈ C1 x ∈ [a, b] that G ki := Dx0 +;g Ai , k = 1, 2, i = 1, 2, fulfill G 1,1 0      x0 x0 1 [x0 , b] , B p (H ) , G 2,1 ∈ C [x0 , b] , B p (H ) , G 1,2 ∈ C [x0 , b] , Bq (H ) ,   x0 ∈ C [x0 , b] , Bq (H ) . Here  ( A1 , A2 ) is as in (8.92), W7 (A1 , A2 ) is as in G 2,2 (8.112), W8 (A1 , A2 ) is as in (8.115), and W9 (A1 , A2 ) as in (8.118). Then (i) if γ, δ > 1 : γ1 + 1δ = 1; 2δ1 < α < 1, we have  (A1 , A2 )1 ≤ 1

W7 (A1 , A2 ) (g (b) − g (a))2α− δ (b − a)

&

b

1

 (2α) (γ (2α − 1) + 1) γ (ii) if

1 2

≤ α < 1, we have

)   A1 (x) p + A2 (x)q d x ,

a

(8.127)

 (A1 , A2 )1 ≤

W8 (A1 , A2 ) (g (b) − g (a))2α−1 (b − a)  (2α)

&

b

)   A1 (x) p + A2 (x)q d x ,

a

(8.128)

and (iii)  (A1 , A2 )1 ≤ W9 (A1 , A2 ) (g (b) − g (a))2α (b − a)  (2α + 1)

& a

b

)   A1 (x) p + A2 (x)q d x . (8.129)

190

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

8.6 Applications We give the following sequential fractional Ostrowski inequalities: Corollary 8.39 (to Corollary 8.31) All as in Corollary 8.31, with g (t) = et . Then (1) for γ, δ > 1 : γ1 + 1δ = 1; 2δ1 < α < 1, we have 1

 (A1 , A2 ) (x0 )1 ≤ &        Dx2α0 −;et A1 ◦ log p 



δ,[ea ,e x0 ]

1

 (2α) (γ (2α − 1) + 1) γ

)  2α− 1δ A2 (x)q e x0 − e x d x + (8.130)

x0

a

&        Dx2α0 +;et A1 ◦ log p 



&        Dx2α0 −;et A2 ◦ logq 



&        Dx2α0 +;et A2 ◦ logq 



δ,[e x0 ,eb ]

x0

)  x0  1 x 2α− δ  A1 (x) p e − e dx +

a

δ,[e x0 ,eb ]

1 2

)  2α− 1δ A2 (x)q e x − e x0 dx +

x0

δ,[ea ,e x0 ]

(2) if

b

b

 2α− 1δ A1 (x) p e x − e x0 dx

) ,

x0

≤ α < 1, we have  ( A1 , A2 ) (x0 )1 ≤

&        Dx2α0 −;et A1 ◦ log p  &        Dx2α0 +;et A1 ◦ log p  &        Dx2α0 −;et A2 ◦ logq  &        Dx2α0 +;et A2 ◦ logq 

 L 1 ([ea ,e x0 ])

a

 L 1 ([e x0 ,eb ])

) (8.131)  x  2α−1 A2 (x)q e − e x0 dx +

x0

)  2α−1 A1 (x) p e x0 − e x dx +

a

 L 1 ([e x0 ,eb ])

b

)  x0  x 2α−1 A2 (x)q e − e dx +

x0

 L 1 ([ea ,e x0 ])

x0

1  (2α)

b x0

 2α−1 A1 (x) p e x − e x0 dx

) ,

8.6 Applications

191

and (3)  (A1 , A2 ) (x0 )1 ≤ &        Dx2α0 −;et A1 ◦ log p 



∞,[ea ,e x0 ]

1  (2α + 1)

)  2α A2 (x)q e x0 − e x dx +

x0

(8.132)

a

&        Dx2α0 +;et A1 ◦ log p 



&        Dx2α0 −;et A2 ◦ logq 



&        Dx2α0 +;et A2 ◦ logq 



∞,[e x0 ,eb ]

∞,[ea ,e x0 ]

∞,[e x0 ,eb ]

b

)  x  x0 2α A2 (x)q e − e dx +

x0 x0

)  2α  A1 (x) p e x0 − e x dx +

a b

 2α A1 (x) p e x − e x0 dx

) .

x0

We continue with Corollary 8.40 (to Corollary 8.31) All as in Corollary 8.31, with g (t) = t. Then (1) for γ, δ > 1 : γ1 + 1δ = 1; 2δ1 < α < 1, we have  (A1 , A2 ) (x0 )1 ≤ &      Dx2α0 − A1  p 



δ,[a,x0 ]

 A2 (x)q (x0 − x)

2α− 1δ

dx +

(8.133)

a

&    2α   A  D∗x 1 p 0



&      Dx2α0 − A2 q 



&    2α   A   D∗x 2 0 q



b

) 1  A2 (x)q (x − x0 )2α− δ d x +

x0

δ,[a,x0 ]

x0

)  A1 (x) p (x0 − x)

2α− 1δ

dx +

a

δ,[x0 ,b]

1 2

1

 (2α) (γ (2α − 1) + 1) γ

)

x0

δ,[x0 ,b]

(2) if

1

b

) 1

 A1 (x) p (x − x0 )2α− δ d x

,

x0

≤ α < 1, we have  ( A1 , A2 ) (x0 )1 ≤

&      Dx2α0 − A1  p 

 L 1 ([a,x0 ])

a

x0

1  (2α) )

A2 (x)q (x0 − x)

2α−1

dx +

(8.134)

192

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

&    2α   A  D∗x  1 0 p &      Dx2α0 − A2 q  &     2α A2 q   D∗x 0

 L 1 ([x0 ,b])

)

b

 A2 (x)q (x − x0 )

2α−1

dx +

x0

 L 1 ([a,x0 ])

)

x0

 A1 (x) p (x0 − x)

2α−1

dx +

a

 L 1 ([x0 ,b])

b

) A1 (x) p (x − x0 )2α−1 d x

,

x0

and (3)  (A1 , A2 ) (x0 )1 ≤ &      Dx2α0 − A1  p 



∞,[a,x0 ]

1  (2α + 1)

) A2 (x)q (x0 − x)2α d x +

x0

(8.135)

a

&     2α A1  p   D∗x 0



&      Dx2α0 − A2 q 



&    2α   A  D∗x 2 q 0



∞,[x0 ,b]

∞,[a,x0 ]

∞,[x0 ,b]

b

) A2 (x)q (x − x0 )2α d x +

x0 x0

) A1 (x) p (x0 − x)

2α

dx +

a b

)  A1 (x) p (x − x0 )2α d x

.

x0

Next come sequential fractional Grüss inequalities: Corollary 8.41 (to Theorem 8.38) All as in Theorem 8.38, with g (t) = et . Then (1) if γ, δ > 1 : γ1 + 1δ = 1, 2δ1 < α < 1, we have  ( A1 , A2 )1 ≤ &

b

2α− 1δ  W7 (A1 , A2 ) eb − ea (b − a) 1

 (2α) (γ (2α − 1) + 1) γ

)   A1 (x) p + A2 (x)q d x ,

(8.136)

a

(2) if

1 2

≤ α < 1, we have 2α−1  W8 (A1 , A2 ) eb − ea (b − a)  (2α) & b )   A1 (x) p + A2 (x)q d x ,

 (A1 , A2 )1 ≤

a

(8.137)

References

193

and (3)

2α  W9 (A1 , A2 ) eb − ea (b − a)  (A1 , A2 )1 ≤  (2α + 1) & b )   A1 (x) p + A2 (x)q d x ,

(8.138)

a

We finish with Corollary 8.42 (to Theorem 8.38) All as in Theorem 8.38, with g (t) = t. Then (1) if γ, δ > 1 : γ1 + 1δ = 1, 2δ1 < α < 1, we have 1

 (A1 , A2 )1 ≤ &

b

W7 (A1 , A2 ) (b − a)2α+ γ

1

 (2α) (γ (2α − 1) + 1) γ

)   A1 (x) p + A2 (x)q d x ,

(8.139)

a

(2) if

1 2

≤ α < 1, we have  ( A1 , A2 )1 ≤ &

b

W8 (A1 , A2 ) (b − a)2α  (2α)

)   A1 (x) p + A2 (x)q d x ,

(8.140)

a

and (3)  ( A1 , A2 )1 ≤ &

b

W9 (A1 , A2 ) (b − a)2α+1  (2α + 1)

)   A1 (x) p + A2 (x)q d x .

(8.141)

a

References 1. Aliprantis, C., Border, K.: Infinite Dimensional Analysis, 3rd edn. Springer, New York (2006) 2. Anastassiou, G.A.: Fractional Differentiation Inequalities. Research Monograph. Springer, New York (2009) 3. Anastassiou, G.A.: Advances on Fractional Inequalities. Research Monograph. Springer, New York (2011)

194

8

p-Schatten Norm Iterated Generalized Fractional Ostrowski …

4. Anastassiou, G.A.: Intelligent Comparisons: Analytic Inequalities. Springer, Heidelberg (2016) 5. Anastassiou, G.A.: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations. Springer, Heidelberg (2018) 6. Anastassiou, G.A.: p-Schatten norm sequential generalized fractional Ostrowski and Grüss type inequalities for several functions. Submitted (2021) 7. Bellman, R.: Some inequalities for positive definite matrices. In: Beckenbach, E.F. (ed.), General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, pp. 89–90. Birkhauser, Basel (1980) ˇ 8. Cebyšev: Sur les expressions approximatives des intégrales définies par les aures prises entre les mêmes limites. Proc. Math. Soc. Charkov 2), 93–98 (1882 9. Chang, D.: A matrix trace inequality for products of Hermitian matrices. J. Math. Anal. Appl. 237, 721–725 (1999) 10. Coop, I.D.: On matrix trace inequalities and related topics for products of Hermitian matrix. J. Math. Anal. Appl. 188, 999–1001 (1994) 11. Dragomir, S.S.: p-Schatten norm inequalities of Ostrowski’s type. RGMIA Res. Rep. Coll. 24, Art. 108, 19 pp (2021) 12. Dragomir, S.S.: p-Schatten norm inequalities of Grüss type. RGMIA Res. Rep. Coll. 24, Art. 115, 16 pp (2021) 13. Mikusinski, J.: The Bochner Integral. Academic, New York (1978) 14. Neudecker, H.: A matrix trace inequality. J. Math. Anal. Appl. 166, 302–303 (1992) 15. Ostrowski, A.: Über die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938) 16. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991) 17. Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979) 18. Zagrebvov, V.A.: Gibbs Semigroups. Operator Theory: Advances and Applications, vol. 273. Birkhauser (2019)

Chapter 9

p-Schatten Norm Generalized Canavati Fractional Ostrowski, Opial and Grüss Inequalities for Multiple Functions

Using generalized Canavati fractional left and right vectorial Taylor formulae we establish generalized fractional Ostrowski, Opial and Grüss inequalities for several functions that take values in the von Neumann-Schatten class B p (H ), 1 ≤ p < ∞. The estimates are with respect to all p-Schatten norms, 1 ≤ p < ∞. We finish with applications. It follows [7].

9.1 Introduction The following results inspire this chapter. Theorem 9.1 (1938, Ostrowski [17]) Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, derivative f  : (a, b) → R is bounded on (a, b),     b) whose  sup    i.e., f ∞ := sup f (t) < +∞. Then t∈(a,b)

   1  b − a

a

b

 2       sup x − a+b 1 2 f  , + f (t) dt − f (x) ≤ − a) (b ∞ 4 (b − a)2

for any x ∈ [a, b]. The constant

1 4

(9.1)

is the best possible.

Ostrowski type inequalities have great applications to integral approximations in Numerical Analysis. We mention ˇ Theorem 9.2 (1882, Cebyšev [9]) Let f, g : [a, b] → R be absolutely continuous   functions with f , g ∈ L ∞ ([a, b]). Then © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_9

195

196

9

   1  b − a

b

f (x) g (x) d x −

a

≤

p-Schatten Norm Generalized Canavati Fractional …

1 b−a





b

f (x) d x a



1 b−a

b

a

  g (x) d x 

    1 (b − a)2  f  ∞ g  ∞ . 12

(9.2)

The above integrals are assumed to exist. The related Grüss type inequalities have many applications to Probability Theory. We presented also ([3], Chaps. 8 and 9) mixed fractional Ostrowski and GrüssCebysev type inequalities for several functions, acting to all possible directions. The estimates involve the left and right Caputo fractional derivatives. See also the monographs written by the author [1], Chaps. 24–26 and [2], Chaps. 2–6. We are motivated also by Dragomir [12] recent work: An operator A ∈ B (H ) is said to belong to the von Neumann-Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite    1 A p := tr |A| p p < ∞. Assume that A : [a, b] → B p (H ), B : [a, b] → Bq (H ), p, q > 1 with are continuous and B is strongly differentiable on (a, b), then    

b

 A (t) B (t) dt −

a

  sup  B  (t)q ×

t∈[a,b]

a

b

1 p

+

1 q

= 1,



 A (s) ds B (u)  ≤ 1

⎧    b 1 a+b   ⎪ A (t) p dt, ⎪ a ⎪ 2 (b − a) + u − 2 ⎪ ⎪ ⎪ ⎪ ⎪  β1   α1 ⎪ β+1 β+1 b ⎪ ⎨ (u−a) +(b−u) A (t)αp , β+1

a

for α, β > 1 with α1 + β1 = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪    ⎪ 2 1 a+b 2 ⎪ ⎪ sup A (t) p , + u − − a) (b ⎩ 4 2

(9.3)

t∈[a,b]

for all u ∈ [a, b], an Ostrowski type inequality. Further inspiration comes from Dragomir [13] recent work on Grüss inequalities: For two continuous functions A, B : [a, b] → B (H ) we define the noncommutative Cebysev fractional  D (A, B) := (b − a) a

b

 A (t) B (t) dt −



b

A (t) dt a

b

B (t) dt. a

If p, q > 1 with 1p + q1 = 1, let A : [a, b] → B p (H ), B : [a, b] → Bq (H ) be strongly differentiable functions on the interval (a, b), then

9.2 Background on Vectorial Generalized Canavati Fractional Calculus



b

D (A, B)1 ≤ D a

1 (b − a)2 4



b

a

    A (u) du, p

    A (u) du p

 a

b

 a

b

    B (u) du q

197

≤

(9.4)

    B (u) du. q

We are also inspired by Opial [16], 1960, famous inequality. Theorem 9.3 Let x (t) ∈ C 1 ([0, h]) be such that x (0) = x (h) = 0, and x (t) > 0 in (0, h). Then  h  h     2 x (t) x  (t) dt ≤ h (9.5) x (t) dt. 4 0 0 In (9.5), the constant optimal function

h 4

is the best possible. Inequality (9.5) holds as equality for the  x (t) =

ct, 0 ≤ t ≤ h2 , c (h − t) , h2 ≤ t ≤ h,

(9.6)

where c > 0 is an arbitrary constant. Opial-type inequalities are used a lot in proving uniqueness of solutions to differential equations and also to give upper bounds to their solutions. For an extensive study about fractional Opial inequalities see the author’s monograph [1]. In this chapter we generalize [3], Ch. 8,9 for several Banach algebra B p (H ) valued functions, in the sense of developing fractional Ostrowski, Opial and Grüss type inequalities. Now our left and right generalized Canavati fractional derivatives are for Banach space valued functions and our integrals are of Bochner type [14]. Applications finish the chapter.

9.2 Background on Vectorial Generalized Canavati Fractional Calculus All in this section come from [5], pp. 109–115 and [4]. Let g : [a, b] → R be a strictly increasing function. such that g ∈ C 1 ([a, b]), and −1 g ∈ C n ([g(a), g(b)]), n ∈ N, (X, ·) is a Banach space. Let f ∈ C n ([a, b] , X ), and call l := f ◦ g −1 : [g (a) , g (b)] → X . It is clear that l, l  , ..., l (n) are continuous functions from [g (a) , g (b)] into f ([a, b]) ⊆ X. Let ν ≥ 1 such that [ν] = n, n ∈ N as above, where [·] is the integral part of the number. Clearly when 0 < ν < 1, [ν] = 0. (I) Let h ∈ C ([g (a) , g (b)] , X ), we define the left Riemann-Liouville Bochner fractional integral as

198

9



 Jνz0 h (z) :=

p-Schatten Norm Generalized Canavati Fractional …

1  (ν)



z

(z − t)ν−1 h (t) dt,

(9.7)

z0

∞ for g (a) ≤ z 0 ≤ z ≤ g (b), where  is the gamma function;  (ν) = 0 e−t t ν−1 dt. We set J0z0 h = h. ν Let α := ν − [ν] (0 < α < 1). We define the subspace C g(x ([g (a) , g (b)] , X ) 0) of C [ν] ([g (a) , g (b)] , X ), where x0 ∈ [a, b] as: ν C g(x ([g (a) , g (b)] , X ) = 0)



 g(x ) h ∈ C [ν] ([g (a) , g (b)] , X ) : J1−α0 h ([ν]) ∈ C 1 ([g (x0 ) , g (b)] , X ) .

(9.8)

ν So let h ∈ C g(x ([g (a) , g (b)] , X ), we define the left g-generalized X -valued 0) fractional derivative of h of order ν, of Canavati type, over [g (x0 ) , g (b)] as

  g(x0 ) ([ν]) ν Dg(x h := J h . 1−α 0)

(9.9)

ν Clearly, for h ∈ C g(x ([g (a) , g (b)] , X ) , there exists 0)



 ν Dg(x h (z) = 0)

d 1  (1 − α) dz



z g(x0 )

(z − t)−α h ([ν]) (t) dt,

(9.10)

for all g (x0 ) ≤ z ≤ g (b) . ν In particular, when f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), we have that 0) 

ν Dg(x 0)



f ◦g

−1



d 1 (z) =  (1 − α) dz



z g(x0 )

 ([ν]) (z − t)−α f ◦ g −1 (t) dt, 

−1





(9.11) and

 −1 (n)

for all g (x0 ) ≤ z ≤ g (b). We have that f ◦g = f ◦g   0 −1 −1 , see [4]. f ◦ g = f ◦ g Dg(x 0) ν By [4], we have for f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ) , where x0 ∈ [a, b] the 0) following left generalized g-fractional, of Canavati type,X -valued Taylor’s formula: n Dg(x 0)

ν Theorem 9.4 Let f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), where x0 ∈ [a, b] is fixed. 0) (i) If ν ≥ 1, then

f (x) − f (x0 ) =

[ν]−1  k=1

1  (ν) for all x0 ≤ x ≤ b.



g(x) g(x0 )



f ◦ g −1

(k) k!

(g (x0 ))

(g (x) − g (x0 ))k +

 ν   f ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(9.12)

9.2 Background on Vectorial Generalized Canavati Fractional Calculus

199

(ii) If 0 < ν < 1, we get 1 f (x) =  (ν)



g(x) g(x0 )

 ν   f ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(9.13)

for all x0 ≤ x ≤ b. (II) Let h ∈ C ([g (a) , g (b)] , X ), we define the right Riemann-Liouville Bochner fractional integral as 

Jzν0 − h



1 (z) :=  (ν)



z0

(t − z)ν−1 h (t) dt,

(9.14)

z

for g (a) ≤ z ≤ z 0 ≤ g (b) . We set Jz00 − h = h. Let α := ν − [ν] (0 < α < 1). We define the subspace ν [ν] ∈ b] as: C g(x [, g , X of C , g , X where x [a, ([g (a) (b)] ) ([g (a) (b)] ), 0 )− 0 ν C g(x ([g (a) , g (b)] , X ) := 0 )−



 1−α 1 ([ν]) h ∈ C , g , X . h ∈ C [ν] ([g (a) , g (b)] , X ) : Jg(x ([g (a) (x )] ) 0 )− 0

(9.15)

ν So let h ∈ C g(x ([g (a) , g (b)] , X ), we define the right g-generalized X -valued 0 )− fractional derivative of h of order ν, of Canavati type, over [g (a) , g (x0 )] as

  1−α ν n−1 ([ν]) J h := h . Dg(x (−1) g(x0 )− 0 )−

(9.16)

ν Clearly, for h ∈ C g(x ([g (a) , g (b)] , X ) , there exists 0 )−



ν h Dg(x 0 )−



(−1)n−1 d (z) =  (1 − α) dz



g(x0 )

(t − z)−α h ([ν]) (t) dt,

(9.17)

z

for all g (a) ≤ z ≤ g (x0 ) ≤ g (b) . ν In particular, when f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), we have that 0 )− 

  (−1)n−1 d ν f ◦ g −1 (z) = Dg(x 0 )−  (1 − α) dz



g(x0 )

 ([ν]) (t − z)−α f ◦ g −1 (t) dt,

z

(9.18)

for all g (a) ≤ z ≤ g (x0 ) ≤ g (b). We get that 

   (n) n f ◦ g −1 (z) = (−1)n f ◦ g −1 Dg(x (z) 0 )−

     0 −1 f ◦ g and Dg(x (z) = f ◦ g −1 (z), all z ∈ [g (a) , g (b)] , see [4]. 0 )−

(9.19)

200

9

p-Schatten Norm Generalized Canavati Fractional …

ν By [4], we have for f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ) , where x0 ∈ [a, b] is 0 )− fixed, the following right generalized g-fractional, of Canavati type, X -valued Taylor’s formula: ν Theorem 9.5 Let f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), where x0 ∈ [a, b] is fixed. 0 )− (i) If ν ≥ 1, then

f (x) − f (x0 ) =

[ν]−1  k=1

1  (ν)



g(x0 ) g(x)



f ◦ g −1

(k) k!

(g (x0 ))

(g (x) − g (x0 ))k +

 ν   f ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

(9.20)

for all a ≤ x ≤ x0 , (ii) If 0 < ν < 1, we get f (x) =

1  (ν)



g(x0 ) g(x)

 ν   f ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

(9.21)

all a ≤ x ≤ x0 . (III) Denote by mν ν ν ν Dg(x = Dg(x Dg(x ...Dg(x (m-times), m ∈ N. 0) 0) 0) 0)

(9.22)

We mention the following modified and generalized left X -valued fractional Taylor’s formula of Canavati type: −1 Theorem 9.6 Let f ∈ C 1 ([a, b] ,  X ), g ∈ C 1 ([a, b]),  strictly increasing: g ∈   iν ν C 1 ([g (a) , g (b)]). Assume that Dg(x f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), 0) 0) 0 < ν < 1, x0 ∈ [a, b], for i = 0, 1, ..., m. Then

f (x) =

1  ((m + 1) ν)



g(x) g(x0 )

  (m+1)ν  −1 f ◦ g (z) dz, (g (x) − z)(m+1)ν−1 Dg(x 0) (9.23)

all x0 ≤ x ≤ b. (IV) Denote by mν ν ν ν Dg(x = Dg(x Dg(x ...Dg(x (m times), m ∈ N. 0 )− 0 )− 0 )− 0 )−

(9.24)

We mention the following modified and generalized right X -valued fractional Taylor’s formula of Canavati type:

9.3 Basic Banach Algebras Background

201

Theorem 9.7 Let f ∈ C 1 ([a, b], X ), g ∈ C 1 ([a, b]), strictly increasing: g −1 ∈   iν ν C 1 ([g (a) , g (b)]). Assume that Dg(x f ◦ g −1 ∈ C g(x ([g (a) , g (b)] , X ), 0 )− 0 )− 0 < ν < 1, x0 ∈ [a, b], for all i = 0, 1, ..., m. Then f (x) =

1  ((m + 1) ν)



g(x0 ) g(x)

  (m+1)ν  −1 f ◦ g (z − g (x))(m+1)ν−1 Dg(x (z) dz, 0 )− (9.25)

all a ≤ x ≤ x0 ≤ b.

9.3 Basic Banach Algebras Background All here come from [18]. We need Definition 9.8 ([18], p. 245) A complex algebra is a vector space A over the complex filed C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(9.26)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(9.27)

α (x y) = (αx) y = x (αy) ,

(9.28)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A)

(9.29)

and if A contains a unit element e such that xe = ex = x (x ∈ A)

(9.30)

e = 1,

(9.31)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 9.9 Commutativity of A is explicited stated when needed.

202

9

p-Schatten Norm Generalized Canavati Fractional …

There exists at most one e ∈ A that satisfies (9.30). Inequality (9.29) makes multiplication to be continuous, more precisely left and right continuous, see [18], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [18], pp. 247–248, Sect. 10.3. We also make Remark 9.10 Next we mention about integration of A-valued functions, see [18], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some  compact Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [18], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   f dμ = x f ( p) dμ ( p) (9.32) x Q

and



Q

 f dμ x = Q

f ( p) x dμ ( p) .

(9.33)

Q

The Bochner integrals we will involve in our chapter follow (9.32) and (9.33). Also, let f ∈ C ([a, b] , X ), where [a, b] ⊂ R, (X, ·) is a Banach space. By [5], p. 3, f is Bochner integrable.

9.4

p-Schatten Norms Background

In this advanced section all come from [12]. Let (H, ·, · ) be a complex Hilbert space and B (H ) the Banach algebra of all bounded linear operators on H . If {ei }i∈I an orthonormal basis of H , we say that A ∈ B (H ) is of trace class if A1 :=



|A| ei , ei < ∞.

(9.34)

i∈I

The definition of A1 does not depend on the choice of the orthornormal basis {ei }i∈I . We denote by B1 (H ) the set of trace class operators in B (H ). We define the trace of a trace class operator A ∈ B1 (H ) to be tr (A) :=

 i∈I

Aei , ei ,

(9.35)

9.4 p-Schatten Norms Background

203

where {ei }i∈I an orthonormal basis of H . Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series (9.35) converges absolutely and it is independent from the choice of basis. The following result collects some properties of the trace: Theorem 9.11 We have: (i) If A ∈ B1 (H ) then A∗ ∈ B1 (H ) and   tr A∗ = tr (A);

(9.36)

(ii) If A ∈ B1 (H ) and T ∈ B (H ), then AT, T A ∈ B1 (H ) and tr (AT ) = tr (T A) and |tr (AT )| ≤ A1 T  ;

(9.37)

(iii) tr (·) is a bounded linear functional on B1 (H ) with tr  = 1; (iv) If A, B ∈ B2 (H ) then AB, B A ∈ B1 (H ) and tr (AB) = tr (B A) ; (v) B f in (H ) (finite rank operators) is a dense subspace of B1 (H ) . An operator A ∈ B (H ) is said to belong to the von Neumann-Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite [20, pp. 60–64]  1   A p := tr |A| p p < ∞, |A| p is an operator notation and not a power. For 1 < p < q < ∞ we have that B1 (H ) ⊂ B p (H ) ⊂ Bq (H ) ⊂ B (H )

(9.38)

A1 ≥  A p ≥  Aq ≥ A .

(9.39)

and For p ≥ 1 the functional · p is a norm on the ∗-ideal B p (H ), which is a Banach   algebra, and B p (H ) , · p is a Banach space. Also, see for instance [20, p. 60–64], for p ≥ 1,   A p =  A∗  p , A ∈ B p (H )

(9.40)

AB p ≤  A p B p , A, B ∈ B p (H )

(9.41)

and AB p ≤ A p B , B A p ≤ B A p , A ∈ B p (H ) , B ∈ B (H ) . (9.42) This implies that

204

9

p-Schatten Norm Generalized Canavati Fractional …

C AB p ≤ C  A p B , A ∈ B p (H ) , B, C ∈ B (H ) .

(9.43)

In terms of p-Schatten norm we have the Hölder inequality for p, q > 1 with 1 =1: q (|tr (AB)| ≤) AB1 ≤ A p Bq , A ∈ B p (H ) , B ∈ Bq (H ) .

1 p

+

(9.44)

For the theory of trace functionals and their applications the interested reader is referred to [19, 20]. For some classical trace inequalities see [10, 11, 15], which are continuations of the work of Bellman [8].

9.5 Main Results We start with 1-Schatten norm weighted mixed generalized Canavati fractional Ostrowski type inequalities involving several functions taking values in the Banach algebra B2 (H ) ⊂ B (H ): Theorem 9.12 Let the ∗-ideal B2 (H ), which (B2 (H ) , ·2 ) is a Banach algebra; x0 ∈ [a, b] ⊂ R, ν ≥ 1, n = [ν]; f i ∈ C n ([a, b] , B2 (H )), i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing such that g −1 ∈ C n ([g (a) , g (b)]), with (k)  f i ◦ g −1 (g (x0 )) = 0, k = 1, ..., n − 1; i = 1, ..., r. Assume further that ν ν f i ◦ g −1 ∈ C g(x ([g (a) , g (b)] , B2 (H )) ∩ C g(x ([g (a) , g (b)] , B2 (H )), 0 )− 0) i = 1, ..., r. Denote by K ( f 1 , ..., fr ) (x0 ) := ⎡

⎛  r  ⎢ b⎜ ⎢ ⎜ ⎣ a ⎝ i=1

⎛ ⎛  b ⎜ ⎟ ⎜ ⎜ ⎜ f j (x)⎟ ⎠ f i (x) d x − ⎝ a ⎝ ⎞

r

j=1 j=i

⎞ r

j=1 j=i

⎤

⎞

⎟ ⎟ ⎥ ⎥ ⎟ f j (x)⎟ ⎠ d x ⎠ f i (x0 )⎦ .

(9.45)

Then  1 K ( f 1 , ..., fr ) (x0 )1 ≤  (ν + 1) i=1 r

⎛ ⎜ (g (x0 ) − g (a))ν ⎜ ⎝

 a

''     ν −1   f ◦ g  Dg(x  i )− 0 2

∞,[g(a),g(x0 )]

⎛ x0

⎜ ⎜ ⎝

⎞ r

j=1 j=i

⎞⎤

 ⎟ ⎟⎥   f j (x) ⎟ d x ⎟⎥ + 2⎠ ⎠⎦

(9.46)

9.5 Main Results

205 ⎛

⎡     ⎢  ν −1   ⎢   ⎣Dg(x0 ) f i ◦ g

2 ∞,[g(x 0 ),g(b)]

⎜ (g (b) − g (x0 ))ν ⎜ ⎝

⎛



b x0

⎜ ⎜ ⎝

⎞ r j=1 j=i

⎞ ⎤⎤

  ⎟ ⎟ ⎥⎥  f j (x) ⎟ d x ⎟⎥⎥ . 2⎠ ⎠ ⎦⎦

(k)  Proof Since f i ◦ g −1 (g (x0 )) = 0, k = 1, ..., [ν] − 1; i = 1, ..., r ; we have by Theorem 9.4 that f i (x) − f i (x0 ) =

1  (ν)



g(x) g(x0 )

 ν   f i ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(9.47)

∀ x ∈ [x0 , b] , and by Theorem 9.5 that f i (x) − f i (x0 ) =

1  (ν)



g(x0 ) g(x)

 ν   f i ◦ g −1 (t) dt, (9.48) (t − g (x))ν−1 Dg(x 0 )−

∀ x ∈ [a, x0 ] , for all i = 1, ..., r.



