New Perspectives on the Theory of Inequalities for Integral and Sum [1st ed. 2021] 3030905624, 9783030905620

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Nazia Irshad Asif R. Khan Faraz Mehmood Josip E. Pečarić

New Perspectives on the Theory of Inequalities for Integral and Sum

New Perspectives on the Theory of Inequalities for Integral and Sum

Nazia Irshad • Asif R. Khan • Faraz Mehmood • Josip Peˇcari´c

New Perspectives on the Theory of Inequalities for Integral and Sum

Nazia Irshad Department of Mathematics Dawood University of Engineering and Technology Karachi, Pakistan

Asif R. Khan Department of Mathematics University of Karachi Karachi, Pakistan

Faraz Mehmood Department of Mathematics Dawood University of Engineering and Technology Karachi, Pakistan

Josip Peˇcari´c Croatian Academy of Sciences and Arts Zagreb, Croatia

ISBN 978-3-030-90562-0 ISBN 978-3-030-90563-7 (eBook) https://doi.org/10.1007/978-3-030-90563-7 Mathematics Subject Classification: 26A16, 26A42, 26A45, 26A46, 26A51, 39B22, 39B62, 26D07, 26D10, 26D15, 26D20, 26D99 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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 book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The discipline of mathematical inequalities has continued to grow rapidly since the beginning of the twentieth century. Jensen, Cauchy, Schwartz, Hölder, Hadamard, Minkowski and Hardy of that era are mathematicians known for their work in inequalities. Inequalities are essential to study mathematics and many related fields. Various applications of inequalities have been established in the field of differential and integral equations, calculus, probability theory, interpolation theory, optimization theory, control theory, game theory, spectral theory, functional analysis, harmonic analysis, economics, physics and geometry. A number of authors use integral inequalities in studying existence, uniqueness, boundedness, stability and asymptotic behaviour of solutions of ordinary and partial differential equations. There are numerous known inequalities, and the list is ongoing. The database of MathSciNet contains over 23,000 references of inequalities and their applications. Primarily the most relevant target audience of this book are all mathematicians either from pure or applied side. Secondary audience may consist of physicists, statisticians, engineers, researchers and decision makers working in various research and government departments. Research scholars, students, faculty members and professionals who are working and doing research in different field and areas including (but not limited to) statistics, physics, optimization theory, numerical integration, probability theory, convex analysis, mathematical analysis, integration and measure theory, and linear inequalities may got help from this book. Our book shows new and latest results and methods with new findings and applications as compared to the book titled General linear inequalities and positivity: Higher order convexity [101]. To be more specific Chap. 1 of this book is similar to Chapter 5 of [101]. In Chapter 5 of another book, the authors have studied general linear inequalities of Popoviciu type via interpolation polynomials with and without Green function. In Chap. 1 of our book we have studied the same general linear inequalities via interpolation polynomials using 4 new Green functions. Furthermore, in Chap. 4 of our book, we have studied Popoviciu and ˇ Cebyšev-Popoviciu type identities and inequalities using ∇ operator and completely monotonic functions, while in monograph [101], the authors have studied similar results for  operator. In [101], the authors have stated some results related to v

vi

Preface

functions with nondecreasing increments, while in our book, we not only state some results for functions with nondecreasing increments but also have compared and linked many important concepts including arithmetic integral mean, Wright convex functions, convex functions, ∇-convex functions, Jensen m-convex functions, m-convex functions, m-∇-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator as different cases of m h f . We can find other similarities in both the books as well, for instance ˇ both books contain results related to Montgomery identities, Cebyšev, Grüss and Ostrowski inequalities, but our book discusses all these identities and inequalities in greater detail. Now, we put some light on the chapter-wise organization of the book as follows: In the first chapter, we would like to discuss linear inequalities via interpolation polynomials with Green functions. To be more specific, our focus would be to obtain Popoviciu type inequalities via • • • • •

Extension of the Montgomery identity. The Taylor’s formula. Hermite interpolation polynomials. The Fink identity. Abel-Gontscharoff’s interpolation polynomials.

We would like to discuss all the above-mentioned results with effect of newly established Green’s functions of a certain dynamical system. For Popoviciu type inequalities, we would also like to obtain results for: • • • • •

n-convex functions at a point. ˇ New upper bounds for remainders through Cebyšev functional. Mean value theorems of Lagrange and Cauchy type. n-exponential and logarithmic convexity. Construction of generalized means and mixed symmetric means, etc.

Some non-trivial examples in all above-mentioned topics will also be presented. Regarding applications, in our proposed research, we would obtain mean value theorems which provide good estimates for physical quantities. We would also obtain special means (averages) which play an important role in weather forecasting and stock exchange (in terms of moving averages) and help us to study dynamical systems and all the systems that do not have any specific pattern or are unpredictable or have high level of uncertainty. In all these cases, we mostly deal with averages. As far as we are concerned with the exponential convexity, exponentially convex functions have many nice properties, for example, these functions are analytic in their domain. These functions also provide us positive-semidefinite matrices. Moreover, they play an important role in studying the properties of Stolarsky and Cauchy means, such as monotonicity of these means. Additionally, a number of important inequalities arise from the logarithmic convexity of some functions. Logarithmic convexity plays an important role in various fields like reliability theory

Preface

vii

and survival analysis, economics, statistics, social sciences, information theory and optimization. Its applications can also be found in applied mathematics. In the second chapter, our focus is on Ostrowski type inequalities. In 1937, a Ukrainian mathematician Alexander Markowich Ostrowski first presented this interesting and useful inequality [141]. It can be used to find out the absolute deviation of functional value from its integral mean. It also approximates area under the curve of a function by a rectangle. It has great importance because of its number of applications in statistics, probability theory, integral operator theory, numerical integration and special means. We present new extensions and generalized results of Ostrowski type integral inequalities including Grüss, ˇ Ostrowski-Grüss and Cebyšev involving parameters with and without weights. We estimate the bounds on the deviation of function values from its mean value for different functions including bounded differentiable functions (also for the case of bounded below only differentiable function and bounded above only differentiable function), functions from Lp spaces for p ≥ 1, function of bounded variation and absolutely continuous function. We generalize Ostrowski-Grüss inequality via weighted Korkine’s identity and Peano kernel approach. We also present new results of fractional Ostrowski inequalities using Riemann Liouville fractional integral. ˇ In addition, we modify Cebyšev and Ostrowski inequality for two independent variables involving parameters. In the process of generalizing new inequalities, we achieve some refinements of Montgomery identities. These inequalities have some valuable applications related to some standard (including midpoint, trapezoidal, perturbed trapezoidal and Simpson’s type) and non-standard numerical quadrature rules and probability theory. Not only they have generalized and better results but also recapture previously obtained inequalities. The third chapter deals with functions with nondecreasing increments and related results. Functions with nondecreasing increments were introduced by Brunk in 1964. We extend this for mth order by using finite difference operator with equally spaced interval. With the help of this special definition of function with nondecreasing increments, we get relationship among functions with nondecreasing increments and arithmetic integral mean, Wright convex functions, convex functions, ∇-convex functions, Jensen m-convex functions, m-convex functions, m-∇-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator as different cases of m h f . We also discuss some examples in each above-stated relation. Generalizations of the Levinson’s-type inequality and Jensen-Mercer’s-type inequality by using Jensen-Boas inequality for function with nondecreasing increments of third order are also deduced. In the final chapter, we introduce a new notion of ∇-convex function and completely monotonic functions in two dimension case. We obtain discrete and integral identities for sequences and functions in two dimensions involving higherorder ∇-convex functions and completely monotonic functions. Further, we discuss the Popoviciu type characterization of positivity of sums and integrals of obtained identities. Further, by recalling higher-order completely monotonic function of one and two variables, we use variety of classes of completely monotonic functions and

viii

Preface

give examples and applications of completely monotonic functions and also recall some generalized results. In order to prove main results of the current chapter, we use various techniques including double induction and Taylor’s series expansion. We also discuss some generalized Lagrange and Cauchy-type’s mean value theorems for ∇-convex functions of higher order for two independent variables. Further that we construct some nonnegative functionals to study exponential convexity of special type and discuss some properties and we present examples and applications of completely monotonic, exponentially convex functions by using various classes ˇ of functions. We also discuss generalization of discrete Cebyšev and Ky Fan’s identities for ∇-convex functions of higher order with two independent variables and discuss similar result for discrete case as well. Last but not least, we also obtain the more generalized Montgomery’s identities for differentiable function of higher order with two independent variables. Generalized Montgomery’s identities support us for contribution in the generalized Ostrowski and Grüss type inequalities for double weighted integrals. We also provide more generalized Ostrowski and Grüss type inequalities for differentiable functions of higher order with two variables as compared to the existing results related to the subject. Karachi, Pakistan Karachi, Pakistan Karachi, Pakistan Zagreb, Croatia

Nazia Irshad Asif R. Khan Faraz Mehmood Josip Peˇcari´c

Contents

1

Linear Inequalities via Interpolation Polynomials and Green Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Linear Inequalities and the Extension of Montgomery Identity with New Green Functions .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Results Obtained by the Extension of Montgomery Identity and New Green Functions.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 Inequalities for n-Convex Functions at a Point .. . . . . . . . . . . . . . . 1.1.3 Bounds for Remainders and Functionals . .. . . . . . . . . . . . . . . . . . . . 1.1.4 Mean Value Theorems .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Linear Inequalities and the Taylor Formula with New Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Results Obtained by the Taylor Formula and New Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.2 Inequalities for n-Convex Functions at a Point .. . . . . . . . . . . . . . . 1.2.3 Bounds for Remainders and Functionals . .. . . . . . . . . . . . . . . . . . . . 1.2.4 Mean Value Theorems and Exponential Convexity .. . . . . . . . . . 1.2.5 Examples with Applications . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Linear Inequalities and Hermite Interpolation with New Green Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 Results Obtained by the Hermite Interpolation Polynomial and Green Functions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.2 Inequalities for n-Convex Functions at a Point .. . . . . . . . . . . . . . . 1.3.3 Bounds for Remainders and Functionals . .. . . . . . . . . . . . . . . . . . . . 1.4 Linear Inequalities and the Fink Identity with New Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 Results Obtained by the Fink identity and New Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.2 Inequalities for n-Convex Functions at a Point .. . . . . . . . . . . . . . . 1.4.3 Bounds for Remainders and Functionals . .. . . . . . . . . . . . . . . . . . . .

1 5 7 13 18 22 24 24 30 32 36 42 44 47 61 62 63 64 68 69

ix

x

Contents

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.1 Results Obtained by the Abel−Gontscharoff’s Interpolation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.2 Results Obtained by the Abel−Gontscharoff’s Interpolation Polynomial and Green Functions . . . . . . . . . . . . . . . 1.5.3 Inequalities for n-Convex Functions at a Point .. . . . . . . . . . . . . . . 1.5.4 Bounds for Remainders and Functionals . .. . . . . . . . . . . . . . . . . . . . 2 Ostrowski Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Generalized Ostrowski Type Inequalities with Parameter . . . . . . . . . . . . 2.1.1 Ostrowski Type Inequality for Bounded Differentiable Functions .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Ostrowski Type Inequalities for Bounded Below Only and Bounded Above Only Differentiable Functions .. . . 2.1.3 Applications to Numerical Integration .. . . .. . . . . . . . . . . . . . . . . . . . 2.2 Generalized Ostrowski Type Inequalities for Functions of Lp Spaces and Bounded Variation . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Ostrowski Type Inequality for Functions of Lp Spaces . . . . . . 2.2.2 Ostrowski Type Inequality for Functions of Bounded Variation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.3 Applications to Numerical Integration .. . . .. . . . . . . . . . . . . . . . . . . . 2.3 Generalized Weighted Ostrowski Type Inequality with Parameter .. . 2.3.1 Weighted Ostrowski Type Inequality with Parameter .. . . . . . . . 2.3.2 Applications to Numerical Integration .. . . .. . . . . . . . . . . . . . . . . . . . 2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 Weighted Ostrowski-Grüss Type Inequality with Parameter by Using Korkine’s Identity . . . .. . . . . . . . . . . . . . . . . . . . 2.4.2 Applications to Probability Theory . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.3 Applications to Numerical Integration .. . . .. . . . . . . . . . . . . . . . . . . . 2.5 Generalized Fractional Ostrwoski Type Inequality with Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.1 Fractional Ostrowski Type Inequality Involving Parameter .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery Identity with Parameters .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.1 Montgomery Identity for Functions of Two Variables involving Parameters . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.2 Generalized Ostrowski Type Inequality . . .. . . . . . . . . . . . . . . . . . . . 2.6.3 Generalized Grüss Type Inequalities.. . . . . .. . . . . . . . . . . . . . . . . . . .

71 72 76 84 86 87 90 91 108 115 118 118 122 126 129 131 141 145 148 157 159 162 163 172 173 177 183

3 Functions with Nondecreasing Increments. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 193 3.1 Functions with Nondecreasing Increments in Real Life . . . . . . . . . . . . . . 197 3.2 Relationship Among Functions with Nondecreasing Increments and Many Others .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 199

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xi

3.3 Functions with Nondecreasing Increments of Order 3 . . . . . . . . . . . . . . . . 207 3.3.1 On Levinson Type Inequalities .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 208 3.3.2 On Jensen-Mercer Type Inequalities .. . . . . .. . . . . . . . . . . . . . . . . . . . 211 ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities . . . 4.1 Linear Inequalities for Higher Order ∇-Convex and Completely Monotonic Functions .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Discrete Identity for Two Dimensional Sequences.. . . . . . . . . . . 4.1.2 Discrete Identity and Inequality for Functions of Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.3 Integral Identity and Inequality for Functions of One Variable.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.4 Integral Identity and Inequality for Functions of Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.5 Mean Value Theorems and Exponential Convexity .. . . . . . . . . . ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . ˇ 4.2.1 Generalized Discrete Cebyšev Identity and Inequality . . . . . . . ˇ 4.2.2 Generalized Integral Cebyšev Identity and Inequality . . . . . . . . 4.2.3 Generalized Integral Ky Fan Identity and Inequality . . . . . . . . . 4.3 Weighted Montgomery Identities for Higher Order Differentiable Function of Two Variables and Related Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 Montgomery Identities for Double Weighted Integrals of Higher Order Differentiable Functions .. . . . . . . . . . 4.3.2 Ostrowski Type Inequalities for Double Weighted Integrals of Higher Order Differentiable Functions .. . . . . . . . . . 4.3.3 Grüss Type Inequalities for Double Weighted Integrals of Higher Order Differentiable Functions .. . . . . . . . . .

213 213 216 219 226 227 245 251 257 263 273

275 279 290 292

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 299 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 307

Notations and Terminologies

R = (−∞, ∞) R+ = (0, ∞) R∗ = [0, ∞) [a, b] (a, b) [a, b) or (a, b] C[a, b] I Io AC[a, b]

f p

Set of all real numbers Set of positive real numbers Set of nonnegative real numbers A closed interval of R with end points a and b An open interval of R with end points a and b A semi-closed interval of R with end points a and b Class of all continuous functions defined on [a, b] An interval of R The interior of the interval I Class of all absolutely continuous real-valued functions defined on [a, b] Class of all continuously differentiable real-valued functions of order n defined on [a, b] Class of integrable functions defined on [a, b] Function spaces with Lp norm defined on [a, b] Function spaces with L∞ norm defined on [a, b]  b  p1 p The norm on Lp defined as |f (x)| dx 0. For small enough we define f (t) by

f (t) =

⎧ ⎪ ⎨

a ≤ t ≤ t0 ; t0 ≤ t ≤ t0 + ; t0 + ≤ t ≤ b.

0, − t0 )n , ⎪ ⎩ 1 (t − t0 )n−1 , (n−1)! 1

n! (t

So, we have    

b

a

    λl (t)f (n) (t)dt  = 

t0 + t0

  1  1 t0 + λl (t) dt  = λl (t)dt

t0

Now from inequality (1.2.45) we have 1



t0 +

t0

1 λl (t)dt ≤ λl (t0 )



t0 +

dt = λl (t0 )

t0

Since 1

→0



t0 +

lim

λl (t)dt = λl (t0 )

t0

the statement follows. In the case λl (t0 ) < 0, we define f (t) by ⎧ 1 n−1 , ⎪ ⎨ (n−1)! (t − t0 − ) 1 f (t) = − n! (t − t0 − )n , ⎪ ⎩ 0, and the rest of the proof is the same as above.

a ≤ t ≤ t0 ; t0 ≤ t ≤ t0 + ; t0 + ≤ t ≤ b; 

36

1 Linear Inequalities via Interpolation Polynomials and Green Functions

1.2.4 Mean Value Theorems and Exponential Convexity Mean Value Theorems Now we give mean value theorems for Al (f ) = A[·,·] l (·, ·, ·, ·, f ), l ∈ {5, 6, 7, 8}. xn Here f0 (x) = n! . Since proving techniques are same as done in previous sections so we omit the details in following theorems (see [5]). Theorem 1.2.15 Let f ∈ C (n) [a, b] and let Al : C (n) [a, b] → R for l ∈ {5, 6, 7, 8} be linear functionals as defined in Theorems 1.2.3 and 1.2.6 respectively. Then there exists ξl ∈ [a, b] for l ∈ {5, 6, 7, 8} such that Al (f ) = f (n) (ξl )Al (f0 ),

l ∈ {5, 6, 7, 8}.

(1.2.46)

Theorem 1.2.16 Let f, h ∈ C (n) [a, b] and let Al : C (n) [a, b] → R for l ∈ {5, 6, 7, 8} be linear functionals as defined in Theorems 1.2.3 and 1.2.6 respectively. Then there exists ξl ∈ [a, b] for l ∈ {5, 6, 7, 8} such that f (n) (ξl ) Al (f ) = (n) Al (h) h (ξk ) assuming that both the denominators are non-zero. (n)

Remark 1.2.2 If the inverse of fh(n) exists, then from the above mean value theorems we can give generalized means  ξl =

f (n) h(n)

−1 

 Al (f ) , Al (h)

l ∈ {5, 6, 7, 8}.

(1.2.47)

Logarithmically Convex Functions A number of important inequalities arise from the logarithmic convexity of some functions as one can see in [114]. Now, we recall some definitions. Here I is an interval in R. Definition 1.2.1 A function f : I → (0, ∞) is called log-convex in J -sense if the following inequality holds for each x1 , x2 ∈ I ,  f

2

x1 + x2 2

 ≤ f (x1 ) f (x2 ) .

Definition 1.2.2 ([158, p. 7]) A function f : I → (0, ∞) is called log-convex if the following inequality holds for each x1 , x2 ∈ I and λ ∈ [0, 1], f (λx1 + (1 − λ)x2 ) ≤ [f (x1 )]λ [f (x2 )](1−λ) .

1.2 Linear Inequalities and the Taylor Formula with New Green Functions

37

Remark 1.2.3 A function log-convex in the J -sense is log-convex if it is continuous as well. n-Exponentially Convex Functions Widder [186] and Bernstein [22] independently introduced an important class of functions named “exponentially convex functions” which is a sub-class of convex functions. These exponential convex functions possess a lot of important characteristics, for example the functions are analytic in their domain. Positivesemidefinite matrices are also provided by these functions. Further, these act as an effective part in studying the characteristics of Cauchy and Stolarsky means, such as monotonicity of their means etc. For more detailed information see [3, 81] and [135]. In [153], Peri´c and Peˇcari´c introduced the notion of n-exponentially convex functions, which is the concept to generalize the exponentially convex functions. In recent given sub-section of this chapter, we would like to describe the same notion of n-exponential convexity using related definitions and some significant consequences with few special remarks from [153]. Here I is an interval in R. Definition 1.2.3 A function f : I → R is n-exponentially convex in the J −sense if the following inequality holds for each ti ∈ I and ui ∈ R, i ∈ {1, . . . , n}, n

i,j =1

 ui uj f

ti + tj 2

 ≥ 0.

Definition 1.2.4 A function f : I → R is n-exponentially convex if it is nexponentially convex in the J -sense and continuous on I . Remark 1.2.4 We can see from the definition that 1-exponentially convex functions in the J -sense are in fact nonnegative functions. Also, n-exponentially convex functions in the J -sense are k-exponentially convex in the J -sense for every k ∈ N such that k ≤ n. It follows that a positive function is log-convex in the J -sense iff it is 2-exponentially convex in the J -sense. Also, using basic theory of convex functions, it follows that a positive function is log-convex iff it is 2-exponentially convex. Definition 1.2.5 A function f : I → R is exponentially convex in the J -sense, if it is n- exponentially convex in the J -sense for each n ∈ N. Remark 1.2.5 A function f : I → R is exponentially convex if it is n-exponentially convex in the J -sense and continuous on I . Example 1.2.1 A function x → cekx is an example of exponentially convex function for k ∈ R and constant c ≥ 0.

38

1 Linear Inequalities via Interpolation Polynomials and Green Functions

Theorem 1.2.17 If function f : I → R is n-exponentially convex in the J -sense, then the matrix    ti + tj m f 2 i,j =1 is positive-semidefinite. Particularly    ti + tj m det f ≥0 2 i,j =1 for each m ∈ N, m ≤ n and ti ∈ I for i ∈ {1, 2, . . . , m}. Corollary 1.2.18 If function f : I → R is exponentially convex, then the matrix    ti + tj m f 2 i,j =1 is positive-semidefinite. Particularly    ti + tj m det f ≥ 0 2 i,j =1 for each m ∈ N and ti ∈ I for i ∈ {1, 2, . . . , m}. Corollary 1.2.19 If function f : I → (0, ∞) is exponentially convex, then f is log-convex, where ∀ t1 , t2 ∈ I ; ∀ λ ∈ [0, 1], we have f (λt1 + (1 − λ)t2 ) ≤ f λ (t1 )f 1−λ (t2 ). Remark 1.2.6 A function f : I → (0, ∞) is log-convex in J -sense iff the inequality  u21 f (t1 ) + 2u1 u2 f

t1 + t2 2

 + u22 f (t2 ) ≥ 0

holds for each t1 , t2 ∈ I and u1 , u2 ∈ R. It follows that a positive function is logconvex in the J -sense iff it is 2-exponentially convex in the J -sense. Also, using basic convexity theory it follows that a positive function is log-convex iff it is 2exponentially convex. Here, we get our results concerning the n-exponential convexity and exponential convexity for our functionals Al , l ∈ {5, 6, 7, 8} as defined in Theorems 1.2.3 and 1.2.6 . Throughout the section I is an interval in R. Theorem 1.2.20 Let D1 = {ft : t ∈ I } be a class of functions such that the function t → [z0 , z1 , . . . , zn ; ft ] is n-exponentially convex in the J -sense on I for

1.2 Linear Inequalities and the Taylor Formula with New Green Functions

39

any n + 1 mutually distinct points z0 , z1 , . . . , zn ∈ [a, b]. Let Al be the linear functionals for l ∈ {5, 6, 7, 8} as defined in Theorems 1.2.3 and 1.2.6 . Then the following statements are valid: (a) The function t → Al (ft ) is n-exponentially convex function in the J -sense on I. (b) If the function t → Al (ft ) is continuous on I , then the function t → Al (ft ) is n-exponentially convex on I . Proof (a) Fix l ∈ {5, 6, 7, 8}. Let us define the function p for ti ∈ I, ui ∈ R, i ∈ In as follows p=

n

ui uj f ti +tj . 2

i,j =1

Since the function t → [z0 , z1 , . . . , zn ; ft ] is n-exponentially convex in the J -sense, therefore [z0 , z1 , . . . , zn ; p] =

n

ui uj [z0 , z1 , . . . , zn ; f ti +tj ] ≥ 0 2

i,j =1

which implies that p is n-convex function on I and therefore Al (p) ≥ 0. Hence n

i,j =1

ui uj Al (f ti +tj ) ≥ 0. 2

We conclude that the function t → Al (ft ) is an n-exponentially convex function on I in J -sense. (b) This part easily follows from definition of n-exponentially convex function.  As a consequence of the above theorem we give the following corollaries: Corollary 1.2.21 Let D2 = {ft : t ∈ I } be a class of functions such that the function t → [z0 , z1 , . . . , zn ; ft ] is an exponentially convex in the J -sense on I for any n + 1 mutually distinct points z0 , z1 , . . . , zn ∈ [a, b]. Let Al be the linear functionals for l ∈ {5, 6, 7, 8} as defined in Theorems 1.2.3 and 1.2.6. Then the following statements are valid: (a) The function t → Al (ft ) is exponentially convex in the J -sense on I . (b) If the function t → Al (ft ) is continuous on I, then the function t → Al (ft ) is exponentially convex on I .