Left multiplying (9.47) and (9.48) with ⎛ ⎜ ⎜ ⎝

⎞ r

j=1 j=i

(r j=1 j=i

⎞

⎛

⎟ ⎜ ⎜ f j (x)⎟ ⎠ f i (x) − ⎝

f j (x) we get, respectively,

r

j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x0 ) =



(r j=1 j=i

f j (x) 

 (ν) ∀ x ∈ [x0 , b] , and

g(x) g(x0 )

 ν   f i ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

⎛ ⎜ ⎜ ⎝

⎞ r

j=1 j=i

⎞

⎛

⎟ ⎜ ⎜ f j (x)⎟ f − (x) i ⎠ ⎝

r

j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x0 ) =



(r j=1 j=i

f j (x) 

 (ν)

g(x0 ) g(x)

(9.49)

 ν   f i ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

∀ x ∈ [a, x0 ] , for all i = 1, ..., r. Adding (9.49) and (9.50) as separate groups, we obtain

(9.50)

206

9

⎞

⎛ r  i=1

⎜ ⎜ ⎝

r

j=1 j=i

1  (ν)

i=1

⎜ ⎜ ⎝

r

j=1 j=i

∀ x ∈ [x0 , b] , and

⎟ f j (x)⎟ ⎠



g(x0 )

i=1

⎜ ⎜ ⎝

r

i=1

j=1 j=i

r r ⎟ 1 ⎜ ⎜ f j (x)⎟ ⎠ ⎝  (ν) i=1 j=1



g(x0 ) g(x)

⎜ ⎜ ⎝

r

j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

 ν   f i ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

r  i=1

⎜ ⎜ ⎝

(9.51)

⎞

⎛

⎟ f j (x)⎟ ⎠ f i (x) − ⎞

⎛

r 

⎞

⎛ r 

g(x)

⎞

⎛

⎟ f j (x)⎟ ⎠ f i (x) − ⎞

⎛ r 

p-Schatten Norm Generalized Canavati Fractional …

r

j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

 ν   f i ◦ g −1 (t) dt, (9.52) (t − g (x))ν−1 Dg(x 0 )−

j=i

∀ x ∈ [a, x0 ] . Next, we integrate (9.51) and (9.52) with respect to x ∈ [a, b]. We have r  b  x0

i=1

⎛ ⎜ ⎜ ⎝

⎡

⎛

⎞ r

j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x) d x −

r  i=1

⎜ ⎜ ⎝

⎛



b x0

⎜ ⎜ ⎝

⎞ r

j=1 j=i

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) =

⎞

⎛

 ⎜ r r 1 ⎢ ⎢ b⎜ ⎢ ⎜ ⎣ x0 ⎝  (ν) i=1 j=1

⎤ ) * ⎟ ⎥    g(x) ⎟ ⎥ ν −1 f dt d x f j (x)⎟ ◦ g (t) (g (x) − t)ν−1 Dg(x ⎥, i ) 0 ⎠ g(x0 ) ⎦

j=i

(9.53) and r   i=1

a

⎞

⎛ x0

⎜ ⎜ ⎝

r

j=1 j=i

⎛

r  ⎟ ⎜ ⎜ f f j (x)⎟ d x − (x) i ⎠ ⎝ i=1

 a

⎞

⎛ x0

⎜ ⎜ ⎝

r

j=1 j=i

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ d x ⎠ f i (x0 ) =

9.5 Main Results ⎡

207 ⎛

 x0 r ⎜ 1 ⎢ ⎢ ⎜ ⎣ ⎝  (ν) a i=1

⎞ r j=1 j=i

⎟ f j (x)⎟ ⎠

)

g(x 0 ) g(x)





ν f i ◦ g −1 (t − g (x))ν−1 Dg(x 0 )−



⎤

*

⎥ (t) dt d x ⎥ ⎦,

(9.54) Finally, adding (9.53) and (9.54) we obtain the useful identity K ( f 1 , ..., fr ) (x0 ) := r  i=1

⎡

⎛

⎢ ⎢ ⎣

⎜ ⎜ ⎝



b

a

⎡⎡ 1  (ν)

⎡ ⎢ +⎢ ⎣

r  i=1



⎢⎢ ⎢⎢ ⎣⎣

 a

⎞

r

j=1 j=i

⎛  ⎟ ⎜ ⎜ f f j (x)⎟ d x − (x) ⎠ i ⎝

a

⎛ x0

⎜ ⎜ ⎝

⎞ r j=1 j=i

⎞

⎛ b x0

⎜ ⎜ ⎝

⎟ f j (x)⎟ ⎠

r

j=1 j=i

⎟ f j (x)⎟ ⎠



)

g(x) g(x0 )

g(x 0 ) g(x)

⎞

⎛ b

⎜ ⎜ ⎝

r

j=1 j=i

⎞

⎤

⎟ ⎟ ⎥ ⎟ ⎥ f j (x)⎟ ⎠ d x ⎠ f i (x0 )⎦ =





ν f i ◦ g −1 (t − g (x))ν−1 Dg(x 0 )−



⎤

*

⎥ (t) dt d x ⎥ ⎦

⎤⎤



⎥⎥  ν   ⎥ f i ◦ g −1 (t) dt d x ⎥ (g (x) − t)ν−1 Dg(x 0) ⎦⎦ . (9.55)

Therefore, we get that K ( f 1 , ..., fr ) (x0 )1 =  ⎞ ⎞ ⎞ ⎤ ⎡ ⎛ ⎛ ⎛     r    ⎢ b ⎜ r ⎟ ⎟ ⎟ ⎥ ⎜ b⎜ r 1  ⎟ ⎟ ⎥ ⎟ ⎢ ⎜ ⎜ ⎜ f j (x)⎠ f i (x) d x − ⎝ f j (x)⎠ d x ⎠ f i (x0 )⎦  ≤  (ν)  ⎣ a ⎝ ⎝ a   i=1 j=1 j=1   j=i j=i 1

⎡⎡ ⎛ ⎤ ⎞   * )    r    g(x 0 )  ⎢⎢ x0 ⎜ r ⎥ ⎟ ν−1 ν −1 ⎢⎢ ⎜ ⎥  ⎟ D f dt d x − g f ◦ g (t) (x) (t (x)) j i g(x 0 )− ⎣⎣ ⎝ ⎦ ⎠ a g(x)  i=1  j=1   j=i ⎡ ⎤ ⎞ ⎛   * )    b r    g(x) ⎥  ⎟ ⎜ ⎢ ν−1 ν −1 ⎥  ⎟ ⎜ ⎢ D f dt d x − t) + f ◦ g (t) (x) (g (x) j i g(x 0 ) ⎣ ⎦ ⎠ ⎝ g(x0 )   x0 j=1   j=i

1

⎤ (9.56) ⎥ ⎥≤ ⎦ 1

208

9 ⎡⎡

1  (ν)

p-Schatten Norm Generalized Canavati Fractional …

 ⎛ ⎞ ⎤    ) *  ⎟  g(x0 ) ⎥     ⎜ ⎟ ⎥ ν f j (x)⎟ f i ◦ g −1 (t) dt  d x ⎥ (t − g (x))ν−1 Dg(x ⎜ )− 0  ⎝ ⎠ g(x) ⎦   j=1   j=i

r ⎢⎢ x0 ⎜ r 

⎢⎢ ⎢⎢ ⎣⎣ a i=1

1

(9.57)  ⎛ ⎤ ⎞ ⎤     ) *  ⎟  g(x) ⎢ b ⎜ r ⎥⎥     ⎜ ⎟ ⎢ ⎥⎥ ν −1 f dt +⎢ f j (x)⎟ ◦ g d x (t) (g (x) − t)ν−1 Dg(x  ⎜ ⎥⎥ ≤ i 0)  ⎠ g(x0 ) ⎣ x0 ⎝ ⎦⎦   j=1   j=i ⎡

1

⎡⎡

⎛

 r ⎢ x0 ⎜ 1 ⎢ ⎢⎢ ⎜ ⎣⎣ ⎝  (ν) a i=1

⎡ ⎢ +⎢ ⎣



⎛ b x0

⎜ ⎜ ⎝

⎞ r j=1 j =i

  ⎟  f j (x) ⎟ 2⎠ ⎞

r j=1 j =i

  ⎟  f j (x) ⎟ 2⎠

)

)

g(x0 ) g(x)

⎤ *    ⎥     ν f i ◦ g −1 (t) dt d x ⎥ (t − g (x))ν−1  Dg(x ⎦ 0 )− 2

⎤⎤ (9.58) *     g(x) ⎥⎥     ν ⎥ f i ◦ g −1 (t) dt d x ⎥ (g (x) − t)ν−1  Dg(x ⎦⎦ =: (∗) . 0) 2

g(x0 )

Hence it holds K ( f 1 , ..., fr ) (x0 )1 ≤ (∗) . We have that (∗) ≤

1  (ν + 1)

⎡⎡ r  i=1



    ⎢⎢  ν   −1 ⎢⎢ D f ◦ g     i g(x 0 )− ⎣⎣

⎛ x0

2 ∞,[g(a),g(x 0 )] a

⎡



    ⎢    ν −1 +⎢   ⎣ Dg(x0 ) f i ◦ g

2 ∞,[g(x 0 ),g(b)]

 1  (ν + 1) i=1 r

(9.59)

⎜ ⎜ ⎝

⎛ b x0

⎜ ⎜ ⎝

⎞ r j=1 j=i

  ⎟ ⎥  f j (x) ⎟ (g (x0 ) − g (x))ν d x ⎥ 2⎠ ⎦ ⎞

r j=1 j=i

⎤

⎤⎤

  ⎟ ⎥⎥  f j (x) ⎟ (g (x) − g (x0 ))ν d x ⎥⎥ ≤ 2⎠ ⎦⎦

(9.60)

''  ν     −1   ◦ g f   Dg(x i 0 )− 2

∞,[g(a),g(x0 )]

⎛ ⎜ (g (x0 ) − g (a))ν ⎜ ⎝

 a

⎛ x0

⎜ ⎜ ⎝

⎞ r

j=1 j=i

⎞⎤

  ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ + 2⎠ ⎠⎦

(9.61)

9.5 Main Results

209

⎡

⎛  ⎜ (g (b) − g (x0 ))ν ⎜ ⎝ ∞,[g(x 0 ),g(b)]

    ⎢    ν −1 ⎢   ⎣ Dg(x0 ) f i ◦ g 2

⎛ b x0

⎜ ⎜ ⎝

⎞ r j=1 j=i

⎞ ⎤⎤

  ⎟ ⎟ ⎥⎥  f j (x) ⎟ d x ⎟⎥⎥ , 2⎠ ⎠ ⎦⎦



proving (9.46). Next comes an L 1 estimate. Theorem 9.13 All as in Theorem 9.12. Then |K ( f 1 , ..., fr ) (x0 )1 ≤ ⎡⎡

 r   ⎢⎢     ν −1   ⎢⎢ D f ◦ g     i g(x0 )− ⎣⎣ 2 L 1 ([g(a),g(x0 )])

⎛ ⎜ ⎜ ⎝

x0

a

i=1

⎡     ⎢    ν −1 +⎢   ⎣ Dg(x0 ) f i ◦ g

b x0

⎜ ⎜ ⎝

⎤

⎞ r j=1 j =i

⎥  ⎟   f j (x) ⎟ (g (x0 ) − g (x))ν−1 d x ⎥ 2⎠ ⎦

⎛



2 L 1 ([g(x0 ),g(b)])

1  (ν)

(9.62) ⎤⎤

⎞ r j=1 j=i

  ⎟ ⎥⎥  f j (x) ⎟ (g (x) − g (x0 ))ν−1 d x ⎥⎥ . 2⎠ ⎦⎦

Proof We observe that (by (9.58), (9.59)) 1  (ν)

(∗) ≤ r  i=1

⎡⎡  ⎢⎢     ν −1   ⎢⎢ D f ◦ g     i g(x )− ⎣⎣ 0



⎛

⎡

⎛

x0

2 L 1 ([g(a),g(x0 )]) a

 ⎢     ν −1   +⎢   ⎣ Dg(x0 ) f i ◦ g

2 L 1 ([g(x0 ),g(b)])

proving (9.62).



b x0

⎜ ⎜ ⎝

⎜ ⎜ ⎝

⎞ r j=1 j =i

⎤

 ⎟  ⎥  f j (x) ⎟ (g (x0 ) − g (x))ν−1 d x ⎥ 2⎠ ⎦

(9.63) ⎤ ⎤

⎞ r j=1 j =i

⎥⎥  ⎟   f j (x) ⎟ (g (x) − g (x0 ))ν−1 d x ⎥⎥ , 2⎠ ⎦⎦



An L p estimate follows. Theorem 9.14 All as in Theorem 9.12. Let now p, q > 1 : K ( f 1 , ..., fr ) (x0 )1 ≤

1 p

+

1 q

1 1

( p (ν − 1) + 1) p  (ν)

= 1. Then

210

9

p-Schatten Norm Generalized Canavati Fractional …

⎡⎡ ⎛  r     ⎢⎢ ⎜    ν −1   ⎢⎢ ⎜ D f ◦ g     i g(x0 )− ⎣⎣ 2 q,[g(a),g(x0 )] ⎝

⎛ x0

(g (x0 ) − g (x))

⎜ ⎜ ⎝

ν− q1

a

i=1

⎡

⎛

 ⎢     ν −1   f +⎢ D ◦ g     i g(x ) ⎣ 0

⎜ ⎜



⎞ r j=1 j =i

⎛ b

2 q,[g(x0 ),g(b)] ⎝ x0

(g (x) − g (x0 ))

⎜ ⎜ ⎝

ν− q1

⎞ r j=1 j =i

⎞⎤

  ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ 2⎠ ⎠⎦ ⎞(9.64) ⎤⎤

 ⎟ ⎟⎥⎥   f j (x) ⎟ d x ⎟⎥⎥ . 2⎠ ⎠⎦⎦

Proof We have that (by (9.58), (9.59)) ⎡⎡ ⎢ 1 ⎢ ⎢⎢ (∗) ≤ ⎣  (ν) i=1 ⎣ r



g(x0 ) g(x)

⎡

x0

a

b x0

⎜ ⎜ ⎝



⎜ ⎜ ⎝

i=1

⎡ ⎢ +⎢ ⎣



x0

j=1 j=i

  ⎟  f j (x) ⎟ 2⎠



g(x0 )−

⎞ r

j=1 j=i

g(x0 )

  ⎟  f j (x) ⎟ 2⎠

 ν  D

g(x0 )



fi ◦ g 

−1

g(x) g(x0 )

fi ◦ g

−1





g(x0 )

(t − g (x))

dt

g(x)

q (t)2 dt

q1

(g (x) − t)



1p p(ν−1)

q (t) dt 2

 dx +

1p

p(ν−1)

dt



q1 dx

≤

(9.65)

⎤ ⎞ 1     ⎥  ⎟ (g (x0 ) − g (x))ν−1+ p    ν    f j (x) ⎟ dx⎥  Dg(x0 )− f i ◦ g −1   1 2⎠ ⎦ 2 q,[g(a),g(x )] 0 ( p (ν − 1) + 1) p j=1 r

j =i

⎛ b

r

 ν  D

g(x)

⎡⎡ ⎛  x0 r  ⎢ ⎢ ⎜ 1 ⎢⎢ ⎜ ⎣⎣ ⎝  (ν) a

⎞

⎛



⎢ ⎢ ⎣

⎛



⎜ ⎜ ⎝

⎤⎤ ⎞ 1     ⎥⎥   ⎟ (g (x) − g (x0 ))ν−1+ p       ν ⎥  f j (x) ⎟ dx⎥  Dg(x0 ) f i ◦ g −1 (z)  1 2⎠ ⎦⎦ 2 q,[g(x ),g(b)] 0 p p − 1) + 1) ( (ν j=1 r

j =i

= ⎡

1 1

( p (ν − 1) + 1) p  (ν) ⎛

 r      ⎢ ⎜    ν −1 ⎢ ⎜ ◦ g f D     i g(x 0 )− ⎣ 2 q,[g(a),g(x 0 )] ⎝ i=1

a

⎛ x0

(g (x0 ) − g (x))

ν− q1

⎜ ⎜ ⎝

⎞ r j=1 j=i

⎞

  ⎟ ⎟  f j (x) ⎟ d x ⎟ 2⎠ ⎠

(9.66)

9.5 Main Results

211 ⎛

       ν f i ◦ g −1   +  Dg(x 0)

⎜ ⎜

⎛



b

2 q,[g(x 0 ),g(b)] ⎝ x 0

(g (x) − g (x0 ))

ν− q1

⎜ ⎜ ⎝

⎞ r j=1 j=i

⎞⎤

  ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ , 2⎠ ⎠⎦



proving (9.64).

We continue with γ-Schatten norm related Ostrowski fractional inequalities:   Theorem 9.15 Let γ ≥ 1, the ∗-ideal Bγ (H ), which Bγ (H ) , ·γ is a Banach  algebra; x0 ∈ [a, b] ⊂ R, ν ≥ 1, n = [ν]; f i ∈ C n [a, b] , Bγ (H ) , i = 1, ..., r ∈ N − {1}; g ∈ C 1 ([a, b]), strictly increasing such that g −1 ∈ C n ([g (a) , g (b)]), (k)  with f i ◦ g −1 (g (x0 ))  = 0, k = 1, ..., n − 1; iν = 1,  ..., r. Assume further  ν , g , B ∩ C that f i ◦ g −1 ∈ C g(x [g (a) (b)] (H ) γ g(x0 ) [g (a) , g (b)] , Bγ (H ) , 0 )− i = 1, ..., r. Here K ( f 1 , ..., fr ) (x0 ) is as in (9.45). Then  1  (ν + 1) i=1 r

K ( f 1 , ..., fr ) (x0 )γ ≤

''     ν −1   ◦ g f  Dg(x  i 0 )− γ

∞,[g(a),g(x0 )]

⎛ ⎛  x0 ⎜ ⎜ ⎜ (g (x0 ) − g (a))ν ⎜ ⎝ a ⎝ ⎡     ⎢  ν −1   ⎢   ⎣Dg(x0 ) f i ◦ g γ

j=1 j=i

  ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ + γ⎠ ⎠⎦ ⎛

∞,[g(x 0 ),g(b)]

⎞⎤

⎞ r

⎜ (g (b) − g (x0 ))ν ⎜ ⎝



⎛ b x0

⎜ ⎜ ⎝

(9.67) ⎞

r j=1 j=i

Proof As similar to Theorem 9.12 is omitted. Use of (9.41).

⎞ ⎤⎤

  ⎟ ⎟ ⎥⎥  f j (x) ⎟ d x ⎟⎥⎥ . γ⎠ ⎠ ⎦⎦



An L 1 estimate follows: Theorem 9.16 All as in Theorem 9.15. Then |K ( f 1 , ..., fr ) (x0 )γ ≤ ⎡⎡    r  ⎢⎢    ν −1   ⎢⎢ D f ◦ g   i g(x0 )− ⎣⎣ γ i=1

⎡

    ⎢  −1  ν +⎢  ⎣ Dg(x0 ) f i ◦ g

   γ



⎛ x0

L 1 ([g(a),g(x0 )]) a

 L 1 ([g(x0 ),g(b)])

⎜ ⎜ ⎝

⎛ b x0

1  (ν)

⎜ ⎜ ⎝

⎞ r j=1 j =i

⎞ r j=1 j=i

⎤

 ⎟  ⎥  f j (x) ⎟ (g (x0 ) − g (x))ν−1 d x ⎥ γ⎠ ⎦

(9.68) ⎤⎤

  ⎟ ⎥⎥  f j (x) ⎟ (g (x) − g (x0 ))ν−1 d x ⎥⎥ . γ⎠ ⎦⎦

212

9

p-Schatten Norm Generalized Canavati Fractional …



Proof As similar to Theorem 9.13 is omitted. An L p estimate follows. Theorem 9.17 All as in Theorem 9.15. Let now p, q > 1 :

   γ

i=1

⎡   ⎢   −1   ν +⎢  ⎣ Dg(x0 ) f i ◦ g

q,[g(a),g(x0 )]

   γ

q,[g(x0 ),g(b)]

1 q

= 1. Then

1

( p (ν − 1) + 1) p  (ν)

⎛  ⎜ ⎜ ⎝

⎛  ⎜ ⎜ ⎝

+

1

K ( f 1 , ..., fr ) (x0 )γ ≤ ⎡⎡   r  ⎢⎢   ν −1  ⎢⎢  ⎣⎣ Dg(x0 )− f i ◦ g

1 p

⎛ x0

⎜ ⎜ ⎝

ν− q1

(g (x0 ) − g (x))

a

⎞ r j=1 j =i

⎛ b

ν− q1

(g (x) − g (x0 ))

x0

⎜ ⎜ ⎝

⎞ r j=1 j =i

⎞⎤

 ⎟ ⎟⎥   f j (x) ⎟ d x ⎟⎥ γ⎠ ⎠⎦ ⎞⎤⎤(9.69)

 ⎟ ⎟⎥⎥   f j (x) ⎟ d x ⎟⎥⎥ . γ⎠ ⎠⎦⎦



Proof As similar to Theorem 9.14 is omitted.

When r = 2 we derive the following p-Schatten norm operator related Ostrowski type Canavati fractional inequalities. Theorem 9.18 Let p, q > 1 : 1p + q1 = 1, and let the ∗-ideals B p (H ), Bq (H ), for     which B p (H ) , · p , Bq (H ) , ·q are Banach algebras; x0 ∈ [a, b] ⊂     R, α ≥ 1, n = [α]; A1 ∈ C n [a, b] , B p (H ) , A2 ∈ C n [a, b] , Bq (H ) ; g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]), with (k)  Ai ◦ g −1 (g (x0 ))  = 0, k = 1, ..., n −1; iα = 1,  2. Assume further that α , g , B ∩ C A1 ◦ g −1 ∈ C g(x [g [g (a) , g (b)] , B p (H ) , (a) (b)] (H ) p 0 )−  g(x0 ) α   α −1 and A2 ◦ g ∈ C g(x0 )− [g (a) , g (b)] , Bq (H ) ∩ C g(x0 ) [g (a) , g (b)] , Bq (H ) . Then (1) it holds   (A1 , A2 ) (x0 ) :=

b

 A2 (x) A1 (x) d x +

a



A2 (x) d x

 A1 (x0 ) −

'

'

x0



g(x0 ) g(x)

a

 A2 (x)

x0

A2 (x0 ) =

a

A2 (x)

b



b

A1 (x) d x

a

1  (α)

A1 (x) A2 (x) d x− a



b

b

g(x) g(x0 )

+  α   −1 A dz dx + ◦ g (z) (z − g (x))α−1 Dg(x 1 0 )−

(g (x) − z)

α−1



α Dg(x 0)



A1 ◦ g

−1





+

(z) dz d x +

(9.70)

9.5 Main Results

'

x0

213

 A1 (x)

(z − g (x))



α−1

g(x)

a

'

g(x0 )



b

A1 (x)

g(x0 )

x0

(2) for γ, δ > 1 :

g(x)

1 γ

+

α Dg(x 0 )−



A2 ◦ g

−1





+

(z) dz d x +

+,  α   −1 A dz dx , ◦ g (z) (g (x) − z)α−1 Dg(x 2 0) = 1, we have that

1 δ

1

 (A1 , A2 ) (x0 )1 ≤ '     α −1   A ◦ g  Dg(x  1 0 )− p

1

 (α) (γ (α − 1) + 1) γ



δ,[g(a),g(x0 )]

x0

α− 1δ

A2 (x)q (g (x0 ) − g (x))

+ dx +

a

'      α A1 ◦ g −1  p   Dg(x 0)



'     α −1   A ◦ g  Dg(x  2 0 )− q



'     α −1   A ◦ g   Dg(x 2 ) 0 q



+ 1 A2 (x)q (g (x) − g (x0 ))α− δ d x +

b

δ,[g(x0 ),g(b)]

x0

δ,[g(a),g(x0 )]

x0

A1 (x) p (g (x0 ) − g (x))

α− 1δ

(9.71) + dx +

a

δ,[g(x0 ),g(b)]

b

A1 (x) p (g (x) − g (x0 ))

α− 1δ

+, ,

dx

x0

(3) we also obtain  (A1 , A2 ) (x0 )1 ≤ '     α −1   A ◦ g  Dg(x  1 0 )− p '     α −1   A ◦ g  Dg(x  1 ) 0 p

L 1 ([g(a),g(x0 )])

x0

L 1 ([g(x0 ),g(b)])

b

 A2 (x)q (g (x) − g (x0 ))

α−1

+ dx +

x0

 L 1 ([g(a),g(x0 )])

 L 1 ([g(x0 ),g(b)])

+ A2 (x)q (g (x0 ) − g (x))α−1 d x +

a



'     α −1   A ◦ g  Dg(x  2 0 )− q '     α −1   A ◦ g   Dg(x 2 0) q



1  (α)

x0

a b

+ A1 (x) p (g (x0 ) − g (x))α−1 d x +

(9.72) +, α−1  A1 (x) p (g (x) − g (x0 )) dx ,

x0

and (4)  ( A1 , A2 ) (x0 )1 ≤

1  (α + 1)

214

9

p-Schatten Norm Generalized Canavati Fractional …

'     α −1   ◦ g A   Dg(x 1 0 )− p



∞,[g(a),g(x0 )]

x0

+  A2 (x)q (g (x0 ) − g (x))α d x +

a

'     α −1   A ◦ g   Dg(x 1 ) 0 p



'     α −1   A ◦ g  Dg(x  2 0 )− q



'     α −1   ◦ g A  Dg(x  2 0) q



∞,[g(x0 ),g(b)]

b

+

α

A2 (x)q (g (x) − g (x0 )) d x +

x0

∞,[g(a),g(x0 )]

x0

a

∞,[g(x0 ),g(b)]

b

+ A1 (x) p (g (x0 ) − g (x))α d x +

(9.73) +, A1 (x) p (g (x) − g (x0 ))α d x .

x0

Proof Here we have that (acting as in the proof of Theorem 9.12 for r = 2) 

b

 (A1 , A2 ) (x0 ) :=



a





'

x0

a



g(x)



b

A2 (x) x0

 A1 (x)

g(x0 ) g(x)

a

'

g(x) g(x0 )

x0

'

g(x0 )

A2 (x)

a

'



b

(9.55)

A2 (x0 ) =

A1 (x) d x

a

1  (α)



b

A1 (x0 ) −

A2 (x) d x

A1 (x) A2 (x) d x− a



b

b

A2 (x) A1 (x) d x +

A1 (x)

g(x) g(x0 )

x0

+  α   −1 A dz dx + ◦ g (z) (z − g (x))α−1 Dg(x 1 0 )−

(g (x) − z)

α−1



α Dg(x 0)



A1 ◦ g

−1





+

(z) dz d x +

+  α   −1 A dz dx + ◦ g (z) (z − g (x))α−1 Dg(x 2 0 )−

+,  α   −1 A dz dx . ◦ g (z) (g (x) − z)α−1 Dg(x 2 0)

(9.74)

Therefore it holds by taking the 1-Schatten norm that    (A1 , A2 ) (x0 )1 =  

b

 A2 (x) A1 (x) d x +

a





b

A2 (x) d x a

1  (α)

b

A1 (x) A2 (x) d x− a

 A1 (x0 ) −



b

A1 (x) d x a

  A2 (x0 )  ≤ 1

*   ) -  x0     g(x0 )   α−1 α −1 Dg(x0 )− A1 ◦ g A2 (x) (z) dz d x  + (z − g (x))   a  g(x) 1

9.5 Main Results

'    '   

215



b

A2 (x) 

 A1 (x)



A1 ◦ g

−1



 +  (z) dz d x   +

1

g(x) g(x0 )

x0

α Dg(x 0)

 +   α   −1 A dz dx ◦ g (z) (z − g (x))α−1 Dg(x 2  + 0 )−

g(x) b



α−1

1

g(x0 )

A1 (x)

a

'   

(g (x) − z)

g(x0 )

x0

x0

g(x)

 +,   α   −1 ≤ A dz dx ◦ g (z) (g (x) − z)α−1 Dg(x 2  0) 1

* (9.75)  ) -      x0  g(x0 ) 1   α−1 D α −1 A A dz ◦ g d x + − g (z) (x) (z (x))   2 1 g(x0 )−   (α) g(x) a  1

'

b x0

'

x0

a

    A2 (x) 

    A1 (x) 

g(x) g(x0 ) g(x0 )

g(x)

    A1 (x) 

(g (x) − z)



α−1

α Dg(x 0)



A1 ◦ g

−1



 +   (z) dz  d x + 1

 +   α   −1  A dz ◦ g d x + (z) (z − g (x))α−1 Dg(x 2  0 )− 1

 +,   α   −1  dx A dz ◦ g ≤ (z) (g (x) − z)α−1 Dg(x 2  0) x0 g(x0 ) 1 (9.76) (by using the p-Schatten norm and Hölder’s type inequality (9.44) for p, q > 1 : 1 + q1 = 1) p '

b

g(x)

⎧⎡ ⎤ ) *     g(x0 )   1 ⎨⎣ x0   α−1 α −1 A2 (x)q  Dg(x0 )− A1 ◦ g (z) dz  d x ⎦ + (z − g (x))   g(x)  (α) ⎩ a p



b x0



   A1 (x) p  

x0

a



b x0

1  (α)

   A2 (x)q  

   A1 (x) p  

- a

x0

g(x)

(g (x) − z)

g(x0 )

α−1

(z − g (x))

α−1

g(x)



A1 ◦ g

−1





  (z) dz   dx +



α Dg(x 0 )−



A2 ◦ g

−1





   (z) dz  d x + q

g(x) g(x0 )

A2 (x)q

α Dg(x 0)

p

g(x0 )

)



.

     α A2 ◦ g −1 (z) dz  ≤ (g (x) − z)α−1 Dg(x  dx 0) q

g(x0 ) g(x)

* (9.77)       α −1 (z) dz d x +  Dg(x0 )− A1 ◦ g

α−1 

(z − g (x))

p

216

9

'

b

 A2 (x)q 

x0

(g (x) − z)

g(x0 )

x0

'

g(x)

g(x0 )

A1 (x) p

g(x)

a

'



b

A1 (x) p

g(x) g(x0 )

x0

p-Schatten Norm Generalized Canavati Fractional …

 α  D

α−1



g(x0 )

A1 ◦ g

−1



+   (z) p dz d x +

+  α    −1  A ◦ g dz dx + (z) (z − g (x))α−1  Dg(x 2 0 )− q

+,  α    −1  dz d x . A ◦ g (z) (g (x) − z)α−1  Dg(x 2 0) q (9.78)

We have proved, so far, that  (A1 , A2 ) (x0 )1 ≤ -

1  (α)

a

'

b

x0

) A2 (x)q

 A2 (x)q

x0

 A1 (x) p

*       α −1 (z) dz d x +  Dg(x0 )− A1 ◦ g

α−1 

(z − g (x))

p

g(x)

g(x) g(x0 )

x0

'

g(x0 )

g(x0 )

(g (x) − z)

α−1

 α  D

α−1

 α  D

(z − g (x))

g(x0 )

g(x)

a



b x0

) A1 (x) p

g(x) g(x0 )



g(x0 )−

A1 ◦ g 

−1

A2 ◦ g



−1

+   (z) p dz d x +



+   (z) q dz d x +

* .       α −1 A ◦ g dz dx =: (λ) . (z) (g (x) − z)α−1  Dg(x  2 0) q

Let now γ, δ > 1 such that γ1 + in (9.79). Then we have that

1 δ

(9.79) = 1, and we apply the usual Hölder’s inequality

 (A1 , A2 ) (x0 )1 ≤ (λ) ≤

1 1

 (α) (γ (α − 1) + 1) γ

⎧⎡ ⎤ ) *1 δ ⎨  x0   δ g(x 0 )  γ(α−1)+1   α ⎣ A2 (x)q (g (x0 ) − g (x)) γ dx⎦ +  Dg(x0 )− A1 ◦ g −1 (z) dz ⎩ a p g(x) ⎡ ⎣



b x0

⎡ ⎣

 a

x0

A2 (x)q (g (x) − g (x0 ))

A1 (x) p (g (x0 ) − g (x))

γ(α−1)+1 γ

γ(α−1)+1 γ

)

g(x)  g(x 0 )

)

  δ  α   Dg(x0 ) A1 ◦ g −1 (z) dz p

g(x 0 )  g(x)

  δ  α   Dg(x0 )− A2 ◦ g −1 (z) dz q

⎤

*1 δ

dx⎦ +

*1 δ

⎤ dx⎦ +

(9.80)

9.5 Main Results ⎡ ⎣



b x0

217

A1 (x) p (g (x) − g (x0 ))

≤

γ(α−1)+1 γ

)

g(x) 

  δ  α   Dg(x0 ) A2 ◦ g −1 (z) dz

1 1

 (α) (γ (α − 1) + 1) γ 

δ,[g(a),g(x0 )]

x0



A2 (x)q (g (x0 ) − g (x))

'     α −1   A ◦ g  Dg(x  2 0 )− q



'     α −1   A ◦ g  Dg(x  2 ) 0 q



x0

δ,[g(a),g(x0 )]

x0

A1 (x) p (g (x0 ) − g (x))

α− 1δ

(9.81) + dx +

a

δ,[g(x0 ),g(b)]

b

A1 (x) p (g (x) − g (x0 ))

α− 1δ

+, dx

,

x0

proving (9.71). We also obtain  (A1 , A2 ) (x0 )1 ≤ (λ) ≤

'      α A2 ◦ g −1 q   Dg(x 0)

dx +

+ 1 A2 (x)q (g (x) − g (x0 ))α− δ d x +

b

δ,[g(x0 ),g(b)]

 L 1 ([g(a),g(x0 )])

x0

1  (α)

A2 (x)q (g (x0 ) − g (x))

α−1

+ dx +

a

 L 1 ([g(x0 ),g(b)])

'      α A2 ◦ g −1 q   Dg(x 0 )−

+

α− 1δ

a

'      α A1 ◦ g −1  p   Dg(x 0)

'     α −1   A ◦ g  Dg(x  1 0) p

δ

q

g(x 0 )

'     α −1   A ◦ g  Dg(x  1 0 )− p

'     α −1   A ◦ g   Dg(x 1 )− 0 p

⎤⎫ ⎬ dx⎦ ⎭

*1

+  A2 (x)q (g (x) − g (x0 ))α−1 d x +

b x0

 L 1 ([g(a),g(x0 )])

 L 1 ([g(x0 ),g(b)])

x0

A1 (x) p (g (x0 ) − g (x))

α−1

a b

 A1 (x) p (g (x) − g (x0 ))

α−1

+ dx +

(9.82) +, dx ,

x0

proving (9.72). At last we derive  (A1 , A2 ) (x0 )1 ≤ (λ) ≤ '      α A1 ◦ g −1  p   Dg(x 0 )−

∞,[g(a),g(x0 )]

 a

x0

1  (α + 1)

+  A2 (x)q (g (x0 ) − g (x))α d x +

218

9

p-Schatten Norm Generalized Canavati Fractional …

'     α −1   A ◦ g  Dg(x  1 ) 0 p



'     α −1   A ◦ g  Dg(x  2 0 )− q



'      α A2 ◦ g −1 q   Dg(x 0)



∞,[g(x0 ),g(b)]

+

α

A2 (x)q (g (x) − g (x0 )) d x +

x0

∞,[g(a),g(x0 )]

∞,[g(x0 ),g(b)]

b

x0

α

+

A1 (x) p (g (x0 ) − g (x)) d x +

a b

A1 (x) p (g (x) − g (x0 ))α d x

x0

+, , (9.83)

proving (9.73). The theorem is proved.