40

1 Linear Inequalities via Interpolation Polynomials and Green Functions



 m (c) The matrix Al f ti +tj 2

is positive-semidefinite. Particularly,

i,j =1

  m det Al f ti +tj 2

i,j =1

≥0

for each m ∈ N and ti ∈ I where i ∈ {1, 2, . . . , m}. Proof Proof follows directly from Theorem 1.2.20 by using definition of exponential convexity and Corollary 1.2.18.  Corollary 1.2.22 Let D3 = {ft : t ∈ I } be a class of functions such that the function t → [z0 , z1 , . . . , zn ; ft ] is 2-exponentially convex in the J -sense on I for any n + 1 mutually distinct points z0 , z1 , . . . , zn ∈ [a, b]. Let Al be the linear functionals for l ∈ {5, 6, 7, 8} as defined in Theorems 1.2.3 and 1.2.6. Then the following statements are valid: (a) If the function t → Al (ft ) is continuous on I , then it is 2-exponentially convex on I . If the function t → Al (ft ) is additionally positive, then it is also logconvex on I . Moreover, the following Lyapunov’s inequality holds for r < s < t; r, s, t ∈ I [Al (fs )]t −r ≤ [Al (fr )]t −s [Al (ft )]s−r .

(1.2.48)

(b) If the function t → Al (ft ) is positive and differentiable on I , then for every s, t, u, v ∈ I such that s ≤ u and t ≤ v, we have μs,t (Al , D3 ) ≤ μu,v (Al , D3 ),

(1.2.49)

where μs,t is defined as ⎧  1 ⎪ ⎨ Al (fs ) s−t , Al (ft ) μs,t (Al , D3 ) =  d A (f ) ⎪ s l ⎩exp ds , Al (fs )

s = t;

(1.2.50)

s = t,

for fs , ft ∈ D3 . Proof (a) It follows directly form Theorem 1.2.20 and Corollary 1.2.19. As the function t → Al (ft ) is log-convex, i.e., ln Al (ft ) is convex, so we have ln[Al (fs )]t −r ≤ ln[Al (fr )]t −s + ln[Al (ft )]s−r , which gives us (1.2.48).

l ∈ {5, 6, 7, 8}

1.2 Linear Inequalities and the Taylor Formula with New Green Functions

41

(b) For convex function f , the inequality f (s) − f (t) f (u) − f (v) ≤ s −t u−v

(1.2.51)

holds ∀ s, t, u, v ∈ I ⊂ R such that s ≤ u, t ≤ v, s = t, u = v. Since by (a), k (ft ) are log-convex for l ∈ {5, 6, 7, 8}, so set f (t) = ln k (ft ) in (1.2.51) we have ln Al (fu ) − ln Al (fv ) ln Al (fs ) − ln Al (ft ) ≤ , s−t u−v

l ∈ {5, 6, 7, 8} (1.2.52)

for s ≤ u, t ≤ v, s = t, u = v, which is equivalent to (1.2.49). The cases for s = t and/or u = v are easily followed from (1.2.52) by taking respective limits.  Remark 1.2.7 The results from Theorem 1.2.20 and Corollaries 1.2.21 and 1.2.22 still hold when any two (all) points z0 , z1 , . . . , zn ∈ [a, b] coincide for a family of differentiable (n-times differentiable) functions ft such that the function t → [z0 , z1 , . . . , zn ; ft ] is n-exponentially convex, exponentially convex and 2expoenetially convex in the J -sense respectively. Now, we give two important remarks and one useful corollary from [81], which we will use in some examples in next section. Remark 1.2.8 For μs,t (Al , F ) defined with (1.2.49) we will refer as mean if a ≤ μs,t (Al , F ) ≤ b for s, t ∈ I and l ∈ {5, 6, 7, 8} where F = {ft : t ∈ I } be a family of functions and [a, b] ⊂ Dom(ft ).  Theorem 1.2.20 gives us the following corollary. Corollary 1.2.23 Let a, b ∈ R and Al be linear functionals for l ∈ {5, 6, 7, 8}. Let F = {ft : t ∈ I } be a family of functions in C (2) [a, b]. If ⎛ a ≤ ⎝

d 2 fs dx 2 d 2 ft dx 2

⎞ ⎠

1 s−t

(ξ ) ≤ b,

for ξ ∈ [a, b], s, t ∈ I , then μs,t (Al , F ) is a mean for l ∈ {5, 6, 7, 8}.

42

1 Linear Inequalities via Interpolation Polynomials and Green Functions

Remark 1.2.9 In some examples, we will get means of this type: ⎛ ⎝

d 2 fs dx 2 d 2 ft dx 2

⎞

1 s−t

⎠

(ξ ) = ξ, ξ ∈ [a, b],

s = t.

1.2.5 Examples with Applications In this section, we use various classes of functions F = {ft : t ∈ I } for any open interval I ⊂ R to construct different examples of exponentially convex functions and applications to Stolarsky-type means. Let us consider some examples: Example 1.2.2 Let F1 = {ψt : [a, b] ⊂ R → [0, ∞) : t ∈ R} be a family of functions defined by " ψt (x) =

etx tn xn n!

, ,

t = 0; t = 0.

n

d t x > 0 for x ∈ [a, b] ⊂ R, the function ψ (x) is a nSince dx n ψt (x) = e t dn convex on R for every t ∈ R and t → dx ψ (x) is exponentially convex by n t definition. Using analogous arguing as in the proof of Theorems 1.2.20, we have that t → [z0 , z1 , . . . , zn ; ψt ] is exponentially convex (and so exponentially convex in the J -sense). Using Corollary 1.2.21 we conclude that t → Al (ψt ), l ∈ {5, 6, 7, 8} are exponentially convex in the J -sense. It is easy to see that these mappings are continuous, so they are exponentially convex. Assume that t → Al (ψt ) > 0 for l ∈ {5, 6, 7, 8}. By introducing convex functions ψt in (1.2.47), we obtain the following means: for l ∈ {5, 6, 7, 8}

Ms,t (Al , F1 ) =

⎧   Al (ψs ) 1 ⎪ ln ⎪ ⎨ s−t Al (ψt ) , ⎪ ⎪ ⎩

Al (id.ψs ) n Al (ψs ) − s Al (id.ψ0 ) (n+1)Al (ψ0 )

, ,

s = t; s=t =

0; s = t = 0;

where id stands for identity function on [a, b] ⊂ R. Here Ms,t (Al , F1 ) = ln(μs,t (Al , F1 )), l ∈ {5, 6, 7, 8} are in fact means. 1  d n ψ  s−t s

Remark 1.2.10 We observe here that [a, b] where a, b ∈ (0, ∞).

dx n d n ψt dx n

(ln ξ ) = ξ is a mean for ξ ∈

1.2 Linear Inequalities and the Taylor Formula with New Green Functions

43

Example 1.2.3 Let F2 = {ϕt : [a, b] ⊂ (0, ∞) → R : t ∈ R} be a family of functions defined as " xt t ∈ {0, . . . , n − 1}; t (t −1)···(t −n+1) , ϕt (x) = (x)j ln(x) , t = j ∈ {0, . . . , n − 1}. (−1)n−1−j j !(n−1−j )! 2

d Since ϕt (x) is n-convex function for x ∈ [a, b] ⊂ (0, ∞) and t → dx 2 ϕt (x) is exponentially convex, so by the same arguments given in previous example we conclude that Al (ϕt ), l ∈ {5, 6, 7, 8} are exponentially convex. We assume that Al (ϕt ) > 0 for l ∈ {5, 6, 7, 8}. For this family of convex functions we obtain the following means: for l ∈ {5, 6, 7, 8}, J = {0, 1, . . . , n − 1}

⎧  1 Al (ϕs ) s−t ⎪ ⎪ ,s =

t, ⎪ ⎨ Al (ϕ t ) n−3 1  A (ϕ ϕ ) n−1 s l 0 Ms,t (Al , F2 ) = exp (−1) (n − 1)! , s = t ∈ J, k=0 k−t Al (ϕs ) + ⎪   ⎪ ⎪  ⎩ exp (−1)n−1 (n − 1)! Al (ϕ0 ϕs ) + n−1 1 k=0,k =t k−t , s = t ∈ J. 2Al (ϕs ) Here Ms,t (Al , F2 ) = μs,t (Al , F2 ), l ∈ {5, 6, 7, 8} are in fact means. Remark 1.2.11 Further, in this choice of family F2 , we have 1  d n ϕ  s−t s

dx n d n ϕt dx n

(ξ ) = ξ, ξ ∈ [a, b], s = t, where a, b ∈ (0, ∞).

So, using Remark 1.2.9 we have an important conclusion that μs,t (Al , F2 ) is in fact mean for l ∈ {1, 2, 3, 4}. Example 1.2.4 Let F3 = {θt : [a, b] ⊂ (0, ∞) → (0, ∞) : t ∈ (0, ∞)} be a family of functions defined by √

e−x t θt (x) = n/2 . t n

√

d −x t is exponentially convex for x ∈ [a, b] ⊂ (0, ∞), Since t → dx n θt (x) = e being the Laplace transform of a nonnegative function [81]. So, by same argument given in Example 1.2.2 we conclude that Al (θt ), l ∈ {5, 6, 7, 8} are exponentially convex. We assume that Al (θt ) > 0 for l ∈ {5, 6, 7, 8}. For this family of functions we have the following possible cases of μs,t ( k , F3 ): for l ∈ {5, 6, 7, 8}

⎧   1 ⎪ ⎨ Al (θs ) s−t Al (θ Ms,t (Al , F3 ) = t ) ⎪ ⎩ exp − A√l (id.θs ) − 2 s A (θ ) l

s

n 2s



,

s = t,

,

s = t.

44

1 Linear Inequalities via Interpolation Polynomials and Green Functions

√ √ By (1.2.47), Ms,t (Al , F3 ) = −( s + t) ln μs,t (Al , F3 ), l ∈ {5, 6, 7, 8} defines a class of means. Example 1.2.5 Let F4 = {φt : [a, b] ⊂ (0, ∞) → (0, ∞) : t ∈ (0, ∞)} be a family of functions defined by " φt (x) =

t −x (ln t )n xn n

, ,

t=

1; t = 1.

n

d −x = e −xlnt > 0 for x ∈ [a, b] ⊂ (0, ∞), so by same Since dx n φt (x) = t argument given in Example 1.2.2 we conclude that t → Al (φt ), l ∈ {5, 6, 7, 8} are exponentially convex. We assume that Al (φt ) > 0 for l ∈ {5, 6, 7, 8}. For this family of functions we have the following possible cases of μs,t ( k , F4 ): for l ∈ {5, 6, 7, 8}

⎧  1 Al (φs ) s−t ⎪ ⎪ , s=

t; ⎪ ⎨ Al (φ  t ) A (id.φ ) n s Ms,t (Al , F4 ) = exp − l , s = t = 1; − s ln ⎪  sAl (φs ) s ⎪ ⎪ ⎩ exp − 1 Al (id.φ1 ) , s = t = 1. (n+1) Al (φ1 ) By (1.2.47), Ms,t (Al , F4 ) = −L(s, t) ln μs,t , (Al , F4 ), l ∈ {5, 6, 7, 8} defines a class of means, where L(s, t) is Logarithmic mean defined as: " L(s, t) =

s = t; s, s = t.

s−t ln s−ln t ,

Remark 1.2.12 Monotonicity of μs,t (Al , Fj ) follow form (1.2.49) for l {5, 6, 7, 8}, j ∈ {1, 2, 3, 4}.

∈

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions In this section, we state new general linear identities and inequalities involving nconvex functions using Hermite interpolation polynomials and Green functions. We will proceed in the same manner as done in the last two sections of this chapter. It is worth mentioning that this section is mainly based on results collected from [90]. Here we recall some basic definitions, facts and results from [101]. Let −∞ < a ≤ a1 < a2 < · · · < ar ≤ b < ∞, r ≥ 2. The Hermite interpolation of a function H ∈ C (n) [a, b] is of the form H (x) = PH (x) + eH (x)

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions

45

where PH is the unique polynomial of degree n−1, called the Hermite interpolating polynomial of H , satisfying (i)

PH (aj ) = H (i) (aj ),

0 ≤ i ≤ kj , 1 ≤ j ≤ r,

r

kj + r = n.

j =1

The associated error eH (x) can be represented in terms of the Green function GH,n (x, s) for the multipoint boundary value problem z(n) (x) = 0,

z(i) (aj ) = 0,

0 ≤ i ≤ kj ,

1 ≤ j ≤ r,

that is, the following result holds (see also [1]): Theorem 1.3.1 Let H ∈ C (n) [a, b] and let PH be its Hermite interpolating polynomial. Then H (x) = PH (x) + eH (x) =

kj r



b Hij (x)H

(i)

(aj ) +

j =1 i=0

GH,n (x, s)H (n) (s) ds,

(1.3.1)

a

where Hij are the fundamental polynomials of the Hermite basis defined by kj −i

1 dk w(x) 1 Hij (x) = i! (x − aj )kj +1−i k! dx k k=0



(x − aj )kj +1 w(x)



  

x=aj

(x − aj )k , (1.3.2)

where w(x) =

r #

(x − aj )kj +1

(1.3.3)

j =1

and GH,n is the Green function defined by

GH,n (x, s) =

⎧ j k j 1

⎪ (aj − s)n−i−1 ⎪ ⎪ ⎪ Hij (x), ⎪ ⎨ (n − i − 1)! j =1 i=0 r ⎪

⎪ ⎪ ⎪ ⎪ ⎩−

s ≤ x;

kj

(aj − s)n−i−1 Hij (x), s ≥ x; (n − i − 1)!

j =j1 +1 i=0

for all aj1 ≤ s ≤ aj1 +1 , j1 ∈ {0, 1, . . . , r} (ar+1 = b).

(1.3.4)

46

1 Linear Inequalities via Interpolation Polynomials and Green Functions

The following are some special cases of the Hermite interpolation of functions: (i) (m, n − m) conditions: r = 2, a1 = a, a2 = b, 1 ≤ m ≤ n − 1, k1 = m − 1 and k2 = n − m − 1. In this case H (x) =

m−1

τi (x)H

(i)

i=0

(a) +

n−m−1

 ηi (x)H

(i)

b

(b) +

Gm,n (x, s)H (n) (s)ds,

a

i=0

where n−m m−1−i 

n−m+k −1 x −a k 1 i x −b τi (x) = (x −a) , i! a −b b−a k

(1.3.5)

k=0

 m n−m−1−i

m + k − 1 x − b k 1 i x−a , (1.3.6) ηi (x) = (x − b) i! b−a a−b k k=0

and the Green function Gm,n is of the form ⎡ ⎤ ⎧ m−1

m−1−j

n − m + p − 1 x − a p ⎪ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ p b−a ⎪ ⎪ j =0 p=0 ⎪ ⎪ ⎪ ⎪ (x − a)j (a − s)n−j −1  b − x n−m ⎪ ⎪ , s ≤ x; × ⎨ j !(n − j − 1)! b−a ⎡ ⎤ Gm,n (x, s) = ⎪ n−m−1 ⎪

n−m−1−i

m + q − 1 b − x q ⎪ ⎪ ⎣ ⎦ ⎪ − ⎪ ⎪ b − a q ⎪ ⎪ i=0 q=0 ⎪ ⎪ ⎪ ⎪ (x − b)i (b − s)n−i−1  x − a m ⎪ ⎩ , s ≥ x. × i!(n − i − 1)! b−a (ii) Taylor’s two-point condition: m ∈ N, n = 2m, r = 2, a1 = a, a2 = b and k1 = k2 = m − 1. In this case H (x) =

m−1

m−i−1

 i=0

+

k=0

 m + k − 1  (x − a)i  x − b m  x − a k (i) H (a) i! a−b b−a k

(x − b)i  x − a m  x − b k (i)  H (b) + i! b−a a−b



b a

G2T ,m (x, s)H (2m) (s)ds,

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions

47

where the Green function G2T ,m is of the form G2T ,m (x, s) =

(−1)m (2m − 1)! ⎧ m−1

m + k − 1 ⎪ ⎪ m ⎪ (x − s)m−1−k q k (x, s), s ≤ x; ⎪ ⎨ p (x, s) k k=0 m−1 ⎪

m + k − 1 ⎪ ⎪ ⎪ q m (x, s) (s − x)m−1−k pk (x, s), x ≤ s; ⎩ k k=0

where p(x, s) =

(s − a)(b − x) and q(x, s) = p(s, x). (b − a)

In this section, we would state and prove new results involving new Green functions and inequalities of type (1.0.3) and (1.0.7) for n-convex functions by making use of the Hermite interpolation. Similar results can be found in [103]. The following lemma yields the sign of the Green function (1.3.4) on certain intervals (see Lemma 2.3.3, page 75, in [1]). Lemma 1.3.1 The Green function GH,n given by (1.3.4) and w given by (1.3.3) satisfy GH,n (x, s) > 0, w(x)

for a1 ≤ x ≤ ar , a1 < s < ar .

1.3.1 Results Obtained by the Hermite Interpolation Polynomial and Green Functions We start this subsection with our first main discrete identity. Theorem 1.3.2 Let all the assumptions of Theorem 1.3.1 be valid with additional conditions x ∈ [a, b]m and p ∈ Rm . Then m

pk H (xk ) = (H (a) − aH (b))

k=1

m

pk + H (b)

k=1

+

kj r



 H

(i+2)

j =1 i=0



b

+ a



m b

a k=1

(aj )

m

pk xk

k=1 m b

pk G1 (xk , s)Hij (s) ds

a k=1

pk G1 (xk , s)GH,n−2 (s, t)H (n) (t) dt ds, (1.3.7)

48

1 Linear Inequalities via Interpolation Polynomials and Green Functions m

pk H (xk ) = (H (b) − bH (a))

k=1

m

k=1

+

kj r



 H

(i+2)

(aj )



b

+



a

pk xk

k=1 m b

pk G2 (xk , s)Hij (s) ds

m b

pk G2 (xk , s)GH,n−2 (s, t)H (n) (t) dt ds, (1.3.8)

a k=1

pk H (xk ) = (H (b) − bH (b) + (H (b) − H (a))a)

k=1

m

pk + H (a)

k=1

+

kj r



 H (i+2) (aj )

 +

b



a

m b

m b

pk G3 (xk , s)Hij (s) ds

pk G3 (xk , s)GH,n−2 (s, t)H (n) (t) dt ds,



pk H (xk ) = (H (a) − aH (a) − (H (b) − H (a))b)

k=1

(1.3.9)

m



pk + H (b)

k=1 kj r



 H

(i+2)

(aj )

+ a

b



m b

m b

m

pk xk

k=1

pk G4 (xk , s)Hij (s) ds

a k=1

j =1 i=0



pk xk

k=1

a k=1

+

m

a k=1

j =1 i=0

m

m

a k=1

j =1 i=0

m

pk − H (a)

pk G4 (xk , s)GH,n−2 (s, t)H (n) (t) dt ds.

(1.3.10)

a k=1

Proof Applying identities (1.1.5), (1.1.10), (1.1.11) and (1.1.12) for f = H and x = xk , multiplying it by pk and summing up over k from 1 to m, we obtain respectively m

pk H (xk ) = (H (a) − aH (b))

k=1

m

k=1

 +

m b

a k=1

pk + H (b)

m

pk xk

k=1

pk G1 (xk , s)H

(s) ds,

(1.3.11)

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions m

pk H (xk ) = (H (b) − bH (a))

k=1

m

pk − H (a)

k=1

 +

m b

m

49

pk xk

k=1

pk G2 (xk , s)H

(s) ds,

(1.3.12)

a k=1 m

pk H (xk ) = (H (b) − bH (b) + (H (b) − H (a))a)

k=1

m

pk + H (a)

k=1



m b

+

m

pk xk

k=1

pk G3 (xk , s)H

(s) ds,

(1.3.13)

a k=1 m

pk H (xk ) = (H (a) − aH (a) − (H (b) − H (a))b)

k=1

m

pk + H (b)

k=1



m b

+

m

pk xk

k=1

pk G4 (xk , s)H

(s) ds.

(1.3.14)

a k=1

By Theorem 1.3.1, H

(s) can be expressed as

H (s) =

kj r



b Hij (s)H

(i+2)

j =1 i=0

(aj ) +

GH,n−2 (s, t)H (n) (t) dt.

(1.3.15)

a

Inserting one by one expression from (1.3.15) in (1.3.11)−(1.3.14) we get (1.3.7)− (1.3.10) respectively.  Under the assumptions of Theorem 1.3.2 here we define some linear functional as follows: A[a,b] (m, x, p, H ) 9

=

m



pk H (xk ) − (H (a) − aH (b))

k=1

−

kj r



 H

(i+2)

j =1 i=0



b

= a



m b

a k=1

(aj )

m b

m

k=1

pk + H (b)

m

pk xk

k=1

pk G1 (xk , s)Hij (s) ds

a k=1

pk G1 (xk , s)GH,n−2 (s, t)H (n) (t) dt ds,

(1.3.16)

50

1 Linear Inequalities via Interpolation Polynomials and Green Functions

A[a,b] 10 (m, x, p, H ) =

m

pk H (xk ) − (H (b) − bH (a))

k=1

−

kj r



 H

(i+2)



b a



m

pk xk

k=1

pk G2 (xk , s)Hij (s) ds

a k=1

j =1 i=0

=

pk − H (a)

k=1

m b

(aj )

m

m b

pk G2 (xk , s)GH,n−2 (s, t)H (n) (t) dt ds,

(1.3.17)

a k=1

A[a,b] 11 (m, x, p, H ) = m

pk H (xk ) − (H (b) − bH (b) + (H (b) − H (a))a)

k=1

−

m

pk + H (a)

k=1

kj r



 H (i+2)(aj )



b



a

pk xk

k=1

pk G3 (xk , s)Hij (s) ds

a k=1

j =1 i=0

=

m b

m

m b

pk G3 (xk , s)GH,n−2 (s, t)H (n) (t) dt ds,

(1.3.18)

a k=1

A[a,b] 12 (m, x, p, H ) = m

pk H (xk ) − (H (a) − aH (a) − (H (b) − H (a))b)

k=1

−

m

pk + H (b)

k=1

kj r



 H (i+2)(aj )



b a



pk xk

k=1

pk G4 (xk , s)Hij (s) ds

a k=1

j =1 i=0

=

m b

m

m b

pk G4 (xk , s)GH,n−2 (s, t)H (n) (t) dt ds.

(1.3.19)

a k=1

Now we would use the identities stated above to prove our next results. Here we also need a remark as well as follows: Remark 1.3.1 It is known that the Bernstein polynomials Bn defined as in [158] Pn (H, x) =

n  

n H (a + ih) (x − a)i (b − x)n−i , i i=0

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions

51

(where h = b−a n ) converges uniformly to H on [a, b] as n → ∞ provided that H is continuous. Further, if H is m-convex function these polynomials have nonnegative dm derivatives of order m, i.e., Pn ≥ 0 which can be prove by induction by using dx x the following formula:     n−m dm n n−m m Pn (H, x) = m! h H (a + ih) (x − a)i (b − x)n−m−i , d mx i m i=0

where the finite difference of the function H defined on [a, b], of order m for m ∈ {0, 1, . . . }, is defined by 0h H (x) = H (x),

m−1 m H (x + h) − m−1 H (x), h H (x) = h h

where h is a non-zero real number. It is easy to see that m h H (x) =

  m

m (−1)m−i H (x + ih) i i=0

for x ∈ [a, b] and h ∈ R provided x + ih ∈ [a, b] for i ∈ {0, . . . , m}. Theorem 1.3.3 Let all the assumptions of Theorem 1.3.2 be valid. Further let H : [a, b] → R be n-convex and for l ∈ {1, 2, 3, 4} m

pk Gl (xk , s) ≥ 0

for all s ∈ [a, b],

(1.3.20)

k=1

and consider the inequality A[a,b] l+8 (m, x, p, H ) ≥ 0,

(1.3.21)

where functionals Al for l ∈ {1, 2, 3, 4} are defined in (1.3.16)–(1.3.19) respectively. (i) If kj for j ∈ {2, . . . , r} are odd, then inequality (1.3.21) holds for each l ∈ {1, 2, 3, 4}. (ii) If kj for j ∈ {2, . . . , r − 1} are odd and kr is even, then the reverse inequality hold in (1.3.21) for each l ∈ {1, 2, 3, 4}. Proof Fix l ∈ {1, 2, 3, 4}. (i) First we consider the case if H ∈ C (n) [a, b], then by given assumptions we have w satisfies w(x) ≥ 0 for all x where w is defined in (1.3.3) and hence, by Lemma 1.3.1, GH,n−2 (s, t) ≥ 0 for all s, t ∈ [a, b]. Therefore, the last terms on the right hand side of (1.3.7)−(1.3.10) are nonnegative, so inequality (1.3.21) holds. By Remark 1.3.1 the inequality for general H follows since every n-convex function can be obtained, by making use of the Bernstein

52

1 Linear Inequalities via Interpolation Polynomials and Green Functions

polynomials, as a uniform limit of n-convex functions with a continuous n-th derivative. (ii) Under these assumptions w(x) ≤ 0, so GH,n−2 (s, t) ≤ 0. The rest of the proof is same as in (i).  In the case of the (m, n − m) conditions we have the following corollary. Corollary 1.3.4 Let τi and ηi be given by (1.3.5) and (1.3.6) and let x ∈ [a, b]m and p ∈ Rm be such that (1.3.20) holds. Let H : [a, b] → R be n-convex and consider the inequalities m

pk H (xk ) ≥ (H (a) − aH (b))

k=1

pk + H (b)

k=1



b

+

 m

a

+

m

n−j

−1

⎞

pk xk

k=1

 ⎛j −1

pk G1 (xk , s) ⎝ τi (s)H (i+2)(a)

k=1

m

i=0

ηi (s)H (i+2) (b)⎠ ds,

(1.3.22)

i=0

m

pk H (xk ) ≥ (H (b) − bH (a))

k=1

pk − H (a)

k=1



b

+

 m

a

+

m

 ⎛j −1

pk G2 (xk , s) ⎝ τi (s)H (i+2)(a)

k=1

n−j

−1

⎞

m

pk xk

k=1

i=0

ηi (s)H (i+2) (b)⎠ ds,

(1.3.23)

i=0

m

pk H (xk ) ≥ (H (b) − bH (b) + (H (b) − H (a))a)

k=1



b

+



a

+

m

k=1

n−j

−1 i=0

 ⎛j −1

pk G3 (xk , s) ⎝ τi (s)H (i+2) (a) ⎞

m

k=1

pk + H (a)

m

pk xk

k=1

i=0

ηi (s)H (i+2) (b)⎠ ds,

(1.3.24)

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions m

pk H (xk ) ≥ (H (a) − aH (a) − (H (b) − H (a))b)

k=1



b

+



a

+

m

 ⎛j −1

pk G4 (xk , s) ⎝ τi (s)H (i+2) (a)

k=1

n−j

−1

⎞

m

53

pk + H (b)

k=1

m

pk xk

k=1

i=0

ηi (s)H (i+2) (b)⎠ ds.

(1.3.25)

i=0

(i) If n − j is even, then (1.3.22)–(1.3.25) hold. (ii) If n − j is odd, then the reverse of (1.3.22)–(1.3.25) hold. In the case of Taylor’s two point conditions we have the following corollary. Corollary 1.3.5 Let x ∈ [a, b]m and p ∈ Rm be such that (1.3.20) holds. Let H : [a, b] → R be n-convex and consider the inequalities m

pk H (xk ) ≥ (H (a) − aH (b))

k=1



b

+ a

m

pk + H (b)

k=1

 m

k=1

m

pk xk

k=1

 ⎛j −1 j −i−1 

j + k − 1 pk G1 s(xk , s) ⎝ k (

i=0 k=0

(s − a)i  s − b j  s − a k (i+2) H (a) i! a−b b−a ) (s − b)i  s − a j  s − b k (i+2) + H (b) ds, (1.3.26) i! b−a a−b

×

m

pk H (xk ) ≥ (H (b) − bH (a))

k=1

m

k=1



b

+ a

 m

k=1

pk − H (a)

m

pk xk

k=1

 ⎛j −1 j −i−1 

j + k − 1 ⎝ pk G2 (xk , s) k (

i=0 k=0

(s − a)i  s − b j  s − a k (i+2) H (a) i! a−b b−a ) (s − b)i  s − a j  s − b k (i+2) + H (b) ds, (1.3.27) i! b−a a−b

×

54

1 Linear Inequalities via Interpolation Polynomials and Green Functions

m

pk H (xk ) ≥ (H (b) − bH (b) + (H (b) − H (a))a)

k=1



b

+



a

m

k=1

 ⎛j −1 j −i−1 

j + k − 1 ⎝ pk G3 (xk , s) k

m

pk + H (a)

k=1

m

pk xk

k=1

i=0 k=0

(

(s − a)i  s − b j  s − a k (i+2) H (a) i! a−b b−a ) (s − b)i  s − a j  s − b k (i+2) + H (b) ds, i! b−a a−b

×

m

pk H (xk ) ≥ (H (a) − aH (a) − (H (b) − H (a))b)

k=1

 +

b

a



m

k=1

 ⎛j −1 j −i−1 

j + k − 1 pk G4 (xk , s) ⎝ k

m

pk + H (b)

k=1

(1.3.28)

m

pk xk

k=1

i=0 k=0

(

(s − a)i  s − b j  s − a k (i+2) H (a) i! a−b b−a ) (s − b)i  s − a j  s − b k (i+2) H (b) ds. + i! b−a a−b

×

(1.3.29)

(i) If j is even, then (1.3.26)–(1.3.29) hold. (ii) If j is odd, then the reverse of (1.3.26)–(1.3.29) hold. Theorem 1.3.6 Let all the assumptions of Theorem 1.3.1 be valid and let x ∈ [a, b]m and p ∈ Rm satisfy (1.0.6). Further let H : [a, b] → R be n-convex and consider the inequality for l ∈ {1, 2, 3, 4} m

pk H (xk ) ≥

kj r



 H

(i+2)

(aj )

pk Gl (xk , s)Hij (s) ds

(1.3.30)

a k=1

j =1 i=0

k=1

m b

and the function F (x) =

kj r

j =1 i=0

 H (i+2)(aj )

b

Gl (x, s)Hij (s) ds. a

(1.3.31)

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions

55

(i) If kj for j ∈ {2, . . . , r} are odd, then (1.3.30) holds. Furthermore, if the m

function F is convex, then inequality pk H (xk ) ≥ 0 holds. k=1

(ii) If kj for j ∈ {2, . . . , r − 1} are odd and kr is even, then the reverse of (1.3.30) holds. Furthermore, if the function F is concave, then inequality  m k=1 pk H (xk ) ≤ 0 holds. Proof Fix l ∈ {1, 2, 3, 4}. The functions Gl (x, s) are convex in the first variable, so assumption (1.3.20) is satisfied by Remark 1.0.4. Now, the claims of the theorem follow from Theorem 1.3.3.  Here we state the integral version of all the results stated in the start of this subsection. Since proving techniques are quit similar so we omit the details. Theorem 1.3.7 Let all the assumptions of Theorem 1.3.1 be valid with additional conditions g : [α, β] → [a, b], p : [α, β] → R. Then for l ∈ {1, 2, 3, 4} 

β



p(x)H (g(x)) dx = (H (a) − aH (b))

α

+

β



p(x) dx + H (b)

p(x)g(x) dx

α

kj r

j =1 i=0



b

+ a



b

 H

(i+2)

b



a

β

 p(x)G1 (g(x), s) dx Hij (s) ds

a



α

β

(aj ) α

 p(x)G1 (g(x), s) dx GH,n−2 (s, t)H (n) (t) dt ds,

α



β

p(x)H (g(x)) dx =

α

(H (b) − bH (a))



β

p(x) dx − H (a)

α

+

kj r



H



b

+ a

β

a

β

p(x)g(x) dx 

b  β

(aj )

p(x)G2 (g(x), s) dx Hij (s) ds a

b  β



 α

 (i+2)

j =1 i=0



β

α

 p(x)G2 (g(x), s) dx GH,n−2 (s, t)H (n) (t) dt ds,

α

p(x)H (g(x)) dx =

α

(H (b) − bH (b) + (H (b) − H (a))a)

 α

β

p(x) dx + H (a)



β

p(x)g(x) dx α

56

1 Linear Inequalities via Interpolation Polynomials and Green Functions

+

kj r



 H (i+2)(aj ) a

j =1 i=0



b

+ a



β

b  β



b  β

a

 p(x)G3 (g(x), s) dx Hij (s) ds

α

 p(x)G3 (g(x), s) dx GH,n−2 (s, t)H (n) (t) dt ds,

α

p(x)H (g(x)) dx =

α





β

(H (a) − aH (a) − (H (b) − H (a))b) +



p(x) dx + H (b)

p(x)g(x) dx

α

+

kj r



H



b

+ a

b  β



(aj )

a

 p(x)G4 (g(x), s) dx Hij (s) ds

a

j =1 i=0

α

b  β

 (i+2)

β

α

 p(x)G4 (g(x), s) dx GH,n−2 (s, t)H (n) (t) dt ds.

α

Under the assumptions of Theorem 1.3.7 here we introduce some further linear functional as follows: A[a,b] 13 ([α, β], g, p, H ) =  β  p(x)H (g(x)) dx − (H (a) − aH (b)) α

−

p(x) dx − H (b)

α

kj r

j =1 i=0



b

=



a

b

 H (i+2)(aj )

b



a



a

β

β

 p(x)G1 (g(x), s) dx GH,n−2 (s, t)H (n) (t) dt ds,

j =1 i=0

a



b a

(1.3.32)

α

kj r



b

p(x)g(x) dx α

β

 H

(i+2)

α

β



β

(aj )

β

p(x)g(x) dx α

 p(x)G2 (g(x), s) dx Hij (s) ds

a



b



p(x) dx + H (a)

α



β

α

α

=



 p(x)G1 (g(x), s) dx Hij (s) ds

A[a,b] 14 ([α, β], g, p, H ) =   β

p(x)H (g(x)) dx − (H (b) − bH (a)) −

β

α

 p(x)G2 (g(x), s) dx GH,n−2 (s, t)H (n) (t) dt ds,

(1.3.33)

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions



A[a,b] 15 ([α, β], g, p, H ) =



p(x)H (g(x)) dx − (H (b) − bH (b)

α

β

+(H (b) − H (a))a)

β

+

 H



b b

a



a

β

β

p(x)g(x) dx α

b





β

(aj )

p(x)G3 (g(x), s) dx Hij (s) ds a

j =1 i=0

=

(i+2)



p(x) dx − H (a)

α kj r



57

α

 p(x)G3 (g(x), s) dx GH,n−2 (s, t)H (n) (t) dt ds, (1.3.34)

α



A[a,b] 16 ([α, β], g, p, H )

β

=

p(x)H (g(x)) dx α

−(H (a) − aH (a) − (H (b) − H (a))b)  β  β

+ p(x) dx − H (b) p(x)g(x) dx α

α



kj

−

r

j =1 i=0



b

= a



a

b

H

(i+2)

β





β

(aj )

p(x)G4 (g(x), s) dx Hij (s) ds a



b

α

 p(x)G4 (g(x), s) dx GH,n−2 (s, t)H (n) (t) dt ds.(1.3.35)

α

Theorem 1.3.8 Let all the assumptions of Theorem 1.3.7 be valid. Further let H : [a, b] → R be n-convex and for l ∈ {1, 2, 3, 4} 

β

p(x)Gl (g(x), s) dx ≥ 0

for all s ∈ [a, b],

(1.3.36)

α

and consider the inequality A[a,b] l+12 ([α, β], g, p, H ) ≥ 0

(1.3.37)

where functionals Al+12 for l ∈ {1, 2, 3, 4} are defined in (1.3.32)–(1.3.35) respectively. (i) If kj for j ∈ {2, . . . , r} are odd, then inequality (1.3.37) holds for each l ∈ {1, 2, 3, 4}. (ii) If kj for j ∈ {2, . . . , r − 1} are odd and kr is even, then the reverse inequality holds in (1.3.37) for each l ∈ {1, 2, 3, 4}.

58

1 Linear Inequalities via Interpolation Polynomials and Green Functions

Corollary 1.3.9 Let τi and ηi be given by (1.3.5) and (1.3.6) and let g : [α, β] → R, p : [α, β] → R be such that (1.3.36) holds. Let H : [a, b] → R be n-convex and consider the inequalities 

β

p(x)H (g(x)) dx ≥

α

(H (a) − aH (b)) 

b

+





β

p(x) dx + H (b)

α

a

×⎝

α

j −1

τi (s)H

(i+2)

(a) +

i=0



β

n−j

−1

⎞ ηi (s)H

(i+2)

(b)⎠ ds,

(1.3.38)

i=0

p(x)H (g(x)) dx ≥

α

(H (b) − bH (a)) 

b

+





β

p(x) dx − H (a)

α

β

p(x)g(x) dx α



β



p(x)G2 (g(x), s) dx a

⎛ ×⎝

j −1

α

τi (s)H (i+2) (a) +

i=0

β

p(x)g(x) dx

p(x)G1 (g(x), s) dx ⎛



β

α



β



n−j

−1

⎞ ηi (s)H (i+2)(b)⎠ ds,

(1.3.39)

i=0

p(x)H (g(x)) dx ≥

α





(H (b) − bH (b) + (H (b) − H (a))a) 

b

+





β

β



p(x) dx + H (a)

α

β

p(x)g(x) dx α

p(x)G3 (g(x), s) dx a

⎛ ×⎝

j −1

i=0

α

τi (s)H (i+2) (a) +

n−j

−1 i=0

⎞ ηi (s)H (i+2)(b)⎠ ds,

(1.3.40)

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions



β

59

p(x)H (g(x)) dx ≥

α





β

(H (a) − aH (a) − (H (b) − H (a))b) 

b

+



β

p(x) dx + H (b)

α



β



p(x)g(x) dx α

p(x)G4 (g(x), s) dx a

α

⎛

⎞ j −1 n−j

−1 × ⎝ τi (s)H (i+2)(a) + ηi (s)H (i+2) (b)⎠ ds. i=0

(1.3.41)

i=0

(i) If n − j is even, then (1.3.38)–(1.3.41) hold. (ii) If n − j is odd, then the reverse of (1.3.38)–(1.3.41) hold. Corollary 1.3.10 g : [α, β] → R, p : [α, β] → R be such that (1.3.36) holds. Let H : [a, b] → R be n-convex and consider the inequalities 

β



β

p(x)H (g(x)) dx ≥ (H (a) − aH (b))

α

p(x) dx + H (b)

α



b

+



a

β α

β

p(x)g(x) dx α

⎛ 

j −1 j −i−1

j + k − 1 ⎝ p(x)G1 (g(x), s) dx k (



i=0 k=0

(s − a)i  s − b j  s − a k (i+2) H (a) i! a−b b−a ) (s − b)i  s − a j  s − b k (i+2) + H (b) ds, i! b−a a−b

×



β



p(x)H (g(x)) dx ≥ (H (b) − bH (a))

α

β

+ a

b



β α

⎛ 

j −1 j −i−1

j + k − 1 ⎝ p(x)G2 (g(x), s) dx k (



p(x) dx − H (a)

α



(1.3.42)

β

p(x)g(x) dx α

i=0 k=0

(s − a)i  s − b j  s − a k (i+2) H (a) i! a−b b−a ) (s − b)i  s − a j  s − b k (i+2) + H (b) ds, i! b−a a−b

×

(1.3.43)

60

1 Linear Inequalities via Interpolation Polynomials and Green Functions



β

p(x)H (g(x)) dx ≥ (H (b) − bH (b) + (H (b) − H (a))a)

α



β

+H (a)

b  β

 p(x)g(x) dx +



β

p(x) dx α

 p(x)G3 (g(x), s) dx

α

a

α

⎛ ( j −1 j −i−1

j + k − 1 (s − a)i  s − b j  s − a k H (i+2) (a) ×⎝ k i! a−b b−a i=0 k=0

(s − b)i  s − a j  s − b k (i+2) + H (b) i! b−a a−b 

β

)

p(x)H (g(x)) dx ≥ (H (a) − aH (a) − (H (b) − H (a))b)

α

+H (b)



β

α



b  β

 p(x)g(x) dx +

(1.3.44)

ds,



β

p(x) dx α

p(x)G4 (g(x), s) dx a

α

⎛ ( j −1 j −i−1

j + k − 1 (s − a)i  s − b j  s − a k ×⎝ H (i+2) (a) i! a−b b−a k i=0 k=0

(s − b)i  s − a j  s − b k (i+2) + H (b) i! b−a a−b

) (1.3.45)

ds.

(i) If j is even, then (1.3.42)–(1.3.45) hold. (ii) If j is odd, then the reverse of (1.3.42)–(1.3.45) hold. Theorem 1.3.11 Let all the assumptions of Theorem 1.3.1 be valid and let g : [α, β] → R and p : [α, β] → R satisfy (1.0.8). Let H : [a, b] → R be n-convex and consider the inequality for l ∈ {1, 2, 3, 4}  α

β

p(x)H (x) dx ≥

kj r



H (i+2) (aj )

j =1 i=0



b

×





β

p(x)Gl (g(x), s) dx Hij (s) ds (1.3.46) a

α

and the function F given by (1.3.31). (i) If kj for j ∈ {2, . . . , r} are odd, then (1.3.46) holds. Furthermore, if the  β p(x)H (g(x))dx ≥ 0 holds. function F is convex, then inequality α

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions

61

(ii) If kj for j ∈ {2, . . . , r − 1} are odd and kr is even, then the reverse of (1.3.46) holds. Furthermore, if the function F is concave, then inequality  β p(x)H (g(x))dx ≤ 0 holds. α

1.3.2 Inequalities for n-Convex Functions at a Point Theorem 1.3.12 Let x ∈ [a, c]m1 , p ∈ Rm1 , y ∈ [c, b]m2 and q ∈ Rm2 be such that for each l ∈ {1, 2, 3, 4} 

m1 c

pk Gl (xk , s)GH,n−2 (s, t) ds ≥ 0

for all t ∈ [a, c],

(1.3.47)

qk Gl (yk , s)GH,n−2 (s, t) ds ≥ 0

for all t ∈ [c, b],

(1.3.48)

a k=1



m2 b

c k=1

and [a,b] A[a,b] l+8 (m1 , x, p, en ) = Al+8 (m2 , y, q, en )

(1.3.49)

where Gl are Green functions given by (1.1.6), (1.1.7), (1.1.8) and (1.1.9) respectively, GH,n−2 be as defined in (1.3.4), and Al+8 be the linear functionals given by (1.3.16) − (1.3.19). If H : [a, b] → R is (n + 1)-convex at point c, then [c,b] A[a,b] l+8 (m1 , x, p, H ) ≤ Al+8 (m2 , y, q, H ).

(1.3.50)

If the inequalities in (1.3.47) and (1.3.48) are reversed, then (1.3.50) holds with the reversed sign of inequality. Theorem 1.3.13 Let g : [α, β] → [a, c], p : [α, β] → R, h : [γ , δ] → [c, b], q : [γ , δ] → R be such that for each l ∈ {1, 2, 3, 4} 

c



a



p(x)Gl (g(x), s)GH,n−2 (s, t) dx ds ≥ 0

for all t ∈ [a, c],

(1.3.51)

q(x)Gl (h(x), s)GH,n−2 (s, t) dx ds ≥ 0

for all t ∈ [c, b]

(1.3.52)

α b

c

β



δ γ

and [c,b] A[a,c] l+12 ([α, β], g, p, en ) = Al+12 ([γ , δ], h, q, en ),

(1.3.53)

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1 Linear Inequalities via Interpolation Polynomials and Green Functions

where Gl are Green functions given by (1.1.6), (1.1.7), (1.1.8) and (1.1.9) respectively, GH,n−2 be as defined in (1.3.4) and Al+12 be the linear functionals given by (1.3.16)–(1.3.19). If H : [a, b] → R is (n + 1)-convex at point c, then [c,b] A[a,c] l+12 ([α, β], g, p, H ) ≤ Al+12 ([γ , δ], h, q, H ).

(1.3.54)

If the inequalities in (1.3.51) and (1.3.52) are reversed, then (1.3.54) holds with the reversed sign of inequality.