Next we present p-Schatten left and right generalized Canavati fractional Opial type inequalities: Theorem 9.19 Let the ∗-ideal B2 (H ), which (B2 (H ) , ·2 ) is a Banach algebra; x0 ∈ [a, b] ⊂ R, ν ≥ 1, n = [ν]; f ∈ C n ([a, b] , B2 (H )), g ∈ C 1 ([a, b]), strictly (k)  increasing such that g −1 ∈ C n ([g (a) , g (b)]), with f ◦ g −1 (g (x0 )) = 0, k = ν , g , B2 (H )) . 0, 1, ..., n − 1. Assume further that f ◦ g −1 ∈ C g(x ([g (a) (b)] ) 0 Let also p, q > 1 : 1p + q1 = 1. Then 

z g(x0 )

2

− q1

       −1  f ◦ g −1 (w) D ν (w) 1 dw ≤ g(x0 ) f ◦ g  z

ν+ 1p − q1

(z − g (x0 ))

1

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) + 2)] p

g(x0 )

(9.84)

2 q    q   ν −1 D f ◦ g dw , (w)   g(x0 ) 2

for all g (x0 ) ≤ z ≤ g (b) . Proof Very similar to the proof of Theorem 13 of [6]. Use of (9.44) for p = q = 2.  A similar result comex next:   Theorem 9.20 Let γ ≥ 1, the ∗-ideal Bγ (H ), which Bγ (H ) , ·γ is a  Banach algebra; x0 ∈ [a, b] ⊂ R, ν ≥ 1, n = [ν]; f ∈ C n [a, b] , Bγ (H ) , g ∈ C 1 ([a, b]), strictly increasing such that g −1 ∈ C n ([g (a) , g (b)]), with (k)  f ◦ g−1 (g (x0 )) = 0, k = 0, 1, ..., n − 1. Assume further that f ◦ g −1 ∈ ν C g(x0 ) [g (a) , g (b)] , Bγ (H ) . Let also p, q > 1 : 1p + q1 = 1. Then 

z g(x0 )

       −1  f ◦ g −1 (w) D ν (w) γ dw ≤ g(x0 ) f ◦ g

(9.85)

9.5 Main Results

2

− q1

219

(z − g (x0 ))

 z

ν+ 1p − q1 1

g(x0 )

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) + 2)] p

2  q   q  ν  ,  Dg(x0 ) f ◦ g −1 (w) dw γ

for all g (x0 ) ≤ z ≤ g (b) . Proof Very similar to the proof of Theorem 13 of [6]. Use of (9.41) for p = γ.  It follows the corresponding right side fractional Opial type inequalities: Theorem 9.21 All as in Theorem 9.19, however now it is ν C g(x ([g (a) , g (b)] , B2 (H )). Then 0 )− 

g(x0 ) z

2

− q1

f ◦ g −1 ∈

       −1  f ◦ g −1 (w) D ν (w) 1 dw ≤ g(x0 )− f ◦ g

(g (x0 ) − z)

ν+ 1p − q1 1

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) + 2)] p

*2 ) q   q g(x0 )    ν −1 , (t) dt  Dg(x0 )− f ◦ g 2

z

(9.86)

for all g (a) ≤ z ≤ g (x0 ) . Proof Based on (9.20), and as similar to the proof of Theorem 9.19 is omitted.  Next comes another right ride fractional Opial type inequality: Theorem 9.20, however now it is  9.22 All as in Theorem  ν , g , B C g(x . Then [g (a) (b)] (H ) γ )− 0 

g(x0 ) z

2

− q1

f ◦ g −1 ∈

       −1  f ◦ g −1 (w) D ν (w) γ dw ≤ g(x0 )− f ◦ g

(g (x0 ) − z)

)

ν+ 1p − q1

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) +

1 2)] p

  q   ν  Dg(x0 )− f ◦ g −1 (t) dt γ

g(x0 )  z

*2 q

,

(9.87)

for all g (a) ≤ z ≤ g (x0 ) . Proof Based on (9.20), and as similar to the proof of Theorem 9.19 is omitted.  It follows the modified generalized left B2 (H )-valued fractional Opial inequality: Theorem 9.23 All as in Theorem 9.6, where X = B2 (H ) and let p, q > 1 : 1 1 = 1. Here we assume that (m+1)q < ν < 1. Then q 

z g(x0 )

     (m+1)ν     Dg(x0 ) f ◦ g −1 (w)  dw ≤  f ◦ g −1 (w) 1

1 p

+

(9.88)

220

9

p-Schatten Norm Generalized Canavati Fractional …

2− q (z − g (x0 ))(m+1)ν+ p − q 1

1

1 1

 ((m + 1) ν) [( p ((m + 1) ν − 1) + 1) ( p ((m + 1) ν − 1) + 2)] p 

z g(x0 )

q2    q  (m+1)ν  −1 f ◦g , (t) dt  Dg(x0 ) 2

for all g (x0 ) ≤ z ≤ g (b) . 

Proof As in Theorem 9.19.

Next comes another modified generalized left Bγ (H )-valued fractional Opial inequality: Theorem 9.24 All as in Theorem 9.6, where X = Bγ (H ) and let p, q > 1 : 1 1 = 1. Here we assume that (m+1)q < ν < 1. Then q 

z g(x0 )

     (m+1)ν     Dg(x0 ) f ◦ g −1 (w)  dw ≤  f ◦ g −1 (w)

1 p

+

(9.89)

γ

2− q (z − g (x0 ))(m+1)ν+ p − q 1

1

1 1

 ((m + 1) ν) [( p ((m + 1) ν − 1) + 1) ( p ((m + 1) ν − 1) + 2)] p 

z g(x0 )

q2    q  (m+1)ν  −1 D f ◦ g dt , (t)   g(x0 ) γ

for all g (x0 ) ≤ z ≤ g (b) . 

Proof As in Theorem 9.19.

The corresponding modified generalized right B2 (H )-valued fractional Opial inequality comes next: Theorem 9.25 All as in Theorem 9.7, where X = B2 (H ) and let p, q > 1 : 1 1 = 1. Here we assume that (m+1)q < ν < 1. Then q 

g(x0 ) z

     (m+1)ν     Dg(x0 )− f ◦ g −1 (w)  dw ≤  f ◦ g −1 (w) 2− q (g (x0 ) − z)(m+1)ν+ p − q 1

1 1

 ((m + 1) ν) [( p ((m + 1) ν − 1) + 1) ( p ((m + 1) ν − 1) + 2)] p 

g(x0 ) z

q2     (m+1)ν  q −1 , (t) dt  Dg(x0 )− f ◦ g 2

+

(9.90)

1

1

1 p

9.5 Main Results

221

for all g (a) ≤ z ≤ g (x0 ) . 

Proof As in Theorem 9.19.

The corresponding modified generalized right Bγ (H )-valued fractional Opial inequality comes next: Theorem 9.26 All as in Theorem 9.7, where X = Bγ (H ) and let p, q > 1 : 1 1 = 1. Here we assume that (m+1)q < ν < 1. Then q 

g(x0 ) z

     (m+1)ν     Dg(x0 )− f ◦ g −1 (w)  dw ≤  f ◦ g −1 (w)

1 p

+

(9.91)

γ

2− q (g (x0 ) − z)(m+1)ν+ p − q 1

1

1 1

 ((m + 1) ν) [( p ((m + 1) ν − 1) + 1) ( p ((m + 1) ν − 1) + 2)] p 

g(x0 )

q2    q  (m+1)ν  −1 , (t) dt  Dg(x0 )− f ◦ g γ

z

for all g (a) ≤ z ≤ g (x0 ) . 

Proof As in Theorem 9.19. We make Remark 9.27 (to Theorem 9.12) Case of inequality (9.46): Call and assume M1 ( f 1 , ..., fr ) := max

i=1,...,r

(9.92)

     ν −1   f ◦ g sup  Dg(x  i 0 )− 2

∞,[g(a),g(x0 )]

x0 ∈[a,b]

.

      γ  sup Dg(x0 ) f i ◦ g −1  

2 ∞,[g(x0 ),g(b)]

x0 ∈[a,b]

,

< +∞.

Then K ( f 1 , ..., fr ) (x0 )1 ≤ Right hand side (9.46) ≤ ⎛ M1 ( f 1 , ..., fr ) (g (b) − g (a))  (ν + 1)

ν

r  i=1

⎜ ⎜ ⎝

 a

⎛ b

⎜ ⎜ ⎝

⎞ r

j=1 j=i

⎞

  ⎟ ⎟  f j (x) ⎟ d x ⎟ . 2⎠ ⎠

(9.93)

222

9

p-Schatten Norm Generalized Canavati Fractional …

We make Remark 9.28 (to Theorem 9.13) Case of inequality (9.62): Call and assume M2 ( f 1 , ..., fr ) := max

i=1,...,r

(9.94)

     ν −1   f ◦ g sup  Dg(x  i 0 )− 2

x0 ∈[a,b]

    ν  −1   f ◦ g sup  Dg(x  i ) 0 2

x0 ∈[a,b]

,

L 1 ([g(a),g(x0 )])

. L 1 ([g(x0 ),g(b)])

< +∞.

Then K ( f 1 , ..., fr ) (x0 )1 ≤ Right hand side (9.62) ≤ ⎛ M2 ( f 1 , ..., fr ) (g (b) − g (a))  (ν)

ν−1

r  i=1

⎜ ⎜ ⎝

⎛



b

a

⎜ ⎜ ⎝

⎞

⎞

  ⎟ ⎟  f j (x) ⎟ d x ⎟ . 2⎠ ⎠

r

j=1 j=i

(9.95)

We make Remark 9.29 (to Theorem 9.14) Case of inequality (9.64): Call and assume ( p, q > 1 : 1p +

1 q

= 1):

M3 ( f 1 , ..., fr ) := max

i=1,...,r

(9.96)

     ν −1   f ◦ g sup  Dg(x  i 0 )− 2

q,([g(a),g(x0 )])

x0 ∈[a,b]

.

    ν  −1   f ◦ g sup  Dg(x  i 0) 2

q,([g(x0 ),g(b)])

x0 ∈[a,b]

,

< +∞.

Then K ( f 1 , ..., fr ) (x0 )1 ≤ Right hand side (9.64) ≤ M3 ( f 1 , ..., fr ) (g (b) − g (a)) 1 p

( p (ν − 1) + 1)  (ν) We make

ν− q1

r  i=1

⎛  ⎜ ⎜ ⎝

a

⎛ b

⎜ ⎜ ⎝

⎞ r

j=1 j=i

⎞

  ⎟ ⎟  f j (x) ⎟ d x ⎟ . 2⎠ ⎠

(9.97)

9.5 Main Results

223

Remark 9.30 (to Theorem 9.15) (γ ≥ 1) Case of inequality (9.67): Call and assume γ M1 ( f 1 , ..., fr ) := max

i=1,...,r

(9.98)

     ν  f i ◦ g −1 γ  sup  Dg(x 0 )−

∞,[g(a),g(x0 )]

x0 ∈[a,b]

.

    ν  −1   f ◦ g sup  Dg(x  i 0) γ

< +∞.

∞,[g(x0 ),g(b)]

x0 ∈[a,b]

,

Then K ( f 1 , ..., fr ) (x0 )γ ≤ Right hand side (9.67) ≤ ⎛

⎛

⎞

⎞

 b r r γ  ⎟ ⎟ ⎜  M1 ( f 1 , ..., fr ) (g (b) − g (a))ν  ⎜ ⎜ ⎜  f j (x) ⎟ d x ⎟ . γ⎠ ⎝ ⎝ ⎠  (ν + 1) a i=1 j=1

(9.99)

j=i

We make Remark 9.31 (to Theorem 9.16) (γ ≥ 1) Case of inequality (9.68): Call and assume: γ M2 ( f 1 , ..., fr ) := max

i=1,...,r

(9.100)

     ν −1   f ◦ g sup  Dg(x  i )− 0 γ

x0 ∈[a,b]

    ν   f i ◦ g −1 γ  sup  Dg(x 0)

x0 ∈[a,b]

L 1 ([g(a),g(x0 )])

,

. L 1 ([g(x0 ),g(b)])

< +∞.

Then K ( f 1 , ..., fr ) (x0 )γ ≤ Right hand side (9.68) ≤ γ M2

( f 1 , ..., fr ) (g (b) − g (a))  (ν)

ν−1

r  i=1

We make Remark 9.32 (to Theorem 9.17) (γ ≥ 1)

⎛  ⎜ ⎜ ⎝

a

⎛ b

⎜ ⎜ ⎝

⎞ r

j=1 j=i

⎞

  ⎟ ⎟  f j (x) ⎟ d x ⎟ . γ⎠ ⎠

(9.101)

224

9

Case of inequality (9.69): Call and assume ( p, q > 1 :

1 p

+

p-Schatten Norm Generalized Canavati Fractional …

1 q

= 1):

γ

M3 ( f 1 , ..., fr ) := max

i=1,...,r

(9.102)

     ν −1   f ◦ g sup  Dg(x  i 0 )− γ

q,([g(a),g(x0 )])

x0 ∈[a,b]

.

    ν  −1   f ◦ g sup  Dg(x  i ) 0 γ

< +∞.

q,([g(x0 ),g(b)])

x0 ∈[a,b]

,

Then K ( f 1 , ..., fr ) (x0 )γ ≤ Right hand side (9.69) ≤ ⎛ γ M3

( f 1 , ..., fr ) (g (b) − g (a))

ν− q1

1 p

( p (ν − 1) + 1)  (ν)

r  i=1

⎜ ⎜ ⎝

⎛



b

a

⎜ ⎜ ⎝

⎞ r

j=1 j=i

⎞

 ⎟ ⎟   f j (x) ⎟ d x ⎟ . γ⎠ ⎠

(9.103)

Remark 9.33 (to Theorem 9.18) (i) for γ, δ > 1 : γ1 + 1δ = 1, case of inequality (9.71): Call and assume N1 (A1 , A2 ) := max

     ν −1   A ◦ g sup  Dg(x  1 )− 0 p

δ,[g(a),g(x0 )]

x0 ∈[a,b]

    ν  −1   sup  Dg(x A ◦ g  1 0) p

δ,[g(x0 ),g(b)]

x0 ∈[a,b]

,

     ν −1   sup  Dg(x A ◦ g  2 )− 0 q

δ,[g(a),g(x0 )]

x0 ∈[a,b]

    ν  −1   A ◦ g sup  Dg(x  2 0) q

,

.

δ,[g(x0 ),g(b)]

x0 ∈[a,b]

,

< +∞.

(9.104)

Then  ( A1 , A2 ) (x0 )1 ≤ right hand side (9.71) ≤ N1 (A1 , A2 ) (g (b) − g (a))α− δ

1

'

1

 (α) (γ (α − 1) + 1) γ

a

b

  A1 (x) p d x +

b

+ A2 (x)q d x .

a

(9.105)

9.5 Main Results

225

(ii) case of inequality (9.72): Call and assume N2 (A1 , A2 ) := max

(9.106)

     ν −1   A ◦ g sup  Dg(x  1 0 )− p

x0 ∈[a,b]

    ν   sup  Dg(x A1 ◦ g −1  p  0)

x0 ∈[a,b]

x0 ∈[a,b]

x0 ∈[a,b]

,

,

L 1 ([g(x0 ),g(b)])

     ν −1   sup  Dg(x A ◦ g  2 0 )− q

    ν   A2 ◦ g −1 q  sup  Dg(x 0)

L 1 ([g(a),g(x0 )])

,

L 1 ([g(a),g(x0 )])

. < +∞.

L 1 ([g(x0 ),g(b)])

Then  ( A1 , A2 ) (x0 )1 ≤ right hand side (9.72) ≤ N2 (A1 , A2 ) (g (b) − g (a))α−1  (α)

'

b



b

A1 (x) p d x +

a

+ A2 (x)q d x .

a

(9.107)

(iii) case of inequality (9.73): Call and assume N3 (A1 , A2 ) := max

(9.108)

     ν  A1 ◦ g −1  p  sup  Dg(x 0 )−

∞,[g(a),g(x0 )]

x0 ∈[a,b]

    ν  −1   sup  Dg(x A ◦ g  1 ) 0 p

∞,[g(x0 ),g(b)]

x0 ∈[a,b]

     ν  sup  Dg(x A2 ◦ g −1 q  0 )−

,

∞,[g(a),g(x0 )]

x0 ∈[a,b]

,

.

    ν  −1   A ◦ g sup  Dg(x  2 0) q

∞,[g(x0 ),g(b)]

x0 ∈[a,b]

,

< +∞.

Then  ( A1 , A2 ) (x0 )1 ≤ right hand side (9.73) ≤ N3 (A1 , A2 ) (g (b) − g (a))α  (α + 1)

' a

b

 A1 (x) p d x + a

b

+ A2 (x)q d x . (9.109)

226

9

p-Schatten Norm Generalized Canavati Fractional …

We need Remark 9.34 (i) This is regarding Theorems 9.12-9.17. Here K ( f 1 , ..., fr ) (x0 ), x0 ∈ [a, b], is as in (9.45). Next we denote and have (case of 1 ≤ ν < 2): 

b

 ( f 1 , ..., fr ) :=

K ( f 1 , ..., fr ) (x0 ) d x0 =

a

⎡

 r  ⎢ ⎢(b − a) ⎣

a

i=1

⎛ b

⎜ ⎜ ⎝

⎞ r j=1 j =i

⎛

⎟ ⎜ ⎜ f j (x)⎟ ⎠ f i (x) d x − ⎝

⎛



b

a

⎜ ⎜ ⎝

⎞ r j=1 j =i

⎞

⎟ ⎟ ⎟ f j (x)⎟ ⎠ dx⎠



b

a



⎤

⎥ f i (x) d x ⎥ ⎦,

(9.110) (ii) This is regarding Theorem 9.18. Here  (A1 , A2 ) (x0 ), x0 ∈ [a, b], is as in (9.70). Next we denote and have (case of 1 ≤ α < 2): 

b

 ( A1 , A2 ) :=

 (A1 , A2 ) (x0 ) d x0 =

a



b

(b − a)

 A2 (x) A1 (x) d x +

a



A2 (x) d x a

A1 (x) A2 (x) d x −

(9.111)

a



b

b

b

 A1 (x) d x −

a

a



b

A1 (x) d x

b

A2 (x) d x .

a

(iii) for γ ≥ 1, it holds   ( f 1 , ..., fr )γ ≤ and

b

a

  ( A1 , A2 )1 ≤

b

K ( f 1 , ..., fr ) (x)γ d x,

(9.112)

 (A1 , A2 ) (x)1 d x.

(9.113)

a

We give the following set of γ-Schatten norm generalized Canavati type fractional Grüss type inequalities involving several functions over Bγ (H ), γ ≥ 1. Theorem 9.35 All as in Theorem 9.12, with 1 ≤ ν < 2 (i.e. n = 1). Then (i) M1 ( f 1 , ..., fr ) (g (b) − g (a))ν (b − a)2  ( f 1 , ..., fr )1 ≤  (ν + 1)

9.5 Main Results

227

 ⎛    r    ⎜  ⎜  f j (x)  2 ⎝  j=1  i=1  j=i 

r 

⎞ ⎟ ⎟, ⎠

(9.114)

∞,[a,b]

where M1 ( f 1 , ..., fr ) is as in (9.92), (ii) M2 ( f 1 , ..., fr ) (g (b) − g (a))ν−1 (b − a)2  ( f 1 , ..., fr )1 ≤  (ν)  ⎛ ⎞     r  ⎜ r  ⎟   ⎜  f j (x)  ⎟, 2 ⎝ ⎠   i=1  j=1  j=i

(9.115)

∞,[a,b]

where M2 ( f 1 , ..., fr ) is as in (9.94), (iii) when p, q > 1 : 1p + q1 = 1, we have M3 ( f 1 , ..., fr ) (g (b) − g (a))ν− q (b − a)2 1

 ( f 1 , ..., fr )1 ≤

1

( p (ν − 1) + 1) p  (ν)  ⎛ ⎞     r r  ⎜    ⎟ ⎜  f j (x)  ⎟, 2 ⎝ ⎠   i=1  j=1  j=i

(9.116)

∞,[a,b]

where M3 ( f 1 , ..., fr ) is as in (9.96). Proof By Remarks 9.34, 9.27-9.29 and that       r b⎜     ⎟   ⎜  f j (x) ⎟ d x ≤ (b − a)   f j (x)   2⎠ 2 ⎝ a  j=1  j=1   j=i j=i



⎛

⎞

r

.

∞,[a,b]

 We continue with Theorem 9.36 All as in Theorem 9.15, with 1 ≤ ν < 2 (i.e. n = 1), γ ≥ 1. Then (i) γ M ( f 1 , ..., fr ) (g (b) − g (a))ν (b − a)2  ( f 1 , ..., fr )γ ≤ 1  (ν + 1)

228

9

p-Schatten Norm Generalized Canavati Fractional …

 ⎛    r    ⎜  ⎜  f j (x)  γ ⎝  j=1  i=1  j=i 

r 

⎞ ⎟ ⎟, ⎠

(9.117)

∞,[a,b]

γ

where M1 ( f 1 , ..., fr ) is as in (9.98), (ii) γ M ( f 1 , ..., fr ) (g (b) − g (a))ν−1 (b − a)2  ( f 1 , ..., fr )γ ≤ 2  (ν)  ⎛ ⎞     r    ⎜ r  ⎟ ⎜  f j (x)  ⎟, γ ⎝ ⎠   i=1  j=1  j=i

(9.118)

∞,[a,b]

γ

where M2 ( f 1 , ..., fr ) is as in (9.100), (iii) when p, q > 1 : 1p + q1 = 1, we have  ( f 1 , ..., fr )γ ≤

γ

M3 ( f 1 , ..., fr ) (g (b) − g (a))ν− q (b − a)2 1

1

( p (ν − 1) + 1) p  (ν)  ⎛ ⎞     r r  ⎜    ⎟ ⎜  f j (x)  ⎟, γ ⎝ ⎠   i=1  j=1  j=i

(9.119)

∞,[a,b]

γ

where M3 ( f 1 , ..., fr ) is as in (9.102). Proof By Remarks 9.34, 9.30-9.32 and that  a

     r     ⎟   ⎜  ⎜  f j (x) ⎟ d x ≤ (b − a)   f j (x)   γ⎠ γ ⎝  j=1  j=1   j=i j=i ⎛

b

⎞

r

.

∞,[a,b]

 Furthermore we have (r = 2 case of p-Schatten norm Grüss inequalities) Theorem 9.37 All as in Theorem 9.18, with 1 ≤ α < 2 (i.e. [α] = 1). Then (i) for γ, δ > 1 : γ1 + 1δ = 1, we have N1 (A1 , A2 ) (g (b) − g (a))α− δ (b − a) 1

 (A1 , A2 )1 ≤

1

 (α) (γ (α − 1) + 1) γ

9.6 Applications

229

'

b

 A1 (x) p d x +

a

b

+ A2 (x)q d x ,

(9.120)

a

where N1 (A1 , A2 ) is as in (9.104), (ii) N2 (A1 , A2 ) (g (b) − g (a))α−1 (b − a)  (A1 , A2 )1 ≤  (α) '

b

 A1 (x) p d x +

a

b

+ A2 (x)q d x ,

(9.121)

a

where N2 (A1 , A2 ) is as in (9.106), and (iii) N3 (A1 , A2 ) (g (b) − g (a))α (b − a)  ( A1 , A2 )1 ≤  (α + 1) '

b

 A1 (x) p d x +

a

b

+ A2 (x)q d x ,

(9.122)

a

where N3 (A1 , A2 ) is as in (9.108). 

Proof By Remarks 9.34 and 9.33.

9.6 Applications We start with applications on Ostrowski type inequalities: Corollary 9.38 (to Theorems 9.12–9.14) All as in Theorem 9.12 for g (t) = t. Then (i) 1 K ( f 1 , ..., fr ) (x0 )1 ≤  (ν + 1) ⎡⎡ r  i=1

⎛  ⎢⎢ ν ⎜   ν⎜ ⎢⎢ D f i   (x0 − a) ⎝ x0 − ⎣⎣ 2 ∞,[a,x0 ]

⎛ x0

a

⎜ ⎜ ⎝

⎡

⎛ ⎛  b ⎢ ν   ⎜ ⎜ ⎢ D f i   ⎜ (b − x0 )ν ⎜ x0 ⎣ ⎝ x ⎝ 2 ∞,[x0 ,b] 0

⎞ r

j=1 j=i

  ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ + 2⎠ ⎠⎦ ⎞

r

j=1 j=i

⎞⎤

⎞⎤⎤ (9.123)

  ⎟ ⎟⎥⎥  f j (x) ⎟ d x ⎟⎥⎥ , 2⎠ ⎠⎦⎦

230

9

p-Schatten Norm Generalized Canavati Fractional …

(ii) K ( f 1 , ..., fr ) (x0 )1 ≤ ⎡⎡ r  i=1

⎢⎢ ν   ⎢⎢ D f i   x0 − ⎣⎣ 2 L 1 ([a,x0 ])

⎡ ⎢ ν   ⎢ D f i   x0 ⎣ 2 L 1 ([x0 ,b])

(iii) when p, q > 1 :

1 p

+

b x0

1 q

⎛



x0

a

⎜ ⎜ ⎝

⎥   ⎟  f j (x) ⎟ (x0 − x)ν−1 d x ⎥ + 2⎠ ⎦

⎜ ⎜ ⎝

⎤⎤

j=1 j=i

⎥⎥   ⎟  f j (x) ⎟ (x − x0 )ν−1 d x ⎥⎥ , 2⎠ ⎦⎦

(9.124)

= 1, we have

⎡⎡

i=1

j=1 j=i

⎤

⎞ r

K ( f 1 , ..., fr ) (x0 )1 ≤

r 

⎞ r

⎛



1  (ν)

⎛

⎢⎢ ν ⎜   ⎢⎢ D f i   ⎜ x0 − ⎣⎣ 2 q,[a,x0 ] ⎝

⎡

 a

1 ⎛

x0

1 ⎜ (x0 − x)ν− q ⎜ ⎝

⎛  ⎢ ν   ⎜ b 1 ⎢ D f i   ⎜ (x − x0 )ν− q x0 ⎣ 2 q,[x0 ,b] ⎝ x 0

1

( p (ν − 1) + 1) p  (ν) ⎞ r

j=1 j=i

⎛ ⎜ ⎜ ⎝

  ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ + 2⎠ ⎠⎦ ⎞

r

j=1 j=i

⎞⎤

⎞⎤⎤

  ⎟ ⎟⎥⎥  f j (x) ⎟ d x ⎟⎥⎥ . 2⎠ ⎠⎦⎦

(9.125)

It follows: Corollary 9.39 (to Theorems 9.15–9.17) All as in Theorem 9.15 for g (t) = t, γ ≥ 1. Then (i) 1 K ( f 1 , ..., fr ) (x0 )γ ≤  (ν + 1) ⎡⎡

⎛ ⎛  r   ν ⎢⎢ ⎜ x0 ⎜   ν⎜  D fi   ⎢⎢ ⎜ − a) (x   0 − x0 ⎣⎣ ⎝ a ⎝ γ ∞,[a,x ] 0 i=1

⎞ r

j=1 j=i

⎞⎤

  ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ + γ⎠ ⎠⎦ (9.126)

9.6 Applications

231

⎡

⎛ ⎛  b ⎜ ⎜ ⎜ (b − x0 )ν ⎜ ⎝ ⎝ ∞,[x0 ,b] x

 ν   ⎢  D fi   ⎢   x0 ⎣ γ

0

⎞ r

j=1 j=i

⎞⎤⎤

  ⎟ ⎟⎥⎥  f j (x) ⎟ d x ⎟⎥⎥ , γ⎠ ⎠⎦⎦

(ii) 1  (ν)

K ( f 1 , ..., fr ) (x0 )γ ≤ ⎡⎡ r   ν ⎢⎢    D fi   ⎢⎢   − x 0 ⎣⎣ γ L ([a,x ]) 1 0

⎛



x0

a

i=1

⎜ ⎜ ⎝

 b  ν   ⎜ ⎢  D fi   ⎜ ⎢   x0 ⎝ ⎣ γ L ([x ,b]) 1 0 x 0

(iii) when p, q > 1 :

j=1 j=i

1 p

+

1 q

⎤

  ⎟ ⎥  f j (x) ⎟ (x0 − x)ν−1 d x ⎥ + γ⎠ ⎦ ⎞

r

j=1 j=i

⎤⎤

  ⎟ ⎥⎥  f j (x) ⎟ (x − x0 )ν−1 d x ⎥⎥ , γ⎠ ⎦⎦

(9.127)

= 1, we have

K ( f 1 , ..., fr ) (x0 )γ ≤ ⎡⎡

i=1

r

⎛

⎡

r 

⎞

⎛

 ν ⎢⎢    D fi   ⎢⎢   x0 − ⎣⎣ γ

⎜ ⎜ q,[a,x0 ] ⎝



1 1

( p (ν − 1) + 1) p  (ν) ⎛

x0

a

1 ⎜ (x0 − x)ν− q ⎜ ⎝

⎡

⎛   ν   ⎢ ⎜ b  ν− q1   ⎢ ⎜ ⎣ Dx0 f i γ q,[x0 ,b] ⎝ x (x − x0 ) 0

⎛ ⎜ ⎜ ⎝

⎞ r

j=1 j=i

  ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ + γ⎠ ⎠⎦ ⎞

r

j=1 j=i

⎞⎤

⎞⎤⎤

  ⎟ ⎟⎥⎥  f j (x) ⎟ d x ⎟⎥⎥ . γ⎠ ⎠⎦⎦

(9.128)

We continue with Corollary 9.40 (to Theorem 9.18) All as in Theorem 9.18, with g (t) = et . Then (i) for γ, δ > 1 : γ1 + 1δ = 1, we have  (A1 , A2 ) (x0 )1 ≤ '      Deαx0 − (A1 ◦ log) p 

δ,[ea ,e x0 ]

1 1

 (α) (γ (α − 1) + 1) γ

 a

x0

+  x0  1 x α− δ A2 (x)q e − e dx +

232

9

p-Schatten Norm Generalized Canavati Fractional …

'      Deαx0 (A1 ◦ log) p 



'      Deαx0 − (A2 ◦ log)q 



δ,[e x0 ,eb ]

x0

δ,[ea ,e x0 ]

'      Deαx0 (A2 ◦ log)q 



δ,[e x0 ,eb ]

+  x  1 x0 α− δ A2 (x)q e − e dx +

b



x0

 A1 (x) p e − e x0

 1 x α− δ

+ dx +

a

b

 α− 1δ A1 (x) p e x − e x0 dx

+, ,

(9.129)

x0

(ii) it holds  (A1 , A2 ) (x0 )1 ≤ '      Deαx0 − (A1 ◦ log) p  '      Deαx0 (A1 ◦ log) p  '      Deαx0 − (A2 ◦ log)q 

 L 1 ([ea ,e x0 ])

'      Deαx0 (A2 ◦ log)q 

a

L 1 ([e x0 ,eb ])

L 1 ([ea ,e x0 ])

+  α−1 A2 (x)q e x0 − e x dx +

x0

 

1  (α)

+  α−1 A2 (x)q e x − e x0 dx +

b x0

+  x0  x α−1  A1 (x) p e − e dx +

x0

(9.130)

a

 L 1 ([e x0 ,eb ])

b

 α−1  A1 (x) p e x − e x0 dx

+, ,

x0

and (iii)  ( A1 , A2 ) (x0 )1 ≤ '      Deαx0 − (A1 ◦ log) p 



'      Deαx0 (A1 ◦ log) p 



'      Deαx0 − (A2 ◦ log)q 



∞,[ea ,e x0 ]

'      Deαx0 (A2 ◦ log)q 

∞,[e x0 ,eb ]



b

+  x0  x α  A2 (x)q e − e dx +

a

∞,[e x0 ,eb ]

∞,[ea ,e x0 ]

x0

1  (α + 1)

b

+  α A2 (x)q e x − e x0 d x +

x0 x0

+  x0  x α A1 (x) p e − e dx +

a

 α  A1 (x) p e x − e x0 d x

x0

We continue with applications on Opial inequalities

+, .

(9.131)

9.6 Applications

233

Corollary 9.41 (to Theorem 9.19) All as in Theorem 9.19 with g (t) = t. Let p, q > 1 : 1p + q1 = 1. Then  z      f (w) D ν f (w) dw ≤ x0 1 x0



2− q (z − x0 )ν+ p − q 1

1

1

z

1

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) + 2)] p

x0

  ν   D f (w)q dw x0 2

q2

, (9.132)

for all x0 ≤ z ≤ b. It follows: Corollary 9.42 (to Theorem 9.20) All as in Theorem 9.20, γ ≥ 1, with g (t) = et . Let also p, q > 1 : 1p + q1 = 1. Then 

z e x0

    (( f ◦ log) (w)) D νx0 ( f ◦ log) (w)  dw ≤ e γ 

2− q (z − e x0 )ν+ p − q 1

1

1 1

 (ν) [( p (ν − 1) + 1) ( p (ν − 1) + 2)] p

z

e x0

 ν    D x0 ( f ◦ log) (w)q dw e γ

q2

,

(9.133)

for all e x0 ≤ z ≤ eb . We finish with applications on Grüss inequalities: Corollary 9.43 (to Theorem 9.35) All as in Theorem 9.35 with g (t) = t (1 ≤ ν < 2). Then (i)  ⎛ ⎞    r  r ν+2    ⎜  ⎟ M1 ( f 1 , ..., fr ) (b − a) ⎜  f j (x)  ⎟,  ( f 1 , ..., fr )1 ≤   2 ⎝ ⎠  (ν + 1)  j=1  i=1  j=i  ∞,[a,b]

(9.134)

where M1 ( f 1 , ..., fr ) is as in (9.92), (ii)

 ( f 1 , ..., fr )1 ≤

M2 ( f 1 , ..., fr ) (b − a)  (ν)

ν+1

 ⎛    r    ⎜  ⎜  f j (x)  2 ⎝  j=1  i=1  j=i 

⎞

r 

⎟ ⎟, ⎠

∞,[a,b]

(9.135)

where M2 ( f 1 , ..., fr ) is as in (9.94), (iii) when p, q > 1 : 1p + q1 = 1, we have

234

9

 ( f 1 , ..., fr )1 ≤

p-Schatten Norm Generalized Canavati Fractional …

M3 ( f 1 , ..., fr ) (b − a)

ν+1+ 1p

1 p

( p (ν − 1) + 1)  (ν)

 ⎛    r    ⎜  ⎜  f j (x)  2 ⎝  j=1  i=1  j=i 

⎞

r 

⎟ ⎟, ⎠

∞,[a,b]

(9.136)

where M3 ( f 1 , ..., fr ) is as in (9.96). It follows (r = 2 case) Corollary 9.44 (to Theorem 9.37) All as in Theorem 9.37, with [a, b] ⊂ R+ − {0}, and g (t) = log t. Then (i) for γ, δ > 1 : γ1 + 1δ = 1, we have  (A1 , A2 )1 ≤ '

b

α− 1δ  N1 (A1 , A2 ) log ab (b − a) 1

 (α) (γ (α − 1) + 1) γ 

A1 (x) p d x +

a

b

+ A2 (x)q d x ,

(9.137)

a

where N1 (A1 , A2 ) is as in (9.104), (ii)

α−1  N2 (A1 , A2 ) log ab (b − a)  (A1 , A2 )1 ≤  (α) '

b

 A1 (x) p d x +

a

b

+ A2 (x)q d x ,

(9.138)

a

where N2 (A1 , A2 ) is as in (9.106), and (iii)

α  N3 (A1 , A2 ) log ab (b − a)  (A1 , A2 )1 ≤  (α + 1) '

b

 A1 (x) p d x +

a

where N3 (A1 , A2 ) is as in (9.108).

a

b

+ A2 (x)q d x ,

(9.139)

References

235

References 1. Anastassiou, G.A.: Fractional Differentiation Inequalities. Research Monograph, Springer, New York (2009) 2. Anastassiou, G.A.: Advances on Fractional Inequalities. Research Monograph, Springer, New York (2011) 3. Anastassiou, G.A.: Intelligent Comparisons: Analytic Inequalities. Springer, Heidelberg, New York (2016) 4. Anastassiou, G.A.: Strong mixed and generalized fractional calculus for Banach space valued functions. Mat. Vesnik 69(3), 176–191 (2017) 5. Anastassiou, G.A.: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations. Springer, Heidelberg, New York (2018) 6. Anastassiou, G.A.: Generalized Canavati Fractional Ostrowski, Opial and Grüss type inequalities for Banach algebra valued functions, submitted (2021) 7. Anastassiou, G.A.: p-Schatten norm generalized Canavati fractional Ostrowski, Opial and Grüss type inequalities involving several functions. Panamerican Math. journal, accepted for publication (2021) 8. Bellman, R.: Some inequalities for positive definite matrices. In: Beckenbach, E.F. (ed.), General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, Birkhauser, Basel, pp. 89–90 (1980) ˇ 9. Cebyšev: Sur les expressions approximatives des intégrales définies par les aures prises entre les mêmes limites. Proc. Math. Soc. Charkov 2, 93–98 (1882) 10. Chang, D.: A matrix trace inequality for products of Hermitian matrices. J. Math. Anal. Appl. 237, 721–725 (1999) 11. Coop, I.D.: On matrix trace inequalities and related topics for products of Hermitian matrix. J. Math. Anal. Appl. 188, 999–1001 (1994) 12. Dragomir, S.S.: p-Schatten norm inequalities of Ostrowski’s type. RGMIA Res. Rep. Coll. 24, Art. 108, 19 pp (2021) 13. Dragomir, S.S.: p-Schatten norm inequalities of Grüss type. RGMIA Res. Rep. Coll. 24, Art. 115, 16 pp (2021) 14. Mikusinski, J.: The Bochner Integral. Academic Press, New York (1978) 15. Neudecker, H.: A matrix trace inequality. J. Math. Anal. Appl. 166, 302–303 (1992) 16. Opial, Z.: Sur une inegalite. Ann. Polon. Math. 8, 29–32 (1960) 17. Ostrowski, A.: Über die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938) 18. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991) 19. Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979) 20. Zagrebvov, V.A.: Gibbs semigroups. Operator Theory: Advances and Applications, vol. 273, Birkhauser (2019)

Chapter 10

γ-Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte Inequalities with von Neumann–Schatten Class Bγ (H) Valued Functions

Employing generalized Canavati fractional left and right vectorial Taylor formulae we prove corresponding left and right fractional Hilbert–Pachpatte inequalities for von Neumann–Schatten class Bγ (H ) valued functions. We cover also the sequential fractional case. We finish with applications. It follows [4].