1.3.3 Bounds for Remainders and Functionals Consider the linear functionals A9 − A16 as define in (1.3.16) − (1.3.19) and (1.3.32) − (1.3.35) respectively and 9 − 12 are defined as respectively for l ∈ {9, 10, 11, 12} (m, x, p, t) [a,b] l



m b

=

pk Gl−8 (xk , s)GH,n−2 (s, t)ds ≥ 0

for all t ∈ [a, b],

a k=1

(1.3.55) and 13 − 16 are defined as respectively for l ∈ {13, 14, 15, 16} ([α, β], g, p, t) [a,b] l



b

=



β

p(x)Gl−12 (g(x), s)GH,n−2 (s, t)ds dx a

α

≥ 0 ∀ t ∈ [a, b]. For the sake of brevity we consider A[·,·] l (·, , ·, ·, H ) [·,·] l (·, ·, ·, t) = l (t) We state our next result.

(1.3.56) =

Al (H ) and

Theorem 1.3.14 Let l ∈ {9, . . . , 16}. Let H ∈ C (n) [a, b] such that for real numbers γ1 and γ2 we have γ1 ≤ H (n) (η) ≤ γ2 for η ∈ [a, b]. Then in representation  n−1  b H (b) − H n−1 (a) Al (f ) = l (x)dx + (b − a)Rnl , b−a a

(1.3.57)

remainder R4l (f ; a, b, n) satisfies estimation |R4l (f ; a, b, n)| ≤

 1 (γ2 − γ1 ) T ( l , l ). 2

(1.3.58)

1.4 Linear Inequalities and the Fink Identity with New Green Functions

63

Now we give some Ostrowski-type inequalities related to the generalized linear inequalities. Theorem 1.3.15 Let Al () = A[·,·] l (·, ·, ·, ) for (l ∈ {9, . . . , 16}) as defined in (1.3.16)–(1.3.19) and (1.3.32)–(1.3.35) and l (t) = [·,·] l (·, ·, ·, t) for l ∈ {9, . . . , 16} be as defined in (1.3.55) and (1.3.56). Furthermore, let (q, r) be a pair of conjugate exponents, i.e., 1 ≤ q, r ≤ ∞, q1 + 1r = 1. Let H (n) ∈ Lq [a, b] for n ≥ 1. Then we have for l ∈ {9, . . . , 16} |Al (H )| ≤ H (n) q  l r .

(1.3.59)

The constant on right hand side of (1.3.59) is sharp for 1 < q ≤ ∞ and the best possible for q = 1. Remark 1.3.2 For idea of the proof kindly seek help from previous sections or see [98]. Analogous to the mean value theorems stated in last two sections, similar results for this section as well can be drived. Further, We can construct linear functionals by taking differences of the left and right hand sides of the inequalities from Theorems 1.3.3 and 1.3.8. By using similar methods as in [98, 99] can be constructed new families of exponentially convex functions and Cauchy-type means, furthermore by using some known properties of exponentially convex functions, we can derive new inequalities and prove monotonicity of the obtained Cauchy-type means analogously as in [98, 99].

1.4 Linear Inequalities and the Fink Identity with New Green Functions n In this section we consider positivity of sum p f (xi ) involving convex  bi=1 i functions of higher order. Analogous for integral a p(x)f (g(x))dx is also given. Representation of a function f via the Fink identity and the Green’s function leads us to identities for which we obtain conditions for positivity of the mentioned sum and integral. We obtain bound for integral remainders which occur in those identities. Some of the results in this section are given without proof. If someone is interested in idea of the proof, previous section can be revisited or see [92]. In [103] we proved various results related to general linear inequalities via Fink identity with and without Green function (see also [101]). Recently, in [29] authors have introduced new Green type functions. Our main objective of present section is to further extend results of [103] using new definitions stated in [29]. Now we recall the Fink identity to prove many useful results. The following result was proved by Fink in [60].

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1 Linear Inequalities via Interpolation Polynomials and Green Functions

Theorem 1.4.1 Let a, b ∈ R, f : [a, b] → R, n ≥ 1 and f (n−1) is absolutely continuous on [a, b]. Then  b n−1

n n−k f (x) = f (t) dt − b−a a k! k=1   f (k−1) (a) (x − a)k − f (k−1) (b) (x − b)k × b−a  b 1 + (x − t)n−1 K (t, x) f (n) (t) dt, (n − 3)! (b − a) a

(1.4.1)

where K (t, x) is as defined in (1.1.2).

1.4.1 Results Obtained by the Fink identity and New Green functions In this section we will put forward some discrete and integral identities and the corresponding linear inequalities using new Green functions and applying the Fink identity. Theorem 1.4.2 Fix l ∈ {1, 2, 3, 4}. Let f : [a, b] → R be such that for n ≥ 3, continuous. f (n−1) is absolutely  m Let xi , yi ∈ [a, b], pi ∈ R for i ∈ {1, 2, . . . , m} be such that m p = 0 and i i=1 i=1 pi xi = 0 and let K (t, x) be the same as defined in (1.1.2). If Gl are the Green functions as defined in (1.1.6)–(1.1.9), then we have m

pi f (xi ) =

i=1

 n−3 

n−k−2 k=0

k! (b − a)

b a

 m

 pi Gl (xi , s)

i=1

  × f (k+1) (b) (s − b)k − f (k+1) (a) (s − a)k ds + 



b

×

f a

(n)

(t) a

m b

1 (n − 3)! (b − a) 

pi Gl (xi , s) (s − t)n−3 K (t, s) ds dt.

(1.4.2)

i=1

The following theorem is the integral version of Theorem 1.4.2. Theorem 1.4.3 Fix l ∈ {1, 2, 3, 4}. Let f : [a, b] → R be such that for n ≥ 3, f (n−1) is absolutely continuous on [a, b] and let p : [α, β] → R and g : [α, β] → β β [a, b] be integrable functions such that α p(x)dx = 0 and α p(x)g(x)dx = 0.

1.4 Linear Inequalities and the Fink Identity with New Green Functions

65

Let K (t, x) be the same as defined in (1.1.2). If Gl are the Green functions as defined in (1.1.6)–(1.1.9), then we have 

β

p (x) f (g (x)) dx =

 n−3

n−k−2

α

k=0

k! (b − a)

b





β

p (x) Gl (g (x) , s) dx a

α

  × f (k+1) (b) (s − b)k − f (k+1) (a) (s − a)k ds  b 1 + f (n) (t) (n − 3)! (b − a) a   b  β  p (x) Gl (g (x) , s) dx (s − t)n−3 K (t, s) ds dt. × a

(1.4.3)

α

Here we introduce some notations here which will be used in rest of the section: [a,b] 17 (m, x, p, t) = [a,b] 18 ([α, β], g, p, t) =



m b

a



pi Gl (xi , s) (s − t)n−3 K (t, s) ds,

(1.4.4)

i=1



b a

β

p (x) Gl (g (x) , s) dx (s − t)n−3 K (t, s) ds,

α

(1.4.5)

A[a,b] 17 (m, x, p, f ) =

m

i=1

pi f (xi ) −

 n−3 

n−k−2 k=0

k! (b − a)

m b

a

pi Gl (xi , s)

i=1

  × f (k+1) (b) (s − b)k − f (k+1) (a) (s − a)k ds, (1.4.6) A[a,b] 18 ([α, β], g, p, f ) = −



β

p (x) f (g (x)) dx α

 n−3 

n−k−2

b  β

 p (x) Gl (g (x) , s) dx

k! (b − a) a α k=0   × f (k+1) (b) (s − b)k − f (k+1) (a) (s − a)k ds. (1.4.7)

The following theorem is our second main result of this section: Theorem 1.4.4 Let all the assumptions of Theorem 1.4.2 be satisfied and let for n ≥ 3, the inequality [a,b] 17 (m, x, p, t) ≥ 0

(1.4.8)

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1 Linear Inequalities via Interpolation Polynomials and Green Functions

holds. If f is n-convex, then we have A[a,b] 17 (m, x, p, f ) ≥ 0.

(1.4.9)

If opposite inequality holds in (1.4.8), then (1.4.9) holds in the reverse direction. Proof Since f (n−1) is absolutely continuous on [a, b], f (n) exists almost everywhere. As f is n-convex, applying definition, we have, f (n) (x) ≥ 0 for all  x ∈ [a, b]. Now by using f (n) ≥ 0 and (1.4.8) in (1.4.2), we have (1.4.9). Corollary 1.4.1 Let all the assumptions of Theorem 1.4.2 be satisfied. In addition we let m

pi (xi − xk )+ ≥ 0

for k ∈ {1, 2, . . . , m}.

i=1

Let n be even and n > 3. If the function f : [a, b] → R is n-convex, then inequality (1.4.9) is satisfied, i. e. m

pi f (xi ) ≥

i=1

 n−3

n−k−2 k=0

k! (b − a)

a

m b

pi Gl (xi , s)

i=1

  × f (k+1) (b) (s − b)k − f (k+1) (a) (s − a)k ds.

(1.4.10)

Further if f (k+1) (a) ≤ 0 and (−1)k f (k+1) (b) ≥ 0 for k ∈ {0, 1, . . . , n − 3} then  m i=1 pi f (xi ) ≥ 0. Proof We fix l ∈ {1, 2, 3, 4} and n > 3. As x and p are real m-tuples such that they satisfy the assumption (1.0.6), by using the convex function x → Gl (x, s) in (1.0.3), we obtain m

pi Gl (xi , s) ≥ 0.

(1.4.11)

i=1

For a ≤ s ≤ t, it is easy to see that  t

m

pi Gl (xi , s) (s − t)n−3 K (t, s) ds ≥ 0

(1.4.12)

a i=1

holds for even n. Now as f is n-convex for even n, by applying Theorem 1.4.4, we get (1.4.10 ).

1.4 Linear Inequalities and the Fink Identity with New Green Functions

67

If a ≤ s ≤ b and k ∈ In−3 , then from assumptions f (k+1) (a) ≤ 0 and (−1)k f (k+1) (b) ≥ 0 we have that f (k+1)(b) (s − b)k − f (k+1)(a) (s − a)k ≥ 0.

(1.4.13)

So, from inequalities (1.4.10), (1.4.11) and (1.4.13) the non-negativity of the right hand side of (1.4.10) is immediate.  An integral version of our second main result states that: Theorem 1.4.6 Let all the assumptions of Theorem 1.4.3 be satisfied and let for n ≥ 3, the inequality [a,b] 18 ([α, β], g, p, t) ≥ 0

(1.4.14)

holds. If f is n-convex, then we have A[a,b] 18 ([α, β], g, p, f ) ≥ 0.

(1.4.15)

If opposite inequality holds in (1.4.5), then (1.4.15) holds in the reverse direction. Corollary 1.4.7 Let all the assumptions of Theorem 1.4.3 be satisfied. In addition we let 

β α

p(x) (g(x) − t)n−1 dx ≥ 0, +

for every t ∈ [a, b].

Let n be even and n > 3. If the function f : [a, b] → R is n-convex, then we have 

β α

p (x) f (g (x)) dx ≥

 n−3

n−k−2 k=0

k! (b − a)

b





β

p (x) Gl (g (x) , s) dx a

α

 × f (k+1) (b) (s −b)k  −f (k+1) (a) (s −a)k ds.

(1.4.16)

Further if f (k+1) (a) ≤ 0 and (−1)k f (k+1) (b) ≥ 0 for k ∈ {0, 1 . . . , n − 3}, then the right hand side of (1.4.16) is non-negative. Proof The proof is analogous to the proof of Corollary 1.4.1 but instead of Theorem 1.4.4, we apply Theorem 1.4.6. 

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1 Linear Inequalities via Interpolation Polynomials and Green Functions

1.4.2 Inequalities for n-Convex Functions at a Point In this subsection we shall give related results for the class of n-convex functions at a point. First we state main results for discrete case. Theorem 1.4.8 Let c ∈ (a, b), x ∈ [a, c]m , y ∈ [c, b]l , p ∈ Rm , q ∈ Rl and f : [a, b] → R be a function such that f (n−1) is absolutely continuous. Let [·,·] [·,·] 17 (·, ·, ·, t) and A17 (·, ·, ·, f )be defined as in (1.4.4) and (1.4.6) and satisfy the following conditions: [a,c] 17 (m, x, p, t) ≥ 0

for every

t ∈ [a, c],

(1.4.17)

[c,b] 17 (l, y, q, t) ≥ 0

for every t ∈ [c, b],

(1.4.18)

and [c,b] A[a,c] 17 (m, x, p, en ) = A17 (l, y, q, en ).

(1.4.19)

If f is (n + 1)-convex at point c, then [c,b] A[a,c] 17 (m, x, p, f ) ≤ A17 (l, y, q, f ).

(1.4.20)

If inequalities in (1.4.17) and (1.4.18) are reversed, then (1.4.20) holds with the reverse sign of inequality. Corollary 1.4.9 Let j1 , j2 , n ∈ N, 2 ≤ j1 , j2 ≤ n and let f : [a, b] → R be (n + 1)-convex at point c. Let m-tuples x ∈ [a, c]m and p ∈ Rm satisfy (1.0.4) and (1.0.5) with n replaced by j1 , let l-tuples y ∈ [c, b]l and q ∈ Rl satisfy l

qi yik = 0,

for all k ∈ {0, 1, . . . , j2 − 1};

i=1 l

j −1

qi (yi − t)+2

≥ 0,

for every t ∈ [y(1), y(l−n+1) ]

i=1

and let (1.4.19) holds. If n − j1 and n − j2 are even, then (1.4.20) holds. Remark 1.4.1 For idea of the proof see [101, pp. 171–172]. Integral analogous of previous theorem may be stated as: Theorem 1.4.10 Let c ∈ (a, b) and let g : [α, β] → [a, c], p : [α, β] → R, h : [γ , δ] → [c, b], q : [γ , δ] → R be integrable functions. Let f : I → R, [a, b] ⊂

1.4 Linear Inequalities and the Fink Identity with New Green Functions

69

I be a function such that f (n−1) is absolutely continuous. Let [·,·] 18 (·, ·, ·, t) and (·, ·, ·, f ) be defined as in (1.4.5) and (1.4.7) satisfy the following conditions: A[·,·] 18 [a,c] 18 ([α, β], g, p, t) ≥ 0

for every

t ∈ [a, c],

(1.4.21)

[c,b] 18 ([γ , δ], h, q, t) ≥ 0 for every

t ∈ [c, b],

(1.4.22)

[c,b] A[a,c] 18 ([α, β], g, p, en ) = A18 ([γ , δ], h, q, en ).

(1.4.23)

and

If f is (n + 1)-convex at point c (for n ≥ 3), then [c,b] A[a,c] 18 ([α, β], g, p, f ) ≤ A18 ([γ , δ], h, q, f ).

(1.4.24)

If inequalities in (1.4.21) and (1.4.22) are reversed, then (1.4.24) holds with the reverse sign of inequality. Corollary 1.4.11 Let j1 , j2 , n ∈ N, 2 ≤ j1 , j2 ≤ n and let f : [a, b] → R be (n + 1)-convex at point c. Let integrable functions g : [α, β] → [a, c], p : [α, β] → R satisfy (1.0.8) with n replaced by j1 , let h : [γ , δ] → [c, b], q : [γ , δ] → R satisfy 

δ

q(x)h(x)k dx = 0,

for all k ∈ {0, 1, . . . , j2 − 1};

γ



δ γ

j −1

q(x) (h(x) − t)+2

dx ≥ 0,

for every t ∈ [c, b],

and let (1.4.23) holds. If n − j1 and n − j2 are even, then (1.4.20) holds.

1.4.3 Bounds for Remainders and Functionals By using same techniques as used in previous sections we may also proof following results. Remark 1.4.2 For the sake of brevity, in present and next sections at some places [·,·] we will use the notations Al (f ) = A[·,·] l (·, ·, ·, f ) and l (t) = l (·, ·, ·, t) for l ∈ {17, 18} as defined in Theorems 1.4.4 and 1.4.6.

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1 Linear Inequalities via Interpolation Polynomials and Green Functions

Now, we are ready to state main results of this section: Theorem 1.4.12 Let f : [a, b] → R be such that f (n) is an absolutely continuous function for n ∈ N with (· − a)(b − ·)[f (n+1) ]2 ∈ L[a, b]. Then it holds for l ∈ {17, 18} 

f (n−1) (b) − f (n−1) (a) Al (f ) = (n − 3)!(b − a)



b a

l (s)ds + R5l (f ; a, b, n),

where the remainder R5l (f ; a, b, n) satisfies the estimation 1 (n − 3)!  1/2   b  (b − a)  (n+1) 2  × (s − a)(b − s)[f (s)] ds  . T ( l , l )  2 a (1.4.25)

|R5l (f ; a, b, n)| ≤

By using Theorem 1.2.11 we obtain the following Grüss type inequality. Theorem 1.4.13 Let f : [a, b] → R be such that f (n) is an absolutely continuous function for n ∈ N with (· − a)(b − ·)[f (n+1) ]2 ∈ L[a, b] with f (n+1) ≥ 0 on [a, b]. Then we have the representation (1.4.25) and the remainder R6l (f ; a, b, n) satisfies the following condition for l ∈ {17, 18} |R6l (f ; a, b, n)|

 b − a  (n−1) 1

 l ∞ ≤ (b) + f (n−1) (a) f (n − 3)! 2 *  − f (n−2) (b) − f (n−2) (a) .

(1.4.26)

Theorem 1.4.14 For l = 17 we assume that x and p satisfy the assumptions of Theorem 1.4.2 and for l = 18 we assume that x and p satisfy the assumptions of Theorem 1.4.3. (i) Let l ∈ {17, 18}. Let f : I → R, [a, b] ⊂ I , be such that f (n) is an absolutely continuous function and γ ≤ f (n) (x) ≤

for x ∈ [a, b].

Then  b  (n−1) (b) − f (n−1) (a) f l (t)dt + R7l (f ; a, b, n), (1.4.27) Al (f ) = (n − 3)!(b − a) a

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial

71

where the remainder R7l (f ; a, b, n) satisfies the estimation |R7l (f ; a, b, n)| ≤

 b−a ( − γ ) T ( l , l ). 2(n − 3)!

(1.4.28)

Theorem 1.4.15 (i) Fix l ∈ {17, 18}. Let (q, r) be a pair of conjugate exponents, that is, 1 ≤ q, r ≤ ∞, q1 + 1r = 1. Let f (n) ∈ Lq [a, b] for some n ∈ N, n > 1. Further, for l = 17 we assume that x and p satisfy the assumptions of Theorem 1.4.2 and for l = 18 we assume that x and p satisfy the assumptions of Theorem 1.4.3. Then we have |Al (f )| ≤

1 f (n) q  l r . (n − 3)!

(1.4.29)

The constant on the right hand side of (1.4.29) is sharp for 1 < q ≤ ∞ and the best possible for q = 1. Remark 1.4.3 Using the same method as in [98], we can construct new families of exponentially convex functions and Cauchy type means along with mean value theorems, as stated in previous sections.

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial The Abel-Gontscharoff interpolation problem in the real case was introduced in 1935 by Whittaker [185] and subsequently by Gontscharoff [63] and Davis [39]. The Abel-Gontscharoff interpolating polynomial for two points with integral remainder is given in [1]: Theorem 1.5.1 Let n, k ∈ N, n ≥ 2, 0 ≤ k ≤ n − 1 and f ∈ C (n) [a, b]; then we have f (t) = Qn−1 (a, b, f, t) + R (f, t) ,

(1.5.1)

where Qn−1 is the Abel-Gontscharoff interpolating polynomial for two-points of degree n − 1, i.e., k

(t − a)i (i) Qn−1 (a, b, f, t) = f (a) i! i=0

+

j n−k−2



j =0 i=0

(t − a)k+1+i (a − b)j −i (k+1+j ) (b) f (k + 1 + i)! (j − i)!

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1 Linear Inequalities via Interpolation Polynomials and Green Functions

and the remainder is given by 

b

R (f, t) =

Gn (t, s)f (n) (s)ds,

a

where Gn (t, s) is the Green function [12, p. 177] given as ⎧   k ⎪

⎪ n−1 ⎪ ⎪ (t − a)i (a − s)n−i−1 , ⎪ ⎨ i 1 i=0   Gn (t, s) = n−1

(n − 1)! ⎪ n−1 ⎪ ⎪ ⎪ − (t − a)i (a − s)n−i−1 , ⎪ ⎩ i

a ≤ s ≤ t; t ≤ s ≤ b.

i=k+1

(1.5.2) Further, for a ≤ s, t ≤ b the following inequality hold ∂ i Gn (t, s) ≥ 0, 0 ≤ i ≤ k, ∂t i

(1.5.3)

∂ i Gn (t, s) ≥ 0, k + 1 ≤ i ≤ n − 1. ∂t i

(1.5.4)

(−1)n−k−1

(−1)n−i

Before we proceed further it should be noted that all these results related to this section can be found in the article [91].

1.5.1 Results Obtained by the Abel−Gontscharoff’s Interpolation We start this section with the identities of generalizations of Popoviciu type inequality using Abel-Gontscharoff interpolating polynomial for two points. Theorem 1.5.2 Let n, k ∈ N, n ≥ 2, 0 ≤ k ≤ n − 1, x ∈ [a, b]m and p ∈ Rm . Let f ∈ C (n) [a, b] and Gn be the Green function defined as in (1.5.2). Then m

r=1



b

pr f (xr ) = θ1 (f ) + a



m

r=1

 pr Gn (xr , s) f (n) (s)ds,

(1.5.5)

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial

73

where θ1 (f ) =

k m

f (i) (a)

i=0

+

i!

j n−k−2





j =0 i=0

pr (xr − a)i

(1.5.6)

r=1 m

 pr (xr − a)k+1+i

r=1

(−1)j −i (b − a)j −i (k+1+j ) (b). f (k + 1 + i)!(j − i)!

Proof Consider the expression m

(1.5.7)

pr f (xr ).

r=1

By using Theorem 1.5.1 we have

f (t) =

k

(t − a)i (i) f (a) i! i=0

+

j n−k−2



j =0 i=0

(t − a)k+1+i (−1)j −i (b − a)j −i (k+1+j ) (b) f (k + 1 + i)! (j − i)!  +

b

Gn (t, s) f (n) (s)ds.

(1.5.8)

a

Substituting this value of f in (1.5.7) and some arrangements, we get (1.5.5).



Integral version of the above theorem can be stated as: Theorem 1.5.3 Let n, k ∈ N, n ≥ 2, 0 ≤ k ≤ n − 1, and x : [α, β] → [a, b], p : [α, β] → R be continuous functions. Let f ∈ C (n) [a, b] and Gn be the Green function defined as in (1.5.2). Then 

β



b

p(τ ) f (x(τ )) dτ = θ2 (f ) +

α

a



β

 p(τ )Gn (x(τ ), s) dτ f (n) (s)ds,

α

(1.5.9)

where θ2 (f ) =

 k

f (i) (a) i=0

+

i!