10.1 Introduction Here all motivation comes from [3]. All fractional terminology comes from Sect. 10.2 next. We start with a left generalized Canavati fractional Hilbert–Pachpatte type inequality over a Banach algebra. Theorem 10.1 Let p, q > 1, such that 1p + q1 = 1, and (A, ·) is a Banach algebra; and i = 1, 2. Let also x0i ∈ [ai , bi ] ⊂ R, νi ≥ 1, n i = [νi ], f i ∈ C ni ([ai , bi ] , A); gi ∈ C 1 ([ai , bi ]), strictly increasing, such that gi−1 ∈ C ni ([gi (ai ) , gi (bi )]), with (ki )  f i ◦ gi−1 (gi (x0i )) = 0, ki = 0, 1, ..., n i − 1. Assume further that f i ◦ gi−1 ∈ νi Cgi (x0i ) ([gi (ai ) , gi (bi )] , A). Then 

g1 (b1 )

g1 (x01 )



g2 (b2 ) g2 (x02 )

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2   ≤ (z 1 −g1 (x01 )) p(ν1 −1)+1 (z 2 −g2 (x02 ))q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

(g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 ))  (ν1 )  (ν2 )

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_10

(10.1)

237

10 γ-Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte …

238

      ν1  Dg1 (x01 ) f 1 ◦ g1−1 

L q ([g1 (x01 ),g1 (b1 )],A)

      ν2  Dg2 (x02 ) f 2 ◦ g2−1 

L p ([g2 (x02 ),g2 (b2 )],A)

.

Integrals in this chapter are of Bochner type [9]. We continue with a right generalized Canavati fractional Hilbert–Pachpatte type inequality over a Banach algebra. Theorem 10.2 All as in Theorem 10.1, however now it is Cgνii(x0i )− ([gi (ai ) , gi (bi )] , A), for i = 1, 2. Then 

g1 (x01 ) g1 (a1 )



g2 (x02 ) g2 (a2 )

f i ◦ gi−1 ∈

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2   ≤ (g1 (x01 )−z 1 ) p(ν1 −1)+1 (g2 (x02 )−z 2 )q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

(g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 ))  (ν1 )  (ν2 )

(10.2)

         ν1  ν2 . Dg (x )− f 1 ◦ g1−1  Dg (x )− f 2 ◦ g2−1  1 01 2 02 L q ([g1 (a1 ),g1 (x01 )],A) L p ([g2 (a2 ),g2 (x02 )],A)

Next comes a sequential left generalized Canavati fractional Hilbert–Pachpatte type inequality over a Banach algebra. Theorem 10.3 Let p, q > 1, such that 1p + q1 = 1, and (A, ·) is a Banach algebra; and i = 1, 2. Let also f i ∈ C 1 ([ai , bi ] , A); gi ∈ C 1 ([ai , bi ]), strictly increas1 < νi < 1, x0i ∈ ing, such that gi−1 ∈ C 1 ([gi (ai ) , gi (bi )]). Assume that (m i +1)q   ji νi νi −1 [ai , bi ], and Dgi (x0i ) f i ◦ gi ∈ Cgi (x0i ) ([gi (ai ) , gi (bi )] , A) , for ji = 0, 1, ..., m i ∈ N. Then       g1 (b1 )  g2 (b2 )  f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2  ≤ −g2 (x02 ))q ((m 2 +1)ν2 −1)+1 (z 1 −g1 (x01 )) p((m 1 +1)ν1 −1)+1 g1 (x01 ) g2 (x02 ) + (z2 q(q((m p( p((m 1 +1)ν1 −1)+1) +1)ν −1)+1) 2 2 (g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 ))  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 )      (m 1 +1)ν1 f 1 ◦ g1−1  Dg (x ) 1

01

L q ([g1 (x01 ),g1 (b1 )],A)

     (m 2 +1)ν2 f 2 ◦ g2−1  Dg (x ) 2

02

(10.3)

L p ([g2 (x02 ),g2 (b2 )],A)

.

The right side analog of Theorem 10.3 follows: Theorem 10.4 Let p, q > 1, such that 1p + q1 = 1, and (A, ·) is a Banach algebra; and i = 1, 2. Let also f i ∈ C 1 ([ai , bi ] , A); gi ∈ C 1 ([ai , bi ]), strictly increas1 < νi < 1, x0i ∈ ing, such that gi−1 ∈ C 1 ([gi (ai ) , gi (bi )]). Assume that (m i +1)q   ji νi νi −1 [ai , bi ], and Dgi (x0i )− f i ◦ gi ∈ Cgi (x0i )− ([gi (ai ) , gi (bi )] , A) , for ji = 0, 1, ..., m i ∈ N. Then

10.2 Background on Vectorial Generalized Canavati Fractional Calculus



g1 (x01 ) g1 (a1 )



g2 (x02 )

g2 (a2 )



239

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2

1 (g1 (x01 )−z 1 ) p((m 1 +1)ν1 −1)+1 p( p((m 1 +1)ν1 −1)+1)

+

2  (g2 (x02 )−z 2 )q ((m 2 +1)ν2 −1)+1 q(q((m 2 +1)ν2 −1)+1)

(g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 ))  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 )      (m 1 +1)ν1 Dg (x )− f 1 ◦ g1−1  1

01

L q ([g1 (a1 ),g1 (x01 )],A)

     (m 2 +1)ν2 Dg (x )− f 2 ◦ g2−1  2

02

≤

(10.4)

L p ([g2 (a2 ),g2 (x02 )],A)

.

Other related inspiration comes from [2]. Let γ ≥ 1, in this chapter we derive γ-Schatten left and right Hilbert–Pachpatte inequalities for Banach algebra Bγ (H ) valued functions with respect to their Canavati type generalized left and right fractional derivatives. We cover also the sequential fractional case. We finish with applications. For Bγ (H ) definition see Sect. 10.4 later.

10.2 Background on Vectorial Generalized Canavati Fractional Calculus All in this section come from [2], pp. 109–115 and [1]. Let g : [a, b] → R be a strictly increasing function. such that g ∈ C 1 ([a, b]), and −1 g ∈ C n ([g(a), g(b)]), n ∈ N, (X, ·) is a Banach space. Let f ∈ C n ([a, b] , X ), and call l := f ◦ g −1 : [g (a) , g (b)] → X . It is clear that l, l , ..., l (n) are continuous functions from [g (a) , g (b)] into f ([a, b]) ⊆ X. Let ν ≥ 1 such that [ν] = n, n ∈ N as above, where [·] is the integral part of the number. Clearly when 0 < ν < 1, [ν] = 0. (I) Let h ∈ C ([g (a) , g (b)] , X ), we define the left Riemann-Liouville Bochner fractional integral as 

 Jνz0 h (z) :=

1  (ν)



z

(z − t)ν−1 h (t) dt,

(10.5)

z0

∞ for g (a) ≤ z 0 ≤ z ≤ g (b), where  is the gamma function;  (ν) = 0 e−t t ν−1 dt. We set J0z0 h = h. ν Let α := ν − [ν] (0 < α < 1). We define the subspace Cg(x ([g (a) , g (b)] , X ) 0) [ν] of C ([g (a) , g (b)] , X ), where x0 ∈ [a, b] as: ν Cg(x ([g (a) , g (b)] , X ) = 0)



g(x h ∈ C [ν] ([g (a) , g (b)] , X ) : J1−α0 ) h ([ν]) ∈ C 1 ([g (x0 ) , g (b)] , X ) .

(10.6)

10 γ-Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte …

240

ν So let h ∈ Cg(x ([g (a) , g (b)] , X ), we define the left g-generalized X -valued 0) fractional derivative of h of order ν, of Canavati type, over [g (x0 ) , g (b)] as

  g(x0 ) ([ν]) ν Dg(x h := J h . 1−α 0)

(10.7)

ν Clearly, for h ∈ Cg(x ([g (a) , g (b)] , X ) , there exists 0)



 ν Dg(x h (z) = 0)

d 1  (1 − α) dz



z

g(x0 )

(z − t)−α h ([ν]) (t) dt,

(10.8)

for all g (x0 ) ≤ z ≤ g (b) . ν In particular, when f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), we have that 0) 

ν Dg(x 0)



f ◦g

−1



d 1 (z) =  (1 − α) dz



z g(x0 )

 ([ν]) (z − t)−α f ◦ g −1 (t) dt, 

−1





(10.9) and

 −1 (n)

f ◦g = f ◦g for all g (x0 ) ≤ z ≤ g (b). We have that   0 −1 −1 f ◦ g = f ◦ g , see [1]. Dg(x 0) ν By [1], we have for f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ) , where x0 ∈ [a, b] the 0) following left generalized g-fractional, of Canavati type,X -valued Taylor’s formula: n Dg(x 0)

ν Theorem 10.5 Let f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), where x0 ∈ [a, b] is fixed. 0) (i) If ν ≥ 1, then [ν]−1 

f (x) − f (x0 ) =



f ◦ g −1

(k) k!

k=1

1  (ν)



g(x) g(x0 )

(g (x0 ))

(g (x) − g (x0 ))k +

 ν   f ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(10.10)

for all x0 ≤ x ≤ b. (ii) If 0 < ν < 1, we get f (x) =

1  (ν)



g(x) g(x0 )

 ν   f ◦ g −1 (t) dt, (g (x) − t)ν−1 Dg(x 0)

(10.11)

for all x0 ≤ x ≤ b. (II) Let h ∈ C ([g (a) , g (b)] , X ), we define the right Riemann-Liouville Bochner fractional integral as 

Jzν0 − h



1 (z) :=  (ν)



z0 z

(t − z)ν−1 h (t) dt,

(10.12)

10.2 Background on Vectorial Generalized Canavati Fractional Calculus

241

for g (a) ≤ z ≤ z 0 ≤ g (b) . We set Jz00 − h = h. ν Let α := ν − [ν] (0 < α < 1). We define the subspace Cg(x ([g (a) , g (b)] , X ) 0 )− [ν] of C ([g (a) , g (b)] , X ), where x0 ∈ [a, b] as: ν Cg(x ([g (a) , g (b)] , X ) := 0 )−



1−α 1 ([ν]) h ∈ C [ν] ([g (a) , g (b)] , X ) : Jg(x h ∈ C , g , X . (10.13) ([g (a) (x )] ) 0 )− 0

ν So let h ∈ Cg(x ([g (a) , g (b)] , X ), we define the right g-generalized X -valued 0 )− fractional derivative of h of order ν, of Canavati type, over [g (a) , g (x0 )] as

  1−α ν n−1 ([ν]) J h := h . Dg(x (−1) g(x0 )− 0 )−

(10.14)

ν Clearly, for h ∈ Cg(x ([g (a) , g (b)] , X ) , there exists 0 )−



ν Dg(x h 0 )−



(−1)n−1 d (z) =  (1 − α) dz



g(x0 )

(t − z)−α h ([ν]) (t) dt,

(10.15)

z

for all g (a) ≤ z ≤ g (x0 ) ≤ g (b) . ν In particular, when f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), we have that 0 )− 

  (−1)n−1 d ν Dg(x f ◦ g −1 (z) = 0 )−  (1 − α) dz



g(x0 )

 ([ν]) (t − z)−α f ◦ g −1 (t) dt,

z

(10.16)

for all g (a) ≤ z ≤ g (x0 ) ≤ g (b). We get that    (n) n f ◦ g −1 (z) = (−1)n f ◦ g −1 Dg(x (z) 0 )−



(10.17)

     0 −1 f ◦ g and Dg(x (z) = f ◦ g −1 (z), all z ∈ [g (a) , g (b)] , see [1]. 0 )− ν By [1], we have for f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ) , where x0 ∈ [a, b] is 0 )− fixed, the following right generalized g-fractional, of Canavati type, X -valued Taylor’s formula: ν Theorem 10.6 Let f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), where x0 ∈ [a, b] is fixed. 0 )− (i) If ν ≥ 1, then

f (x) − f (x0 ) =

[ν]−1  k=1

1  (ν)



g(x0 )

g(x)



f ◦ g −1

(k) k!

(g (x0 ))

(g (x) − g (x0 ))k +

 ν   f ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

(10.18)

10 γ-Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte …

242

for all a ≤ x ≤ x0 , (ii) If 0 < ν < 1, we get 1 f (x) =  (ν)



g(x0 )

g(x)

 ν   f ◦ g −1 (t) dt, (t − g (x))ν−1 Dg(x 0 )−

(10.19)

all a ≤ x ≤ x0 . (III) Denote by mν ν ν ν Dg(x = Dg(x Dg(x ...Dg(x (m-times), m ∈ N. 0) 0) 0) 0)

(10.20)

We mention the following modified and generalized left X -valued fractional Taylor’s formula of Canavati type: −1 Theorem 10.7 Let f ∈ C 1 ([a,b] , X ), g ∈ C 1 ([a,  b]), strictly increasing: g ∈   iν ν C 1 ([g (a) , g (b)]). Assume that Dg(x f ◦ g −1 ∈ Cg(x ([g (a) , g (b)] , X ), 0 < 0) 0) ν < 1, x0 ∈ [a, b], for i = 0, 1, ..., m. Then

f (x) =

1  ((m + 1) ν)



g(x)

g(x0 )

  (m+1)ν  −1 f ◦ g (z) dz, (g (x) − z)(m+1)ν−1 Dg(x 0) (10.21)

all x0 ≤ x ≤ b. (IV) Denote by mν ν ν ν Dg(x = Dg(x Dg(x ...Dg(x (m times), m ∈ N. 0 )− 0 )− 0 )− 0 )−

(10.22)

We mention the following modified and generalized right X -valued fractional Taylor’s formula of Canavati type: 1 −1 Theorem 10.8 Let f ∈ C 1 ([a, b]  , X ), g ∈ C ([a,b]), strictly increasing: g ∈ iν ν 1 −1 C ([g (a) , g (b)]). Assume that Dg(x0 )− f ◦ g ∈ Cg(x0 )− ([g (a) , g (b)] , X ), 0 < ν < 1, x0 ∈ [a, b], for all i = 0, 1, ..., m. Then

f (x) =

1  ((m + 1) ν)



g(x0 ) g(x)

  (m+1)ν  f ◦ g −1 (z) dz, (z − g (x))(m+1)ν−1 Dg(x 0 )− (10.23)

all a ≤ x ≤ x0 ≤ b.

10.3 Basic Banach Algebras Background All here come from [11]. We need

10.3 Basic Banach Algebras Background

243

Definition 10.9 ([11], p. 245) A complex algebra is a vector space A over the complex filed C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(10.24)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(10.25)

α (x y) = (αx) y = x (αy) ,

(10.26)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A)

(10.27)

and if A contains a unit element e such that xe = ex = x (x ∈ A)

(10.28)

e = 1,

(10.29)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 10.10 Commutativity of A will be explicated stated when needed. There exists at most one e ∈ A that satisfies (10.28). Inequality (10.27) makes multiplication to be continuous, more precisely left and right continuous, see [11], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [11], pp. 247–248, Sect. 10.3. We also make Remark 10.11 Next we mention about integration of A-valued functions, see [11], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some  compact Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [11], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then

244

10 γ-Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte …



 f dμ =

x Q

and



x f ( p) dμ ( p)

(10.30)

Q

 f dμ x = Q

f ( p) x dμ ( p) .

(10.31)

Q

The Bochner integrals we will involve in our chapter follow (10.30) and (10.31). Also, let f ∈ C ([a, b] , X ), where [a, b] ⊂ R, (X, ·) is a Banach space. By [2], p. 3, f is Bochner integrable.

10.4

p-Schatten Norms Background

In this advanced section all come from [8]. Let (H, ·, · ) be a complex Hilbert space and B (H ) the Banach algebra of all bounded linear operators on H . If {ei }i∈I an orthonormal basis of H , we say that A ∈ B (H ) is of trace class if A1 :=



|A| ei , ei < ∞.

(10.32)

i∈I

The definition of A1 does not depend on the choice of the orthornormal basis {ei }i∈I . We denote by B1 (H ) the set of trace class operators in B (H ). We define the trace of a trace class operator A ∈ B1 (H ) to be tr (A) :=



Aei , ei ,

(10.33)

i∈I

where {ei }i∈I an orthonormal basis of H . Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series (10.33) converges absolutely and it is independent from the choice of basis. The following result collects some properties of the trace: Theorem 10.12 We have: (i) If A ∈ B1 (H ) then A∗ ∈ B1 (H ) and   tr A∗ = tr (A);

(10.34)

(ii) If A ∈ B1 (H ) and T ∈ B (H ), then AT, T A ∈ B1 (H ) and tr (AT ) = tr (T A) and |tr (AT )| ≤ A1 T  ; (iii) tr (·) is a bounded linear functional on B1 (H ) with tr  = 1; (iv) If A, B ∈ B2 (H ) then AB, B A ∈ B1 (H ) and tr (AB) = tr (B A) ;

(10.35)

10.4 p-Schatten Norms Background

245

(v) B f in (H ) (finite rank operators) is a dense subspace of B1 (H ) . An operator A ∈ B (H ) is said to belong to the von Neumann–Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite [13, pp. 60–64]  1   A p := tr |A| p p < ∞, |A| p is an operator notation and not a power. For 1 < p < q < ∞ we have that B1 (H ) ⊂ B p (H ) ⊂ Bq (H ) ⊂ B (H )

(10.36)

 A1 ≥  A p ≥  Aq ≥ A .

(10.37)

and For p ≥ 1 the functional · p is a norm on the ∗-ideal B p (H ), which is a Banach   algebra, and B p (H ) , · p is a Banach space. Also, see for instance [13, pp. 60–64], for p ≥ 1,   A p =  A∗  p , A ∈ B p (H )

(10.38)

AB p ≤  A p B p , A, B ∈ B p (H )

(10.39)

and AB p ≤ A p B , B A p ≤ B A p , A ∈ B p (H ) , B ∈ B (H ) . (10.40) This implies that C AB p ≤ C A p B , A ∈ B p (H ) , B, C ∈ B (H ) .

(10.41)

In terms of p-Schatten norm we have the Hölder inequality for p, q > 1 with 1 =1: q (|tr (AB)| ≤)  AB1 ≤ A p Bq , A ∈ B p (H ) , B ∈ Bq (H ) .

1 p

+

(10.42)

For the theory of trace functionals and their applications the interested reader is referred to [12, 13]. For some classical trace inequalities see [6, 7, 10], which are continuations of the work of Bellman [5].

10 γ-Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte …

246

10.5 Main Results We start with a 1-2-Schatten norms left generalized Canavati fractional Hilbert– Pachpatte type inequality over B2 (H ). Theorem 10.13 Let p, q > 1, such that 1p + q1 = 1, and (B2 (H ) , ·2 ) is the ∗-ideal; i = 1, 2. Let also x0i ∈ [ai , bi ] ⊂ R, νi ≥ 1, n i = [νi ], f i ∈ C ni ([ai , bi ] , B2 (H )); gi ∈ C 1 ([ai , bi ]), strictly increasing, such that (ki )  gi−1 ∈ C ni ([gi (ai ) , gi (bi )]), with f i ◦ gi−1 (gi (x0i )) = 0, ki = 0, 1, ..., n i − 1. νi −1 Assume further that f i ◦ gi ∈ Cgi (x0i ) ([gi (ai ) , gi (bi )] , B2 (H )). Then 

g1 (b1 )



g1 (x01 )

g2 (b2 )

g2 (x02 )

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 1   ≤ (z 1 −g1 (x01 )) p(ν1 −1)+1 (z 2 −g2 (x02 ))q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

(g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 ))  (ν1 )  (ν2 )     ν1   D   f 1 ◦ g −1   g1 x01  1



   

 ν2       Dg x 2 02 2 L q g1 x01 ,g1 b1 ,B2 (H ) 









(10.43)

     f 2 ◦ g2−1   

     . 2 L p g2 x02 ,g2 b2 ,B2 (H )

Proof By (10.10) and assumptions we get that 

 zi     1 ν f i ◦ gi−1 (z i ) = (z i − ti )νi −1 Dg i (x ) f i ◦ gi−1 (ti ) dti , i 0i  (νi ) gi (x0i )

(10.44)

for all gi (x0i ) ≤ z i ≤ gi (bi ); i = 1, 2. By Hölder’s inequality we obtain       f 1 ◦ g1−1 (z 1 ) ≤ 2

1  (ν1 )



z1

g1 (x01 )

 z1     1  ν  (z 1 − t1 )ν1 −1  Dg 1(x ) f 1 ◦ g1−1 (t1 ) dt1 ≤ 1 01 2  (ν1 ) g1 (x01 )

(z 1 − t1 ) p(ν1 −1) dt1 p (ν1 −1)+1

1 (z 1 − g1 (x01 )) p  (ν1 ) ( p (ν1 − 1) + 1) 1p

1  p

z1

g1 (x01 )



z1

g1 (x01 )

1  q   q  ν1  =  Dg1 (x01 ) f 1 ◦ g1−1 (t1 ) dt1 2

q1  q    ν1  −1 (t1 ) dt1 .  Dg1 (x01 ) f 1 ◦ g1 2

(10.45)

That is     f 1 ◦ g −1 (z 1 ) ≤ 1 2 

z1 g1 (x01 )

p (ν1 −1)+1

1 (z 1 − g1 (x01 )) p  (ν1 ) ( p (ν1 − 1) + 1) 1p

q1  q    ν1   Dg1 (x01 ) f 1 ◦ g1−1 (t1 ) dt1 , 2

(10.46)

10.5 Main Results

247

for all g1 (x01 ) ≤ z 1 ≤ g1 (b1 ). Similarly, we prove that     f 2 ◦ g −1 (z 2 ) ≤ 2 2 

z2

g2 (x02 )

q (ν2 −1)+1

1 (z 2 − g2 (x02 )) q  (ν2 ) (q (ν2 − 1) + 1) q1

  p 1p    ν2  −1 , (t2 ) dt2  Dg2 (x02 ) f 2 ◦ g2

(10.47)

2

for all g2 (x02 ) ≤ z 2 ≤ g2 (b2 ). Therefore we have     f 1 ◦ g −1 (z 1 ) ≤ 1 2

p (ν1 −1)+1

1 (z 1 − g1 (x01 )) p  (ν1 ) ( p (ν1 − 1) + 1) 1p

       ν1  Dg1 (x01 ) f 1 ◦ g1−1  

2 q,[g1 (x01 ),g1 (b1 )]

for all g1 (x01 ) ≤ z 1 ≤ g1 (b1 ); and     f 2 ◦ g −1 (z 2 ) ≤ 2 2

,

(10.48)

q (ν2 −1)+1

1 (z 2 − g2 (x02 )) q  (ν2 ) (q (ν2 − 1) + 1) q1

     ν2    Dg2 (x02 ) f 2 ◦ g2−1  

2 p,[g2 (x02 ),g2 (b2 )]

,

(10.49)

for all g2 (x02 ) ≤ z 2 ≤ g2 (b2 ). Hence we get that            f 1 ◦ g1−1 (z 1 )  f 2 ◦ g2−1 (z 2 ) ≤ 2

2

(z 1 − g1 (x01 ))

1 1

1

 (ν1 )  (ν2 ) ( p (ν1 − 1) + 1) p (q (ν2 − 1) + 1) q

p (ν1 −1)+1 p

(z 2 − g2 (x02 ))

q (ν2 −1)+1 q

(10.50)

             ν1  ν2 ≤  Dg (x ) f 1 ◦ g1−1    Dg (x ) f 2 ◦ g2−1   1 01 2 02 2 q,[g1 (x01 ),g1 (b1 )] 2 p,[g2 (x02 ),g2 (b2 )] 1

1

(using Young’s inequality for a, b ≥ 0, a p b q ≤ 1  (ν1 )  (ν2 )



a p

+ qb )

(z 2 − g2 (x02 ))q(ν2 −1)+1 (z 1 − g1 (x01 )) p(ν1 −1)+1 + p( p (ν1 − 1) + 1) q(q (ν2 − 1) + 1)



10 γ-Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte …

248

        ν1  Dg1 (x01 ) f 1 ◦ g1−1  

2 L q ([g1 (x01 ),g1 (b1 )],B2 (H ))

       ν2  Dg2 (x02 ) f 2 ◦ g2−1  

2 L p ([g2 (x02 ),g2 (b2 )],B2 (H ))

∀ (z 1 , z 2 ) ∈ [g1 (x01 ) , g1 (b1 )] × [g2 (x02 ) , g2 (b2 )] . So far we have       f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) 1 2 1   (z 1 −g1 (x01 )) p(ν1 −1)+1 (z 2 −g2 (x02 ))q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

(10.42)

,

≤

(10.51)

       f 1 ◦ g −1 (z 1 )  f 2 ◦ g −1 (z 2 ) 1 2 2 2  ≤ (z 1 −g1 (x01 )) p(ν1 −1)+1 (z 2 −g2 (x02 ))q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

(10.52)

    1  ν1    Dg1 (x01 ) f 1 ◦ g1−1   2 L q ([g1 (x01 ),g1 (b1 )],B2 (H ))  (ν1 )  (ν2 )      ν2    Dg2 (x02 ) f 2 ◦ g2−1  

2 L p ([g2 (x02 ),g2 (b2 )],B2 (H ))

,

∀ (z 1 , z 2 ) ∈ [g1 (x01 ) , g1 (b1 )] × [g2 (x02 ) , g2 (b2 )] . The denominators in (10.51), (10.52) can be zero only when both z 1 = g1 (x01 ) and z 2 = g2 (x02 ) . Therefore we obtain (10.43), by integrating (10.51), (10.52) over  [g1 (x01 ) , g1 (b1 )] × [g2 (x02 ) , g2 (b2 )] . We continue with the corresponding right generalized Canavati fractional Hilbert– Pachpatte type inequality over B2 (H ). Theorem 10.14 All as in Theorem 10.13, however now it is Cgνii(x0i )− ([gi (ai ) , gi (bi )] , B2 (H )), for i = 1, 2. Then 

g1 (x01 )



g1 (a1 )

g2 (x02 )

g2 (a2 )

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 1   ≤ (g1 (x01 )−z 1 ) p(ν1 −1)+1 (g2 (x02 )−z 2 )q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

(g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 ))  (ν1 )  (ν2 )     ν1 −1   D     g1 x01 − f 1 ◦ g1 

   



 ν2          Dg x 2 02 − 2 L q g1 a1 ,g1 x01 ,B2 (H ) 

f i ◦ gi−1 ∈



(10.53)

     f 2 ◦ g2−1   

Proof Similar to Theorem 10.13, by using now (10.18).

     . 2 L p g2 a2 ,g2 x02 ,B2 (H )



Next comes a sequential analogous left generalized Canavati fractional Hilbert– Pachpatte type inequality over B2 (H ).

10.5 Main Results

249

Theorem 10.15 Let p, q > 1, such that 1p + q1 = 1, and (B2 (H ) , ·2 ) is the ∗-ideal; i = 1, 2. Let also f i ∈ C 1 ([ai , bi ] , B2 (H )); gi ∈ C 1 ([ai , bi ]), strictly 1 < νi < 1, increasing, such that gi−1 ∈ C 1 ([gi (ai ) , gi (bi )]). Assume that (m i +1)q   ji νi ν −1 i x0i ∈ [ai , bi ], and Dgi (x0i ) f i ◦ gi ∈ Cgi (x0i ) ([gi (ai ) , gi (bi )] , B2 (H )) , for ji = 0, 1, ..., m i ∈ N. Then 

g1 (b1 ) g1 (x01 )



g2 (b2 )

g2 (x02 )



      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 1

(z 1 −g1 (x01 )) p((m 1 +1)ν1 −1)+1 p( p((m 1 +1)ν1 −1)+1)

+

(z 2 −g2 (x02 ))q ((m 2 +1)ν2 −1)+1 q(q((m 2 +1)ν2 −1)+1)

≤

(g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 ))  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 )        (m 1 +1)ν1 f 1 ◦ g1−1   Dg1 (x01 )

2 L q ([g1 (x01 ),g1 (b1 )],B2 (H ))

(10.54)

       (m 2 +1)ν2 f 2 ◦ g2−1   Dg2 (x02 )

2 L p ([g2 (x02 ),g2 (b2 )],B2 (H ))

.



Proof Using (10.21), as similar to Theorem 10.13 the proof is omitted. The right side analog of Theorem 10.15 follows:

Theorem 10.16 Let p, q > 1, such that 1p + q1 = 1, and (B2 (H ) , ·2 ) is the ∗-ideal; i = 1, 2. Let also f i ∈ C 1 ([ai , bi ] , B2 (H )); gi ∈ C 1 ([ai , bi ]), strictly 1 < νi < increasing, such that gi−1 ∈ C 1 ([gi (ai ) , gi (bi )]). Assume that (m i +1)q   ji νi νi −1 1, x0i ∈ [ai , bi ], and Dgi (x0i )− f i ◦ gi ∈ Cgi (x0i )− ([gi (ai ) , gi (bi )] , B2 (H )) , for ji = 0, 1, ..., m i ∈ N. Then 

g1 (x01 ) g1 (a1 )



g2 (x02 )

g2 (a2 )



      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 1

(g1 (x01 )−z 1 ) p((m 1 +1)ν1 −1)+1 p( p((m 1 +1)ν1 −1)+1)

+

(g2 (x02 )−z 2 )q ((m 2 +1)ν2 −1)+1 q(q((m 2 +1)ν2 −1)+1)

(g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 ))  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 )        (m 1 +1)ν1 Dg1 (x01 )− f 1 ◦ g1−1  

2 L q ([g1 (a1 ),g1 (x01 )],B2 (H ))

       (m 2 +1)ν2 Dg2 (x02 )− f 2 ◦ g2−1  

≤

(10.55)

2 L p ([g2 (a2 ),g2 (x02 )],B2 (H ))

Proof Using (10.23), as similar to Theorem 10.13 is omitted.

.



We continue with a γ-Schatten norm left generalized Canavati fractional Hilbert– Pachpatte type inequality over Bγ (H ), γ ≥ 1.   Theorem 10.17 Let γ ≥ 1, p, q > 1, such that 1p + q1 = 1, and Bγ (H ) , ·γ is the ∗-ideal; i = 1,2. Let also x0i ∈ [ai , bi ] ⊂ R, νi ≥ 1, n i = [νi ], f i ∈ C ni [ai , bi ] , Bγ (H ) ; gi ∈ C 1 ([ai , bi ]), strictly increasing, such that (ki )  gi−1 ∈ C ni ([gi (ai ) , gi (bi )]), with f i ◦ gi−1 (gi (x0i )) = 0, ki = 0, 1, ..., n i − 1.   νi −1 Assume further that f i ◦ gi ∈ Cgi (x0i ) [gi (ai ) , gi (bi )] , Bγ (H ) . Then

10 γ-Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte …

250



g1 (b1 )

g1 (x01 )



g2 (b2 )

g2 (x02 )

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 γ   ≤ (z 1 −g1 (x01 )) p(ν1 −1)+1 (z 2 −g2 (x02 ))q (ν2 −1)+1 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)

(g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 ))  (ν1 )  (ν2 )     ν1 −1  D  g1 (x01 ) f 1 ◦ g1 

   γ

L q ([g1 (x01 ),g1 (b1 )],Bγ (H ))

    ν2 −1  D  g2 (x02 ) f 2 ◦ g2 

   γ

(10.56)

L p ([g2 (x02 ),g2 (b2 )],Bγ (H ))

.



Proof Similar to Theorem 10.13, by using norm (10.39).