β

j  β n−k−2



j =0 i=0

p(τ ) (x(τ ) − a)i dt

α

α

 p(τ ) (x(τ ) − a)

k+1+i

dτ

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1 Linear Inequalities via Interpolation Polynomials and Green Functions

×

(−1)j −i (b − a)j −i (k+1+j ) f (b). (k + 1 + i)!(j − i)!

(1.5.10)

If x and p satisfy additional conditions, then we get generalization of Popoviciu type  inequality for n-convex functions, i. e., we give a lower bound for the sum pr f (xr ) which depends only on nodes x1 , . . . , xm , weights p1 , . . . , pm and values of higher derivatives of a function f at points a and b. Theorem 1.5.4 Let all the assumptions of Theorem 1.5.2 be valid. In addition, if for all s ∈ [a, b] m

pr Gn (xr , s) ≥ 0,

(1.5.11)

r=1

then for every n-convex function f : [a, b] → R, following inequality holds m

pr f (xr ) ≥ θ1 (f ),

(1.5.12)

r=1

where θ1 (f ) is given in (1.5.6). If the reverse inequality in (1.5.11) holds, then also the reverse inequality in (1.5.12) holds. Proof Since the function f is n-convex, therefore without loss of generality we can assume that f is n-times differentiable and f (n) (x) ≥ 0, for all x ∈ [a, b]. Hence we can apply Theorem 1.5.2 to get (1.5.12).  Integral version of the above theorem can be stated as: Theorem 1.5.5 Let all the assumptions of Theorem 1.5.3 be valid. In addition, if for all s ∈ [a, b] 

β

p(τ ) Gn (x(τ ), s) dτ ≥ 0,

(1.5.13)

α

then for every n-convex function f : [a, b] → R, it holds 

β

p(τ ) f (x(τ )) dτ ≥ θ2 (f ),

(1.5.14)

α

where θ2 (f ) is defined in (1.5.10). If the reverse inequality in (1.5.13) holds, then also the reverse inequality in (1.5.14) holds.

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial

75

 In some cases the assumption m r=1 pr Gn (xr , s) ≥ 0, s ∈ [a, b] can be replaced with more simpler conditions in which  we recognize assumptions from Popoviciu’s theorem about positivity of sum pr f (xr ) for a convex function f . Namely we have the following statement. Theorem 1.5.6 Let n, k ∈ N, n ≥ 2, 1 ≤ k ≤ n − 1, x ∈ [a, b]m p ∈ Rm be m-tuples such that condition (1.0.6) and let Gn be the Green function defined as in (1.5.2). (i) If k is odd and n is even or k is even and n is odd, then for every n-convex function f : [a, b] → R, it holds m

pr f (xr ) ≥ θ1 (f ),

(1.5.15)

r=1

where θ1 (f ) is given in (1.5.6). Moreover, if f (i) (a) ≥ 0 for i ∈ {2, . . . , k} and f (k+1+j ) (b) ≥ 0 if j − i is even and f (k+1+j ) (b) ≤ 0 if j − i is odd for i ∈ {0, . . . , j } and j ∈ m

pr f (xr ) ≥ 0. {0, . . . , n − k − 2}, then r=1

(ii) If k and n both are even or odd, then for every n-convex function f : [a, b] → R, the reverse inequality in (1.5.15) holds. Moreover, if f (i) (a) ≤ 0 for i ∈ {0, . . . , k} and f (k+1+j ) (b) ≤ 0 if j − i is even, and f (k+1+j ) (b) ≥ 0 if j − i is odd for i ∈ {0, . . . , j } and j ∈ {0, . . . , n − k − 2}, m

pr f (xr ) ≤ 0. then r=1

Proof (i) Let us consider properties (1.5.3) and (1.5.4) for i = 2. If k is odd and n is ∂ 2 Gn (t, s) ∂ 2 Gn (t, s) even, then for k = 1 we get (−1)n−2 ≥ 0 from (1.5.4), i.e. ≥ ∂t 2 ∂t 2 0, i. e. Gn is convex. For k > 1, from (1.5.3) we get the same inequality. If k is even and n is odd, then k ≥ 2 and from (1.5.3) we get that Gn is convex in the first variable. By Remark 1.0.4, applied on the function Gn we get m

pr Gn (xr , s) ≥ 0,

r=1

i.e., the assumptions of Theorem 1.5.4 are fullfilled and inequality (1.5.15) holds. If further assumptions on f (i) (a) and f (k+1+j ) (b) are valid, then the right-hand side of (1.5.15) is nonnegative. The case (ii) is proved in a similar manner.  An integral analogue of the previous theorem is the following theorem.

76

1 Linear Inequalities via Interpolation Polynomials and Green Functions

Theorem 1.5.7 Let n, k ∈ N, n ≥ 2, 1 ≤ k ≤ n − 1, x : [α, β] → [a, b] and p : [α, β] → R be continuous functions satisfying 

β



β

p(τ )dτ = 0,

α

p(τ )x(τ )dτ = 0,

α

and 

β α

p(τ )(x(τ ) − s)+ dτ ≥ 0

for s ∈ [a, b],

and let Gn be the Green function defined as in (1.5.2). (i) If k is odd and n is even or k is even and n is odd, then for every n-convex function f : [a, b] → R, then 

β

p(τ ) f (x(τ )) dτ ≥ θ2 (f ).

(1.5.16)

α

Moreover, if f (i) (a) ≥ 0 for i ∈ {0, . . . , k} and f (k+1+j ) (b) ≥ 0 if j − i is even and f (k+1+j ) (b) ≤ 0 if j − i is odd for i ∈ {0, . . . , j } and j ∈ β {0, . . . , n − k − 2}, then α p(t)f (x(t)) dt ≥ 0. (ii) If k and n both are even or odd, then for every n-convex function f : [a, b] → R, then the reverse inequality holds in (1.5.16). Moreover, if f (i) (a) ≤ 0 for i ∈ {0, . . . , k} and f (k+1+j ) (b) ≤ 0 if j − i is even, and f (k+1+j ) (b) ≥ 0 if j − i is odd for i ∈ {0, . . . , j } and j ∈ {0, . . . , n − k − 2}, β then α p(t)f (x(t)) dt ≤ 0.

1.5.2 Results Obtained by the Abel−Gontscharoff’s Interpolation Polynomial and Green Functions Now we obtain results using the Green function G, (1.5.17), together with the AbelGontscharoff polynomials. Here it is worth mentioning that we would use G0 for Green function G defined in (1.5.17). Now we recall the definition of Green function G which would be used in some of our results. The function G : [a, b] × [a, b] is defined by ⎧ (s − b)(t − a) ⎪ ⎨ −a G(s, t) = (t −bb)(s − a) ⎪ ⎩ b−a

for a ≤ t ≤ s; for s ≤ t ≤ b.

The function G is convex and continuous with respect to both s and t.

(1.5.17)

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial

77

For any function f : [a, b] → R, f ∈ C (2) [a, b], we can obtain the following integral identity by simply using integration by parts f (x) =

x−a b−x f (a) + f (b) + b−a b−a



b

G(x, s)f

(s)ds,

(1.5.18)

a

where the function G is defined as above in (1.5.17) (see also [187]). We begin with some identities related to generalizations of Popoviciu type inequality. Theorem 1.5.8 Let n, k ∈ N, n ≥ 4, 0 ≤ k ≤ n − 1, f ∈ C (n) [a, b] and x ∈ [a, b]m, p ∈ Rm . Also let G and Gn be defined by (1.5.17) and (1.5.2) respectively. Then   b  b 

m m

pr f (xr ) = θ3 (f, G0 ) + pr G(xr , s) Gn−2 (s, t)f (n) (t)dtds, a

r=1

a

r=1

where θ3 (f, G0 ) is defined as f (b) − f (a)

bf (a) − af (b)

pr xr + pr θ3 (f, G0 ) = b−a b−a m

m

r=1

+

k

i=0

+

f (i+2) (a) i!

j n−k−4



j =0 i=0

 ×

m b



r=1

m b

pr G(xr , s) (s − a)i ds

(1.5.19)

a r=1

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)!

pr G(xr , s) (s − a)k+1+i ds.

a r=1

In similar manner we can state further results related to other Green functions G1 − G4 as follows: Theorem 1.5.9 Let n, k ∈ N, n ≥ 4, 0 ≤ k ≤ n − 1, f ∈ C (n) [a, b] and x ∈ [a, b]m, p ∈ Rm . Also let G1 − G4 and Gn be defined by (1.1.6)–(1.1.9) and (1.5.2) respectively. Then for l ∈ {4, 5, 6, 7} m

pr f (xr ) = θl (f, Gl−3 )

r=1

 + a

b



b a

 m

r=1

 pr Gl−3 (xr , s) Gn−2 (s, t)f (n) (t)dtds,

78

1 Linear Inequalities via Interpolation Polynomials and Green Functions

where θ4 (f, G1 ) = (f (a) − af (b))

m

pr + f (b)

r=1

+

 k

f (i+2) (a) i!

i=0

+

j n−k−4



j =0 i=0

 ×

m b

m

pr xr

r=1

m b

pr G1 (xr , s) (s − a)i ds

a r=1

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)! pr G1 (xr , s) (s − a)k+1+i ds,

(1.5.20)

a r=1

θ5 (f, G2 ) = (f (b) − bf (a))

m



pr − f (a)

r=1

+

 k

f (i+2) (a) i!

i=0

+

j n−k−4



j =0 i=0

 ×

m b

m b

m

pr xr

r=1

pr G2 (xr , s) (s − a)i ds

a r=1

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)! pr G2 (xr , s) (s − a)k+1+i ds,

(1.5.21)

a r=1

θ6 (f, G3 ) = (f (b) − bf (b) + (f (b) − f (a))a)

m

pr + f (a)

r=1

+

 k

f (i+2) (a) i=0

+

i!

j n−k−4



j =0 i=0

 ×

m b

a r=1

m b

m

pr xr

r=1

pr G3 (xr , s) (s − a)i ds

a r=1

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)! pr G3 (xr , s) (s − a)k+1+i ds,

(1.5.22)

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial

θ7 (f, G4 ) = (f (a) − af (a) − (f (b) − f (a))b)

m

pr + f (b)

r=1

+

 k

f (i+2) (a) i!

i=0

+

j n−k−4



j =0 i=0

 ×

m b

m b

79 m

pr xr

r=1

pr G4 (xr , s) (s − a)i ds

a r=1

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)! pr G4 (xr , s) (s − a)k+1+i ds.

(1.5.23)

a r=1

Theorem 1.5.10 Let n, k ∈ N, n ≥ 4, 0 ≤ k ≤ n − 1, f ∈ C (n) [a, b], and let x : [α, β] → [a, b], p : [α, β] → R be continuous functions. Also let G1 − G4 and Gn be defined by (1.1.6)–(1.1.9) and (1.5.2) respectively. Then 

β

p(τ )f (x(τ ))dτ = θ8 (f, G0 )

α



b

+



a

b



a

β

p(τ )G(x(τ ), s)Gn−2 (s, t)f (n) (t)dτ ds dt,

α

where  bf (a) − af (b) β p(τ )x(τ )dτ + p(τ )dτ b−a α α   k

f (i+2) (a) b β + p(τ )G(x(τ ), s)dτ (s − a)i ds (1.5.24) i! a α

f (b) − f (a) θ8 (f, G0 ) = b−a



β

i=0

+

j n−k−4



j =0 i=0

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)! 

b

× a



β α

p(τ )G(x(τ ), s) (s − a)k+1+i dτ ds.

80

1 Linear Inequalities via Interpolation Polynomials and Green Functions

Theorem 1.5.11 Let n, k ∈ N, n ≥ 4, 0 ≤ k ≤ n − 1, f ∈ C (n) [a, b], and let x : [α, β] → [a, b], p : [α, β] → R be continuous functions and G0 − G4 , Gn be defined by (1.5.17), (1.1.6)–(1.1.9) and (1.5.2) respectively. Then for l ∈ {9, 10, 11, 12} 

β

p(τ )f (x(τ )) dτ = θl (f, Gl−8 )

α



b

+



a

b β a

p(τ )Gl−8 (x(τ ), s)Gn−2 (s, t)f (n) (t)dτ ds dt,

α



β

θ9 (f, G1 ) = (f (a) − af (b))



p(τ )x(τ ) dτ

α

+

 k

f (i+2) (a) i!

i=0

+

j n−k−4



j =0 i=0

α



b a

β

p(τ )G1 (x(τ ), s)dτ (s − a)i ds

α

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)! 

b

×



a

β

p(τ )G1 (x(τ ), s) (s − a)k+1+i dτ ds,

α



β

θ10(f, G2 ) = (f (b) − bf (a))



 k

f (i+2) (a) i!

i=0

+

j n−k−4



j =0 i=0

b

p(τ )x(τ ) dτ α



a

β

p(τ )G2 (x(τ ), s)dτ (s − a)i ds

α

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)! 

b

× a



β

p(τ )G2 (x(τ ), s) (s − a)k+1+i dτ ds,

α





θ11(f, G3 ) = (f (b) − bf (b) + (f (b) − f (a))a)



+f (a)

β

β

p(τ ) dτ α

p(τ )x(τ ) dτ α

β

p(τ ) dτ − f (a)

α

+

β

p(τ ) dτ + f (b)

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial

+

 k

f (i+2) (a) i=0

+

i!

j n−k−4



j =0 i=0

b



a

β

p(τ )G3 (x(τ ), s)dτ (s − a)i ds

α

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)! 

b

×



a

β

p(τ )G3 (x(τ ), s) (s − a)k+1+i dτ ds,

α







θ12(f, G4 ) = (f (a) − af (a) − (f (b) − f (a))b) + +f (b)



81

β

p(x) dx α

β

p(x)g(x) dx α

+

 k

f (i+2) (a) i=0

+

i!

j n−k−4



j =0 i=0

b



a

β

p(τ )G4 (x(τ ), s)dτ (s − a)i ds

α

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)! 

b

× a



β

p(τ )G4 (x(τ ), s) (s − a)k+1+i dτ ds,

α

Theorem 1.5.12 Let n, k ∈ N, n ≥ 4, 0 ≤ k ≤ n−1, x ∈ [a, b]m and p ∈ Rm . Also let G0 ,G1 − G4 , Gn be defined by (1.5.17), (1.1.6)–(1.1.9) and (1.5.2) respectively. If f : [a, b] → R is n-convex, and 

b a



m

 pr Gl−3 (xr , s) Gn−2 (s, t)ds ≥ 0,

t ∈ [a, b],

(1.5.25)

r=1

then for l ∈ {3, 4, 5, 6, 7} m

pr f (xr ) ≥ θl (f, Gl−3 ).

(1.5.26)

r=1

If the reverse inequality in (1.5.25) holds, then also the reverse inequality in (1.5.26) holds. Proof It follows from n-convexity of a function f and from Theorem 1.5.8. 

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1 Linear Inequalities via Interpolation Polynomials and Green Functions

As from (1.5.3) we have (−1)n−k−3 Gn−2 (s, t) ≥ 0, therefore for the case when n even and k is odd or n is odd and k is even, it is enough to assume that is m r=1 pr Gl−3 (xr , s) ≥ 0, s ∈ [α, β], {3, 4, 5, 6, 7}, instead of the assumption (1.5.25) in Theorem 1.5.12. Similarly we can discuss for the reverse inequality in (1.5.26). Integral version of the above theorem can be stated as: Theorem 1.5.13 Let n, k ∈ N, n ≥ 4, 0 ≤ k ≤ n − 1, x : [α, β] → [a, b], p : [α, β] → R be continuous functions and G0 ,G1 −G4 , Gn be defined by (1.5.17), (1.1.6)–(1.1.9) and (1.5.2) respectively. If f : [a, b] → R is n-convex, and 

b a



β

p(τ )Gl−8 (x(τ ), s)Gn−2 (s, t)dτ ds ≥ 0,

(1.5.27)

α

then l ∈ {8, 9, 10, 11, 12} 

β

p(τ )f (x(τ ))dτ ≥ θl (f, Gl−8 ).

(1.5.28)

α

If the reverse inequality in (1.5.27) holds, then also the reverse inequality in (1.5.28) holds. As from (1.5.3) we have (−1)n−k−3 Gn−2 (s, t) ≥ 0, therefore for the case when n is even and k is odd or n is odd and k is even, it is enough to assume that b a p(τ )Gl−8 (x(τ ), s)dτ ≥ 0, s ∈ [α, β], l ∈ {8, 9, 10, 11, 12}, instead of the assumption (1.5.27) in Theorem 1.5.13. Similarly we can discuss for the reverse inequality in (1.5.28). If we deal with assumptions from Remark 1.0.4, which are equivalent to Popoviciu’s conditions for positivity of sum involving convex function f , then for some combinations of n and k we get result for n-convex function f . Precisely, we get the following theorem. Theorem 1.5.14 Let n, k ∈ N, n ≥ 4, 0 ≤ k ≤ n − 1. Let G0 and G1 − G4 be defined by (1.5.17), (1.1.6)–(1.1.9) and let f : [a, b] → R be n-convex. Let x ∈ [a, b]m and p ∈ R satisfy (1.0.4). (i) If n is even and k is odd or n is odd and k is even, then for l ∈ {0, 1, 2, 3, 4} m

pl f (xr ) ≥

r=1

 k

f (i+2) (a) i=0

+

i!

j n−k−4



j =0 i=0

pr Gl (xr , s) (s − a)i ds

a r=1

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)!

 ×

m b

m b

a r=1

pr Gl (xr , s) (s − a)k+1+i ds.

(1.5.29)

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial

83

Moreover if f (i+2) (a) ≥ 0 for i ∈ {0, . . . , k} and f (k+3+j ) (b) ≥ 0 if j − i is even and f (k+3+j ) (b)  ≤ 0 if j − i is odd for i ∈ {0, . . . , j } and j ∈ {0, . . . , n − k − 4}, then m r=1 pr f (xr ) ≥ 0. (ii) If n and k both are even or both are odd, then reverse inequality holds in (1.5.29). Moreover if f (i+2) (a) ≤ 0 for i ∈ {0, . . . , k} and f (k+3+j ) (b) ≤ 0 if j − i is even and f (k+3+j ) (b)  ≥ 0 if j − i is odd for i ∈ {0, . . . , j } and j ∈ {0, . . . , n − k − 4}, then m r=1 pr f (xr ) ≤ 0. Proof (i) By using (1.5.3) we have (−1)n−k−3 Gn−2 (s, t) ≥ 0, s, t ∈ [a, b], therefore if n is even and k is odd or n is odd and k is even then Gn−2 (s, t) ≥ 0. Since Gl are convex for {0, 1, 2, 3, 4} and Gn−2 is nonnegative, the inequality (1.5.25) holds. Hence by Theorem 1.5.12 the inequality (1.5.29) holds. By using the other conditions the nonnegativity of the right-hand side of (1.5.29) is obvious. Similarly we prove (ii).  The integral version of Theorem 1.5.14 can be stated as: Theorem 1.5.15 Let n, k ∈ N, n ≥ 4, 0 ≤ k ≤ n − 1 and let x : [α, β] → [a, b] and p : [α, β] → R be any continuous functions satisfy: 

β



β

p(τ ) dτ ≥ 0,

α

α

p(τ )(x(τ ) − t)+ dτ ≥ 0 for t ∈ [a, b].

(1.5.30)

Also let G, G1 − G4 be defined by (1.5.17) and (1.1.6)–(1.1.9). Consider f : [a, b] → R is n-convex. (i) If n is even and k is odd or n is odd and k is even, then for l ∈ {0, 1, 2, 3, 4} 

β

p(τ )f (x(τ ))dτ ≥

α

 k

f (i+2) (a) i=0

+

j n−k−4



j =0 i=0



b

× a



β

i!

b a



β

p(τ )Gl (x(τ ), s)(s −a)i dτ ds

α

(−1)j −i (b − a)j −i f (k+3+j ) (b) (k + 1 + i)! (j − i)!

p(τ )Gl (x(τ ), s) (s − a)k+1+i dτ ds.

(1.5.31)

α

Moreover if f (i+2) (a) ≥ 0 for i ∈ {0, . . . , k} and f (k+3+j ) (b) ≥ 0 if j − i is even and f (k+3+j ) (b) ≤ 0 if j − i is odd for i ∈ {0, . . . , j } and j ∈ {0, . . . , n − k − 4}, then the right-hand side of (1.5.31) is nonnegative, that is integral version of (1.0.3) holds. (ii) If n and k both are even or both are odd, then reverse inequality holds in (1.5.31).

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1 Linear Inequalities via Interpolation Polynomials and Green Functions

Moreover if f (i+2) (a) ≤ 0 for i ∈ {0, . . . , k} and f (k+3+j ) (b) ≤ 0 if j −i is even and f (k+3+j ) (b) ≥ 0 if j − i is odd for i ∈ {0, . . . , j } and j ∈ {0, . . . , n − k − 4}, then the right hand side of the reverse inequality in (1.5.31) is nonpositive, that is the reverse inequality in the integral version of (1.0.3) holds.

1.5.3 Inequalities for n-Convex Functions at a Point Under the assumptions of Theorems 1.5.2, 1.5.8 and 1.5.9 here we define some linear functional as follows: m

A[a,b] 19 (m, x, p, f ) =

pr f (xr ) − θ1 (f ).

(1.5.32)

pr f (xr ) − θl−1 (f, Gl−4 ).

(1.5.33)

r=1

For l ∈ {4, 5, 6, 7, 8} A[a,b] l+16 (m, x, p, f ) =

m

r=1

Similarly, under the assumptions of Theorem 1.5.3, 1.5.10 and 1.5.11 here we introduce some further linear functional as follows: A[a,b] 25 ([α, β], x, p, f ) =



β

p(τ )f (x(τ )) dτ − θ2 (f ).

(1.5.34)

α

For l ∈ {10, 11, 12, 13, 14} we have A[a,b] l+16 ([α, β], x, p, f ) =



β

p(τ )f (x(τ )) dτ − θl−1 (f, Gl−10 ). (1.5.35)

α

Here we also define some new functionals 19 and l+16 for l ∈ {4, 5, 6, 7, 8} as follows: [a,b] 19 (m, x, p, t) =

m

pr Gn (xr , t) for all t ∈ [a, b],

(1.5.36)

r=1

[a,b] l+16 (m, x, p, t) =



m b

pr Gl−4 (xr , s)Gn−2 (s, t)ds ≥ 0

for all t ∈ [a, b],

a r=1

(1.5.37)

1.5 Linear Inequalities and the Abel-Gontscharoff’s Interpolation Polynomial

85

and 25 and l+16 for l ∈ {10, 11, 12, 13, 14} are defined as [a,b] 25 ([α, β], x, p, t) =



β

p(τ )Gn (x(τ ), t) dτ dx ≥ 0

α

[a,b] l+16 ([α, β], x, p, t) =

∀ t ∈ [a, b], (1.5.38)



b



β

p(τ )Gl−10 (x(τ ), s)Gn−2 (s, t)dτ ds a

α

≥0 ∀

t ∈ [a, b].

(1.5.39)

[·,·] For the sake of brevity we consider A[·,·] l (·, ·, ·, f ) = Al (f ) and l (·, ·, ·, t) = l (t). We state our next result.

Theorem 1.5.16 Let x ∈ [a, c]m1 , p ∈ Rm1 , y ∈ [c, b]m2 and q ∈ Rm2 be such that for each l ∈ {19, 20, 21, 22, 23, 24} [a,c] (m1 , x, p, t) ≥ 0 l

for all t ∈ [a, c],

(1.5.40)

[c,] l (m2 , y, q, t) ≥ 0

for all t ∈ [c, b],

(1.5.41)

(m1 , x, p, en ) = A[c,b] (m2 , y, q, en ). A[a,c] l l

(1.5.42)

and

where Al and l be the linear functionals given by (1.5.32)–(1.5.35) and (1.5.36)– (1.5.39). If f : [a, b] → R is (n + 1)-convex at point c, then A[a,c] (m1 , x, p, f ) ≤ A[c,b] (m2 , y, q, f ). l l

(1.5.43)

If the inequalities in (1.5.40) and (1.5.41) are reversed, then (1.5.43) holds with the reversed sign of inequality. Theorem 1.5.17 Let x : [α, β] → [a, c], p : [α, β] → R, y : [γ , δ] → [c, b], q : [γ , δ] → R be such that for each l ∈ {25, 26, 27, 28, 29, 30} ([α, β], x, p, t) ≥ 0 [a,c] l

for all t ∈ [a, c],

(1.5.44)

[c,b] ([γ , δ], y, q, t) ≥ 0 l

for all t ∈ [c, b]

(1.5.45)

and ([α, β], x, p, en ) = A[c,b] ([γ , δ], y, p, en ) A[a,c] l l

(1.5.46)

86

1 Linear Inequalities via Interpolation Polynomials and Green Functions

Then A[a,c] ([α, β], x, p, f ) ≤ A[c,b] ([γ , δ], y, p, f ), l l

(1.5.47)

where Al and l be the linear functionals given by (1.5.32)–(1.5.35) and (1.5.36)– (1.5.39) respectively. If the inequalities in (1.5.44) and (1.5.45) are reversed, then (1.5.47) holds with the reversed sign of inequality.

1.5.4 Bounds for Remainders and Functionals [·,·] For the sake of brevity we consider A[·,·] l (·, ·, ·, f ) = Al (f ) and l (·, ·, ·, t) = l (t) which was defined in previous subsection. Now we state our next result.

Theorem 1.5.18 Let l ∈ {19, . . . , 30}. Let f ∈ C (n) [a, b] such that for real numbers γ1 and γ2 we have γ1 ≤ f (n) (η) ≤ γ2 for η ∈ [a, b]. Then in representation 

f n−1 (b) − f n−1 (a) Al (f ) = b−a



b a

l (ξ )dξ + (b − a)R8l (f ; a, b, n),

(1.5.48)

remainder R8l (f ; a, b, n) satisfies estimation |R8l (f ; a, b, n)| ≤

 1 (γ2 − γ1 ) T ( l , l ). 2

(1.5.49)

Now we give some Ostrowski-type inequalities related to the generalized linear inequalities. Theorem 1.5.19 Let for l ∈ {19, . . . , 30} Al and l be linear functionals as defined in previous subsection. Furthermore, let (q, r) be a pair of conjugate exponents, i.e., 1 ≤ q, r ≤ ∞, q1 + 1r = 1. Let f (n) ∈ Lq [a, b] for n ≥ 1. Then we have for l ∈ {19, . . . , 30} |Al (f )| ≤ f (n) q  l r .

(1.5.50)

The constant on right hand side of (1.5.50) is sharp for 1 < q ≤ ∞ and the best possible for q = 1. Remark 1.5.1 For idea of the proof kindly see [98]. Using the same method as given in [5] we can state mean value theorems and results related to exponentially convexity.

Chapter 2

Ostrowski Inequality

The art of doing mathematics consists in finding that special case which contains all the germs of generality. —David Hilbert

According to Hardy’s famous statement “Behind every theorem lies an inequality,” and it is true that many significant and well-known results in mathematics are inequalities, new types of interesting inequalities are discovered every year and number of researchers improve, extend, refine and generalize various mathematical inequalities. Several books on inequalities have been published, interested readers can see [20, 27, 48, 61, 69]. While the topic of inequalities covers many branches of mathematics but this chapter focuses on those associated with Ostrowski inequality. Firstly, we would like to recall the classical Ostrowski inequality. Ostrowski Inequality In 1937, Ostrowski established an inequality in his paper [141]. It is a known fact that this type of inequality could be used to approximate the absolute deviation of functional value from its integral mean. This result is now a days known as Ostrowski inequality. The database of MathSciNet includes more than 360 papers with the keywords “Ostrowski” and “Inequality”. Various generalizations and extensions of the Ostrowski inequality have appeared in the literature and detailed history on Ostrowski inequality, see [6, 8, 17, 19, 36, 52] and references therein. For further literature we can see unpublished doctoral thesis [75, 192] and Dragomir’s monograph [48]. In following theorem, we give this inequality from [48]. Theorem 2.0.1 Let f : I = [a, b] ⊆ R → R be a differentiable function on I o (where I o is the interior of I ) such that f ∈ L[a, b], where a, b ∈ I and a < b.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Irshad et al., New Perspectives on the Theory of Inequalities for Integral and Sum, https://doi.org/10.1007/978-3-030-90563-7_2

87

88

2 Ostrowski Inequality

If |f (t)| ≤ M valid ∀ t ∈ (a, b) where M is positive real constant. Then (  2 )    b   x − a+b 1 1 2 f (x) − + f (t)dt  ≤ M(b − a)  b−a a 4 (b − a)2   (x − a)2 + (b − x)2 = M. 2(b − a)

(2.0.1)

Proof We have Montgomery identity from (1.1.1) 

b

(b − a)f (x) −



x

f (t)dt =

a

(t − a)f (t)dt +

a



b

(t − b)f (t)dt.

x

Taking the modulus, we deduce    (b − a)f (x) − 

b a

  x   b       f (t)dt  ≤  (t − a)f (t)dt  +  (t − b)f (t)dt  

a

x

≤ a

  (t − a) f (t) dt +



x

≤M



b

  (t − b) f (t) dt

x



b

(t − a)dt + M

a

x

(t − b)dt

x

 M (x − a)2 + (b − x)2 2 (  ) x − a+b 2 1 2 = M(b − a) + . 4 b−a

=



This produces our classical Ostrowski inequality (2.0.1). Remark 2.0.2

(1) Here the constant 14 in first inequality is the best possible in the sense that it cannot be replaced by smaller one. (2) In latest versions M is usually replaced by f ∞ = ess sup |f (t)| < ∞. t ∈(a,b)

(3) Since f is bounded so the result is also valid for functions of bounded variation. (4) In this result some assumptions may be relaxed by using absolutely continuous functions instead of differentiable functions. (5) This result may be proved in variety of ways by using different techniques including Lagrange mean value theorem and Montgomery identity, and by using direct calculation etc. (6) Ostrowski’s inequality helps us to estimate the bound of Hermite-Hadamard’s left inequality.

2 Ostrowski Inequality

89

(7) This inequality may be interpreted in the following manners: (a) Estimation of deviation of functional values of function with bounded derivative from the integral mean. (b) It measures the estimate of approximating area under the curve by rectangle. (8) The celebrated inequality has vast applications in statistics, numerical integration, probability theory and special mean(s) among many others. (9) It has close connection with other celebrated inequalities including Ostrowski-Grüss, Grüss and Hermite-Hadamard inequalities. Hermite-Hadamard Inequality In [72], Hermite presented the following result known as Hermite-Hadamard dual inequality for convex function. Theorem 2.0.2 Let f : I → R be a convex function. Then  f

a+b 2

 ≤

1 b−a



b

f (x)dx ≤

a

f (a) + f (b) . 2

It is worth mentioning that, for concave function f , both inequalities would be in reverse order. Standard and Non-Standard Quadrature Rules From [52] and [115], we have some standard quadrature rules of midpoint, trapezoidal, 13 Simpson’s, 38 Simpson’s and average of midpoint and trapezoidal rule as given in I1 (f ), I2 (f ), I3 (f ), I4 (f ) and I5 (f ) respectively. We will also capture some non-standard quadrature rules from our findings as given in I6 (f ) − I9 (f ). 1 I1 (f ) : b−a I2 (f ) :

1 b−a

I3 (f ) :

1 b−a

I4 (f ) :

1 b−a

I5 (f ) :

1 b−a

I6 (f ) :

1 b−a

 

b a

f (x)dx ∼ =f



 a+b , 2

f (a) + f (b) , f (x)dx ∼ = 2 a     b a+b 1 f (a) + f (b) ∼ + 2f f (x)dx = , 3 2 2 a       b 2a + b 3 f (a) + f (b) a + 2b ∼ +f f (x)dx = +f , 8 3 3 3 a     b a+b 1 f (a) + f (b) ∼ +f f (x)dx = , 2 2 2 a      b 1 a+b f (x)dx ∼ −f (a) + 2f + f (b) , = 2 2 a b

90

2 Ostrowski Inequality

1 I7 (f ) : b−a I8 (f ) : I9 (f ) :

1 b−a 1 b−a

  

b a b

f (x)dx ∼ = f (a),

b

f (x)dx ∼ = f (b).

a

a

    1 a+b ∼ f (x)dx = f (a) + 2f − f (b) , 2 2

2.1 Generalized Ostrowski Type Inequalities with Parameter Generalized Ostrowski type inequalities for bounded, bounded below only and bounded above only differentiable functions are given in this section. Some applications to find error bounds of midpoint, trapezoidal, 13 Simpson’s and 38 Simpson’s quadrature rules are discussed. Error bounds are also provided for some non-standard quadratures rules. Our proposed findings would generalize the results of Masjed-Jamei and Dragomir in [116] and [115] and Ujevi´c in [184]. The results of this section have been obtained in [77]. Ostrowski inequality for differentiable functions have been generalized many times as stated in [53, 79, 115, 116, 184]. To establish our main results, we need these two lemmas from [7] as given below. Lemma 2.1.1 Let f : I → R, be absolutely continuous function in I 0 such that x ∈ [a, b] with a+

b−a a+b ≤x≤ . 2 2

Then  a

b

  f (x) + f (a + b − x) f (a) + f (b) + (1 − ) P (x, t)f (t)dt = (b − a) 2 2  b f (t)dt, (2.1.1) − a

∀ ∈ [0, 1], where Peano kernel P (x, t) is defined as ⎧  ⎪ t − a + b−a , t ∈ [a, x]; ⎪ 2 ⎪ ⎪ ⎪ ⎨ P (x, t) = t − a+b 2 , t ∈ (x, a + b − x]; ⎪ ⎪ ⎪ ⎪ ⎪  ⎩ , t ∈ (a + b − x, b]. t − b − b−a 2

2.1 Generalized Ostrowski Type Inequalities with Parameter

91

Lemma 2.1.2 Let f : I → R, be a differentiable function and if φ(t) ≤ f (t) ≤ (t) such that φ,  ∈ C[a, b] and t ∈ [a, b]. Then we have     f (t) − φ(t) + (t)  ≤ (t) − φ(t) . (2.1.2)   2 2 Now we are going to present our main results of this section.

2.1.1 Ostrowski Type Inequality for Bounded Differentiable Functions Theorem 2.1.1 Let f : I → R, be a differentiable function in I 0 . If φ(x) ≤ f (x) ≤ (x) for any φ,  ∈ C[a, b] and x ∈ [a, b], then the following inequality holds f (x) + f (a + b − x) 1 f (a) + f (b) + (1 − ) − m(x, ) ≤ 2 2 b−a ≤ M(x, ),



b

f (t)dt a

(2.1.3)

where ⎡     b−a   λ + |λ| 1 ⎣ x− a+ 2 b−a m(x, ) = φ λ+a+ b−a 2 2 − b−a 2 +

     a+b −x  2 λ + |λ| λ − |λ| b−a a+b dλ +  λ+a+ φ λ+ 2 2 2 2 x− a+b 2

      b−a 2 λ + |λ| λ − |λ| a+b b−a dλ +  λ+ φ λ+b− 2 2 2 2 a+ b−a 2 −x    λ − |λ| b−a dλ +  λ+b− 2 2 +

and ⎡     b−a   λ − |λ| 1 ⎣ x− a+ 2 b−a M(x, ) = φ λ+a+ b−a 2 2 − b−a 2 +

     a+b −x  2 λ − |λ| λ + |λ| b−a a+b  λ+a+ dλ + φ λ+ 2 2 2 2 x− a+b 2

92

2 Ostrowski Inequality

      b−a 2 λ − |λ| λ + |λ| a+b b−a  λ+ dλ + φ λ+b− 2 2 2 2 a+ b−a 2 −x    λ + |λ| b−a +  λ+b− dλ . 2 2

+

φ(t) + (t) in (2.1.1), we get 2    b φ(t) + (t) P (x, t) f (t) − dt 2 a  b   b 1 P (x, t)f (t)dt − P (x, t) (φ(t) + (t)) dt = 2 a a   b  f (x) + f (a + b − x) f (a) + f (b) + (1 − ) − f (t)dt = (b − a) 2 2 a   x   1 b−a − t − a+ (φ(t) + (t)) dt 2 a 2   a+b−x  a+b t− + (φ(t) + (t)) dt 2 x      b b−a (2.1.4) t − b− + (φ(t) + (t)) dt . 2 a+b−x

Proof Replacing f (t) by f (t) −

From (2.1.2) and (2.1.4), we get    b   (b − a) f (a) + f (b) + (1 − ) f (x) + f (a + b − x) − f (t)dt  2 2 a  x      a+b−x  b−a a+b 1 t − a+ t− − (φ(t) + (t)) dt + 2 a 2 2 x      b  b−a × (φ(t) + (t)) dt + t − b− (φ(t) + (t)) dt  2 a+b−x  b      φ(t) + (t) =  P (x, t) f (t) − dt  2 a     b   φ(t) + (t) |P (x, t)|  f (t) − dt  ≤ 2 a    b (t) − φ(t) |P (x, t)| ≤ dt 2 a

2.1 Generalized Ostrowski Type Inequalities with Parameter

93

  x    b − a  1  t − a+ =  ((t) − φ(t)) dt 2 a  2   a+b−x   a + b   + t − 2  ((t) − φ(t)) dt x      b  b − a   +  ((t) − φ(t)) dt . t − b − 2 a+b−x

(2.1.5)

After rearranging the terms of (2.1.5), we get f (a) + f (b) f (x) + f (a + b − x) 1 m(x, ) ≤ + (1 − ) − 2 2 b−a



b

f (t)dt a

≤ M(x, ), where  x        1 b − a  (t) b−a − t − a + t − a+  b−a a 2 2 2         b − a  φ(t) b−a + t − a + dt + t − a+  2 2 2     a+b−x  a + b  (t) a+b a + b  − t − + t− + t− 2 2  2 2 x        b  b−a a + b  φ(t) t − b− dt + + t −  2 2 2 a+b−x        b − a  (t) b−a  − t − b − + t − b−  2 2 2       b − a  φ(t) + t − b − dt  2 2 ⎡     b−a   λ + |λ| b−a λ − |λ| 1 ⎣ x− a+ 2 φ λ+a+ + = b−a b−a 2 2 2 − 2

m(x, ) =

     a+b −x  2 λ + |λ| b−a a+b × λ + a + dλ + φ λ+ 2 2 2 x− a+b 2       b−a 2 λ + |λ| a+b b−a λ − |λ|  λ+ dλ + φ λ+b− 2 2 2 2 a+ b−a 2 −x    λ − |λ| b−a +  λ+b− dλ 2 2 +

94

2 Ostrowski Inequality

and  x        1 b − a  (t) b−a  + t − a + t − a+  b−a a 2 2 2         b − a  φ(t) b−a − t − a + dt + t − a+  2 2 2     a+b−x  a + b  (t) a+b a + b  + t − + t− t− +  2 2 2 2 x        b  b−a a + b  φ(t) dt + t − b− − t − 2  2 2 a+b−x     b − a  (t) + t − b −  2 2          b−a b − a  φ(t) + t − b− − t − b − dt  2 2 2 ⎡     b−a   λ − |λ| b−a λ + |λ| 1 ⎣ x− a+ 2 φ λ+a+ + = b−a 2 2 2 − b−a 2

M(x, ) =

     a+b −x  2 λ − |λ| b−a a+b × λ + a + dλ + φ λ+ 2 2 2 x− a+b 2       b−a 2 λ − |λ| a+b b−a λ + |λ|  λ+ dλ + φ λ+b− 2 2 2 2 a+ b−a 2 −x    λ + |λ| b−a +  λ+b− dλ . 2 2 +

 Remark 2.1.1 By choosing = 1 in (2.1.3), it would be independent of x, we get the bound for trapezoidal rule (Hermite–Hadamard right bound) in the following corollary. Corollary 2.1.2 Let all assumptions of Theorem 2.1.1 be valid. Then m1 ≤

1 f (a) + f (b) − 2 b−a



b

f (t)dt ≤ M1 ,

(2.1.6)

a

where 1 m1 = b−a

(

b−a 2

− b−a 2



    ) a+b a+b λ + |λ| λ − |λ| φ λ+  λ+ + dλ 2 2 2 2

2.1 Generalized Ostrowski Type Inequalities with Parameter

95

and 1 M1 = b−a

(

b−a 2

− b−a 2



    ) a+b λ + |λ| a+b λ − |λ| φ λ+ +  λ+ dλ , 2 2 2 2

which is Corollary 2 of [116]. From now onwards, throughout the section φ0 , φ1 , 0 and 1 are real constants. Special Case 2.1.2.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.6), then (b − a) 1 f (a) + f (b) (φ0 − 0 ) ≤ − 8 2 b−a



b

f (t)dt ≤

a

(b − a) (0 − φ0 ), 8

which is Corollary 2 of [184]. Special Case 2.1.2.(b) If we take, φ(x) = φ1 x+φ0 = 0 and (x) = 1 x+0 = 0 in (2.1.6), then m2 ≤

1 f (a) + f (b) − 2 b−a



b

f (t)dt ≤ M2 ,

a

where m2 =

  (b − a) (b − a) (a + b) (φ1 + 1 ) + (φ1 − 1 ) + φ0 − 0 8 3 2

M2 =

  (b − a) (b − a) (a + b) (φ1 + 1 ) + (1 − φ1 ) + 0 − φ0 . 8 3 2

and

Remark 2.1.2 By choosing x = following corollary.

a+b in (2.1.3), we get the inequality in the 2

Corollary 2.1.3 Let all assumptions of Theorem 2.1.1 be valid. Then m3 ( ) ≤

f (a) + f (b) + (1 − )f 2



a+b 2

 −

1 b−a



b

f (t)dt ≤ M3 ( ),

a

(2.1.7)

96

2 Ostrowski Inequality

where (

  b−a λ + |λ| φ λ+a+ 2 2 − b−a 2   b−a λ − |λ|  λ+a+ + dλ 2 2    b−a  2 b−a λ + |λ| × φ λ + b −

+ 2 2 ( −1) b−a 2    λ − |λ| b−a +  λ+b− dλ 2 2

1 m3 ( ) = b−a

(1− ) b−a 2



and (    (1− ) b−a 2 b−a 1 λ − |λ| φ λ+a+ M3 ( ) = b − a − b−a 2 2 2   λ + |λ| b−a +  λ+a+ dλ 2 2       b−a  2 λ − |λ| b−a b−a λ + |λ| φ λ+b−  λ+b− + + dλ . 2 2 2 2 ( −1) b−a 2

The Corollary 2.1.3 could be more useful to get different quadrature bounds as under. Remark 2.1.3 By choosing = 0 in (2.1.7), we get the bound for midpoint rule (Hermite–Hadamard left bound) in the following corollary. Corollary 2.1.4 Let all assumptions of Theorem 2.1.1 be valid. Then  m4 ≤ f

a+b 2

 −

1 b−a



b

f (t)dt ≤ M4 ,

a

where 1 m4 = b−a 

b−a 2

+ 0

(

0 − b−a 2





 λ − |λ| λ + |λ| φ (λ + b) +  (λ + b) dλ 2 2

  λ − |λ| λ + |λ| φ (λ + a) +  (λ + a) dλ 2 2

(2.1.8)

2.1 Generalized Ostrowski Type Inequalities with Parameter

97

and 1 M4 = b−a 

b−a 2

+ 0

(



0

− b−a 2



 λ + |λ| λ − |λ| φ (λ + b) +  (λ + b) dλ 2 2

  λ + |λ| λ − |λ| φ (λ + a) +  (λ + a) dλ , 2 2

which is Corollary 1 of [116]. Special Case 2.1.4.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.8), then    b (b − a) a+b (b − a) 1 (φ0 − 0 ) ≤ f (0 − φ0 ), f (t)dt ≤ − 8 2 b−a a 8 which is in fact the Special Case 1 of Theorem 1 and Corollary 1 as cited in [116] and [184] respectively. Special Case 2.1.4.(b) If we take, φ(x) = φ1 x+φ0 = 0 and (x) = 1 x+0 = 0 in (2.1.8), then  m5 ≤ f

a+b 2

 −

1 b−a



b

f (t)dt ≤ M5 ,

a

where m5 =

(b − a) 8

M5 =

(b − a) 8



b−a (φ1 + 1 ) + aφ1 − b1 + φ0 − 0 3



and 

 b−a (φ1 + 1 ) + a1 − bφ1 + 0 − φ0 , 3

which is the result of Corollary 1 of [116]. 1 1 Remark 2.1.4 By choosing = in (2.1.7), we obtain bounds for Simpson’s 3 3 rule in the following corollary. Corollary 2.1.5 Let all the assumptions of Theorem 2.1.1 be valid. Then m6 ≤

    b a+b 1 f (a) + f (b) 1 + 2f f (t)dt ≤ M6 , (2.1.9) − 3 2 2 b−a a

98

2 Ostrowski Inequality

where 1 m6 = b−a  +

b−a 6

(

b−a 3



− b−a 6



− b−a 3

    5a + b λ − |λ| 5a + b λ + |λ| φ λ+ +  λ+ dλ 2 6 2 6

     a + 5b a + 5b λ + |λ| λ − |λ| φ λ+  λ+ + dλ 2 6 2 6

and 1 M6 = b−a  +

b−a 6

− b−a 3

(

b−a 3

− b−a 6





    5a + b 5a + b λ − |λ| λ + |λ| φ λ+  λ+ + dλ 2 6 2 6

     a + 5b λ + |λ| a + 5b λ − |λ| φ λ+ +  λ+ dλ . 2 6 2 6

Special Case 2.1.5.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.9), then     b a+b 5(b − a) 1 f (a) + f (b) 1 (φ0 − 0 ) ≤ + 2f f (t)dt − 72 3 2 2 b−a a ≤

5(b − a) (0 − φ0 ), 72

which is Corollay 4 of [184]. Special Case 2.1.5.(b) If we take, φ(x) = φ1 x+φ0 = 0 and (x) = 1 x+0 = 0 in (2.1.9), then m7 ≤

    b a+b 1 f (a) + f (b) 1 + 2f f (t)dt ≤ M7 , − 3 2 2 b−a a

where m7 =

  (b − a) a b (b − a)(φ1 + 1 ) + (7φ1 − 31 ) + (3φ1 − 71 ) + 5(φ0 − 0 ) 72 2 2

and M7 =

  (b − a) a b (b − a)(φ1 + 1 ) + (71 − 3φ1 ) + (31 − 7φ1 ) + 5(0 − φ0 ) . 72 2 2

1 Remark 2.1.5 By choosing = in (2.1.7), we get the bound of average midpoint 2 and trapezoidal rule in the following corollary.