We continue with the corresponding right generalized Canavati fractional Hilbert– Pachpatte type inequality over Bγ (H ). Theorem 10.18 All as in Theorem 10.17, however now it is  Cgνii(x0i )− [gi (ai ) , gi (bi )] , Bγ (H ) , for i = 1, 2. Then 

g1 (x01 ) g1 (a1 )



g2 (x02 )

g2 (a2 )

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 γ   ≤ (g1 (x01 )−z 1 ) p(ν1 −1)+1 (g2 (x02 )−z 2 )q (ν2 −1)+1 + q(q(ν2 −1)+1) p( p(ν1 −1)+1)

(g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 ))  (ν1 )  (ν2 )       ν1  −1   D f ◦ g   1 1  g1 (x01 )− γ

f i ◦ gi−1 ∈

L q ([g1 (a1 ),g1 (x01 )],Bγ (H ))

    ν2 −1  D  g2 (x02 )− f 2 ◦ g2 

   γ

Proof Similar to Theorem 10.13, by using now (10.39).

(10.57)

L p ([g2 (a2 ),g2 (x02 )],Bγ (H ))

.



Next comes a sequential analogous left generalized Canavati fractional Hilbert– Pachpatte type inequality over Bγ (H ).   Theorem 10.19 Let γ ≥ 1, p, q > 1, such that 1p + q1 = 1, and Bγ (H ) , ·γ is   the ∗-ideal; i = 1, 2. Let also f i ∈ C 1 [ai , bi ] , Bγ (H ) ; gi ∈ C 1 ([ai , bi ]), strictly 1 increasing, such that gi−1 ∈ C 1 ([gi (ai ) , gi (bi )]). Assume that (m i +1)q < νi < 1,     ji νi νi −1 x0i ∈ [ai , bi ], and Dgi (x0i ) f i ◦ gi ∈ Cgi (x0i ) [gi (ai ) , gi (bi )] , Bγ (H ) , for ji = 0, 1, ..., m i ∈ N. Then       g1 (b1 )  g2 (b2 )  f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 γ  ≤ p ((m 1 +1)ν1 −1)+1 q ((m 2 +1)ν2 −1)+1 −g −g (z (x )) (z (x )) 1 1 01 2 2 02 g1 (x01 ) g2 (x02 ) + p( p((m 1 +1)ν1 −1)+1) q(q((m 2 +1)ν2 −1)+1) (g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 ))  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 )

(10.58)

10.6 Applications

251

    (m 1 +1)ν1    D f 1 ◦ g1−1    g1 (x01 ) γ

L q ([g1 (x01 ),g1 (b1 )],Bγ (H ))

    (m 2 +1)ν2    D f 2 ◦ g2−1    g2 (x02 ) γ

L p ([g2 (x02 ),g2 (b2 )],Bγ (H ))

.



Proof Using (10.21), as similar to Theorem 10.13 the proof is omitted. The right side analog of Theorem 10.19 follows:

  Theorem 10.20 Let γ ≥ 1, p, q > 1, such that 1p + q1 = 1, and Bγ (H ) , ·γ is   the ∗-ideal; i = 1, 2. Let also f i ∈ C 1 [ai , bi ] , Bγ (H ) ; gi ∈ C 1 ([ai , bi ]), strictly 1 increasing, such that gi−1 ∈ C 1 ([gi (ai ) , gi (bi )]). Assume that (m i +1)q < νi <     ji νi νi −1 1, x0i ∈ [ai , bi ], and Dgi (x0i )− f i ◦ gi ∈ Cgi (x0i )− [gi (ai ) , gi (bi )] , Bγ (H ) , for ji = 0, 1, ..., m i ∈ N. Then 

g1 (x01 ) g1 (a1 )



g2 (x02 )

g2 (a2 )



      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 γ (g1 (x01 )−z 1 ) p((m 1 +1)ν1 −1)+1 p( p((m 1 +1)ν1 −1)+1)

+

(g2 (x02 )−z 2 )q ((m 2 +1)ν2 −1)+1 q(q((m 2 +1)ν2 −1)+1)

≤

(g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 ))  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 )    (m 1 +1)ν1  −1  D  g1 (x01 )− f 1 ◦ g1 

   γ

L q ([g1 (a1 ),g1 (x01 )],Bγ (H ))

   (m 2 +1)ν2  −1  D  g2 (x02 )− f 2 ◦ g2 

   γ

(10.59)

L p ([g2 (a2 ),g2 (x02 )],Bγ (H ))

.

Proof Using (10.23), as similar to Theorem 10.13 is omitted.



10.6 Applications We give the following γ-Schatten Hilbert–Pachpatte fractional inequalities: Corollary 10.21 (to Theorem 10.13) All as in Theorem 10.13 for g1 (t) = g2 (t) = t. Then  b1  b2  f 1 (z 1 ) f 2 (z 2 )1 dz 1 dz 2  ≤ (10.60) (z 1 −x01 ) p(ν1 −1)+1 (z 2 −x02 )q (ν2 −1)+1 x01 x02 + p( p(ν1 −1)+1) q(q(ν2 −1)+1)     ν   (b1 − x01 ) (b2 − x02 )   D ν1 f 1    D 2 f 2   . x01 x02 2 2 L p ([x02 ,b2 ],B2 (H )) L ,b ],B ([x (H )) q 01 1 2  (ν1 )  (ν2 ) We continue with Corollary 10.22 (to Theorem 10.15) All as in Theorem 10.15 for g1 (t) = g2 (t) = t. Then  b1  b2  f 1 (z 1 ) f 2 (z 2 )1 dz 1 dz 2  ≤ (10.61) p ((m 1 +1)ν1 −1)+1 −x (z ) (z 2 −x02 )q ((m 2 +1)ν2 −1)+1 1 01 x01 x02 + p( p((m 1 +1)ν1 −1)+1) q(q((m 2 +1)ν2 −1)+1)

10 γ-Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte …

252

(b1 − x01 ) (b2 − x02 )  ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 )  (m +1)ν   1  D 1 f 1 2  L x01

q ([x 01 ,b1 ],B2 (H ))

 (m +1)ν   2  D 2 f 2 2  L x02

p ([x 02 ,b2 ],B2 (H ))

.

Next we present Corollary 10.23 (to Theorem 10.18) All as in Theorem 10.18 for g1 (t) = g2 (t) = et . Then  ex01  ex02 ( f 1 ◦ log) (z 1 ) ( f 2 ◦ log) (z 2 )γ dz 1 dz 2  x  ≤ (e 01 −z 1 ) p(ν1 −1)+1 (e x02 −z 2 )q (ν2 −1)+1 e a1 e a2 + p( p(ν1 −1)+1) q(q(ν2 −1)+1) (e x01 − ea1 ) (e x02 − ea2 ) (10.62)  (ν1 )  (ν2 )        ν2  ν1    D . x02 − ( f 2 ◦ log)   Dex01 − ( f 1 ◦ log)γ  e γ  L [ea2 ,e x02 ],B (H ) L q ([ea1 ,e x01 ],Bγ (H )) ) p( γ We finish with Corollary 10.24 (to Theorem 10.20) All as in Theorem 10.20, with g1 (t) = g2 (t) = log t, where [ai , bi ] ⊂ R+ − {0}, i = 1, 2. Then 

log x01

log a1



log x02

log a2



      f 1 ◦ et (z 1 ) f 2 ◦ et (z 2 ) dz 1 dz 2 γ (log x01 −z 1 ) p((m 1 +1)ν1 −1)+1 p( p((m 1 +1)ν1 −1)+1)

 log

x01 a1

 log

+ x02 a2

(log x02 −z 2 )q ((m 2 +1)ν2 −1)+1 q(q((m 2 +1)ν2 −1)+1)



 ((m 1 + 1) ν1 )  ((m 2 + 1) ν2 )    (m 1 +1)ν1    t   D  log x01 − f 1 ◦ e γ 

L q ([log a1 ,log x01 ],Bγ (H ))

≤

   (m 2 +1)ν2    t   D  log x02 − f 2 ◦ e γ 

(10.63)

L p ([log a2 ,log x02 ],Bγ (H ))

.

References 1. Anastassiou, G.A.: Strong mixed and generalized fractional calculus for Banach space valued functions. Mat. Vesnik 69(3), 176–191 (2017) 2. Anastassiou, G.A.: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations. Springer, Heidelberg (2018) 3. Anastassiou, G.A.: Generalized Canavati Fractional Hilbert-Pachpatte type inequalities for Banach algebra valued functions. J. Comput. Anal. Appl. 30(1), 66–77 (2022) 4. Anastassiou, G.A.: γ- Schatten norm Generalized Canavati Fractional Hilbert-Pachpatte type inequalities for von Neumann-Schatten class Bγ (H ) valued functions. Analele Universitatii Oradea, Fasc. Matematica, accepted (2021)

References

253

5. Bellman, R.: Some inequalities for positive definite matrices. In: Beckenbach, E.F. (ed.), General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, pp. 89–90. Birkhauser, Basel (1980) 6. Chang, D.: A matrix trace inequality for products of Hermitian matrices. J. Math. Anal. Appl. 237, 721–725 (1999) 7. Coop, I.D.: On matrix trace inequalities and related topics for products of Hermitian matrix. J. Math. Anal. Appl. 188, 999–1001 (1994) 8. Dragomir, S.S.: p-Schatten norm inequalities of Ostrowski’s type. RGMIA Res. Rep. Coll. 24, Art. 108, 19 pp (2021) 9. Mikusinski, J.: The Bochner Integral. Academic, New York (1978) 10. Neudecker, H.: A matrix trace inequality. J. Math. Anal. Appl. 166, 302–303 (1992) 11. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991) 12. Simon, B.: Trace ideals and Their Applications. Cambridge University Press, Cambridge (1979) 13. Zagrebvov, V.A.: Gibbs Semigroups. Operator Theory: Advances and Applications, vol. 273. Birkhauser (2019)

Chapter 11

γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte Inequalities with von Neumann–Schatten Class Bγ (H) Valued Functions Using Ordinary Vectorial Derivatives

Using a generalized vectorial Taylor formula involving ordinary vector derivatives we establish mixed Ostrowski, Opial and Hilbert–Pachpatte type inequalities for several von Neumann–Schatten class Bγ (H ) valued functions. The estimates are with respect to all norms · p , 1 ≤ p ≤ ∞. We finish with applications. It follows [4].

11.1 Introduction Our main motivation is [3]. We mention a uniform mixed generalized Ostrowski type inequality for several functions that are Banach algebra valued. Theorem 11.1 ([3]) Let n ∈ N and f i ∈ C n ([a, b] , A), i = 1, ..., r ∈ N − {1}; where [a, b] ⊂ R and (A, ·) is a Banach algebra. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]). We assume that ( j)  f i ◦ g −1 (g (x0 )) = 0, j = 1, ..., n − 1; i = 1, ..., r ; where x0 ∈ [a, b] be fixed. Denote by E ( f 1 , ..., fr ) (x0 ) := ⎡ r  i=1

⎢ ⎢ ⎣

 a

⎞

⎛ b

⎜ ⎜ ⎝

r j=1 j=i

⎛  ⎟ ⎜ ⎜ f f j (x)⎟ d x − (x) ⎠ i ⎝

a

⎞

⎛ b

⎜ ⎜ ⎝

r j=1 j=i

⎞

⎤

⎟ ⎟ ⎥ ⎟ ⎥ f j (x)⎟ ⎠ d x ⎠ f i (x0 )⎦ .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_11

(11.1)

255

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

256

Then (1) E ( f 1 , ..., fr ) (x0 ) = ⎡

⎡

1 (n − 1)!

⎞

⎛

⎤

  r ⎢ r ⎟  g(x0 ) ⎢ x0 ⎜ ⎥  (n)  ⎟ ⎢ ⎢ ⎜ ⎥ f j (x)⎟ (z − g (x))n−1 f i ◦ g −1 (z) dz d x ⎥ + ⎢(−1)n ⎢ ⎜ ⎠ g(x) ⎣ ⎣ a ⎝ ⎦ i=1

j=1 j=i

⎡



⎢ ⎢ ⎣

⎞

⎛ b x0

r ⎟ ⎜ ⎜ f j (x)⎟ ⎠ ⎝ j=1 j=i



⎤⎤ (11.2)

g(x)

g(x0 )

 ⎥⎥  (n) ⎥ (g (x) − z)n−1 f i ◦ g −1 (z) dz d x ⎥ ⎦⎦ ,

and (2) E ( f 1 , ..., fr ) (x0 ) ≤

1 n!

⎧ ⎞ ⎞⎤ ⎡⎡ ⎛ ⎛ ⎪ ⎪   ⎪  x ⎜ r ⎢⎢ r ⎥ ⎜ ⎨   ⎟ ⎟  (n)  ⎟⎥ ⎢⎢ ⎜ 0⎜  f i ◦ g −1   f j (x)⎟ (g (x0 ) − g (a))n ⎜ ⎟ d x ⎟⎥ + ⎢⎢ ⎜     ⎪ ⎠ ⎠⎦ ⎣⎣ ⎝ a ⎝ ⎪ ∞,[g(a),g(x0 )] ⎪ j=1 ⎩i=1 j =i

⎞ ⎞⎤(11.3) ⎤⎫ ⎡ ⎛ ⎛ ⎪ ⎪      r  ⎬ ⎟ ⎟⎥⎥⎪ ⎢ ⎜ b ⎜ (n)   ⎟ ⎟ ⎥ ⎥ ⎢ ⎜ ⎜ −1 n     f d x f ◦ g − g (g (b) (x )) (x) ⎟ ⎟⎥⎥ . ⎢ i ⎜ ⎜ 0 j   ⎠ ⎠⎦⎦⎪ ⎣ ⎝ x0 ⎝ ⎪ ∞,[g(x0 ),g(b)] ⎪ j=1 ⎭ j=i

We also mention a left generalized Opial type inequality for ordinary vector valued derivatives: Theorem 11.2 ([3]) Let p, q > 1 : 1p + q1 = 1, and n ∈ N, f ∈ C n ([a, b] , A); where [a, b] ⊂ R and (A, ·) is a Banach algebra. Let g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]) . We assume that ( j)  f ◦ g −1 (g (x0 )) = 0, j = 0, 1, ..., n − 1; where x0 ∈ [a, b] be fixed. Then 

g(x) g(x0 )

   (n)    (z) dz ≤  f ◦ g −1 (z) f ◦ g −1 

(g (x) − g (x0 ))n+ p − q 1

1

1 1

2 q (n − 1)! [( p (n − 1) + 1) ( p (n − 1) + 2)] p for all x0 ≤ x ≤ b.

g(x)

g(x0 )

 q  q2    −1 (n) (z) dz ,  f ◦g (11.4)

11.1 Introduction

257

We also mention a left generalized Hilbert–Pachpatte inequality for ordinary vector valued derivatives. Theorem 11.3 ([3]) Let i = 1, 2; p, q > 1 : 1p + q1 = 1, and n i ∈ N, f i ∈ C ni ([ai , bi ] , A); where [ai , bi ] ⊂ R and (A, ·) is a Banach algebra. Let gi ∈ C 1 ([ai , bi ]), strictly increasing, such that gi−1 ∈ C ni ([gi (ai ) , gi (bi )]). We assume ( ji )  that f i ◦ gi−1 (gi (x0i )) = 0, ji = 0, 1, ..., n i − 1; where x0i ∈ [ai , bi ] be fixed. Then       g1 (b1 )  g2 (b2 )   f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2   ≤ (z 1 −g1 (x01 )) p(n 1 −1)+1 (z 2 −g2 (x02 ))q (n 2 −1)+1 g1 (x01 ) g2 (x02 ) + p( p(n 1 −1)+1) q(q(n 2 −1)+1) (g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 )) (n 1 − 1)! (n 2 − 1)!     (n 1 )  (n 2 )       f 1 ◦ g1−1   f 2 ◦ g2−1  L q ([g1 (x01 ),g1 (b1 )],A)

(11.5)

L p ([g2 (x02 ),g2 (b2 )],A)

.

We are motivated also by Dragomir [8] recent work: Let (H, ·, · ) be a complex Hilbert space and B (H ) the Banach algebra of all bounded linear operators on H . An operator A ∈ B (H ) is said to belong to the von Neumann–Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite " 1 !  A p := tr |A| p p < ∞. Assume that A : [a, b] → B p (H ), B : [a, b] → Bq (H ), p, q > 1 with are continuous and B is strongly differentiable on (a, b), then    

b

 A (t) B (t) dt −

a

  sup  B (t)q ×

t∈[a,b]

a

b

+

1 q

= 1,

  A (s) ds B (u)  ≤ 

1

⎧! #" $ b # 1 # ⎪ A (t) p dt, − a) + #u − a+b ⎪ a 2 (b 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ & β1 $  α1 % ⎪ β+1 β+1 b ⎪ ⎨ (u−a) +(b−u) A (t)αp , β+1

1 p

a

for α, β > 1 with α1 + β1 = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ & % ⎪ ⎪ ⎪ 1 (b − a)2 + u − a+b 2 sup A (t) , ⎪ p ⎩ 4 2

(11.6)

t∈[a,b]

for all u ∈ [a, b], an Ostrowski type inequality. Ostrowski type inequalities have great applications to integral approximations in Numerical Analysis.

258

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

We presented ([1], Chaps. 8, 9) mixed fractional Ostrowski inequalities for several functions for various norms. In this chapter we generalize [1], Chaps. 8, 9 for several B p (H ) valued functions by using ordinary vector valued derivatives and our integrals here are of Bochner type [9]. Opial-type inequalities are used a lot in proving uniqueness of solutions to differential equations and also to give upper bounds to their solutions. In this chapter we also derive Opial type inequalities for B p (H ) valued functions with respect to ordinary vector valued derivatives. Additionally we include in this article related to B p (H ) Hilbert–Pachpatte type inequalities, [11]. We finish the chapter with selective applications to Ostrowski, Opial and Hilbert–Pachpatte inequalities.

11.2 Background We use the following generalized vector Taylor’s formula: Theorem 11.4 ([2], p. 97) Let n ∈ N and f ∈ C n ([a, b] , X ), where [a, b] ⊂ R and (X, ·) is a Banach space. Let g ∈ C 1 ([a, b]), strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]). Let any x, y ∈ [a, b]. Then f (x) = f (y) +

n−1  (g (x) − g (y))i  i=1

1 + (n − 1)!



g(x)

g(y)

i!

f ◦ g −1

(i)

(g (y))

(11.7)

 (n) (g (x) − z)n−1 f ◦ g −1 (z) dz.

The derivatives here are defined similarly to the numerical ones, see [13], pp. 83–86. The above integral is of Bochner type [9], and so are the integrals in this work. By [2], p. 3, if f ∈ C ([a, b] , X ) then f is Bochner integrable.

11.3 About Basic Banach Algebras All here come from [12]. We need Definition 11.5 ([12], p. 245) A complex algebra is a vector space A over the complex field C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(11.8)

11.3 About Basic Banach Algebras

259

(x + y) z = x z + yz, x (y + z) = x y + x z,

(11.9)

α (x y) = (αx) y = x (αy) ,

(11.10)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A)

(11.11)

and if A contains a unit element e such that xe = ex = x (x ∈ A)

(11.12)

e = 1,

(11.13)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 11.6 Commutativity of A will be explicated stated when needed. There exists at most one e ∈ A that satisfies (11.12). Inequality (11.11) makes multiplication to be continuous, more precisely left and right continuous, see [12], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [12], pp. 247–248, Sect. 10.3. We also make Remark 11.7 Next we mention about integration of A-valued functions, see [12], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some $ compact Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [12], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   x f dμ = x f ( p) dμ ( p) (11.14) Q

Q

260

and

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …



  f dμ x = Q

f ( p) x dμ ( p) .

(11.15)

Q

The Bochner integrals we will involve in our chapter follow (11.14) and (11.15).

11.4

p-Schatten Norms Background

In this advanced section all come from [8]. Let (H, ·, · ) be a complex Hilbert space and B (H ) the Banach algebra of all bounded linear operators on H . If {ei }i∈I an orthonormal basis of H , we say that A ∈ B (H ) is of trace class if A1 :=



|A| ei , ei < ∞.

(11.16)

i∈I

The definition of A1 does not depend on the choice of the orthornormal basis {ei }i∈I . We denote by B1 (H ) the set of trace class operators in B (H ). We define the trace of a trace class operator A ∈ B1 (H ) to be tr (A) :=



Aei , ei ,

(11.17)

i∈I

where {ei }i∈I an orthonormal basis of H . Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series (11.17) converges absolutely and it is independent from the choice of basis. The following result collects some properties of the trace: Theorem 11.8 We have: (i) If A ∈ B1 (H ) then A∗ ∈ B1 (H ) and   tr A∗ = tr (A);

(11.18)

(ii) If A ∈ B1 (H ) and T ∈ B (H ), then AT, T A ∈ B1 (H ) and tr (AT ) = tr (T A) and |tr (AT )| ≤ A1 T  ;

(11.19)

(iii) tr (·) is a bounded linear functional on B1 (H ) with tr  = 1; (iv) If A, B ∈ B2 (H ) then AB, B A ∈ B1 (H ) and tr (AB) = tr (B A) ; (v) B f in (H ) (finite rank operators) is a dense subspace of B1 (H ) . An operator A ∈ B (H ) is said to belong to the von Neumann–Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite [15, pp. 60–64]

11.5 Main Results

261

" 1 !  A p := tr |A| p p < ∞, |A| p is an operator notation and not a power. For 1 < p < q < ∞ we have that B1 (H ) ⊂ B p (H ) ⊂ Bq (H ) ⊂ B (H )

(11.20)

A1 ≥  A p ≥  Aq ≥ A .

(11.21)

and For p ≥ 1 the functional · p is a norm on the ∗-ideal B p (H ), which is a Banach   algebra, and B p (H ) , · p is a Banach space. Also, see for instance [15, pp. 60–64], for p ≥ 1,   A p =  A∗  p , A ∈ B p (H )

(11.22)

 AB p ≤ A p B p , A, B ∈ B p (H )

(11.23)

and AB p ≤ A p B , B A p ≤ B A p , A ∈ B p (H ) , B ∈ B (H ) . (11.24) This implies that C AB p ≤ C A p B , A ∈ B p (H ) , B, C ∈ B (H ) .

(11.25)

In terms of p-Schatten norm we have the Hölder inequality for p, q > 1 with 1 =1: q (|tr (AB)| ≤) AB1 ≤ A p Bq , A ∈ B p (H ) , B ∈ Bq (H ) .

1 p

+

(11.26)

For the theory of trace functionals and their applications the interested reader is referred to [14, 15]. For some classical trace inequalities see [6, 7, 10], which are continuations of the work of Bellman [5].

11.5 Main Results We start with 1-2-Schatten norms mixed generalized Ostrowski type inequalities for several functions that are Banach algebra B2 (H ) ⊂ B (H ) valued. A uniform estimate follows.

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

262

Theorem 11.9 Let n ∈ N and f i ∈ C n ([a, b] , B2 (H )), i = 1, ..., r ∈ N − {1}; where [a, b] ⊂ R and B2 (H ) is a ∗-ideal, which (B2 (H ) , ·2 ) is a Banach algebra. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]). We ( j)  assume that f i ◦ g −1 (g (x0 )) = 0, j = 1, ..., n − 1; i = 1, ..., r ; where x0 ∈ [a, b] be fixed. Denote by E ( f 1 , ..., fr ) (x0 ) := ⎞ ⎞ ⎞ ⎤ ⎛ ⎛ ⎛   r r r  ⎟ ⎟ ⎟ ⎥ ⎢ b ⎜ ⎜ b ⎜ ⎟ f i (x) d x − ⎜ ⎥ ⎟ ⎢ ⎜ ⎜ f f j (x)⎟ (x) j ⎠ ⎠ d x ⎠ f i (x0 )⎦ . (11.27) ⎣ a ⎝ ⎝ a ⎝ ⎡

i=1

j=1 j=i

j=1 j=i

Then E ( f 1 , ..., fr ) (x0 )1 ≤

1 n!

⎧ ⎡⎡ ⎛ ⎛ ⎞ ⎞⎤ ⎪  ⎪   x0 r r ⎨ (n)     ⎢⎢ ⎜ ⎜ ⎟ ⎟⎥     f j (x) ⎟ d x ⎟⎥ + ⎢⎢ f i ◦ g −1 ⎜ (g (x0 ) − g (a))n ⎜   2⎠ ⎣⎣ ⎝ ⎝ ⎠⎦ ⎪ a ⎪ 2 ∞,[g(a),g(x0 )] j=1 ⎩ i=1 j =i

⎤⎤⎫ ⎡ ⎛ ⎛ ⎞ ⎞(11.28) ⎪ ⎪     r  ⎬ ⎢ ⎜ b ⎜ ⎟ ⎟⎥⎥⎪ (n)    ⎟ ⎥ ⎥ ⎢ ⎜ ⎜ ⎟ −1 n     f j (x) 2 ⎟ d x ⎟⎥⎥ . (g (b) − g (x0 )) ⎜ ⎢ f i ◦ g ⎜   ⎣ ⎝ x0 ⎝ ⎠ ⎠⎦⎦⎪ ⎪ 2 ∞,[g(x0 ),g(b)] ⎪ j=1 ⎭ j=i

( j)  Proof Let x0 ∈ [a, b] such that f i ◦ g −1 (g (x0 )) = 0, j = 1, ..., n − 1; i = 1, ..., r. Let x ∈ [a, x0 ], then by Theorem 11.4 we have 

1 f i (x) − f i (x0 ) = (n − 1)! =

(−1)n (n − 1)!



g(x)

g(x0 )

g(x0 ) g(x)

 (n) (g (x) − z)n−1 f i ◦ g −1 (z) dz

(11.29)

 (n) (z − g (x))n−1 f i ◦ g −1 (z) dz,

for i = 1, ..., r. And for x ∈ [x0 , b], then again by Theorem 11.4 we get 1 f i (x) − f i (x0 ) = (n − 1)! for i = 1, ..., r. We multiply (11.29) by





g(x) g(x0 )

'r j=1 j=i

 (n) (g (x) − z)n−1 f i ◦ g −1 (z) dz,

 f j (x) to get:

(11.30)

11.5 Main Results

263

⎞

⎛ ⎜ ⎜ ⎝ 

r j=1 j=i

⎞

⎛

⎟ ⎜ ⎜ f j (x)⎟ ⎠ f i (x) − ⎝

r j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x0 ) =



'r

f j (x) (−1)n 

j=1 j=i

(n − 1)!

g(x0 )

g(x)

 (n) (z − g (x))n−1 f i ◦ g −1 (z) dz,

(11.31)

∀ x ∈ [a, x0 ] ; for i = 1, ..., r. Similarly, we get (by (11.30)) ⎛ ⎜ ⎜ ⎝  'r j=1 j=i

⎞ r j=1 j=i

⎟ ⎜ ⎜ f j (x)⎟ ⎠ f i (x) − ⎝

 f j (x) 

(n − 1)!

g(x) g(x0 )

⎞

⎛ r j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

 (n) (g (x) − z)n−1 f i ◦ g −1 (z) dz,

(11.32)

∀ x ∈ [x0 , b] ; for i = 1, ..., r. Adding (11.31) and (11.32) as separate groups, we obtain ⎞

⎛

⎞

⎛

r r r r   ⎟ ⎟ ⎜ ⎜ ⎟ f i (x) − ⎜ ⎜ f f j (x)⎟ (x) j ⎠ ⎠ f i (x0 ) = ⎝ ⎝ i=1

j=1 j=i

i=1

⎞

⎛

r r ⎟ (−1)n  ⎜ ⎜ f j (x)⎟ ⎠ ⎝ (n − 1)! i=1 j=1



g(x0 )

g(x)

j=1 j=i

 (n) (z − g (x))n−1 f i ◦ g −1 (z) dz,

j=i

∀ x ∈ [a, x0 ] , and

⎞

⎛ r  i=1

⎜ ⎜ ⎝

r j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x) −

⎞

⎛ r  i=1

⎜ ⎜ ⎝

r j=1 j=i

⎟ f j (x)⎟ ⎠ f i (x0 ) =

(11.33)

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

264

⎞

⎛ r 

1 (n − 1)!

i=1

⎜ ⎜ ⎝

r j=1 j=i

⎟ f j (x)⎟ ⎠



g(x) g(x0 )

 (n) (g (x) − z)n−1 f i ◦ g −1 (z) dz,

(11.34)

∀ x ∈ [x0 , b] . Next, we integrate (11.33) and (11.34) with respect to x ∈ [a, b]. We have r  

⎛ x0

a

i=1

⎛

⎞

r r  ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ f i (x) d x − f (x) j ⎝ ⎝ ⎠ j=1 j=i

⎡

i=1

 a

⎛ x0

⎞

⎞

r ⎜ ⎟ ⎟ ⎜ ⎟ f j (x)⎟ ⎝ ⎠ d x ⎠ f i (x0 ) = j=1 j=i

(11.35) ⎤

⎞

⎛

   ⎜ r r ⎟  g(x0 ) ⎥  (n) (−1)n  ⎢ ⎟ ⎢ x0 ⎜ ⎥ n−1 −1 fi ◦ g f j (x)⎟ (z − g (x)) (z) dz d x ⎥ , ⎢ ⎜ ⎠ g(x) ⎣ a ⎝ ⎦ (n − 1)! i=1 j=1 j=i

and r   i=1

⎞

⎞ ⎞ ⎛  r r r b ⎜  ⎟ ⎟ ⎟ ⎜ b ⎜ ⎟ f i (x) d x − ⎟ ⎜ ⎜ ⎜ f f j (x)⎟ (x) j ⎠ ⎠ d x ⎠ f i (x0 ) = (11.36) ⎝ ⎝ ⎝ x x ⎛

0

j=1 j=i

i=1

⎡ 1 (n − 1)!

⎛

r  i=1

⎢ ⎢ ⎣

⎞

⎛



b x0

⎜ ⎜ ⎝

r j=1 j=i

⎟ f j (x)⎟ ⎠



g(x)

g(x0 )

0

j=1 j=i



⎤

⎥  (n) (g (x) − z)n−1 f i ◦ g −1 (z) dz d x ⎥ ⎦.

Finally, adding (11.35) and (11.36) we obtain the useful identity E ( f 1 , ..., fr ) (x0 ) = ⎡

⎞

⎛

⎛

⎞

⎛

⎞

⎤

r ⎢ b ⎜ r r ⎟ ⎜ b ⎜ ⎟ ⎟ ⎥  1 ⎟ ⎜ ⎟ ⎟ ⎥ ⎢ ⎜ ⎜ f j (x)⎟ f i (x) d x − ⎜ f j (x)⎟ d x ⎟ f i (x0 )⎥ = ⎢ ⎜ ⎜ ⎠ ⎝ a ⎝ ⎠ ⎠ ⎦ (n − 1)! ⎣ a ⎝ i=1 j=1 j=1 j=i

j=i

⎤   r ⎢ r ⎟  g(x0 ) ⎥ ⎢ x0 ⎜  (n)  ⎟ ⎥ ⎢ ⎢ ⎜ f j (x)⎟ (z − g (x))n−1 f i ◦ g −1 (z) dz d x ⎥ + ⎢(−1)n ⎢ ⎜ ⎠ g(x) ⎦ ⎣ ⎣ a ⎝ ⎡

i=1

⎡

⎞

⎛

j=1 j=i

11.5 Main Results

⎡ ⎢ ⎢ ⎣

⎞

⎛



b x0

⎜ ⎜ ⎝

265

r j=1 j=i

⎟ f j (x)⎟ ⎠



g(x) g(x0 )



⎤⎤

⎥⎥  (n) ⎥ (g (x) − z)n−1 f i ◦ g −1 (z) dz d x ⎥ ⎦⎦ , (11.37)

see also (11.2). Therefore, we get that E ( f 1 , ..., fr ) (x0 )1 =  ⎞ ⎞ ⎞ ⎤ ⎡ ⎛ ⎛ ⎛     r  r r  ⎢ b ⎜ ⎟ ⎟ ⎟ ⎥ ⎜ b ⎜ 1  ⎟ ⎟ ⎟ ⎥ ⎢ ⎜ ⎜ ⎜ f j (x)⎟ f i (x) d x − ⎜ f j (x)⎟ d x ⎟ f i (x0 )⎥ ≤  ⎢ ⎜ ⎜   ⎠ ⎠ ⎠ ⎦ ⎣ ⎝ ⎝ a ⎝ (n − 1)!  i=1 a j=1 j=1   j=i j=i 1

⎧ ⎞ ⎡⎡ ⎛ ⎤   ⎪   ⎪    r r ⎨ ⎢⎢ x0 ⎜ g(x0 )  ⎟ ⎥   n−1 −1 (n)  ⎟ ⎢⎢ ⎜ ⎥ f f ◦ g − g dz d x (x) (z (x)) (z) j i   ⎣ ⎠ ⎣ ⎝ ⎦ ⎪ g(x) a   ⎪ i=1 j=1 ⎩   j=i

⎡ ⎞ ⎛ ⎤      b   g(x)  ⎢ ⎟ ⎜ r ⎥   (n) n−1 −1 ⎢  ⎟ ⎜ ⎥ fi ◦ g + ⎣ f j (x)⎠ (g (x) − z) (z) dz d x ⎦  ⎝ g(x0 )   x0 j=1   j=i

1

⎤⎫ ⎪ ⎪ ⎥⎬ ⎥ ≤ ⎦⎪ ⎪ ⎭ 1

⎧  ⎛ ⎤ ⎡⎡ ⎞  ⎪    ⎪   r r ⎨   x g(x ) 0  ⎢⎢ 0  ⎜ (n)  ⎥ ⎟ 1 ⎜ ⎥ ⎢⎢ f j (x)⎟ (z − g (x))n−1 f i ◦ g −1 (z) dz   dx⎦ ⎝ ⎣⎣ ⎠ (n − 1)! ⎪ g(x) a   ⎪ j=1 ⎩ i=1   j =i

1

 ⎛ ⎞ ⎤⎤⎫   ⎪   ⎪   r  ⎬ ⎟  g(x) ⎢ b ⎜ ⎥⎥⎪  (n)  ⎜ ⎟ ⎢ ⎥⎥ (11.20) n−1 −1 fi ◦ g + ⎢ f j (x)⎟ ≤ (g (x) − z) (z) dz  d x ⎥⎥ ⎜  ⎠ g(x0 ) ⎣ x0 ⎝ ⎦⎦⎪ ⎪   j=1 ⎪ ⎭   j=i ⎡

1 (11.38) ⎧ ⎤ ⎡⎡ ⎛ ⎞ ⎪   ⎪    g(x0 )  x0 r r ⎨    (n)   ⎥  ⎢⎢ ⎜  ⎟ 1 −1 ⎥  f j (x) ⎟ ⎢⎢ ⎜ (z − g (x))n−1  (z)  dz d x ⎦  fi ◦ g 2⎠ ⎣⎣ ⎝ (n − 1)! ⎪ g(x) a ⎪ 2 j=1 ⎩ i=1 j =i

⎡ ⎢ + ⎢ ⎣



⎛ b x0

⎞

r   ⎟ ⎜  f j (x) ⎟ ⎜ 2⎠ ⎝ j=1 j =i



g(x) g(x0 )

⎤⎤⎫ ⎪   ⎪ (n)    ⎥⎥⎬ −1  dz d x ⎥⎥ =: (ξ) . f ◦ g (g (x) − z)n−1  (z) i   ⎦⎦⎪ ⎪ 2 ⎭

(11.39)

Hence it holds E ( f 1 , ..., fr ) (x0 )1 ≤ (ξ) .