2.1 Generalized Ostrowski Type Inequalities with Parameter

99

Corollary 2.1.6 Let all the assumptions of Theorem 2.1.1 be valid. Then m8 ≤

    b a+b 1 f (a) + f (b) 1 +f f (t)dt ≤ M8 , − 2 2 2 b−a a

(2.1.10)

where 1 m8 = b−a  +

b−a 4

(

b−a 4



− b−a 4



− b−a 4

    3a + b λ − |λ| 3a + b λ + |λ| φ λ+ +  λ+ dλ 2 4 2 4

     a + 3b a + 3b λ + |λ| λ − |λ| φ λ+  λ+ + dλ 2 4 2 4

and 1 M8 = b−a  +

b−a 4

− b−a 4

(

b−a 4

− b−a 4





    3a + b λ + |λ| 3a + b λ − |λ| φ λ+ +  λ+ dλ 2 4 2 4

     a + 3b a + 3b λ − |λ| λ + |λ| φ λ+  λ+ + dλ . 2 4 2 4

Special Case 2.1.6.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.10), then     b a+b (b − a) 1 f (a) + f (b) 1 (φ0 − 0 ) ≤ +f f (t)dt − 16 2 2 2 b−a a ≤

(b − a) (0 − φ0 ), 16

which is Corollary 3 of [184]. Special Case 2.1.6.(b) If we take, φ(x) = φ1 x+φ0 = 0 and (x) = 1 x+0 = 0 in (2.1.10), then     b a+b 1 f (a) + f (b) 1 +f m9 ≤ f (t)dt ≤ M9 , − 2 2 2 b−a a where m9 =

  (b − a) (b − a) a b (φ1 + 1 ) + (φ1 − 1 ) + (φ1 − 1 ) + φ0 − 0 16 6 2 2

100

2 Ostrowski Inequality

and   (b − a) (b − a) a b (φ1 + 1 ) + (1 − φ1 ) + (1 − φ1 ) + 0 − φ0 . M9 = 16 6 2 2 Remark 2.1.6 By choosing x = a in (2.1.3), then for any value of ∈ [0, 1], we get the bound of trapezoidal rule (Hermite–Hadamard right bound) in the following corollary. Corollary 2.1.7 Let all the assumptions of Theorem 2.1.1 be valid. Then m10 ≤

1 f (a) + f (b) − 2 b−a



b

f (t)dt ≤ M10 ,

(2.1.11)

a

where m10

1 = b−a

(

b−a 2



     a+b a+b λ + |λ| λ − |λ| φ λ+  λ+ + dλ 2 2 2 2



     a+b λ + |λ| a+b λ − |λ| φ λ+ +  λ+ dλ , 2 2 2 2

− b−a 2

and M10

1 = b−a

(

b−a 2

− b−a 2

which is Corollary 2 cited in [116]. If we take, φ(x) = φ0 = 0 and (x) = 0 = 0, and φ(x) = φ1 x + φ0 = 0 and (x) = 1 x + 0 = 0 in (2.1.11), then we obtain the results as stated in Special Case 2.1.2.(a) and Special Case 2.1.2.(b) respectively. Remark 2.1.7 By choosing x = b and = 0 in (2.1.3), we get the bound for trapezoidal rule (Hermite–Hadamard right bound) in following corollary. Corollary 2.1.8 Let all the assumptions of Theorem 2.1.1 be valid. Then m11 ≤

f (a) + f (b) 1 − 2 b−a



b

f (t)dt ≤ M11 ,

(2.1.12)

a

where m11

 0   1 λ − |λ| λ + |λ| = φ (λ + b) +  (λ + b) dλ b − a −(b−a) 2 2      b−a  2 a+b a+b λ + |λ| λ − |λ| φ λ+  λ+ − + dλ 2 2 2 2 − b−a 2    b−a  λ − |λ| λ + |λ| φ (λ + a) +  (λ + a) dλ + 2 2 0

2.1 Generalized Ostrowski Type Inequalities with Parameter

101

and   0  λ + |λ| λ − |λ| 1 φ (λ + b) +  (λ + b) dλ b − a −(b−a) 2 2      b−a  2 a+b λ + |λ| a+b λ − |λ| − φ λ+ +  λ+ dλ 2 2 2 2 − b−a 2    b−a  λ + |λ| λ − |λ| φ (λ + a) +  (λ + a) dλ . + 2 2 0

M11 =

Special Case 2.2.8.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.12), then    b f (a) + f (b) 3(b − a) 3(b − a) 1 f (t)dt ≤ (φ0 − 0 ) ≤ − (0 − φ0 ). 8 2 b−a a 8 Special Case 2.2.8.(b) If we take, φ(x) = φ1 x+φ0 = 0 and (x) = 1 x+0 = 0 in (2.1.12), then  m12 ≤

  b f (a) + f (b) 1 f (t)dt ≤ M12 , − 2 b−a a

where m12 =

  (b − a) 7(b − a) a b 3 (φ1 + 1 ) + (7φ1 + 1 ) − (φ1 + 71 ) + (φ0 − 0 ) 2 12 8 8 4

and M12 =

  (b − a) 7(b − a) a b 3 (φ1 + 1 ) + (φ1 + 71 ) − (7φ1 + 1 ) + (0 − φ0 ) . 2 12 8 8 4

1 in (2.1.3), we get the bound for Remark 2.1.8 By choosing x = b and = 2 trapezoidal rule (Hermite–Hadamard right bound) in the following corollary. Corollary 2.1.9 Let all the assumptions of Theorem 2.1.1 be valid. Then 

m13

  b f (a) + f (b) 1 ≤ f (t)dt ≤ M13 , − 2 b−a a

(2.1.13)

102

2 Ostrowski Inequality

where m13

1 = b−a  −

b−a 2

(





−(b−a) 4



b−a 4

    λ − |λ| λ + |λ| 3a + b 3a + b + dλ φ λ+  λ+ 2 4 2 4

    λ + |λ| a+b λ − |λ| a+b φ λ+ +  λ+ dλ 2 2 2 2



− b−a 2

+

3(b−a) 4

−3(b−a) 4

     λ + |λ| a + 3b λ − |λ| a + 3b φ λ+ +  λ+ dλ 2 4 2 4

and M13

1 = b−a  −

3(b−a) 4 −(b−a) 4



− b−a 2

 +

b−a 2

(

−3(b−a) 4

    λ − |λ| 3a + b λ + |λ| 3a + b φ λ+ +  λ+ dλ 2 4 2 4

    λ − |λ| a+b a+b λ + |λ| φ λ+  λ+ + dλ 2 2 2 2



b−a 4



     λ + |λ| λ − |λ| a + 3b a + 3b + dλ . φ λ+  λ+ 2 4 2 4

Special Case 2.1.9.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.13), then    b f (a) + f (b) 3(b − a) 3(b − a) 1 (φ0 − 0 ) ≤ (0 − φ0 ). f (t)dt ≤ − 16 2 b−a a 16 Special Case 2.1.9.(b) If we take, φ(x) = φ1 x+φ0 = 0 and (x) = 1 x+0 = 0 in (2.1.13), then  m14 ≤

  b f (a) + f (b) 1 f (t)dt ≤ M14 , − 2 b−a a

where m14 =

  (b − a) 5 a b (b − a)(φ1 + 1 ) + (5φ1 − 1 ) + (φ1 − 51 ) + 3(φ0 − 0 ) 16 3 2 2

and M14 =

  (b − a) 5 a b (b − a)(φ1 + 1 ) + (51 − φ1 ) + (1 − 5φ1 ) + 3(0 − φ0 ) . 16 3 2 2

2.1 Generalized Ostrowski Type Inequalities with Parameter

103

1 in (2.1.3), we get the bound for Remark 2.1.9 By choosing x = b and = 3 trapezoidal rule (Hermite–Hadamard right bound) in the following corollary. Corollary 2.1.10 Let all the assumptions of Theorem 2.1.1 be valid. Then  m15 ≤

  b f (a) + f (b) 1 f (t)dt ≤ M15 , − 2 b−a a

(2.1.14)

where m15

(

1 = b−a  −

5(b−a) 6 −(b−a) 6

b−a 2





b−a 6

    λ + |λ| 5a + b λ − |λ| 5a + b φ λ+ +  λ+ dλ 2 6 2 6

    λ − |λ| λ + |λ| a+b a+b + dλ φ λ+  λ+ 2 2 2 2



− b−a 2

+



−5(b−a) 6

     λ − |λ| λ + |λ| a + 5b a + 5b + dλ φ λ+  λ+ 2 6 2 6

and M15

1 = b−a  −

5(b−a) 6 −(b−a) 6

b−a 2



− b−a 2

 +

(

b−a 6 −5(b−a) 6



    λ − |λ| 5a + b 5a + b λ + |λ| φ λ+  λ+ + dλ 2 4 2 6

    a+b λ + |λ| a+b λ − |λ| φ λ+ +  λ+ dλ 2 2 2 2



     λ − |λ| a + 5b a + 5b λ + |λ| φ λ+  λ+ + dλ . 2 6 2 6

Special Case 2.1.10.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.14), then    b 17(b − a) 1 f (a) + f (b) 17(b − a) (φ0 − 0 ) ≤ − (0 − φ0 ). f (t)dt ≤ 72 2 b−a a 72

Special Case 2.1.10.(b) If we take, φ(x) = φ1 x +φ0 = 0 and (x) = 1 x +0 = 0 in (2.1.14), then  m16 ≤

  b 1 f (a) + f (b) − f (t)dt ≤ M16 , 2 b−a a

104

2 Ostrowski Inequality

where m16

 (b − a) 3a 3b 11(b − a)(φ1 + 1 ) + (33φ1 − 1 ) + (φ1 − 331 ) = 72 2 2 +17(φ0 − 0 )]

and M16 =

 (b − a) 3a 3b (b − a)(φ1 + 1 ) + (331 − φ1 ) + (1 − 33φ1) 72 2 2 + 17(0 − φ0 )] .

1 in (2.1.3), we get the bound for Remark 2.1.10 By choosing x = b and = 4 trapezoidal rule (Hermite–Hadamard right bound) in following corollary. Corollary 2.1.11 Let all the assumptions of Theorem 2.1.1 be valid. Then  m17 ≤

  b f (a) + f (b) 1 f (t)dt ≤ M17 , − 2 b−a a

(



(2.1.15)

where m17

1 = b−a  −

b−a 2

−(b−a) 8





     λ − |λ| λ + |λ| a + 7b a + 7b + dλ φ λ+  λ+ 2 8 2 8



b−a 8

    λ − |λ| λ + |λ| 7a + b 7a + b + dλ φ λ+  λ+ 2 8 2 8

    λ + |λ| a+b λ − |λ| a+b φ λ+ +  λ+ dλ 2 2 2 2

− b−a 2

+

7(b−a) 8

−7(b−a) 8

and M17

1 = b−a  −

7(b−a) 8

− (b−a) 8



− b−a 2

 +

b−a 2

(

b−a 8 −7(b−a) 8



    λ − |λ| 7a + b λ + |λ| 7a + b φ λ+ +  λ+ dλ 2 8 2 8

    λ + |λ| λ − |λ| a+b a+b + dλ φ λ+  λ+ 2 2 2 2



     λ + |λ| λ − |λ| a + 7b a + 7b + dλ . φ λ+  λ+ 2 8 2 8

2.1 Generalized Ostrowski Type Inequalities with Parameter

105

Special Case 2.1.11.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.15), then    b 17(b − a) 1 f (a) + f (b) 17(b − a) (φ0 − 0 ) ≤ − (0 − φ0 ). f (t)dt ≤ 64 2 b−a a 64

Special Case 2.1.11.(b) If we take, φ(x) = φ1 x +φ0 = 0 and (x) = 1 x +0 = 0 in (2.1.15), then  m18 ≤

  b 1 f (a) + f (b) − f (t)dt ≤ M18 , 2 b−a a

where m18 =

 (b − a) 35 a b (b − a)(φ1 + 1 ) + (35φ1 + 1 ) − (φ1 + 351) 64 3 2 2 +17(φ0 − 0 )]

and M18

 (b − a) 35 a b (b − a)(φ1 + 1 ) + (351 + φ1 +) − (1 + 35φ1 ) = 64 3 2 2 +17(0 − φ0 )] .

3a + b and = 0 in (2.1.3), we get the bound Remark 2.1.11 If we choose x = 4 for trapezoidal type rule in the following corollary. Corollary 2.1.12 Let all the assumptions of Theorem 2.1.1 be valid. Then m19

      b 1 3a + b a + 3b 1 ≤ f (t)dt ≤ M19 , (2.1.16) f +f − 2 4 4 b−a a

where m19

1 = b−a  +

b−a 4

− b−a 4

 + 0

b−a 4

(

0 − b−a 4



 λ − |λ| λ + |λ| φ (λ + b) +  (λ + b) dλ 2 2

    a+b a+b λ + |λ| λ − |λ| φ λ+  λ+ + dλ 2 2 2 2  )  λ − |λ| λ + |λ| φ (λ + a) +  (λ + a) dλ 2 2 

106

2 Ostrowski Inequality

and M19

1 = b−a  +

b−a 4

− b−a 4

 + 0

b−a 4

(



0 − b−a 4

 

 λ + |λ| λ − |λ| φ (λ + b) +  (λ + b) dλ 2 2

    a+b a+b λ − |λ| λ + |λ| φ λ+  λ+ + dλ 2 2 2 2

  λ + |λ| λ − |λ| φ (λ + a) +  (λ + a) dλ . 2 2

Special Case 2.1.12.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.16), then       b 1 (b − a) 3a + b a + 3b 1 (φ0 − 0 ) ≤ f (t)dt f +f − 16 2 4 4 b−a a ≤

(b − a) (0 − φ0 ). 16

Special Case 2.1.12.(b) If we take, φ(x) = φ1 x +φ0 = 0 and (x) = 1 x +0 = 0 in (2.1.16), then m20

      b 1 3a + b a + 3b 1 ≤ f (t)dt ≤ M20 , f +f − 2 4 4 b−a a

where m20

  (b − a) (b − a) a b (φ1 + 1 ) + (3φ1 − 1 ) + (φ1 − 31 ) + φ0 − 0 = 16 6 4 4

and M20

  (b − a) (b − a) a b (φ1 + 1 ) + (31 − φ1 ) + (1 − 3φ1 ) + 0 − φ0 . = 16 6 4 4

1 2a + b and = in (2.1.3), then we get the Remark 2.1.12 If we choose x = 3 4 3 bound of Simpson’s rule in the following corollary. 8 Corollary 2.1.13 Let all the assumptions of Theorem 2.1.1 be valid. Then m21 ≤

      b 2a + b a + 2b 3 f (a) + f (b) 1 f (t)dt ≤ M21 , +f +f − 8 3 3 3 b−a a

(2.1.17)

2.1 Generalized Ostrowski Type Inequalities with Parameter

107

where m21

1 = b−a  +

b−a 6

5(b−a) 24



− b−a 8



− b−a 6

 +

(

    λ + |λ| a+b λ − |λ| a+b φ λ+ +  λ+ dλ 2 2 2 2 

b−a 8

    λ − |λ| λ + |λ| 7a + b 7a + b + dλ φ λ+  λ+ 2 8 2 8

− 5(b−a) 24

     λ + |λ| a + 7b a + 7b λ − |λ| φ λ+  λ+ + dλ 2 8 2 8

and M21

1 = b−a  +

5(b−a) 24

− b−a 8



− b−a 6

 +

b−a 6

(

b−a 8

− 5(b−a) 24



    λ − |λ| 7a + b 7a + b λ + |λ| φ λ+  λ+ + dλ 2 8 2 8

    a+b λ + |λ| a+b λ − |λ| φ λ+ +  λ+ dλ 2 2 2 2 

     λ + |λ| λ − |λ| a + 7b a + 7b + dλ . φ λ+  λ+ 2 8 2 8

Special Case 2.1.13.(a) If we take, φ(x) = φ0 = 0 and (x) = 0 = 0 in (2.1.17), then 25(b − a) (0 − φ0 ) 576       b 2a + b a + 2b 1 3 f (a) + f (b) +f +f − f (t)dt ≤ 8 3 3 3 b−a a ≤

25(b − a) (φ0 − 0 ). 576

Special Case 2.1.13.(b) If we take, φ(x) = φ1 x +φ0 = 0 and (x) = 1 x +0 = 0 in (2.1.17), then m22 ≤

      b 1 3 f (a) + f (b) 2a + b a + 2b +f +f − f (t)dt ≤ M22 , 8 3 3 3 b−a a

where m22 =

 (b − a) 31 19 (b − a)(φ1 + 1 ) + (aφ1 − b1 ) + (bφ1 − a1 ) 192 6 6  25 + (φ0 − 0 ) 3

108

2 Ostrowski Inequality

and M22

 (b − a) 31 19 (b − a)(φ1 + 1 ) + (b1 − aφ1 ) + (a1 − bφ1 ) = 192 6 6  25 + (0 − φ0 ) . 3

2.1.2 Ostrowski Type Inequalities for Bounded Below Only and Bounded Above Only Differentiable Functions The condition φ(x) ≤ f (x) ≤ (x) is true in Theorem 2.1.1, but sometimes we are not able to find both bounds of a function. Now we define two theorems. The first one would be helpful when f is bounded from above and the second one would be helpful when f is bounded from below. Theorem 2.1.14 Let f : I → R be differentiable function in I 0 . If f is bounded from below, then φ(x) ≤ f (x) such that φ ∈ C[a, b], x ∈ [a, b], ∀ ∈ [0, 1] we have    b f (x) + f (a + b − x) 1 f (a) + f (b) + (1 − ) − m23 (x, ) ≤ f (t)dt 2 2 b−a a ≤ M23 (x, ),

(2.1.18)

where m23 (x, ) =

  a+b t− φ(t)dt 2 a   x  b b−a + (1 − ) φ(t)dt − (1 − ) φ(t)dt 2 a a+b−x   (2 − )a + b b−a , x− − max , 2 2      b a+b −x f (b) − f (a) − φ(t)dt 2 a 1 b−a



b

2.1 Generalized Ostrowski Type Inequalities with Parameter

109

and 1 M23 (x, ) = b−a + −



b a

b−a 2  b



  a+b t− φ(t)dt 2 x

(1 − ) φ(t)dt

a



(1 − ) φ(t)dt

a+b−x

  (2 − )a + b b−a , x − , + max 2 2      b a+b −x f (b) − f (a) − φ(t)dt . 2 a

Proof Since    f (a) + f (b) f (x) + f (a + b − x) P (x, t) f (t) − φ(t) dt = (b − a) + (1 − ) 2 2 a  b  b − f (t)dt − P (x, t)φ(t)dt

 b

a



a

f (x) + f (a + b − x) f (a) + f (b) + (1 − ) = (b − a) 2 2  x     b b−a − f (t)dt − φ(t)dt t − a+ 2 a a   a+b−x  a+b φ(t)dt t− + 2 x )     b b−a φ(t)dt . t − b− + 2 a+b−x

Applying modulus property, we get    b   (b − a) f (a) + f (b) + (1 − ) f (x) + f (a + b − x) − f (t)dt  2 2 a  x      a+b−x  b−a a+b − t − a+ t− φ(t)dt + φ(t)dt 2 2 a x



110

2 Ostrowski Inequality

     b−a + t − b− φ(t)dt  2 a+b−x  b      = P (x, t) f (t) − φ(t) dt  



b

a

b

≤

 |P (x, t)| f (t) − φ(t) dt

a



b

≤ max |P (x, t)| t ∈[a,b]

 f (t) − φ(t) dt

a

    (2 − )a + b b−a a+b , x− −x = max , 2 2 2    b × f (b) − f (a) − φ(t)dt .

(2.1.19)

a



After re-arranging (2.1.19), we get the required inequality (2.1.18).

Remark 2.1.13 The inequality (2.1.18) is the generalized case of Theorem 2 which is presented in [116] and Theorem 2 which is presented in [115]. Remark 2.1.14 If we select x = corollary.

a+b in (2.1.18), then we get the following 2

Corollary 2.1.15 Let all the assumptions of Theorem 2.1.14 be valid. Then     b a+b f (a) + f (b) 1 m24 ( ) ≤ f (t)dt ≤ M24 ( ), + (1 − )f − 2 2 b−a a

(2.1.20) where m24 ( ) =

1 b−a



b

 a+b φ(t)dt 2  (1 − ) φ(t)dt −

t−



a a+b

2 b−a + 2 a  b−a b−a , (1 − ) − max 2 2    b φ(t)dt f (b) − f (a) −

a

b a+b 2

 (1 − ) φ(t)dt

2.1 Generalized Ostrowski Type Inequalities with Parameter

111

and 

1 M24 ( ) = b−a

b a



  a+b t− φ(t)dt 2 a+b

2 b−a + (1 − ) φ(t)dt − 2 a  b−a b−a , (1 − ) + max 2 2    b φ(t)dt . f (b) − f (a) −



b a+b 2

 (1 − ) φ(t)dt

a

Theorem 2.1.16 Let f : I → R be differentiable function in I 0 . If f is bounded above, i.e., f (x) ≤ (x) such that  ∈ C[a, b], x ∈ [a, b], ∀ ∈ [0, 1] we have 

  b f (x) + f (a + b − x) f (a) + f (b) 1 + (1 − ) m25 (x, ) ≤ f (t)dt − 2 2 b−a a ≤ M25 (x, ),

(2.1.21)

where   a+b t− (t)dt 2 a   x  b b−a + (1 − ) (t)dt − (1 − ) (t)dt 2 a a+b−x     (2 − )a + b b−a a+b , x − − x − max , 2 2 2  b  × (t)dt − f (b) + f (a)

1 m25 (x, ) = b−a



b



b

a

and   a+b (t)dt t− 2 a  x   b b−a + (1 − ) (t)dt − (1 − ) (t)dt 2 a a+b−x

1 M25 (x, ) = b−a

112

2 Ostrowski Inequality

    (2 − )a + b a+b b−a , x− , −x + max 2 2 2  b  × (t)dt − f (b) + f (a) . a

Proof Since 

b

 P (x, t) f (t) − (t) dt

a

  f (a) + f (b) f (x) + f (a + b − x) = (b − a) + (1 − ) 2 2  b  b − f (t)dt − P (x, t)(t)dt a

a

  b  f (x) + f (a + b − x) f (a) + f (b) + (1 − ) − f (t)dt = (b − a) 2 2 a     x   a+b−x  b−a a+b (t)dt + (t)dt t − a+ t− − 2 2 a x      b b−a (t)dt , t − b− + 2 a+b−x so we get     b  (b − a) f (a) + f (b) + (1 − ) f (x) + f (a + b − x) − f (t)dt  2 2 a  x      a+b−x  b−a a+b − t − a+ t− (t)dt + (t)dt 2 2 a x      b  b−a (t)dt  t − b− + 2 a+b−x  b      = P (x, t) f (t) − (t) dt  

a

b

≤

 |P (x, t)| (t) − f (t) dt

a



b

≤ max |P (x, t)| t ∈[a,b]

 (t) − f (t) dt

a

     b  (2 − )a + b a+b b−a , x− , −x = max (t)dt − f (b) + f (a) . 2 2 2 a

(2.1.22)

2.1 Generalized Ostrowski Type Inequalities with Parameter

113



After rearranging (2.1.22), we get the inequality (2.1.21).