(11.40)

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

266

We have that ⎧ ⎤ ⎡⎡ ⎛ ⎞ ⎪ ⎪ r    x0 r (n)    ⎥  ⎢ ⎜ ⎟  1 ⎨ ⎢    f j (x) ⎟ (g (x0 ) − g (x))n d x ⎥ ⎢⎢ f i ◦ g −1 ⎜ (ξ) ≤   2⎠ ⎦ ⎣⎣ ⎝ n! ⎪ a ⎪ 2 ∞,[g(a),g(x0 )] j=1 ⎩ i=1 j =i

⎡ ⎤⎤⎫ ⎛ ⎞ ⎪ ⎪    b ⎜ r  ⎬ ⎢ ⎥⎥⎪ ⎟ (n)       ⎢ ⎥⎥ ⎜ ⎟ −1 n     ≤ f − g + ⎢ f i ◦ g d x (g (x) (x (x) )) ⎥ ⎥ ⎜ ⎟ 0 j   2⎠ ⎣ ⎦⎦⎪ ⎪ 2 ∞,[g(x0 ),g(b)] x0 ⎝ j=1 ⎪ ⎭ j=i

(11.41) ⎧ ⎤ ⎡⎡ ⎛ ⎛ ⎞ ⎞ ⎪   ⎪    r r ⎨   x  0 (n)  ⎥ ⎢  ⎟ ⎢ ⎜ ⎜ ⎟     1 ⎢⎢ f i ◦ g −1 ⎜  f j (x) ⎟ d x ⎟⎥   (g (x0 ) − g (a))n ⎜   2⎠ ⎠⎦ ⎣⎣ ⎝ ⎝ n! ⎪ a ⎪ 2 ∞,[g(a),g(x0 )] j=1 ⎩ i=1 j =i

⎡

 (n)  ⎢   f i ◦ g −1  +⎢  ⎣

⎛

   

2 ∞,[g(x0 ),g(b)]

⎜ (g (b) − g (x0 ))n ⎜ ⎝

⎞ ⎞⎤⎤⎫ ⎪ ⎪ r ⎜   ⎟ ⎟⎥⎥⎬ ⎜  f j (x) ⎟ d x ⎟⎥⎥ , 2⎠ ⎝ ⎠⎦⎦⎪ ⎪ j=1 ⎭ ⎛



b x0

(11.42)

j =i



proving (11.28). Next comes an L 1 estimate. Theorem 11.10 All as in Theorem 11.9. Then E ( f 1 , ..., fr ) (x0 )1 ≤

1 (n − 1)!

⎧ ⎤ ⎡⎡ ⎛ ⎞ ⎪  ⎪   x0 r r ⎨ (n)    ⎥  ⎢⎢ ⎜ ⎟     f j (x) ⎟ (g (x0 ) − g (x))n−1 d x ⎥ ⎢⎢ f i ◦ g −1 ⎜   2⎠ ⎦ ⎣⎣ ⎝ ⎪ a ⎪ 2 L 1 ([g(a),g(x0 )]) j=1 ⎩ i=1 j =i

⎞ ⎤⎤⎫ ⎪ ⎪      r  ⎢ ⎥⎥⎪ ⎬ (n)   b ⎜  ⎟ ⎜ ⎟ ⎢ ⎥ ⎥ −1 n−1    f j (x) ⎟ (g (x) − g (x0 )) + ⎢ f i ◦ g d x . ⎜ ⎥ ⎥   2 ⎠ ⎣ ⎦⎦⎪ ⎪ 2 L 1 ([g(x0 ),g(b)]) x0 ⎝ j=1 ⎪ ⎭ ⎛

⎡

j =i

(11.43) Proof By (11.39), (11.40), we get that E ( f 1 , ..., fr ) (x0 )1 ≤ (ξ) ≤

1 − (n 1)!

⎧ ⎡⎡ ⎛ ⎞ ⎤ ⎪ ⎪     ⎪  r ⎢⎢ r  ⎨ ⎥ (n)   x0 ⎜  ⎟ ⎢⎢ ⎜ ⎥    f j (x) ⎟ (g (x0 ) − g (x))n−1 d x ⎥ ⎢⎢ f i ◦ g −1 ⎜   2⎟ ⎪ ⎣ ⎣ ⎝ ⎠ ⎦ a ⎪ 2 L g(a),g(x ) ]) ([ 1 0 ⎪ j=1 ⎩i=1 j =i

11.5 Main Results

267

⎛ ⎞ ⎤⎤⎫ ⎪ ⎪     b ⎜ r ⎬ ⎟ ⎢ ⎥⎥⎪       (n) ⎥ ⎜ ⎟ ⎢ ⎥ −1 n−1       f j (x) 2 ⎟ (g (x) − g (x0 )) + ⎢ f i ◦ g d x ⎥⎥ , ⎜   ⎠ ⎣ ⎦⎦⎪ ⎪ 2 L 1 ([g(x0 ),g(b)]) x0 ⎝ j=1 ⎪ ⎭ ⎡

j =i

(11.44) 

proving (11.43). An L p estimate follows. Theorem 11.11 All as in Theorem 11.9, and let p, q > 1 : E ( f 1 , ..., fr ) (x0 )1 ≤

1 p

= 1. Then

1 q

1 1

(n − 1)! ( p (n − 1) + 1) p

⎡

⎛   ⎜ x 0 (n)   n− 1 ⎢ ⎜   (g (x0 ) − g (x)) q ⎢ f i ◦ g −1 ⎜   ⎣ ⎝ a 2 L q ([g(a),g(x0 )]) i=1

 r ⎢  

⎛

⎞

   

⎜ b n− 1 ⎜ (g (x) − g (x0 )) q ⎜ ⎝ x0 2 L q ([g(x0 ),g(b)])

⎞

r  ⎜  ⎟ ⎟ ⎟ ⎜  f j (x) ⎟ dx⎟ ⎜ 2⎟ ⎝ ⎠ ⎠ j=1 j =i

⎛

⎛      −1 (n)   +  f i ◦ g 

+

⎞

⎞⎤

r  ⎜ ⎥  ⎟ ⎟ ⎜ ⎟⎥  f j (x) ⎟ d x ⎜ ⎟ ⎟ ⎥. 2⎠ ⎝ ⎠⎦ j=1 j =i

(11.45) Proof By (11.39), (11.40), we get that E ( f 1 , ..., fr ) (x0 )1 ≤ (ξ) ≤

1 (n − 1)!

⎧ ⎡⎡ ⎛ ⎞ ⎪ ⎪  g(x0 )  1p  x0 r r ⎨  ⎟ ⎢⎢ ⎜  p(n−1) ⎢⎢ ⎜  f j (x) ⎟ dz (z − g (x)) 2⎠ ⎣⎣ a ⎝ ⎪ g(x) ⎪ j=1 ⎩ i=1 j=i



g(x0 ) g(x)

⎡ ⎢ ⎢ ⎣



⎛ b x0

⎜ ⎜ ⎝



(  q  q1    −1 (n) (z) dz d x +  fi ◦ g 2

⎞ r j=1 j=i

  ⎟  f j (x) ⎟ 2⎠

g(x)

g(x0 )



g(x) g(x0 )

(g (x) − z)

 q  q1    −1 (n) (z) dz d x  fi ◦ g 2

 1p p(n−1)

dz

(() =

1 (n − 1)!

(11.46)

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

268

⎧ ⎡⎡ ⎛ ⎞ ⎪ ⎪ p(n−1)|+1 ⎪ r ⎢⎢ x0 ⎜ r  ⎨ p  ⎟ (g (x0 ) − g (x)) ⎢⎢ ⎜  f j (x) ⎟ ⎢⎢ ⎜ ⎟ 1 2 ⎪ ⎣⎣ a ⎝ ⎠ ⎪ ⎪ j=1 ( p (n − 1) + 1) p ⎩i=1

⎤  (n)     f i ◦ g −1   

⎥ ⎥ dx⎥ ⎦ 2 L q ([g(a),g(x0 )])

j =i

⎡

⎛

⎤⎤⎫ ⎪ ⎪ ⎥⎥⎪ ⎬ ⎥⎥ d x ⎥⎥ ⎦⎦⎪ ⎪ 2 L q ([g(x0 ),g(b)]) ⎪ ⎭

⎞

r  ⎢ b ⎜  ⎟ (g (x) − g (x0 )) ⎢ ⎜  f j (x) ⎟ +⎢ ⎜ 1 2⎟ ⎣ x0 ⎝ ⎠ j=1 ( p (n − 1) + 1) p

p(n−1)|+1 p

 (n)     f i ◦ g −1   

j =i

=

   

1 1

(n − 1)! ( p (n − 1) + 1) p

⎧ ⎡ ⎛ ⎪ ⎪  ⎪    r ⎨ ⎢ ⎜ x0 (n)   n− 1 ⎢ ⎜   (g (x0 ) − g (x)) q ⎢ f i ◦ g −1 ⎜   ⎪ ⎣ ⎝ a ⎪ 2 L q ([g(a),g(x0 )]) ⎪ ⎩i=1

⎛

⎞

   

⎜ b n− 1 ⎜ (g (x) − g (x0 )) q ⎜ ⎝ x0 2 L q ([g(x0 ),g(b)])

⎞

r  ⎜ ⎜

⎜ ⎝

j=1 j =i

 ⎟ ⎟ ⎟  f j (x) ⎟ dx⎟ 2⎟ ⎠ ⎠

⎞⎤⎫ ⎪ ⎪ ⎟⎥⎪ ⎬  ⎟ ⎟ ⎟⎥   f j (x) 2 ⎟ d x ⎟⎥ , ⎜ ⎝ ⎠ ⎠⎦⎪ ⎪ ⎪ j=1 ⎭ ⎛

⎛  (n)     f i ◦ g −1  +  

   

⎞

r  ⎜ ⎜ j =i

(11.47) 

proving (11.45).

We continue with mixed generalized Ostrowski type inequalities for several functions that are Banach algebra Bγ (H ) ⊂ B (H ), γ ≥ 1, valued. A uniform estimate follows.   Theorem 11.12 Let γ ≥ 1, n ∈ N and f i ∈ C n [a, b] , Bγ (H ) , i = 1,  ..., r ∈ N − {1}; where [a, b] ⊂ R and Bγ (H ) is a ∗-ideal, which Bγ (H ) , ·γ is a Banach algebra. Let g ∈ C 1 ([a, b]) , strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]). ( j)  We assume that f i ◦ g −1 (g (x0 )) = 0, j = 1, ..., n − 1; i = 1, ..., r ; where x0 ∈ [a, b] be fixed. Denote again by E ( f 1 , ..., fr ) (x0 ) := ⎞ ⎞ ⎞ ⎤ ⎛ ⎛ ⎛   r r r  ⎟ ⎟ ⎟ ⎥ ⎢ b ⎜ ⎜ b ⎜ ⎟ f i (x) d x − ⎜ ⎟ ⎥ ⎢ ⎜ ⎜ f f j (x)⎟ (x) j ⎠ ⎠ d x ⎠ f i (x0 )⎦ . (11.48) ⎣ a ⎝ ⎝ a ⎝ ⎡

i=1

j=1 j=i

j=1 j=i

Then E ( f 1 , ..., fr ) (x0 )γ ≤

1 n!

11.5 Main Results

269

⎧ ⎡⎡ ⎛ ⎛ ⎞ ⎞⎤  ⎪  ⎪  x0 r r  ⎨  (n)     ⎢⎢ ⎜ ⎜ ⎟ ⎟⎥   f i ◦ g −1    f j (x) ⎟ d x ⎟⎥ + ⎢⎢ ⎜ (g (x0 ) − g (a))n ⎜    γ⎠ ⎣⎣ ⎝ ⎝ ⎠⎦   ⎪ a ⎪ γ ∞,[g(a),g(x )] j=1 ⎩ i=1 0 j =i

⎫ ⎞⎤⎤(11.49) ⎪ ⎪ ⎪ r  ⎟ ⎥ ⎥ ⎜ b ⎜ ⎟  ⎟ ⎟⎥⎥⎬ ⎜ ⎜ n   f j (x) γ ⎟ d x ⎟⎥⎥ . (g (b) − g (x0 )) ⎜ ⎜ ⎝ x0 ⎝ ⎠ ⎠⎦⎦⎪ ⎪ ⎪ ∞,[g(x0 ),g(b)] j=1 ⎭

⎡

⎛

  ⎢  (n)     ⎢ −1   ⎢ f i ◦ g   ⎣ γ

⎛

⎞

j =i



Proof As similar to Theorem 11.9 is omitted, use of (11.23). An L 1 estimate follows: Theorem 11.13 All as in Theorem 11.12. Then E ( f 1 , ..., fr ) (x0 )γ ≤

1 (n − 1)!

⎧ ⎡⎡ ⎛ ⎞ ⎤ ⎪  ⎪    x0 r r ⎨     (n)     ⎟ ⎢⎢  ⎜ ⎥  f j (x) ⎟ (g (x0 ) − g (x))n−1 d x ⎥   ⎢⎢ f i ◦ g −1 ⎜   γ⎠ ⎣⎣  ⎝ ⎦ ⎪ ⎪ γ L ([g(a),g(x )]) a j=1 ⎩ i=1 1 0 j=i

⎡   ⎢  (n)    ⎢  −1   + ⎢ f i ◦ g   ⎣ γ

⎤⎤⎫ ⎛ ⎞ ⎪ ⎪ ⎪  b ⎜ r  ⎥⎥⎬ ⎟  ⎟ ⎥⎥ ⎜ n−1   d x ⎥⎥ . f j (x) γ ⎟ (g (x) − g (x0 )) ⎜ ⎦⎦⎪ ⎝ ⎠ ⎪ ⎪ L 1 ([g(x0 ),g(b)]) x0 j=1 ⎭ j =i

(11.50) 

Proof As similar to Theorem 11.10 is omitted. An L p estimate follows. Theorem 11.14 All as in Theorem 11.12, and let p, q > 1 : E ( f 1 , ..., fr ) (x0 )γ ≤ ⎡ r  i=1

 (n)  ⎢   ⎢  f i ◦ g −1    ⎣ 

    γ

   (n)      −1  +  f i ◦ g    γ

⎛

L q ([g(a),g(x 0 )])

L q ([g(x0 ),g(b)])

⎜ ⎜ ⎝



1 p

+

1 q

= 1. Then

1 1

(n − 1)! ( p (n − 1) + 1) p ⎛ x0

(g (x0 ) − g (x))

n− q1

a

⎜ ⎜ ⎝

⎞ r j=1 j=i

⎛

⎛

⎜ b n− 1 ⎜ (g (x) − g (x0 )) q ⎜ ⎝ x0

r  ⎜ ⎜

⎜ ⎝

  ⎟ ⎟  f j (x) ⎟ d x ⎟ γ⎠ ⎠ ⎞

j=1 j =i

⎞

⎞⎤

⎥  ⎟ ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ . γ⎟ ⎠ ⎠⎦

(11.51)

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

270



Proof As similar to Theorem 11.11 is omitted.

When r = 2 we derive the following p-Schatten norm operator related Ostrowski type inequalities. Theorem 11.15 Let p, q > 1 : 1p + q1 = 1, and let the ∗-ideals B p (H ), Bq (H ), for     which B p (H ) , · p , Bq (H ) , ·q are Banach algebras; n ∈ N, x0 ∈ [a, b] ⊂     R, A1 ∈ C n [a, b] , B p (H ) , A2 ∈ C n [a, b] , Bq (H ) ; g ∈ C 1 ([a, b]), strictly (k)  increasing, such that g −1 ∈ C n ([g (a) , g (b)]), with Ai ◦ g −1 (g (x0 )) = 0, k = 1, ..., n − 1; i = 1, 2. Then (1) 

b

 (A1 , A2 ) (x0 ) :=

 A2 (x) A1 (x) d x +

A1 (x) A2 (x) d x−

a



a



b

A2 (x) d x

b





b

A1 (x0 ) −

A1 (x) d x

a

A2 (x0 ) =

a

 g(x0 ) + x0 *  ,   1 n n−1 −1 (n) A1 ◦ g A2 (x) (z − g (x)) (z) dz d x + (−1) (n − 1)! g(x) a +



b

A2 (x)

g(x0 )

x0

+ (−1)n

x0

 A1 (x)

g(x0 ) g(x)

a

+



b

A1 (x) 1 γ

g(x)

 ,  (n) (g (x) − z)n−1 A1 ◦ g −1 (z) dz d x +  ,  (n) (z − g (x))n−1 A2 ◦ g −1 (z) dz d x + (11.52)

(g (x) − z)

n−1

g(x0 )

x0

(2) for γ, δ > 1 :

g(x)

+



A2 ◦ g

 −1 (n)



,.

(z) dz d x

= 1, we obtain

1 δ

 (A1 , A2 ) (x0 )1 ≤ ./       −1 (n)    A ◦ g   1  p /       −1 (n)       A 1 ◦ g p

 δ,[g(a),g(x0 )]

δ,[g(x0 ),g(b)]

1 1

(n − 1)! (γ (n − 1) + 1) γ x0

(  A2 (x)q (g (x0 ) − g (x))

n− 1δ

dx +

a



b

(  A2 (x)q (g (x) − g (x0 ))

n− 1δ

dx +

x0

(11.53)

11.5 Main Results

271

/       −1 (n)       A2 ◦ g q



/       −1 (n)       A2 ◦ g q



δ,[g(a),g(x0 )]

(

x0

A1 (x) p (g (x0 ) − g (x))

n− 1δ

dx +

a

δ,[g(x0 ),g(b)]

()

b

A1 (x) p (g (x) − g (x0 ))

n− 1δ

,

dx

x0

(3)  (A1 , A2 ) (x0 )1 ≤ ./       −1 (n)       A1 ◦ g p /       −1 (n)    A ◦ g   1  p /       −1 (n)    A ◦ g    2 q

 L 1 ([g(a),g(x0 )])

(

x0

A2 (x)q (g (x0 ) − g (x))n−1 d x +

a

 L 1 ([g(x0 ),g(b)])

(

b

A2 (x)q (g (x) − g (x0 ))

n−1

dx +

x0

 L 1 ([g(a),g(x0 )])

/       −1 (n)    A ◦ g   2  q

1 (n − 1)!

((11.54)

x0

A1 (x) p (g (x0 ) − g (x))n−1 d x +

a

 L 1 ([g(x0 ),g(b)])

b

() A1 (x) p (g (x) − g (x0 ))

n−1

,

dx

x0

and (4)  (A1 , A2 ) (x0 )1 ≤ ./       −1 (n)       A1 ◦ g p



∞,[g(a),g(x0 )]

/       −1 (n)    A ◦ g    1 p



b

x0

1 n! (

 A2 (x)q (g (x0 ) − g (x))n d x +

a

( A2 (x)q (g (x) − g (x0 )) d x + (11.55) n

∞,[g(x0 ),g(b)]

x0

/       −1 (n)    A ◦ g   2  q



/       −1 (n)       A2 ◦ g q



∞,[g(a),g(x0 )]

∞,[g(x0 ),g(b)]

x0

( A1 (x) p (g (x0 ) − g (x)) d x + n

a b

()  A1 (x) p (g (x) − g (x0 )) d x n

x0

Proof Writing (11.37) for r = 2 and for A1 , A2 , we derive

.

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

272





b

 ( A1 , A2 ) (x0 ) =

A2 (x) A1 (x) d x +

A1 (x) A2 (x) d x−

a



a



b

b

 A1 (x0 ) −

A2 (x) d x



b

A2 (x0 ) =

A1 (x) d x

a

a

 g(x0 ) + x0 *  ,  (n) 1 A2 (x) (z − g (x))n−1 A1 ◦ g −1 (z) dz d x + (−1)n (n − 1)! g(x) a +



b

A2 (x)

g(x0 )

x0

+ (−1)

x0

n

 A1 (x)

g(x0 ) g(x)

a

+



b

 ,  (n) (g (x) − z)n−1 A1 ◦ g −1 (z) dz d x +

g(x)

A1 (x)

g(x)

(z − g (x))

n−1



(g (x) − z)

n−1

g(x0 )

x0



A2 ◦ g

A2 ◦ g

 −1 (n)

 −1 (n)



,

(z) dz d x + (11.56) 

,,

(z) dz d x

proving (11.52). Hence it holds    (A1 , A2 ) (x0 )1 =  

b



a







b

A1 (x0 ) −

A2 (x) d x

A1 (x) A2 (x) d x− a



b

b

A2 (x) A1 (x) d x +

A1 (x) d x

a

a

  A2 (x0 )  = 1

.   ( /   (n) x0 g(x0 ) 1  A2 (x) (z − g (x))n−1 A1 ◦ g −1 (z) dz d x +  (−1)n (n − 1)!  g(x) a

+



b

A2 (x)

g(x0 )

x0

+ (−1)

x0

n

 A1 (x)

a

+



b

A1 (x) x0

g(x)

g(x0 ) g(x)

g(x) g(x0 )

 ,  (n) (g (x) − z)n−1 A1 ◦ g −1 (z) dz d x +

(z − g (x))

(g (x) − z)

n−1

n−1





A2 ◦ g

A2 ◦ g

 −1 (n)

 −1 (n)



,

(z) dz d x + (11.57) 

,-  (z) dz d x   ≤ 1

 g(x0 ) *+ x0  ,     1 n−1 −1 (n)  A A ◦ g − g dz dx  (z (x)) (z) 2 (x) 1   + (n − 1)! g(x) a 1

11.5 Main Results

+    +   

273



b

A2 (x)

A1 (x)



b

A1 (x)



A1 ◦ g

 −1 (n)



,  (z) dz d x   + 1

(11.58)

1

g(x)

g(x0 )

x0

(g (x) − z)

n−1

 ,   (n) (z − g (x))n−1 A2 ◦ g −1 (z) dz d x   +

g(x0 ) g(x)

a

+   

g(x0 )

x0



x0

g(x)

 ,   (n) (g (x) − z)n−1 A2 ◦ g −1 (z) dz d x   ≤ 1

 g(x0 ) *+ x0  ,      1 n−1 −1 (n)  A2 (x)  dx + A ◦ g − g dz (z (x)) (z) 1   (n − 1)! g(x) a 1 +

b x0

+

x0

a

    A2 (x) 

    A1 (x) 

+

b x0

g(x) g(x0 )

g(x0 )



A1 ◦ g

 −1 (n)

,    (z) dz  d x + 1

(z − g (x))

n−1

g(x)

    A1 (x) 

(g (x) − z)

n−1



A2 ◦ g

 −1 (n)

,    (z) dz  d x +

(11.59)

1

g(x)

g(x0 )

,   (n) d x ≤ (g (x) − z)n−1 A2 ◦ g −1 (z) dz   1

(by using the p-Schatten norm and Hölder’s type inequality (11.26 ) for p, q > 1 : 1 + q1 = 1) p 1 (n − 1)!

./ a

/

b x0

/

x0

a

x0

  A2 (x)q  

  A2 (x)q  

g(x0 )

  A1 (x) p  

/

b x0

1 (n − 1)!

g(x)

g(x0 ) g(x)

  A1 (x) p  

*+ a

x0

g(x0 ) g(x)

(z − g (x))

n−1

A1 ◦ g

 −1 (n)

(    (z) dz  d x + p

(g (x) − z)

n−1



A1 ◦ g

 −1 (n)

(   (z) dz   dx + p

(z − g (x))

n−1



A2 ◦ g

 −1 (n)

(   (z) dz   d x + (11.60) q

g(x)

(g (x) − z)

n−1

g(x0 )

 A2 (x)q





A2 ◦ g

 −1 (n)

()    ≤ (z) dz  d x q

g(x0 ) g(x)

 ,  (n)    (z − g (x))n−1  A1 ◦ g −1 (z) dz d x + p

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

274

+

b

 A2 (x)q

g(x0 )

x0

+

x0

 A1 (x) p b

g(x0 )

(g (x) − z)

n−1

 ,      −1 (n) (z) dz d x +  A1 ◦ g p

 ,  (n)    (z − g (x))n−1  A2 ◦ g −1 (z) dz d x + (11.61) q

g(x)

a

+

g(x)

 A1 (x) p

g(x)

g(x0 )

x0

 , (n)    (g (x) − z)n−1  A2 ◦ g −1 (z) dz d x . q

So far we have proved that  (A1 , A2 ) (x0 )1 ≤ 1 (n − 1)! +

*+

x0

 A2 (x)q 

A2 (x)q

g(x0 )

x0

+

x0

 A1 (x) p

+

b x0

 A1 (x) p

g(x0 )

g(x)

a

g(x)

g(x) g(x0 )

 ,  (n)    (z − g (x))n−1  A1 ◦ g −1 (z) dz d x + p

g(x)

a b

g(x0 )

(g (x) − z)

(z − g (x))

n−1

n−1

p

 ,      −1 (n) (z) dz d x + (11.62)  A2 ◦ g q

 , (n)    =: (λ) . (g (x) − z)n−1  A2 ◦ g −1 (z) dz d x

Let now γ, δ > 1 such that γ1 + in (11.62). Then, we have that

q

= 1, and we apply the usual Hölder’s inequality

1 δ

1

 ( A1 , A2 ) (x0 )1 ≤ (λ) ≤ ./

x0

A2 (x)q (g (x0 ) − g (x))

γ(n−1)+1 γ

1

(n − 1)! (γ (n − 1) + 1) γ



b

A2 (x)q (g (x) − g (x0 ))

γ(n−1)+1 γ



a

x0

A1 (x) p (g (x0 ) − g (x))

g(x) g(x0 )

x0

/

g(x0 )

γ(n−1)+1 γ



(  δ  1δ    −1 (n) (z) dz d x +  A1 ◦ g

g(x0 )

g(x)

(  δ  1δ    −1 (n) (z) dz d x +  A1 ◦ g p

g(x)

a

/

 ,      −1 (n) (z) dz d x +  A1 ◦ g

p

(11.63)  1δ (   (n) δ  (z) dz d x +  A2 ◦ g −1 q

11.5 Main Results

/

b

275

A1 (x) p (g (x) − g (x0 ))



γ(n−1)+1 γ

g(x0 )

x0

≤

g(x)

 δ  1δ    −1 (n) A ◦ g (z)  2  dz d x

1

(n − 1)! (γ (n − 1) + 1) γ 

/       −1 (n)       A1 ◦ g p



/       −1 (n)       A2 ◦ g q



/       −1 (n)    A ◦ g   2  q



δ,[g(a),g(x0 )]

(

x0

n− 1δ

 A2 (x)q (g (x0 ) − g (x))

dx +

a

δ,[g(x0 ),g(b)]

( (11.64)

b

A2 (x)q (g (x) − g (x0 ))

n− 1δ

dx +

x0

δ,[g(a),g(x0 )]

(

x0

A1 (x) p (g (x0 ) − g (x))

n− 1δ

dx +

a

δ,[g(x0 ),g(b)]

()

b

A1 (x) p (g (x) − g (x0 ))

 (A1 , A2 ) (x0 )1 ≤ (λ) ≤ ./       −1 (n)       A 1 ◦ g p



/       −1 (n)    A ◦ g    1 p



∞,[g(a),g(x0 )]

x0

b

1 (n − 1)! (

 A2 (x)q (g (x0 ) − g (x))n−1 d x + ( A2 (x)q (g (x) − g (x0 ))

n−1

dx +

x0



x0

((11.65) A1 (x) p (g (x0 ) − g (x))n−1 d x +

a

 L 1 ([g(x0 ),g(b)])

,

dx

a

L 1 ([g(x0 ),g(b)])

L 1 ([g(a),g(x0 )])

n− 1δ

x0

proving (11.53). We also obtain

/       −1 (n)    A ◦ g   2  q

q

1

./       −1 (n)       A1 ◦ g p

/       −1 (n)       A 2 ◦ g q

()

b

() A1 (x) p (g (x) − g (x0 ))

x0

proving (11.54). At last we derive  (A1 , A2 ) (x0 )1 ≤ (λ) ≤

1 n!

n−1

dx

,

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

276

./       −1 (n)       A1 ◦ g p



∞,[g(a),g(x0 )]

/       −1 (n)       A1 ◦ g p



b

x0

(  A2 (x)q (g (x0 ) − g (x)) d x + n

a

( A2 (x)q (g (x) − g (x0 )) d x + (11.66) n

∞,[g(x0 ),g(b)]

x0

/       −1 (n)    A ◦ g    2 q



/       −1 (n)    A ◦ g   2  q



∞,[g(a),g(x0 )]

x0

( A1 (x) p (g (x0 ) − g (x)) d x + n

a b

()  A1 (x) p (g (x) − g (x0 )) d x

,

n

∞,[g(x0 ),g(b)]

x0

proving (11.55). The theorem is proved.



Next we present a left generalized Opial type inequality involving 1-2-Schatten norms. Theorem 11.16 Let p, q > 1 : 1p + q1 = 1, and n ∈ N, f ∈ C n ([a, b] , B2 (H )); where [a, b] ⊂ R and B2 (H ) is the ∗-ideal. Let g ∈ C 1 ([a, b]), strictly increas( j)  ing, such that g −1 ∈ C n ([g (a) , g (b)]) . We assume that f ◦ g −1 (g (x0 )) = 0, j = 0, 1, ..., n − 1; where x0 ∈ [a, b] be fixed. Then 

g(x)

g(x0 )

   (n)    (z) dz ≤  f ◦ g −1 (z) f ◦ g −1 1



(g (x) − g (x0 ))n+ p − q 1

1

1

1

2 q (n − 1)! [( p (n − 1) + 1) ( p (n − 1) + 2)] p

g(x)

g(x0 )

 q  q2    −1 (n) (z) dz ,  f ◦g 2

(11.67)

for all x0 ≤ x ≤ b.  ( j) Proof Let x0 ∈ [a, b] such that f ◦ g −1 (g (x0 )) = 0, j = 0, 1, ..., n − 1. For x ∈ [x0 , b] by Theorem 11.4 we have 

 f ◦ g −1 (g (z)) =

1 (n − 1)!



g(x)

g(x0 )

 (n) (g (x) − z)n−1 f ◦ g −1 (z) dz.

(11.68)

By Hölder’s inequality we obtain     f ◦ g −1 (g (x)) ≤ 2

1 (n − 1)!



g(x)

g(x0 )

 (n)    (g (x) − z)n−1  f ◦ g −1 (z) dz ≤ 2

(11.69)

11.5 Main Results



1 (n − 1)!

277 g(x)

g(x0 )

(g (x) − z) p(n−1) dt p(n−1)+1

1 (g (x) − g (x0 )) p (n − 1)! ( p (n − 1) + 1) 1p Call

 ϕ (g (x)) :=

ϕ (g (x0 )) = 0. Thus

and

g(x) g(x0 )

 1p 

g(x) g(x0 )



g(x) g(x0 )

 q  q1    −1 (n) f ◦ g = (z)   dz 2

 q  q1    −1 (n) (z) dz .  f ◦g 2

 (n)   q (z) dz,  f ◦ g −1 2

q (n) dϕ (g (x))    =  f ◦ g −1 (g (x)) ≥ 0, 2 dg (x) 

dϕ (g (x)) dg (x)

 q1

  (n)   =  f ◦ g −1 (g (x)) ≥ 0, 2

(11.70)

(11.71)

(11.72)

∀ g (x) ∈ [g (x0 ) , g (b)] . Consequently, we get        −1 (n)  f ◦ g −1 (g (w))  f ◦ g (g (w))   ≤ 2 2

 1 dϕ (g (w)) q ϕ (g (w)) , 1 dg (w) (n − 1)! ( p (n − 1) + 1) p (g (w) − g (x0 ))

p(n−1)+1 p

(11.73)

∀ g (w) ∈ [g (x0 ) , g (b)] . Then we observe that 

g(x)

g(x0 )



g(x)

g(x0 )

    (n) (11.26)   (g (w)) dg (w) ≤  f ◦ g −1 (g (w)) f ◦ g −1 1

       −1 (n)  f ◦ g −1 (g (w))  f ◦ g (g (w))   dg (w) ≤ 2 2

1 1

(n − 1)! ( p (n − 1) + 1) p 

g(x) g(x0 )

(g (w) − g (x0 ))

p(n−1)+1 p

 1 dϕ (g (w)) q ϕ (g (w)) dg (w) ≤ dg (w)

(11.74)

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

278

1 1

(n − 1)! ( p (n − 1) + 1) p  g(x) g(x0 )

(g (w) − g (x0 )) p(n−1)+1 dg (w)

 1  p g(x)

dϕ (g (w)) dg (w) ϕ (g (w)) dg (w) g(x0 )

1 q

=

1 1

1

(n − 1)! ( p (n − 1) + 1) p ( p (n − 1) + 2) p (g (x) − g (x0 ))



p(n−1)+2 p

g(x)

g(x0 )

ϕ (g (w)) dϕ (g (w)) 

(g (x) − g (x0 ))n+ p − q 1

1

1

1

(n − 1)! ( p (n − 1) + 1) p ( p (n − 1) + 2) p 

(g (x) − g (x0 ))n+ p − q 1

1

1

1

2 q (n − 1)! (( p (n − 1) + 1) ( p (n − 1) + 2)) p

g(x) g(x0 )

 q1

=

ϕ2 (g (x)) 2

(11.75)  q1

=

 q  q2    −1 (n) (z) dz ,  f ◦g 2

(11.76) 

for all g (x0 ) ≤ g (x) ≤ g (b), proving (11.67).

The corresponding B2 (H ) right generalized Opial type inequality follows: Theorem 11.17 All as in Theorem 11.16. Then 

g(x0 ) g(x)

   (n)    (z) dz ≤  f ◦ g −1 (z) f ◦ g −1 1



(g (x0 ) − g (x))n+ p − q 1

1

1 1

2 q (n − 1)! (( p (n − 1) + 1) ( p (n − 1) + 2)) p

g(x0 )

g(x)

 q  q2    −1 (n) (z) dz ,  f ◦g 2

(11.77)

for all a ≤ x ≤ x0 . 

Proof As similar to Theorem 11.16 is omitted. A Bγ (H ), γ ≥ 1, left Opial inequality follows:

n ∈ N, f ∈ Theorem 11.18 Let γ ≥ 1, p, q > 1 : 1p + q1 = 1, and   n 1 C [a, b] , Bγ (H ) ; where [a, b] ⊂ R and Bγ (H ) is the ∗-ideal. Let g ∈ C ([a, b]), strictly increasing, such that g −1 ∈ C n ([g (a) , g (b)]) . We assume that ( j)  f ◦ g −1 (g (x0 )) = 0, j = 0, 1, ..., n − 1; where x0 ∈ [a, b] be fixed. Then 

g(x) g(x0 )

   (n)    (z) dz ≤  f ◦ g −1 (z) f ◦ g −1 γ

11.5 Main Results

279



(g (x) − g (x0 ))n+ p − q 1

1

1

1

2 q (n − 1)! [( p (n − 1) + 1) ( p (n − 1) + 2)] p

g(x)

g(x0 )

 q  q2    −1 (n) (z) dz ,  f ◦g γ

(11.78)

for all x0 ≤ x ≤ b. 

Proof As similar to Theorem 11.16 is omitted. Use of (11.23). A Bγ (H ), γ ≥ 1, right Opial inequality follows: Theorem 11.19 All as in Theorem 11.18. Then 

g(x0 )

g(x)

   (n)    (z) dz ≤  f ◦ g −1 (z) f ◦ g −1 γ



(g (x0 ) − g (x))n+ p − q 1

1

1

1

2 q (n − 1)! (( p (n − 1) + 1) ( p (n − 1) + 2)) p

g(x0 )

g(x)

 q  q2    −1 (n) (z) dz ,  f ◦g γ

(11.79)

for all a ≤ x ≤ x0 . 

Proof As similar to Theorem 11.16 is omitted.

Next we present a B2 (H ) left generalized Hilbert–Pachpatte inequality for ordinary derivatives. Theorem 11.20 Let i = 1, 2; p, q > 1 : 1p + q1 = 1, and n i ∈ N, f i ∈ C ni ([ai , bi ] , B2 (H )); where [ai , bi ] ⊂ R and B2 (H ) is the ∗-ideal. Let gi ∈ C 1 ([ai , bi ]), strictly increasing, such that gi−1 ∈ C ni ([gi (ai ) , gi (bi )]). We ( ji )  assume that f i ◦ gi−1 (gi (x0i )) = 0, ji = 0, 1, ..., n i − 1; where x0i ∈ [ai , bi ] be fixed. Then 

g1 (b1 ) g1 (x01 )



g2 (b2 )

g2 (x02 )

      f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 1   ≤ (z 1 −g1 (x01 )) p(n 1 −1)+1 (z 2 −g2 (x02 ))q (n 2 −1)+1 + p( p(n 1 −1)+1) q(q(n 2 −1)+1)

(g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 )) (n 1 − 1)! (n 2 − 1)!     (n 1 )  (n 2 )         f 1 ◦ g1−1    f 2 ◦ g2−1   2 L q ([g1 (x01 ),g1 (b1 )],B2 (H ))

Proof Let i = 1, 2; x0 ∈ [ai , bi ], such that 0, 1, ..., n i − 1. For xi ∈ [x0i , bi ] by Theorem 11.4 we have

(11.80)

2 L p ([g2 (x02 ),g2 (b2 )],B2 (H ))



f i ◦ gi−1

( ji )

.

(gi (x0i )) = 0, ji =

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

280



fi ◦

gi−1



1 (gi (xi )) = (n i − 1)!



gi (xi )

gi (x0i )

 (ni ) (gi (xi ) − z i )ni −1 f i ◦ gi−1 (z i ) dz i . (11.81)

As in (11.69) we have     f 1 ◦ g −1 (g1 (x1 )) ≤ 1 2 

g1 (x1 ) g1 (x01 )

p (n 1 −1)+1

1 (g1 (x1 ) − g1 (x01 )) p 1 (n 1 − 1)! ( p (n 1 − 1) + 1) p

 q  q1    −1 (n 1 ) ≤ (z) dz  f 1 ◦ g1 2

p (n 1 −1)+1

  (n 1 )      f 1 ◦ g1−1  

1 (g1 (x1 ) − g1 (x01 )) p 1 (n 1 − 1)! ( p (n 1 − 1) + 1) p

2 L q ([g1 (x01 ),g1 (b1 )])

,

(11.82)

for all x1 ∈ [x01 , b1 ] . Similarly, we obtain that     f 2 ◦ g −1 (g2 (x2 )) ≤ 2 2

q (n 2 −1)+1

1 (g2 (x2 ) − g2 (x02 )) q 1 (n 2 − 1)! (q (n 2 − 1) + 1) q

  (n 2 )      f 2 ◦ g2−1  

2 L p ([g2 (x02 ),g2 (b2 )])

,

(11.83)

for all x2 ∈ [x02 , b2 ] . By (11.82) and (11.83) we get       f 1 ◦ g −1 (g1 (x1 )) f 2 ◦ g −1 (g2 (x2 )) 1

2

       f 1 ◦ g −1 (g1 (x1 ))  f 2 ◦ g −1 (g2 (x2 )) ≤ 1 2 2 2 (g1 (x1 ) − g1 (x01 ))

p (n 1 −1)+1 p

2 L q ([g1 (x01 ),g1 (b1 )])

≤

1 (n 1 − 1)! (n 2 − 1)!

(g2 (x2 ) − g2 (x02 ))

1

( p (n 1 − 1) + 1) p   (n 1 )      f 1 ◦ g1−1  

(11.26)

1

q (n 2 −1)+1 q 1

(q (n 2 − 1) + 1) q   (n 2 )      f 2 ◦ g2−1  

2 L p ([g2 (x02 ),g2 (b2 )])

1

1

(using Young’s inequality for a, b ≥ 0, a p b q ≤

a p

+ qb )

(11.84) ≤

11.5 Main Results

281

 (g2 (x2 ) − g2 (x02 ))q(n 2 −1)+1 (g1 (x1 ) − g1 (x01 )) p(n 1 −1)+1 + p ( p (n 1 − 1) + 1) q (q (n 2 − 1) + 1) (11.85)           −1 (n 1 )   −1 (n 2 )   ,    f 2 ◦ g2    f 1 ◦ g1

1 (n 1 − 1)! (n 2 − 1)!



2 L q ([g1 (x01 ),g1 (b1 )])

2 L p ([g2 (x02 ),g2 (b2 )])

∀ (x1 , x2 ) ∈ [x01 , b1 ] × [x02 , b2 ] . So far we have       f 1 ◦ g −1 (g1 (x1 )) f 2 ◦ g −1 (g2 (x2 )) 1 2 1  ≤ (g1 (x1 )−g1 (x01 )) p(n 1 −1)+1 (g2 (x2 )−g2 (x02 ))q (n 2 −1)+1 + p( p(n 1 −1)+1) q(q(n 2 −1)+1)

(11.86)

  (n 1 )  1     f 1 ◦ g1−1   2 L q ([g1 (x01 ),g1 (b1 )],B2 (H )) (n 1 − 1)! (n 2 − 1)!   (n 2 )      f 2 ◦ g2−1  

2 L p ([g2 (x02 ),g2 (b2 )],B2 (H ))

,

∀ (x1 , x2 ) ∈ [x01 , b1 ] × [x02 , b2 ] . The denominator in (11.86) can be zero, only when both g1 (x1 ) = g1 (x01 ) and g2 (x2 ) = g2 (x02 ) . Therefore we obtain (11.80), by integrating (11.86) over [g1 (x01 ) , g1 (b1 )] ×  [g2 (x02 ) , g2 (b2 )] . It follows the B2 (H ) right generalized Hilbert–Pachpate inequality for ordinary derivatives. Theorem 11.21 All as in Theorem 11.20. Then       g1 (x01 )  g2 (x02 )   f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 1   ≤ (g1 (x01 )−z 1 ) p(n 1 −1)+1 (g2 (x02 )−z 2 )q (n 2 −1)+1 g1 (a1 ) g2 (a2 ) + p( p(n 1 −1)+1) q(q(n 2 −1)+1) (g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 )) (n 1 − 1)! (n 2 − 1)!      (n 1 )   (n 2 )       f 1 ◦ g1−1    f 2 ◦ g2−1   2 L q ([g1 (a1 ),g1 (x01 )],B2 (H ))

Proof As similar to Theorem 11.20 is omitted.

(11.87)

2 L p ([g2 (a2 ),g2 (x02 )],B2 (H ))

.



Next we present a Bγ (H ), γ ≥ 1, left generalized Hilbert–Pachpatte inequality for ordinary derivatives. Theorem 11.22 Let γ ≥ 1, i = 1, 2; p, q > 1 : 1p + q1 = 1, and n i ∈ N, f i ∈   C ni [ai , bi ] , Bγ (H ) ; where [ai , bi ] ⊂ R and Bγ (H ) is the ∗-ideal. Let gi ∈

282

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

C 1 ([ai , bi ]), strictly increasing, such that gi−1 ∈ C ni ([gi (ai ) , gi (bi )]). We assume ( ji )  that f i ◦ gi−1 (gi (x0i )) = 0, ji = 0, 1, ..., n i − 1; where x0i ∈ [ai , bi ] be fixed. Then       g1 (b1 )  g2 (b2 )   f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 γ   ≤ p (n 1 −1)+1 q (n 2 −1)+1 −g −g (z (x )) (z (x )) 1 1 01 2 2 02 g1 (x01 ) g2 (x02 ) + q(q(n 2 −1)+1) p( p(n 1 −1)+1) (g1 (b1 ) − g1 (x01 )) (g2 (b2 ) − g2 (x02 )) (n 1 − 1)! (n 2 − 1)!        −1 (n 1 )     f 1 ◦ g1    γ

       −1 (n 2 )     f 2 ◦ g2    γ

 L q [g1 (x01 ),g1 (b1 )],Bγ (H )

(11.88)

  L p [g2 (x02 ),g2 (b2 )],Bγ (H )

.



Proof As similar to Theorem 11.20 is omitted. Use of (11.23).

It follows the Bγ (H ), γ ≥ 1, right generalized Hilbert–Pachpate inequality for ordinary derivatives. Theorem 11.23 All as in Theorem 11.22. Then       g1 (x01 )  g2 (x02 )   f 1 ◦ g −1 (z 1 ) f 2 ◦ g −1 (z 2 ) dz 1 dz 2 1 2 γ   ≤ (g1 (x01 )−z 1 ) p(n 1 −1)+1 (g2 (x02 )−z 2 )q (n 2 −1)+1 g1 (a1 ) g2 (a2 ) + p( p(n 1 −1)+1) q(q(n 2 −1)+1) (g1 (x01 ) − g1 (a1 )) (g2 (x02 ) − g2 (a2 )) (n 1 − 1)! (n 2 − 1)!   (n 1 )      f 1 ◦ g1−1  

    γ

L q ([g1 (a1 ),g1 (x 01 )],Bγ (H ))

    γ

  (n 2 )      f 2 ◦ g2−1  

(11.89)

. L p ([g2 (a2 ),g2 (x 02 )],Bγ (H ))



Proof As similar to Theorem 11.20 is omitted.

11.6 Applications We start with B2 (H ) Ostrowski type inequalities. Corollary 11.24 (to Theorem 11.15) All as in Theorem 11.15, with g (t) = t. Then (1) for γ, δ > 1 : γ1 + 1δ = 1, we have  (A1 , A2 ) (x0 )1 ≤

1 1

(n − 1)! (γ (n − 1) + 1) γ

11.6 Applications

283

./     (n)     A1  p 



δ,[a,x0 ]

(

x0

A2 (x)q (x0 − x)

n− 1δ

dx +

(11.90)

a

/     (n)     A1  p 



/     (n)    A  2 q 



/     (n)    A   2 q 



δ,[x0 ,b]

(

b

A2 (x)q (x − x0 )

n− 1δ

dx +

x0

δ,[a,x0 ]

(

x0

A1 (x) p (x0 − x)

n− 1δ

dx +

a

δ,[x0 ,b]

()

b

A1 (x) p (x − x0 )

n− 1δ

,

dx

x0

(2)  (A1 , A2 ) (x0 )1 ≤ ./     (n)    A1  p  /     (n)    A1  p 

 L 1 ([a,x0 ])

 L 1 ([x0 ,b])

/     (n)    A2 q 

(

x0

A2 (x)q (x0 − x)

n−1

dx +

a

(

b

A2 (x)q (x − x0 )

n−1

dx +

(11.91)

x0

 L 1 ([a,x0 ])

/     (n)    A )   2 q 

1 (n − 1)!

(

x0

 A1 (x) p (x0 − x)

n−1

dx +

a

 L 1 ([x0 ,b])

()

b

 A1 (x) p (x − x0 )

n−1

dx

,

x0

and (3)  (A1 , A2 ) (x0 )1 ≤ ./     (n)     A1  p 



/     (n)    A  1  p 



∞,[a,x0 ]

∞,[x0 ,b]

x0

1 n! (

 A2 (x)q (x0 − x) d x + n

a b

(  A2 (x)q (x − x0 ) d x + n

x0

(11.92)

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

284

/     (n)     A2 q 



/     (n)     A2 )q 



∞,[a,x0 ]

(

x0

A1 (x) p (x0 − x) d x + n

a

()

b

A1 (x) p (x − x0 ) d x n

∞,[x0 ,b]

.

x0

We continue with Bγ (H ), γ ≥ 1, Ostrowski type inequalities. Corollary 11.25 (to Theorems 11.12–11.14) All as in Theorem 11.12, with g (t) = et . Then (1) . r ++   1     E ( f 1 , ..., fr ) (x0 )γ ≤ ( f i ◦ log)(n) γ  ∞,[ea ,e x0 ] n! i=1 ⎛  x0 n ⎜ e − ea ⎜ ⎝

 a

⎛ x0

⎜ ⎜ ⎝

⎞ r j=1 j=i

⎞⎤

 ⎟ ⎟⎥   f j (x) ⎟ d x ⎟⎥ + γ⎠ ⎠⎦

⎞⎤⎤⎫ ⎪ ⎪    ⎢  ⎟ ⎟⎥⎥⎬ ⎜ b⎜   b  n   x (n) 0 ⎢( f i ◦ log)   ⎜ ⎜  f j (x) ⎟ d x ⎟⎥⎥ , e − e γ ∞, e x0 ,eb γ⎠ ⎣ ⎝ x ⎝ ⎠⎦⎦⎪ [ ] ⎪ 0 j=1 ⎭ ⎡

⎛



⎛

⎞

r

j=i

(11.93) (2) . r ++     1   E ( f 1 , ..., fr ) (x0 )γ ≤ ( f i ◦ log)(n) γ  L 1 ([ea ,e x0 ]) − 1)! (n i=1 ⎛



x0

a

⎡    ⎢ ( f i ◦ log)(n)   ⎢  γ ⎣

⎞

⎤

r   ⎟ ⎜ ⎥  ⎜  f j (x) ⎟ e x0 − e x n−1 d x ⎥ + γ⎠ ⎝ ⎦ j=1 j=i

⎤⎤⎫ ⎪ ⎪  b  ⎟ ⎜ r  ⎥⎥⎬  ⎜  f j (x) ⎟ e x − e x0 n−1 d x ⎥⎥ , γ⎠ ⎦⎦⎪ L 1 ([e x0 ,eb ]) x0 ⎝ ⎪ j=1 ⎭ ⎛

⎞

j=i

(11.94) and (3) if p, q > 1 :

1 p

+

1 q

= 1, we have

11.6 Applications

285

E ( f 1 , ..., fr ) (x0 )γ ≤

1 1

(n − 1)! ( p (n − 1) + 1) p

⎡

⎛  r   ⎢ ⎜ x0  x   n− q1 ( f i ◦ log)(n)   ⎢ ⎜ e 0 − ex   γ L ([ea ,e x0 ]) ⎝ ⎣ q a

⎛

⎛ L q ([

e x0 ,eb

⎜ ⎜ ]) ⎝



j=1 j=i

⎛ b x0

⎞

r ⎜   ⎟ ⎟ ⎜  f j (x) ⎟ d x ⎟ + γ⎠ ⎠ ⎝

i=1

      ( f i ◦ log)(n) γ 

⎞

 x n− q1 ⎜ ⎜ e − e x0 ⎝

⎞ r j=1 j=i

⎞⎤

  ⎟ ⎟⎥  f j (x) ⎟ d x ⎟⎥ . γ⎠ ⎠⎦ (11.95)

We continue with a B2 (H ) Opial type inequality. Corollary 11.26 (to Theorem 11.16) All as in Theorem 11.16 with g (t) = t. Then 

x x0

   f (z) f (n) (z) dz ≤ 1 

(x − x0 )n+ p − q 1

1

1

z

1

2 q (n − 1)! [( p (n − 1) + 1) ( p (n − 1) + 2)] p

 (n) q  f (z) dz 2

x0

 q2

, (11.96)

for all x0 ≤ x ≤ b. Next comes a Bγ (H ), γ ≥ 1, Opial type inequality. Corollary 11.27 (to Theorem 11.18) All as in Theorem 11.18 with g (t) = et . Then 

ex e x0

  (( f ◦ log) (z)) ( f ◦ log)(n) (z) dz ≤ γ 

(e x − e x0 )n+ p − q 1

1

1 1

2 q (n − 1)! [( p (n − 1) + 1) ( p (n − 1) + 2)] p

ex e x0

  ( f ◦ log)(n) (z)q dz γ

 q2 ,

(11.97)

for all x0 ≤ x ≤ b. A B2 (H ) Hilbert–Pachpatte type inequality follows. Corollary 11.28 (to Theorem 11.20) All as in Theorem 11.20 for g1 (t) = g2 (t) = t. Then  b1  b2  f 1 (z 1 ) f 2 (z 2 )1 dz 1 dz 2  ≤ p (n 1 −1)+1 −x (z (z 2 −x02 )q (n 2 −1)+1 1 01 ) x01 x02 + p( p(n 1 −1)+1) q(q(n 2 −1)+1)

11 γ-Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte …

286

      (b1 − x01 ) (b2 − x02 )   (n 2 )    (n 1 )   .  f 2    f 1   2 L q ([x01 ,b1 ],B2 (H )) 2 L p ([x02 ,b2 ],B2 (H )) (n 1 − 1)! (n 2 − 1)! (11.98) We finish with a Bγ (H ), γ ≥ 1, Hilbert–Pachpatte type inequality. Corollary 11.29 (to Theorem 11.22) All as in Theorem 11.22 for g1 (t) = g2 (t) = log t, and [ai , bi ] ⊂ R+ − {0}, i = 1, 2. Then 

log b1 log x01



log b2

log x02

      f 1 ◦ et (z 1 ) f 2 ◦ et (z 2 ) dz 1 dz 2 γ  ≤ (z 1 −log x01 ) p(n 1 −1)+1 (z 2 −log x02 )q (n 2 −1)+1 + p( p(n 1 −1)+1) q(q(n 2 −1)+1)  log

b1 x01

 log

b2 x02



(11.99) (n 1 − 1)! (n 2 − 1)!             t (n 2 )    f 1 ◦ et (n 1 )    .       f 2 ◦ e γ L γ L q ([log x 01 ,log b1 ],Bγ (H )) p ([log x 02 ,log b2 ],Bγ (H ))

References 1. Anastassiou, G.A.: Intelligent Comparisons: Analytic Inequalities. Springer, Heidelberg (2016) 2. Anastassiou, G.A.: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations, Springer, Heidelberg (2018) 3. Anastassiou, G.A.: Generalized Ostrowski, Opial and Hilbert-Pachpatte type inequalities fot Banach algebra valued functions involving integer vectorial derivatives. J. Comput. Anal. Appl. 30(1), 78–94 (2022) 4. Anastassiou, G.A.: γ-Schatten norm Generalized Ostrowski, Opial and Hilbert-Pachpatte type inequalities for von Neumann-Schatten class Bγ (H ) valued functions with integer vectorial derivatives. J. Appl. Math. Comput. 2(1), 32–57 (2021) 5. Bellman, R.: Some inequalities for positive definite matrices. In: Beckenbach, E.F. (ed.), General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, pp. 89-90. Birkhauser, Basel (1980) 6. Chang, D.: A matrix trace inequality for products of Hermitian matrices. J. Math. Anal. Appl. 237, 721–725 (1999) 7. Coop, I.D.: On matrix trace inequalities and related topics for products of Hermitian matrix. J. Math. Anal. Appl. 188, 999–1001 (1994) 8. Dragomir, S.S.: p-Schatten norm inequalities of Ostrowski’s type. RGMIA Res. Rep. Coll. 24, Art. 108, 19 pp (2021) 9. Mikusinski, J.: The Bochner Integral. Academic, New York (1978) 10. Neudecker, H.: A matrix trace inequality. J. Math. Anal. Appl. 166, 302–303 (1992) 11. Pachpatte, B.G.: Inequalities similar to the integral analogue of Hilbert’s inequalities. Tamkang J. Math. 30(1), 139–146 (1999) 12. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991) 13. Shilov, G.E.: Elementary Functional Analysis. Dover Publications Inc., New York (1996) 14. Simon, B.: Trace ideals and Their Applications. Cambridge University Press, Cambridge (1979) 15. Zagrebvov, V.A.: Gibbs Semigroups. Operator Theory: Advances and Applications, vol 273. Birkhauser (2019)

Chapter 12

γ-Schatten Norm Multivariate Ostrowski Inequalities for Multiple Neumann–Schatten Class Bγ (H) Valued Functions

Here we are dealing with several smooth functions from a compact convex set of Rk , k ≥ 2 to a Neumann–Schatten class Bγ (H ), γ ≥ 1, which is a Banach algebra. For these we prove general multivariate Ostrowski inequalities with estimates in norms · p , for all 1 ≤ p ≤ ∞. We provide also interesting applications. It follows [2].

12.1 Introduction Our main motivation comes from [1]. We mention a general Ostrowski type inequality result regarding several Banach algebra valued functions. Theorem 12.1 ([1]) Let p, q > 1 : 1p + q1 = 1; (A, ·) a Banach algebra and f i ∈ → C n+1 (Q, A), i = 1, ..., r ; r ∈ N, n ∈ Z+ , and fixed − x0 ∈ Q ⊂ Rk , k ≥ 2, where α Q is a compact and convex subset. Here all vector partial derivatives f iα := ∂∂z αfi , k  αλ = j, j = 1, ..., n, fulfill where α = (α1 , ..., αk ), αλ ∈ Z+ , λ = 1, ..., k, |α| = λ=1 −  f iα → x0 = 0, i = 1, ..., r. Denote (12.1) Dn+1 ( f i ) := max  f iα ∞,Q , α:|α|=n+1

i = 1, ..., r, and

k  −  → → |z λ − x0λ | . z −− x0 l1 :=

(12.2)

λ=1

Then © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_12

287

12 γ-Schatten Norm Multivariate Ostrowski Inequalities …

288

  ⎛ ⎛ ⎛ ⎞ ⎞ ⎞      r  r r r     ⎜ − ⎜ ⎜ ⎟ ⎟ ⎟        → − → − → − → − → − →   ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ z f z d z − z d z f x f f ρ i ρ i 0 ≤  ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ Q  i=1 Q ρ=1  ρ=1 i=1   ρ =i ρ =i ⎞ ⎛ ⎛ ⎞ (12.3)  max Dn+1 ( f i )  r r  →⎟ − n+1 →⎟ ⎜ ⎜ → → ⎜ ⎜  fρ − z −− x0 l1 d − z ⎟ z⎟ ⎠ ⎝ ⎝ ⎠≤ (n + 1)! Q ρ=1 i=1

i∈{1,...,r }

ρ =i

⎧ ⎪ ⎪ ⎨

max Dn+1 ( f i )

i∈{1,...,r }

(n + 1)!

min

⎪ ⎪ ⎩

Q

⎛ ⎞⎤ ⎡   r r − n+1 → ⎢ ⎜   ⎟⎥ → → ⎜  f ρ  ⎟⎥ , z −− x0 l1 d − z ⎢ ∞,Q ⎠⎦ ⎝ ⎣ i=1

⎛ ⎞    r ⎜  ⎟ ⎢ ⎜  f ρ ⎟ ⎢ ⎝ ⎠ ∞,Q ⎣  i=1  ρ=1   ρ =i ⎡

   − →n+1   · − x0 l1 

ρ=1 ρ =i

⎤

r  

⎞ ⎡⎛     r r     ⎜      ⎟ ⎢ ⎢ n+1   − → ⎜     ⎟ ⎢ ⎢ f ρ ⎠  · − x 0 l1    ⎝ ⎣ ⎣ L p (Q,A)  i=1  ρ=1   ρ =i

⎥ ⎥, ⎦ L 1 (Q,A)

⎡

L q (Q,A)

⎤⎤⎫ ⎪ ⎪ ⎥⎥⎬ ⎥⎥ . ⎦⎦⎪ ⎪ ⎭

(12.4)

In particular we obtained the following Ostrowski type inequality. Theorem 12.2 ([1]) Let (A, ·) a Banach algebra and f i ∈ C n+1  k  k   → x0 ∈ [aλ , bλ ] , A , i = 1, ..., r ; r ∈ N, n ∈ Z+ , and fixed − [aλ , bλ ] ⊂ Rk , λ=1

λ=1

α

k ≥ 2. Here all vector partial derivatives f iα := ∂∂z αfi , where α = (α1 , ..., αk ), αλ ∈ k →  αλ = j, j = 1, ..., n, fulfill f iα − x0 = 0, i = 1, ..., r. Z+ , λ = 1, ..., k, |α| = Denote

λ=1

Dn+1 ( f i ) := i = 1, ..., r. Then

max  f iα 

α:|α|=n+1

∞,

k 

λ=1

[aλ ,bλ ]

,

 ⎛ ⎞   r  r  ⎜ − ⎟ −  → →  ⎜ ⎟ fi → z z d− z− f ρ k   ⎝ ⎠ [aλ ,bλ ] ρ=1  i=1 λ=1  ρ =i

(12.5)

12.2 About Banach Algebras

⎛ r  i=1

⎜ ⎜ ⎝

289

⎛

 k 

[aλ ,bλ ]

λ=1

⎜ ⎜ ⎝

    ⎟ ⎟ −    − → →  ⎟ fi − z ⎟ d z x fρ → 0 ≤ ⎠ ⎠   ⎞

r  ρ=1 ρ =i

⎞

(12.6)

⎡ ⎛ ⎞⎤   r r ⎢ ⎜   ⎟⎥ ⎜  f ρ  k ⎟⎥ max Dn+1 ( f i ) ⎢ ⎣ ⎝ i∈{1,...,r } ∞, [aλ ,bλ ] ⎠⎦



i=1

ρ=1 ρ =i

λ=1

⎡ ⎢ ⎢ ⎢ ⎣

⎤  k 

λ=1

k  ρλ =n+1

λ=1

ρλ !

1 k  λ=1

k   ⎥ ⎥ (bλ − x0λ )ρλ +1 + (x0λ − aλ )ρλ +1 ⎥ . ⎦

(ρλ + 1) λ=1

Ostrowski type inequalities have a lot of important applications in Numerical Analysis and Probability. We are also inspired by [6]. In this chapter we establish multivariate Ostrowski type inequalities for several smooth functions from a compact convex subset of Rk , k ≥ 2, to von Neumann– Schatten class Bγ (H ), γ ≥ 1, which is a Banach algebra. These involve the norms · p , 1 ≤ p ≤ ∞.

12.2 About Banach Algebras All here come from [10]. We need Definition 12.3 ([10], p. 245) A complex algebra is a vector space A over the complex field C in which a multiplication is defined that satisfies x (yz) = (x y) z,

(12.7)

(x + y) z = x z + yz, x (y + z) = x y + x z,

(12.8)

α (x y) = (αx) y = x (αy) ,

(12.9)

and for all x, y and z in A and for all scalars α. Additionally if A is a Banach space with respect to a norm that satisfies the multiplicative inequality x y ≤ x y (x ∈ A, y ∈ A)

(12.10)

12 γ-Schatten Norm Multivariate Ostrowski Inequalities …

290

and if A contains a unit element e such that xe = ex = x (x ∈ A)

(12.11)

e = 1,

(12.12)

and

then A is called a Banach algebra. A is commutative iff x y = yx for all x, y ∈ A. We make Remark 12.4 Commutativity of A will be explicated stated when needed. There exists at most one e ∈ A that satisfies (12.11). Inequality (12.10) makes multiplication to be continuous, more precisely left and right continuous, see [10], p. 246. Multiplication in A is not necessarily the numerical multiplication, it is something more general and it is defined abstractly, that is for x, y ∈ A we have x y ∈ A, e.g. composition or convolution, etc. For nice examples about Banach algebras see [10], p. 247–248, Sect. 10.3. We also make Remark 12.5 Next we mention about integration of A-valued functions, see [10], p. 259, Sect. 10.22: If A is a Banach algebra and f is a continuous A-valued function on some  compact Hausdorff space Q on which a complex Borel measure μ is defined, then f dμ exists and has all the properties that were discussed in Chap. 3 of [10], simply because A is a Banach space. However, an additional property can be added to these, namely: If x ∈ A, then   f dμ = x f ( p) dμ ( p) (12.13) x Q

and



Q

  f dμ x = Q

f ( p) x dμ ( p) .

(12.14)

Q

The vector integrals we will involve in our chapter follow (12.13) and (12.14).

12.3

p-Schatten Norms Background

In this advanced section all come from [6]. Let (H, ·, ·) be a complex Hilbert space and B (H ) the Banach algebra of all bounded linear operators on H . If {ei }i∈I an orthonormal basis of H , we say that A ∈ B (H ) is of trace class if

12.3 p-Schatten Norms Background

291

A1 :=



|A| ei , ei  < ∞.

(12.15)

i∈I

The definition of A1 does not depend on the choice of the orthornormal basis {ei }i∈I . We denote by B1 (H ) the set of trace class operators in B (H ). We define the trace of a trace class operator A ∈ B1 (H ) to be tr (A) :=



Aei , ei  ,

(12.16)

i∈I

where {ei }i∈I an orthonormal basis of H . Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series (12.16) converges absolutely and it is independent from the choice of basis. The following result collects some properties of the trace: Theorem 12.6 We have: (i) If A ∈ B1 (H ) then A∗ ∈ B1 (H ) and   tr A∗ = tr (A);

(12.17)

(ii) If A ∈ B1 (H ) and T ∈ B (H ), then AT, T A ∈ B1 (H ) and tr (AT ) = tr (T A) and |tr (AT )| ≤ A1 T  ;

(12.18)

(iii) tr (·) is a bounded linear functional on B1 (H ) with tr  = 1; (iv) If A, B ∈ B2 (H ) then AB, B A ∈ B1 (H ) and tr (AB) = tr (B A) ; (v) B f in (H ) (finite rank operators) is a dense subspace of B1 (H ) . An operator A ∈ B (H ) is said to belong to the von Neumann–Schatten class B p (H ), 1 ≤ p < ∞ if the p-Schatten norm is finite [14, pp. 60–64]  ! 1 A p := tr |A| p p < ∞, |A| p is an operator notation and not a power. For 1 < p < q < ∞ we have that B1 (H ) ⊂ B p (H ) ⊂ Bq (H ) ⊂ B (H )

(12.19)

 A1 ≥  A p ≥  Aq ≥ A .

(12.20)

and For p ≥ 1 the functional · p is a norm on the ∗-ideal B p (H ), which is a Banach   algebra, and B p (H ) , · p is a Banach space. Also, see for instance [14, pp. 60–64], for p ≥ 1,   A p =  A∗  p , A ∈ B p (H )

(12.21)

12 γ-Schatten Norm Multivariate Ostrowski Inequalities …

292

 AB p ≤ A p B p , A, B ∈ B p (H )

(12.22)

and AB p ≤ A p B , B A p ≤ B A p , A ∈ B p (H ) , B ∈ B (H ) . (12.23) This implies that C AB p ≤ C A p B , A ∈ B p (H ) , B, C ∈ B (H ) .

(12.24)

In terms of p-Schatten norm we have the Hölder inequality for p, q > 1 with 1 = 1: q (|tr (AB)| ≤) AB1 ≤ A p Bq , A ∈ B p (H ) , B ∈ Bq (H ) .

1 p

+

(12.25)

For the theory of trace functionals and their applications the interested reader is referred to [13, 14]. For some classical trace inequalities see [4, 5, 9], which are continuations of the work of Bellman [3].

12.4 Vector Analysis Background (see [12], pp. 83–94) Let f (t) be a function defined on [a, b] ⊆ R taking values in a real or complex normed linear space (X, ·), Then f (t) is said to be differentiable at a point t0 ∈ [a, b] if the limit f (t0 + h) − f (t0 ) (12.26) f  (t0 ) = lim h→0 h exists in X , the convergence is in ·. This is called the derivative of f (t) at t = t0 . We call f (t) differentiable on [a, b], iff there exists f  (t) ∈ X for all t ∈ [a, b]. Similarly and inductively are defined higher order derivatives of f , denoted f  , f (3) , ..., f (k) , k ∈ N, just as for numerical functions. For all the properties of derivatives see [12], pp. 83–86. Let now (X, ·) be a Banach space, and f : [a, b] → X. b We define the vector valued Riemann integral a f (t) dt ∈ X as the limit of the vector valued Riemann sums in X , convergence is in ·. The definition is as for the numerical  b valued functions. If a f (t) dt ∈ X we call f integrable on [a, b]. If f ∈ C ([a, b] , X ), then f is integrable, [12], p. 87. For all the properties of vector valued Riemann integrals see [12], pp. 86–91.

12.4 Vector Analysis Background

293

We define the space C n ([a, b] , X ), n ∈ N, of n-times continuously differentiable functions from [a, b] into X ; here continuity is with respect to · and defined in the usual way as for numerical functions·. Let (X, ·) be a Banach space and f ∈ C n ([a, b] , X ), then we have the vector valued Taylor’s formula, see [12], pp. 93–94, and also [11], (IV, 9; 47). It holds f (y) − f (x) − f  (x) (y − x) −

1  1 f (n−1) (x) (y − x)n−1 f (x) (y − x)2 − ... − 2 (n − 1)!

(12.27)

=

1 (n − 1)!



y

(y − t)n−1 f (n) (t) dt, ∀x, y ∈ [a, b] .

x

In particular (12.27) is true when X = Rm , Cm , m ∈ N, etc. A function f (t) with values in a normed linear space X is said to be piecewise continuous (see [12], p. 85) on the interval a ≤ t ≤ b if there exists a partition a = t0 < t1 < t2 < ... < tn = b such that f (t) is continuous on every open interval tk < t < tk+1 and has finite limits f (t0 + 0) , f (t1 − 0) , f (t1 + 0) , f (t2 − 0) , f (t2 + 0) , ..., f (tn − 0) . Here f (tk − 0) = lim f (t) , f (tk + 0) = lim f (t) . t↑tk

t↓tk

The values of f (t) at the points tk can be arbitrary or even undefined. A function f (t) with values in normed linear space X is said to be piecewise smooth on [a, b], if it is continuous on [a, b] and has a derivative f  (t) at all but a finite number of points of [a, b] , and if f  (t) is piecewise continuous on [a, b] (see [12], p. 85). Let u (t) and v (t) be two piecewise smooth functions on [a, b], one a numerical function and the other a vector function with values in Banach space X . Then we have the following integration by parts formula  a

b

 u (t) dv (t) = u

(t) v (t) |ab

−

b

v (t) du (t) ,

(12.28)

a

see [12], p. 93. We mention also the mean value theorem for Banach space valued functions. Theorem 12.7 (see [8], p. 3) Let  f ∈ C ([a, b] , X ), where X is a Banach space. Assume f  exists on [a, b] and  f  (t) ≤ K , a < t < b, then  f (b) − f (a) ≤ K (b − a) .

(12.29)

Here the multiple Riemann integral of a function from a real box or a real compact and convex subset to a Banach space is defined similarly to numerical one however convergence is with respect to ·. Similarly are defined the vector valued partial derivatives as in the numerical case. We mention the equality of vector valued mixed partials derivatives.

12 γ-Schatten Norm Multivariate Ostrowski Inequalities …

294

Proposition 12.8 (see Proposition 4.11 of [7], p. 90) Let Q = (a, b) × (c, d) ⊆ R2 and f ∈ C (Q, X ), where (X, ·) is a Banach space. Assume that ∂t∂ f (s, t), ∂ ∂2 ∂2 f t) and ∂t∂s f (s, t) exist and are continuous for (s, t) ∈ Q, then ∂s∂t f (s, t) ∂s (s, exists for (s, t) ∈ Q and ∂2 ∂2 f (s, t) = f (s, t) , for (s, t) ∈ Q. ∂s∂t ∂t∂s

(12.30)

12.5 Main Results We present a general Ostrowski type inequality result regarding several B2 (H )Banach algebra valued functions. Theorem 12.9 Let p, q > 1 : 1p + q1 = 1; B2 (H ) is a ∗-ideal, which (B2 (H ) , ·2 ) is a Banach algebra, and f i ∈ C n+1 (Q, B2 (H )), i = 1, ..., r ; r ∈ N, n ∈ Z+ , and → Q is a compact and convex subset. Here all vecfixed − x0 ∈ Q ⊂ Rk , k ≥ 2, where α tor partial derivatives f iα := ∂∂z αfi , where α = (α1 , ..., αk ), αλ ∈ Z+ , λ = 1, ..., k, k →  |α| = x0 = 0, i = 1, ..., r. αλ = j, j = 1, ..., n, fulfill f iα − λ=1

Denote Dn+1 ( f i ) := i = 1, ..., r, and

  max  f iα 2 ∞,Q ,

α:|α|=n+1

k  −  → → |z λ − x0λ | . z −− x0 l1 :=

(12.31)

(12.32)

λ=1

Then   ⎛ ⎛ ⎛ ⎞ ⎞ ⎞     r   r r r    ⎜ − ⎜ ⎜ ⎟ ⎟ ⎟  −  − −  −  → → → → − → →  ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ fρ z ⎠ fi z d z − f ρ z ⎠ d z ⎠ fi x0   ≤  ⎝ ⎝ Q⎝   i=1 Q ρ=1 ρ=1 i=1   ρ =i ρ =i 1 ⎛ ⎛ ⎞ ⎞ (12.33)  max Dn+1 ( f i )  r r  → ⎟ − n+1 →⎟ ⎜ ⎜ → → ⎜ ⎜  fρ − z −− x0 l1 d − z 2 ⎟ z⎟ ⎝ ⎝ ⎠ ⎠≤ (n + 1)! Q ρ=1 i=1

i∈{1,...,r }

ρ =i

max Dn+1 ( f i )

i∈{1,...,r }

(n + 1)!

⎧ ⎛ ⎞⎤ ⎡ ⎪ ⎪   r r ⎨ − n+1 → ⎢ ⎜    ⎟⎥ → → ⎜  f ρ   ⎟⎥ , z −− x0 l1 d − min z ⎢ 2 ⎝ ⎠⎦ ⎣ ∞,Q ⎪ ⎪ ρ=1 i=1 ⎩ Q ρ =i

12.5 Main Results

295

⎛ ⎞    r ⎜   ⎟ ⎢ ⎜  f ρ  ⎟ ⎢ ⎝ 2 ⎠ ∞,Q ⎣  i=1  ρ=1   ρ =i ⎡

   − →n+1   · − x0 l1 

⎤

r  

⎡⎛ ⎞     r r     ⎜   ⎢ ⎢ ⎟    n+1   − → ⎜  ⎢ ⎢ ⎟    f ρ 2 ⎠  · − x 0 l1    ⎝ ⎣ ⎣ L p (Q)  i=1  ρ=1  ρ =i 

⎥ ⎥, ⎦ L 1 (Q)

⎡

L q (Q)

⎤⎤⎫ ⎪ ⎪ ⎥⎥⎬ ⎥⎥ . ⎦⎦⎪ ⎪ ⎭

(12.34)

→ − →  → Proof Take gi − z −→ x0 , 0 ≤ t ≤ 1; i = 1, ..., r . Notice that x0 + t − := f i − z (t) −    → − → → → → gi − z (0) = f i x 0 and gi − z (1) = f i z . The jth derivative of gi − z (t), based on Proposition 12.8, is given by ⎡" #j ⎤ k  ∂ ( j) f i ⎦ (x01 + t (z 1 − x01 ) , ..., x0k + t (z k − x0k )) gi − (t) = ⎣ (z λ − x0λ ) → z ∂z λ λ=1

(12.35) and ( j) gi − → z

⎡" #j ⎤ k   → ∂ fi ⎦ − x0 , (0) = ⎣ (z λ − x0λ ) ∂z λ

(12.36)

λ=1

for j = 1, ..., n + 1; i = 1, ..., r. n+1 Let f iα be a partial derivative (Q, B2 (H )). Because by assumption −  of f i ∈ C → of the theorem we have f iα x0 = 0 for all α : |α| = j, j = 1, ..., n, we find that ( j)

gi − (0) = 0, j = 1, ..., n; i = 1, ..., r. → z Hence by vector Taylor’s theorem (12.27) we see that ( j)

n gi − (0)  →  →  →  →  → z + Rin − z − fi − x0 = z , 0 = Rin − z ,0 , fi − j! j=1

(12.37)

where →  Rin − z , 0 :=

 0

i = 1, ..., r. Therefore,

1

 0

t1



tn−1

... 0

$

gi(n) − → z

(tn ) −

gi(n) − → z

  (0) dtn ... dt1 , %

(12.38)

12 γ-Schatten Norm Multivariate Ostrowski Inequalities …

296

 →   Rin − z , 0 2 ≤



1



0

t1



tn−1

...

0

      (n+1)   (ξ (tn ))  tn dtn ... dt1 , gi − → z 2 ∞

0

(12.39) over t by the vector mean value Theorem 12.7 applied on gi(n) Moreover, we (0, ). − → n z get   1  t1  tn−1     →      Rin − z , 0 2 ≤ gi(n+1) ... t dt   − → n n ...dt1 z 2 ∞,[0,1]

0

0

0

    (n+1)     gi − → z

2 ∞,[0,1]

. (n + 1)!       However, there exists a ti0 ∈ [0, 1] such that gi(n+1)   − → z =

2 ∞,[0,1]

That is     (n+1)     gi − → z

2 ∞,[0,1]

(12.40)     = gi(n+1) (t ) − → i0  . z 2

 ⎡" #n+1 ⎤   k      − ∂ − → − → →  ⎣ ⎦ = fi x0 + ti0 z − z 0i  (z λ − x0λ )  ∂z λ   λ=1

⎡"  k   ∂ ⎣ |z λ − x0λ |  ≤  ∂z λ=1

2

⎤

λ

 #n+1  → −  →  z −→ z 0i . fi ⎦ − x0 + ti0 −  2

I.e.,     (n+1)   gi −   → z

2 ∞,[0,1]

⎡"  k   ∂  |z λ − x0λ |  ≤⎣  ∂z λ=1

λ

   

⎤  #n+1   fi ⎦ , 

(12.41)

2 ∞

i = 1, ..., r. Hence by (12.41) we get &  →   Rin − z , 0 2 ≤

k 

λ=1

'    n+1  ∂   |z λ − x0λ |  ∂zλ   fi 2 ∞

(n + 1)!

#n+1 " k n+1 Dn+1 ( f i )  Dn+1 ( f i )  → → − |z λ − x0λ | z −− x0 l1 , = (n + 1)! λ=1 (n + 1)! i = 1, ..., r. Therefore it holds

≤

(12.42)

12.5 Main Results

297

max Dn+1 ( f i )   n+1 →  i∈{1,...,r } → →   Rin − − z −− x0 l1 , z ,0 2 ≤ (n + 1)!

(12.43)

for i = 1, ..., r. By (12.37) we get that ⎛ ⎜ ⎜ ⎝

⎞ r  ρ=1 ρ =i

⎛

⎞

r 

→⎟ − → ⎜ ⎜ z ⎟ fρ − ⎠ fi z − ⎝

⎛

→⎟ − → ⎜ ⎜ z ⎟ fρ − ⎠ fi x0 = ⎝

ρ=1 ρ =i

⎞ r  ρ=1 ρ =i

→⎟ − →  z ⎟ fρ − ⎠ Rin z , 0 , (12.44)

for all i = 1, ..., r. Hence ⎛ r  i=1

⎜ ⎜ ⎝

⎛

⎞ r  ρ=1 ρ =i

r 

→⎟ − → z ⎟ fρ − ⎠ fi z −

i=1

⎛ =

r  i=1

ρ=1 ρ =i

→⎟ − → z ⎟ fρ − ⎠ fi x0

⎞

r 

⎜ ⎜ ⎝

⎜ ⎜ ⎝

⎞ r 

ρ=1 ρ =i

→⎟ − →  z ⎟ fρ − ⎠ Rin z , 0 .

(12.45)

Therefore we find E ( f 1 , ..., fr ) (x0 ) := ⎛

⎛  r  r r   ⎜ ⎜ − ⎟ −  − → → ⎜ ⎜ ⎟ fi → z z d z − f ρ ⎝ ⎝ ⎠ i=1

Q

⎛

⎞

ρ=1 ρ =i

i=1

r   i=1

⎛

⎞

r ⎜ →⎟ − − → →⎟ ⎜ ⎟ z ⎟ fρ − ⎝ ⎠ d z ⎠ fi x0 = ρ=1 ρ =i

⎞

r 

⎜ ⎜ ⎝ Q

Q

⎞

ρ=1 ρ =i

→⎟ − →  − → z ⎟ fρ − ⎠ Rin z , 0 d z .

(12.46)

Consequently, we have that E ( f 1 , ..., fr ) (x0 )1 =   ⎛ ⎛ ⎛ ⎞ ⎞ ⎞       r r r    r  ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ −        → − → − → − → − → − →  ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ fρ z ⎠ fi z d z − f ρ z ⎠ d z ⎠ fi x0    = ⎝ ⎝ ⎝ Q  i=1 Q ρ=1  ρ=1 i=1   ρ =i ρ =i 1

12 γ-Schatten Norm Multivariate Ostrowski Inequalities …

298

  ⎛ ⎞     r  r   ⎜ − ⎟    → − → − →   ≤ ⎜ ⎟ z R z , 0 d z f ρ in   ⎝ ⎠  i=1 Q ρ=1    ρ =i

(12.47)

1

 ⎛  ⎞      r r   ⎜ −  ⎟  −  − → → →  ⎜ ⎟ f ρ z ⎠ Rin z , 0 d z   ⎝  ≤  ρ=1 i=1  Q   ρ =i 1

⎛  ⎞ ⎞      r r  ⎜ ⎜ →⎟ − →  →⎟  − ⎜ ⎜ ⎟ z ⎟ fρ − ⎝ ⎝ ⎠ Rin z , 0  d z ⎠ ≤ Q   i=1  ρ=1  ρ =i ⎛

1

(by (12.25), (12.22)) r  i=1

⎛ ⎛ ⎞ ⎞  r ⎜ ⎜  −  ⎟  − (12.43) →  →⎟   − ⎜ ⎜  fρ → ⎟ z 2 ⎟ ⎝ Q⎝ ⎠ Rin z , 0 2 d z ⎠ ≤

(12.48)

ρ=1 ρ =i

⎛

⎛ ⎞ ⎞  max Dn+1 ( f i )  r r  → ⎟ − n+1 →⎟ ⎜ ⎜ i∈{1,...,r } → ⎜ ⎜  fρ − → z −− x0 l1 d − z 2 ⎟ z⎟ ⎝ ⎝ ⎠ ⎠. (n + 1)! i=1

Q

ρ=1 ρ =i

So far we have proved E ( f 1 , ..., fr ) (x0 )1 ≤ ⎛ ⎛ ⎞ ⎞  max Dn+1 ( f i )  r r n+1 →⎟ ⎜ ⎜  −  ⎟ − i∈{1,...,r } → → ⎜ ⎜  fρ → z −− x0 l1 d − z 2 ⎟ z⎟ ⎝ ⎝ ⎠ ⎠ =: (ξ) . (n + 1)! Q ρ=1 i=1 ρ =i

(12.49) Furthermore it holds ⎛ ⎞⎤ ⎡   max Dn+1 ( f i )   r r ⎜    ⎟⎥ n+1 → ⎢ i∈{1,...,r } → → ⎜  f ρ   ⎟⎥ , − z −− x0 l1 d − z ⎢ (ξ) ≤ 2 ∞,Q ⎠⎦ ⎝ ⎣ (n + 1)! Q ρ=1 i=1 ρ =i

(12.50) and

12.5 Main Results

299

⎛ ⎡ ⎞     max Dn+1 ( f i )  r r    ⎜      ⎢ ⎟ i∈{1,...,r }  − → n+1  ⎜  f ρ  ⎟ ⎢ (ξ) ≤  · − x0 l1    2 ⎝ ⎣ ⎠ ∞,Q (n + 1)!  i=1  ρ=1   ρ =i

⎤ ⎥ ⎥, ⎦ L 1 (Q)

(12.51)

and finally ⎡⎛ ⎞ ⎡      max Dn+1 ( f i ) ⎢ r r ⎜   ⎢ ⎟   i∈{1,...,r } ⎢⎜  f ρ  ⎟ ⎢ (ξ) ≤   2 ⎝ ⎣ ⎠ ⎣ (n + 1)!  i=1  ρ=1  ρ =i 

⎤⎤   ⎥⎥  − →  ⎥⎥  n+1  ⎦⎦  · − x0 l1  L p (Q) , L q (Q)

(12.52) 

proving (12.33), (12.34). We give

Corollary 12.10 (to Theorem 12.9) All as in Theorem 12.9, with f 1 = ... = fr = f , r ∈ N. Then       → −  → − −  → → → r − r −1 −  f z d z − f z d z f x0    ≤ Q

Q

Dn+1 ( f ) (n + 1)! Dn+1 ( f ) min (n + 1)!    − →n+1   · − x0 l1 

∞,Q

 Q

1

  − n+1 → r −1 → − f → z 2 − z −→ x0 l1 d − z ≤

( Q

(12.53)

 %r −1 − n+1 → $  →  f 2  → z −− x0 l d − z , ∞,Q 1

   f r −1  2

 n+1   − →   · − x ,  0 L 1 (Q) l1 

L p (Q)

   f r −1  2

) L q (Q)

.

(12.54) We make Remark 12.11 Of great interest are applications of Theorem 12.9 when Q = k  [aλ , bλ ], where [aλ , bλ ] ⊂ R, λ = 1, ..., k.

λ=1

We observe that by the multinomial theorem we get: " k 

 k  λ=1

[aλ ,bλ ]

 k 

[aλ ,bλ ]

λ=1

λ=1

#n+1 |z λ − x0λ |

dz 1 ...dz k =

 ρ1 +ρ2 +...ρk

(n + 1)! ρ !ρ !...ρk ! =n+1 1 2

|z 1 − x01 |ρ1 |z 2 − x02 |ρ2 ... |z k − x0k |ρk dz 1 ...dz k =

(12.55)

12 γ-Schatten Norm Multivariate Ostrowski Inequalities …

300

 ρ1 +ρ2 +...ρk

 k  bλ (n + 1)!  |z λ − x0λ |ρλ dz λ = ρ !ρ !...ρk ! aλ =n+1 1 2 λ=1

  bλ k  x0λ (n + 1)!  ρλ ρλ (x0λ − z λ ) dz λ + (z λ − x0λ ) dz λ = k  aλ x0λ k λ=1  ρ ! λ ρλ =n+1 

λ=1

λ=1

 k  (n + 1)!  (x0λ − aλ )ρλ +1 + (bλ − x0λ )ρλ +1 . k  ρλ + 1 k λ=1  ρλ ! ρλ =n+1 

(12.56)

λ=1

λ=1



We have found that

k 

[aλ ,bλ ]

− n+1 → → → z −− x0  d − z =

(12.57)

l1

λ=1

 k  (n + 1)!  (bλ − x0λ )ρλ +1 + (x0λ − aλ )ρλ +1 . k  ρλ + 1 k λ=1  ρλ ! ρλ =n+1 

λ=1

λ=1

Based on (12.33), (12.34) and (12.57) we conclude: Theorem 12.12 Let  (B2 (H ) , ·2 ) the ∗-ideal and f i ∈ C n+1  k k   → x0 ∈ [aλ , bλ ] , B2 (H ) , i = 1, ..., r ; r ∈ N, n ∈ Z+ , and fixed − [aλ , bλ ] ⊂ λ=1

λ=1

α

Rk , k ≥ 2. Here all vector partial derivatives f iα := ∂∂z αfi , where α = (α1 , ..., αk ), k  →  x0 = 0, i = αλ = j, j = 1, ..., n, fulfill f iα − αλ ∈ Z+ , λ = 1, ..., k, |α| = 1, ..., r. Denote

λ=1

Dn+1 ( f i ) := i = 1, ..., r. Then

  max  f iα 2 

α:|α|=n+1

∞,

k 

λ=1

[aλ ,bλ ]

,

 ⎛ ⎞   r  r  ⎜ − ⎟ −  → →  ⎜ ⎟ fi → f z z d− z− ρ k   ⎝ ⎠ [aλ ,bλ ] ρ=1  i=1 λ=1  ρ =i

(12.58)

12.5 Main Results

301 r  i=1

⎛  ⎜ ⎜ k ⎝ 

⎛ [aλ ,bλ ]

λ=1

⎜ ⎜ ⎝

    ⎟ ⎟ −    − → →  ⎟ fi − z ⎟ d z x fρ → 0  ≤ ⎠ ⎠   ⎞

r  ρ=1 ρ =i

⎞

(12.59)

1

⎡

⎞⎤   r r ⎢ ⎜    ⎟⎥ ⎜  f ρ   k ⎟⎥ max Dn+1 ( f i ) ⎢ 2 ∞, [a ,b ] ⎠⎦ ⎣ ⎝ i∈{1,...,r } λ λ



⎛

i=1

ρ=1 ρ =i

λ=1

⎡ ⎢ ⎢ ⎢ ⎣

⎤  k 

λ=1

k  ρλ =n+1

λ=1

ρλ !

1 k  λ=1

k   ⎥ ⎥ (bλ − x0λ )ρλ +1 + (x0λ − aλ )ρλ +1 ⎥ . ⎦

(ρλ + 1) λ=1

We continue with a general Ostrowski type inequality result regarding several Bγ (H ), γ ≥ 1, Banach algebra valued functions. Theorem 12.13 Let γ ≥ 1, p, q > 1 : 1p + q1 = 1; Bγ (H ) is a ∗-ideal, which     Bγ (H ) , ·γ is a Banach algebra, and f i ∈ C n+1 Q, Bγ (H ) , i = 1, ..., r ; r ∈ → x0 ∈ Q ⊂ Rk , k ≥ 2, where Q is a compact and convex subN, n ∈ Z+ , and fixed − α set. Here all vector partial derivatives f iα := ∂∂z αfi , where α = (α1 , ..., αk ), αλ ∈ Z+ , k →  x0 = 0, i = 1, ..., r. αλ = j, j = 1, ..., n, fulfill f iα − λ = 1, ..., k, |α| = Denote

λ=1

Dn+1 ( f i ) := i = 1, ..., r, and

  max  f iα γ ∞,Q ,

α:|α|=n+1

k  −  → → |z λ − x0λ | . z −− x0 l1 :=

(12.60)

(12.61)

λ=1

Then   ⎛ ⎛ ⎛ ⎞ ⎞ ⎞     r   r r r     ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ −        → − → − → − → − → − →  ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ fρ z ⎠ fi z d z − f ρ z ⎠ d z ⎠ fi x0   ≤  ⎝ ⎝ ⎝ Q   i=1 Q ρ=1 ρ=1 i=1   ρ =i ρ =i γ

⎞ (12.62)

⎛ ⎛ ⎞  max Dn+1 ( f i )  r r n+1 →⎟ ⎜ ⎜  −  ⎟ − i∈{1,...,r } → → ⎜ ⎜  fρ → z −− x0 l1 d − z⎟ z γ ⎟ ⎝ ⎝ ⎠ ⎠≤ (n + 1)! Q ρ=1 i=1 ρ =i

12 γ-Schatten Norm Multivariate Ostrowski Inequalities …

302

⎧ ⎪ ⎪ ⎨

max Dn+1 ( f i )

i∈{1,...,r }

min

(n + 1)!

⎪ ⎪ ⎩

⎛ ⎞⎤ ⎡   r r   − n+1 → ⎢ ⎜    ⎟⎥ → → ⎜  f ρ   ⎟⎥ , z −− x0 l1 d − z ⎢ γ ∞,Q ⎠⎦ ⎝ ⎣ Q ρ=1 ρ =i

i=1

⎛ ⎞    r    ⎜      ⎢ ⎟ n+1   − → ⎜      ⎢ ⎟ · − x f  0 l1  ρ γ ⎠  ⎝ ⎣ ∞,Q  i=1  ρ=1   ρ =i ⎡

⎤

r  

⎡ ⎡⎛ ⎞     r r     ⎜      ⎢ ⎢ ⎟ n+1   − → ⎜     ⎢ ⎢ ⎟ f ρ γ ⎠  · − x 0 l1    ⎝ ⎣ ⎣ L p (Q)  i=1  ρ=1  ρ =i 

⎥ ⎥, ⎦ L 1 (Q)

L q (Q)

⎤⎤⎫ ⎪ ⎪ ⎥⎥⎬ ⎥⎥ . ⎦⎦⎪ ⎪ ⎭

(12.63)



Proof As similar to Theorem 12.9 is omitted. Use of (12.22). We also give

Corollary 12.14 (to Theorem 12.13) All as in Theorem 12.13, with f 1 = ... = fr = f , r ∈ N. Then    

 → − z d→ z − f −



r

f

Q

r −1

−  → → z d− z



γ

Q

Dn+1 ( f ) (n + 1)! Dn+1 ( f ) min (n + 1)!    − →n+1   · − x0 l1 

∞,Q

 Q

 −  → f x0   ≤

  − n+1 − r −1 − → − → →  f →   z γ z − x 0 l1 d z ≤

( Q

(12.64)

 %r −1 − n+1 → $  →  f  γ  → z −− x0 l d − z , ∞,Q

   f r −1  γ

1

L1

 n+1   − →   · − x ,  0 l1  (Q)

L p (Q)

   f r −1  γ

) L q (Q)

.

(12.65) In particular we obtain:   Theorem 12.15 Let Bγ (H ) , ·γ , γ ≥ 1, the ∗-ideal and f i ∈ C n+1   k k   → x0 ∈ [aλ , bλ ] , Bγ (H ) , i = 1, ..., r ; r ∈ N, n ∈ Z+ , and fixed − [aλ , bλ ] ⊂ λ=1

α

λ=1

Rk , k ≥ 2. Here all vector partial derivatives f iα := ∂∂z αfi , where α = (α1 , ..., αk ), k →  αλ = j, j = 1, ..., n, fulfill f iα − x0 = 0, i = 1, ..., αλ ∈ Z+ , λ = 1, ..., k, |α| = r.

λ=1

12.5 Main Results

303

Denote Dn+1 ( f i ) := i = 1, ..., r. Then

  max  f iα γ 

α:|α|=n+1

∞,

k 

λ=1

[aλ ,bλ ]

,

(12.66)

 ⎛ ⎞   r  r  ⎜ − ⎟ −  → →  ⎜ ⎟ fi → z z d− z− f ρ k   ⎝ ⎠ [aλ ,bλ ] ρ=1  i=1 λ=1  ρ =i ⎛

   r  ⎜ ⎟ ⎟ −    → − → − → ⎜ ⎟ ⎟ f ρ z ⎠ d z ⎠ fi x0   ≤ ⎝ [aλ ,bλ ] ρ=1  λ=1  ρ =i

 r  ⎜ ⎜ k ⎝  i=1

⎛

⎞

⎞

(12.67)

γ

⎡ ⎛ ⎞⎤   r r   ⎢ ⎜    ⎟⎥ ⎜  f ρ   k ⎟⎥ max Dn+1 ( f i ) ⎢ γ ⎣ ⎝ i∈{1,...,r } ∞, [aλ ,bλ ] ⎠⎦



i=1

ρ=1 ρ =i

λ=1

⎡ ⎢ ⎢ ⎢ ⎣

⎤  k 

λ=1

k  ρλ =n+1

λ=1

ρλ !

1 k  λ=1

k   ⎥ ⎥ (bλ − x0λ )ρλ +1 + (x0λ − aλ )ρλ +1 ⎥ . ⎦

(ρλ + 1) λ=1

Proof As similar to Theorem 12.12 is omitted.



When r = 2 we derive the following p-Schatten norm operator related Ostrowski type inequalities. Theorem 12.16 Let p, q > 1 : 1p + q1 = 1; and let the ∗-ideals B p (H ), Bq (H ) ,     → for which B p (H ) , · p , Bq (H ) , ·q are Banach algebras, n ∈ Z+ ; − x ∈Q⊂  0  k n+1 Q, B p (H ) , R , k ≥ 2, where Q is a compact and convex subset, and f 1 ∈ C   α f 2 ∈ C n+1 Q, Bq (H ) . Here all vector partial derivatives f iα := ∂∂z αfi , where α = k  →  x0 = αλ = j, j = 1, ..., n, fulfill f iα − (α1 , ..., αk ), αλ ∈ Z+ , λ = 1, ..., k, |α| = λ=1

0, i = 1, 2. Denote by Dn+1 ( f 1 ) := and Dn+1 ( f 2 ) :=

  max  f 1α  p ∞,Q ,

(12.68)

  max  f 2α q ∞,Q ,

(12.69)

α:|α|=n+1

α:|α|=n+1

12 γ-Schatten Norm Multivariate Ostrowski Inequalities …

304

and

k   − 1 1 → → |z λ − x0λ | ; γ, δ > 1 : + = 1. z −− x0 l1 := γ δ

(12.70)

λ=1

   

Then



→ −  → z f1 → z d− z + f2 −



Q

→ −  → z f2 → z d− z− f1 − Q

 → − z d→ z f2 −



→ f1 − x0 −



Q

 → − z d→ z f1 −



Q

 −  → f 2 x0   ≤ 1

max (Dn+1 ( f 1 ) , Dn+1 ( f 2 )) (n + 1)! ( Q

 − n+1 →  → −  f2 → z −→ x0 l1 d − z q − z +

 Q

)  − n+1 −  − → − → →   f1 →   z p z − x 0 l1 d z ≤ (12.71)

max (Dn+1 ( f 1 ) , Dn+1 ( f 2 )) (n + 1)! ( min Q

* +   − n+1 −   − → →  f 2 q   f 1  p  →  z − x 0 l1 d z , + ∞,Q ∞,Q

   − →n+1   · − x0 l1 

∞,Q

   − →n+1   · − x0 l1 

L γ (Q)

*   f 2 q 

*   f 2 q 

Lδ

L1

  +  f 1  p  L (Q)

  +  f 1  p  L (Q)

+ 1 (Q)

+) δ (Q)

,

.

(12.72) 

Proof Similar to Theorem 12.9. Use of (12.25). We finish with Corollary 12.17 (to Theorem 12.16) All as in Theorem 12.16, with Q = Then

λ=1

[aλ , bλ ].

    −  −  − → −  → → → →  k f 2 z f 1 z d z + k f1 − z f2 → z d− z−   [aλ ,bλ ]  [aλ ,bλ ] λ=1

⎛  ⎝

k 

λ=1

⎞

k 

[aλ ,bλ ]

λ=1

⎛  −    → → z d− z ⎠ f1 − x0 − ⎝ f2 →

k 

[aλ ,bλ ]

λ=1

 ⎞   −    → → z d− z ⎠ f2 − x0  f1 →  ≤  1

References

305

⎡

  max (Dn+1 ( f 1 ) , Dn+1 ( f 2 )) ⎣ f 1  p 

∞,

k 

λ=1

⎡ ⎢ ⎢ ⎢ ⎣

[aλ ,bλ ]

  +  f 2 q 

⎤

∞,

k 

[aλ ,bλ ]

⎦ (12.73)

λ=1

⎤  k 

λ=1

k  ρk =n+1

λ=1

ρk !

1 k  λ=1

k   ⎥ ⎥ (bλ − x0λ )ρλ +1 + (x0λ − aλ )ρλ +1 ⎥ . ⎦

(ρλ + 1) λ=1

Proof Similar to Theorem 12.12.



References 1. Anastassiou, G.A.: Multivariate Ostrowski type inequalities for several Banach algebra valued functions. J. Comput. Anal. Appl. 30(1), 95–106 (2022) 2. Anastassiou, G.A.: γ-Schatten norm Multivariate Ostrowski type inequalities for several Neumann-Schatten class Bγ (H ) valued functions. Adv. Nonlinear Var. Inequal. Accepted (2022) 3. Bellman, R.: Some inequalities for positive definite matrices. In: Beckenbach, E.F. (ed.) General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, pp. 89–90. Birkhauser, Basel (1980) 4. Chang, D.: A matrix trace inequality for products of Hermitian matrices. J. Math. Anal. Appl. 237, 721–725 (1999) 5. Coop, I.D.: On matrix trace inequalities and related topics for products of Hermitian matrix. J. Math. Anal. Appl. 188, 999–1001 (1994) 6. Dragomir, S.S.: p-Schatten norm inequalities of Ostrowski’s type. RGMIA Res. Rep. Coll. 24, Art. 108, 19 (2021) 7. Driver, B.: Analysis Tools with Applications. Springer, New York (2003) 8. Ladas, G., Laksmikantham, V.: Differential Equations in Abstract Spaces. Academic, New York(1972) 9. Neudecker, H.: A matrix trace inequality. J. Math. Anal. Appl. 166, 302–303 (1992) 10. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991) 11. Schwartz, L.: Analyse Mathematique. Hermann, Paris (1967) 12. Shilov, G.E.: Elementary Functional Analysis. Dover Publications Inc., New York (1996) 13. Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979) 14. Zagrebvov, V.A.: Gibbs Semigroups. Operator Theory: Advances and Applications, vol. 273, Birkhauser (2019)

Chapter 13

Conclusion

During the last 50 years fractional calculus due to its wide applications to many applied sciences has become a main trend in mathematics. By using generalized Caputo and Canavati fractional left and right vectorial Taylor formulae we prove a great variety of generalized fractional inequalities. More precisely, we derive generalized Caputo Fractional Ostrowski and Grüss type inequalities involving several Banach algebra valued functions. Also we establish generalized Canavati fractional Ostrowski, Opial, Grüss and Hilbert–Pachpatte type inequalities for multiple Banach algebra valued functions. By applying the p-Schatten norms over the von Neumann– Schatten classes we produce the analogous refined and elegant inequalities. Many applications follow. This monograph’s results are expected to find applications in many areas of pure and applied mathematics, especially in fractional inequalities and fractional differential equations. Other interesting applications can be in applied sciences like geophysics, physics, chemistry, economics and engineering, etc. The central conclusion we derive from this monograph is that generalized fractional calculus penetrates almost any branch of mathematics with great applications, especially in natural phenomena.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Fractional Inequalities In Banach Algebras, Studies in Systems, Decision and Control 441, https://doi.org/10.1007/978-3-031-05148-7_13

307