Remark 2.1.15 The inequality (2.1.21) is the special case of Theorem 3 which is presented in [116] and Theorem 3 which is presented in [115]. Remark 2.1.16 By choosing x = corollary.

a+b 2

in (2.1.21), then we achieve following

Corollary 2.1.17 Let all the assumptions of Theorem 2.1.16 be valid. Then     b a+b f (a) + f (b) 1 + (1 − )f m26 ( ) ≤ f (t)dt ≤ M26 ( ), − 2 2 b−a a

(2.1.23) where 1 m26 ( ) = b−a



b

a

  a+b (t)dt t− 2

 a+b  b 2 b−a + (1 − ) (t)dt − (1 − ) (t)dt a+b 2 a 2    b b−a b−a , (1 − ) (t)dt − f (b) + f (a) − max 2 2 a 

and M26 ( ) =

1 b−a



b a



  a+b t− (t)dt 2

 a+b  b 2 b−a + (1 − ) (t)dt − (1 − ) (t)dt a+b 2 a 2    b b−a b−a , (1 − ) (t)dt − f (b) + f (a) . + max 2 2 a Remark 2.1.17 If φ(x) ≤ f (x) ≤ (x) such that x ∈ [a, b] and φ,  ∈ C[a, b], and if we select = 0, then I6 (f ) can be bounded as m27

     b a+b 1 1 −f (a) + 2f + f (b) − ≤ f (t)dt ≤ M27 , (2.1.24) 2 2 b−a a

where m27

1 = b−a

(

a+b 2

a



(b − a) (t − b)φ(t)dt + (t − a) φ(t)dt + a+b 2 2 b





b

φ(t)dt a

114

2 Ostrowski Inequality

and M27

1 = b−a

(

a+b 2

 (t − a) (t)dt +

a

b a+b 2

(t − b)(t)dt +

(b − a) 2



b

 (t)dt ,

a

which is in fact Corollary 3 and Corollary 4 of [116] and [115] respectively. Proof In order to prove (2.1.24) both the Corollary 2.1.15 and Corollary 2.1.17 results should be used at the same time. First of all by selecting = 0 in (2.1.20), we get (  a+b )  b b 2 1 b−a φ(t)dt − φ(t)dt (t − a) φ(t)dt + a+b b−a a 2 a 2      b 1 a+b 1 ≤ f (t)dt, (2.1.25) −f (a) + 2f + f (b) − 2 2 b−a a provided that φ(t) ≤ f (t) ∀ t ∈ [a, b]. On the other hand, by assuming = 0 in (2.1.23), we obtain      b a+b 1 1 −f (a) + 2f + f (b) − f (t)dt 2 2 b−a a ) (  a+b  b b 2 1 b−a ≤ (t)dt − (t)dt , (t − a) (t)dt + a+b b−a a 2 a 2 (2.1.26) provided that f (t) ≤ (t) ∀ t ∈ [a, b]. Now by combining the above two results (2.1.25) and (2.1.26), the result (2.1.24) is derived.  Remark 2.1.18 If φ(x) ≤ f (x) ≤ (x) such that x ∈ [a, b] and φ,  ∈ C[a, b], and if we select = 0, then I7 (f ) can be bounded as m28 ≤

     b 1 a+b 1 f (t)dt ≤ M28 , (2.1.27) f (a) + 2f − f (b) − 2 2 b−a a

where m28

1 = b−a

(

a+b 2

a

 (t − a) (t)dt +

b a+b 2

(t − b)(t)dt −

(b − a) 2





b

(t)dt a

2.1 Generalized Ostrowski Type Inequalities with Parameter

115

and 1 M28 = b−a

(

)  (b − a) b (t − a) φ(t)dt + (t − b)φ(t)dt − φ(t)dt , a+b 2 a 2  b

a+b 2

a

which is in fact Corollary 4 and Corollary 5 of [116] and [115] respectively. 

Proof The proof of (2.1.27) is similar to Remark 2.1.17.

Remark 2.1.19 If φ(x) ≤ f (x) ≤ (x) for any x ∈ [a, b] and φ,  ∈ C[a, b] then by replacing x = b, and = 0 in (2.1.20) and (2.1.23), respectively, then I8 (f ) can be bounded as 1 (b − a)



b

(t − b)(t)dt ≤ f (a) −

a

1 (b − a)



b

f (t)dt ≤

a

1 (b − a)



b

(t − b)φ(t)dt,

a

which is in fact Corollary 5 and Corollary 2 of [116] and [115] respectively. Remark 2.1.20 If φ(x) ≤ f (x) ≤ (x) such that x ∈ [a, b] and φ,  ∈ C[a, b] then by replacing x = a and = 0 in (2.1.20) and (2.1.23), respectively, then I9 (f ) can be bounded as 1 (b − a)



b

(t − a)φ(t)dt ≤ f (b) −

a

1 (b − a)



b

f (t)dt ≤

a

1 (b − a)



b

(t − a)(t)dt,

a

which is in fact Corollary 6 and Corollary 3 of [116] and [115] respectively. Now, we will discuss certain applications in numerical quadrature rules that can be used to get some sharp bounds.

2.1.3 Applications to Numerical Integration Let In : a = ζ0 < ζ1 < · · · < ζn = b be a partition of the interval [a, b] and ζk = ζk+1 − ζk , k ∈ {0, 1, 2, . . . , n − 1}. Then 

b

f (t)dt = Qn (f, In ) + Rn (f, In ),

(2.1.28)

a

where Qn (f, In ) is defined as Qn (f, In ) :=

  n−1 

ζk + ζk+1 − ξk f (ζk ) + f (ζk+1 ) + (1 − )f (ξk ) + f ζk ,

2 2 k=0

(2.1.29)

116

2 Ostrowski Inequality

∀ ∈ [0, 1] and ζk +

ζk ζk + ζk+1 ≤ ξk ≤ . 2 2

Theorem 2.1.18 Let all the assumptions of Theorem 2.1.1 be valid. Then (2.1.28) is valid and Qn (f, In ) is given in the form of (2.1.29) and the remainder Rn (f, In ) becomes +   , |Rn (f, In )| ≤ sup R , R  , where   ζ ξk − ζk + 2 k  λ k

  + |λk | ζk R= φ λk + ζk + ζ 2 2 − 2 k   λk − |λk | ζk +  λk + ζk + dλk 2 2    ζk +ζk+1 −ξk  2 ζk + ζk+1 λk + |λk | φ λk + + ζ +ζ 2 2 ξk − k 2k+1   λk − |λk | ζk + ζk+1  λk + dλk + 2 2     ζk 2 ζk λk + |λk | φ λk + ζk+1 − + ζ 2 2 ζk + 2 k −ξk   λk − |λk | ζk  λk + ζk+1 − dλk + 2 2 

and   ζ ξk − ζk + 2 k  λ k

  ζk − |λk | φ λk + ζk + R = ζ 2 2 − 2 k   λk + |λk | ζk  λk + ζk + dλk + 2 2    ζk +ζk+1 −ξk  2 λk − |λk | ζk + ζk+1 φ λk + + ζ +ζ 2 2 ξk − k 2k+1   λk + |λk | ζk + ζk+1 +  λk + dλk 2 2



(2.1.30)

2.1 Generalized Ostrowski Type Inequalities with Parameter

117

  λk − |λk | ζk φ λk + ζk+1 − + ζ 2 2 ζk + 2 k −ξk   λk + |λk | ζk +  λk + ζk+1 − dλk , 2 2 

ζk 2



∀ ξk ∈ [ζk , ζk+1 ]. Proof If inequality (2.1.3) is applied on [ζk , ζk+1 ], we have 

ζk+1

Rk (f, Ik ) =

f (t)dt ζk

   f (ζk ) + f (ζk+1 ) ζk + ζk+1 − ξk + (1 − )f (ξk ) + f ζk , − 2 2 we sum up Rk (f, Ik ) from 0 to n − 1 over k. This produces  Rn (f, In ) =

b

f (t)dt a

  n−1 

f (ζk ) + f (ζk+1 ) ζk + ζk+1 − ξk +(1 − )f (ξk ) + f ζk .

− 2 2 k=0

It follows from (2.1.3) that |Rn (f, In )|   = 

b

f (t)dt −

a

  n−1 

f (ζk ) + f (ζk+1 ) ζk + ζk+1 − ξk 

+(1 − )f (ξk ) + f  2 2 k=0

⎧    ⎨  ξk − ζk + ζ2 k  λ + |λ |  ζk k k φ λ + ζ +

≤ sup  k k ⎩ − ζk 2 2 2   ζk λk − |λk |  λk + ζk + dλk + 2 2   λk + |λk | ζk + ζk+1 + φ λk + ζ +ζ 2 2 ξk − k 2k+1   λk − |λk | ζk + ζk+1 dλk +  λk + 2 2     ζk 2 λk + |λk | ζk + + ζ −

φ λ k k+1 ζ 2 2 ζk + 2 k −ξk 

ζk +ζk+1 −ξk 2



118

2 Ostrowski Inequality

    λk − |λk | ζk  λk + ζk+1 − dλk  , + 2 2    ξ −ζ + ζk      k k 2 λk − |λk | ζk  + ζ +

φ λ k k  2 2  − ζ2 k +

  λk + |λk | ζk  λk + ζk + dλk 2 2

  λk − |λk | ζk + ζk+1 φ λ + k ζ +ζ 2 2 ξk − k 2k+1   λk + |λk | ζk + ζk+1  λk + dλk 2 2     ζk 2 λk − |λk | ζk φ λk + ζk+1 − + ζ 2 2 ζk + 2 k −ξk     ζk λk + |λk |  λk + ζk+1 − + dλk  . 2 2 

+

ζk +ζk+1 −ξk 2



 Remark 2.1.21 In similar manner, we can state applications of all corollaries, remarks and special cases presented in the previous section.

2.2 Generalized Ostrowski Type Inequalities for Functions of Lp Spaces and Bounded Variation In current section, some Ostrowski type inequalities for Lp spaces and functions of bounded variations are stated. Applications are also given for obtaining error bounds of some composite quadrature formulae. Result of this section can be found in [76].

2.2.1 Ostrowski Type Inequality for Functions of Lp Spaces Lets recall the result of Hölder’s inequality from Theorem 1.1.13 that will be useful in our next results. In 2010, a generalization of Ostrowski inequality [141] discussed by Milovanovi´c and Cvetkovi´c in [129], which follows:

2.2 Generalized Ostrowski Type Inequalities for Functions of Lp Spaces and. . .

119

Theorem 2.2.1 Let f : [a, b] → R be a function such that f is bounded on (a, b), i.e., f ∞ = sup |f (t)| < ∞. t ∈(a,b)

Then ∀ x ∈ [a, b], following inequality holds     b 2 2   f (x) − 1  ≤ (x − a) + (b − x) f ∞ f (t)dt   b−a a 2(b − a) (  2 ) x − a+b 1 2 + = (b − a)f ∞ . (2.2.1) 4 (b − a)2 To prove our main results in this section, we need the following lemma extracted from [116]. Lemma 2.2.1 Let f : [a, b] → R be absolutely continuous function where kernel P1 (x, t) given as

P1 (x, t) =

⎧ b−a ⎪ ⎪ ⎪ ⎨ t − x + 2 , t ∈ [a, x]; ⎪ ⎪ ⎪ ⎩ t − x − b − a , t ∈ (x, b]. 2

(2.2.2)

Then 1 b−a

 a

b

P1 (x, t)f (t)dt = f (x) −

   b a+b 1 f (b) − f (a) x− − f (t)dt. b−a 2 b−a a

(2.2.3) Proof Using kernel (2.2.2), after some computation, we obtain 

x a

     x a+b b−a b−a

f (x) + x − f (t)dt t −x+ f (t)dt = f (a) − 2 2 2 a

and 

b x

     b a+b b−a b−a f (t)dt = f (x) − x − f (b) − t −x− f (t)dt. 2 2 2 x

By adding above two equations and after some simplifications, we get (2.2.3). Now we shall use the above identity to obtain our results for first derivative bounded functions and prove inequalities for absolutely continuous functions in which f ∈ Lq [a, b] for q ≥ 1.

120

2 Ostrowski Inequality

Theorem 2.2.2 Let f : [a, b] → R be an absolutely continuous function on I o 1 1 where a, b ∈ I and a < b. If f ∈ Lq [a, b], for q ≥ 1 and + = 1, then for any q r x ∈ [a, b], this inequality holds      b   f (x) − f (b) − f (a) x − a + b − 1 f (t)dt   b−a 2 b−a a ( r+1     )1 f q a + b r+1 b − a r+1 r a+b ≤ −x + x− +2 . 1 2 2 2 (b − a)(r + 1) r (2.2.4) Proof Applying absolute on (2.2.3) and then applying Hölder inequality, we obtain      b   f (x) − f (b) − f (a) x − a + b − 1  f (t)dt   b−a 2 b−a a    1  b

 P (x, t)f (t)dt = 1   b−a a 1 ≤ b−a



b a

|P1 (x, t)| dt r

 1r 

b

|f (t)| dt q

 q1

a

   r r  1r  b     t − x − b − a  dt + t − x + b − a  dt f q     2 2 a x ( r+1     )1 f q a + b r+1 b − a r+1 r a+b −x = + x− +2 . 1 2 2 2 (b − a)(r + 1) r 1 = b−a



x

 Remark 2.2.1 If we substitute r = 1 (and q → ∞) in (2.2.4), then we get the following result. Corollary 2.2.3 Let f : [a, b] → R be an absolutely continuous function on I o for a, b ∈ I where a < b. If f ∈ L∞ [a, b], then for any x ∈ [a, b], this inequality holds      b   f (x) − f (b) − f (a) x − a + b − 1  f (t)dt   b−a 2 b−a a (  2 ) x − a+b 1 2 + (2.2.5) (b − a)f ∞ . ≤ 4 (b − a)2

2.2 Generalized Ostrowski Type Inequalities for Functions of Lp Spaces and. . .

121

Remark 2.2.2 The inequality (2.2.5) is the generalization of Ostrowski inequality, i.e., by replacing f (a) = f (b) in (2.2.5) we get (2.2.1) and also by choosing f ∞ = M we get (2.0.1). Remark 2.2.3 If we replace x = a+b 2 in (2.2.4), then we get the midpoint inequality (Hermite-Hadamard left bound) in the following corollary. Corollary 2.2.4 Let all the assumptions of Theorem 2.2.2 be valid. Then      1  b  1 b−a r  ≤ f a + b − 1 f (t)dt f q .  2 r +1  2 b−a a

(2.2.6)

Remark 2.2.4 If we replace x = a or x = b in (2.2.4), we get the trapezoidal inequality (Hermite-Hadamard right bound) in the following corollary. Corollary 2.2.5 Let all the assumptions of Theorem 2.2.2 be valid. Then    b  f (a) + f (b)  1  − f (t)dt   2 b−a a (   )1  a − b r+1 b − a r+1 r 1 +3 f q . (2.2.7) ≤ 1 2 2 r (b − a)(r + 1) 1. If r is odd, then    1  b  f (a) + f (b)  1 2(b − a) r 1  ≤ − f (t)dt f q .   2 2 b−a a r +1 (2.2.8) 2. If r is even, then    1  b  1 b−a r  f (a) + f (b) 1   − f (t)dt  ≤ f q .  2 b−a a 2 r +1 (2.2.9) a+b in (2.2.5), then we get midpoint inequality Remark 2.2.5 If we replace x = 2 (Hermite Hadamard left bound) in the following corollary. Corollary 2.2.6 Let all the assumptions of Corollary 2.2.3 be valid. Then      b   1 f a + b − 1  ≤ (b − a)f ∞ . f (t)dt   4 2 b−a a

(2.2.10)

122

2 Ostrowski Inequality

Remark 2.2.6 By replacing x = a or x = b in (2.2.5), we get trapezoidal inequality (Hermite Hadamard right bound) in the following corollary. Corollary 2.2.7 Let all the assumptions of Corollary 2.2.3 be valid. Then    b  f (a) + f (b)  1 1  − f (t)dt  ≤ (b − a)f ∞ ,  2 b−a a 2

(2.2.11)

1 is better one. 2 Remark 2.2.7 From the inequality (2.2.10) and (2.2.6), we retrieve the result of Corollary 5 and Corollary 8 of [7], respectively. where the constant

2.2.2 Ostrowski Type Inequality for Functions of Bounded Variation The main purpose of present subsection is to obtain some Ostrowski type inequalities for functions of bounded variation and to give some special cases of the obtained results. In start, we need a useful definition stated as under: Definition 2.2.1 Total variation of a continuous real-valued function f on [a, b] ⊆ R is b  a

(f ) =

sup

n

P −1

P ∈P [a,b] i=0

|f (xi+1 ) − f (xi )|,

where P = {x0 , · · · , xnP }, be a partition of [a, b] satisfying xi ≤ xi+1 for 0 ≤ i ≤ nP − 1 and supremum is taken over P[a, b] = {P |P is partition of [a, b]} of all partitions of [a, b]. Definition 2.2.2 Continue real-valued function ρ on R is of bounded variation on [a, b] ⊂ R if its total variation is finite, i. e., f ∈ BV [a, b] ⇐⇒

b 

(f ) < +∞.

a

Function of bounded variation is a real-valued function in which total variation is bounded. Functions of bounded variation of single variable were first presented by Camille Jordan in 1881 dealing with convergence of fourier series. For further details see [8]. First of all, we would like to recall some useful results related to bounded variation.

2.2 Generalized Ostrowski Type Inequalities for Functions of Lp Spaces and. . .

123

Lemma 2.2.2 ([7]) Let f : [a, b] → R be absolutely continuous function and g : [a, b] → R be function of bounded variation, the following inequality holds:    

b a

 b   f (t)dg(t) ≤ sup |f (t)| (g). t ∈[a,b]

(2.2.12)

a

Because of its numerous applications, the Ostrowski inequality is one of the fundamental results in mathematical inequalities. In 2001, Dragomir presented the result of the Ostrowski inequality for bounded variation [43] in the following theorem. Theorem 2.2.8 Let f : [a, b] → R be a function of bounded variation on [a, b] holds for x ∈ [a, b]. Then  ) b   (  b  x − a+b     1 1   2 f (x) − + f (t)dt  ≤ (f ).   b−a a 2  b−a  a The constant smaller one.

1 2

is the best possible in such a way that it cannot be replaced by the

In [46], Dragomir established another result for functions of bounded variation. Theorem 2.2.9 Let all the assumptions of Theorem 2.2.8 be valid. Then  ) b  (   b  x − 3a+b     f (x + f (a + b − x)) 1 1   4  − + f (t)dt  ≤ (f ).   2 b−a a 4  b−a  a The constant smaller one.

1 4

is the best possible in such a way that it cannot be replaced by the

In [37, 40, 44, 111, 155], many authors generalized and improved similar type of inequalities. Now we will discuss generalization of Ostrowski type inequality using the same technique. Theorem 2.2.10 Let all the assumptions of Theorem 2.2.8 be valid. Then      b   f (x) − f (b) − f (a) x − a + b − 1 f (t)dt   b−a 2 b−a a   b    a + b − 2x  1  ,1 (f ), ≤ max   2 b−a a where the constant

1 2

is sharp.

(2.2.13)

124

2 Ostrowski Inequality

Proof Using Lemma 2.2.1 with the result (2.2.12) for f (t) = P (x, t), and g(t) = f (t), t ∈ [a, b], we get    1  b − a

b a

  P1 (x, t)df (t)

   1 1  x P1 (x, t)df (t) + ≤  b−a a b−a ≤

1 sup |P1 (x, t)| b − a t ∈[a,x]

x 

   

b x

  P1 (x, t)df (t)

 1 sup |P1 (x, t)| (f ) b − a t ∈(x,b] x b

(f ) +

a

  b    x a + b a + b  b−a    1 1    max  − x , max  − x  = (f ) + (f ) b−a 2 2 b−a 2 a x : = M(x). We notice that )    ( x b  a + b  b−a 1   M(x) ≤ max  − x , (f ) + (f ) b−a 2 2 a x =

   b  a + b − 2x  1 ,1 max  (f ), 2 b−a  a

which proves the inequality (2.2.13). To show that the 12 constant is sharp in (2.2.13), let (2.2.13) is valid for constant C > 0, we get      b   f (x) − f (b) − f (a) x − a + b − 1  f (t)dt   b−a 2 b−a a   b    a + b − 2x  ,1  (f ), ≤ C max   b−a a for any x ∈ [a, b]. Consider the function f : [a, b] → {0, 1} defined as f (t) =

0, 1,

t ∈ (a, b); t ∈ {a, b}.

(2.2.14)

2.2 Generalized Ostrowski Type Inequalities for Functions of Lp Spaces and. . .

125

For x = a, we have 

b

f (t)dt = 0

a

and b 

(f ) = 2.

a

By using (2.2.14), we obtain, 1 ≤ 2C or 1 ≤C 2 and thus it is proved that the constant

1 2



is sharp.

in (2.2.13), then we get the midpoint Remark 2.2.8 If we replace x = inequality (Hermite-Hadamard left bound) in the following corollary. a+b 2

Corollary 2.2.11 Let all the assumptions of Theorem 2.2.10 be valid. Then      b b   1 f a + b − 1 ≤ f (t)dt (f ),   2 2 b−a a a

(2.2.15)

where 12 is the sharp constant. Inequality (2.2.15) is Corollary 2 of [7] by Alomari and Corollary 2.6 of [46] by of Dragomir. Remark 2.2.9 If we replace x = a or x = b in (2.2.13), then we get the trapezoidal inequality (Hermite-Hadamard right bound) in the following corollary. Corollary 2.2.12 Let all the assumptions of Theorem 2.2.10 be valid. Then    b b  1  f (a) + f (b) 1 ≤  − f (t)dt (f ),  2  2 b−a a a

(2.2.16)

where the constant 12 is sharp. Inequality (2.2.16) is Corollary 2 of [7] by Alomari and Corollary 2.4 of [46] by Dragomir. Now, we will discuss certain applications to numerical quadrature of our results given in this section.

126

2 Ostrowski Inequality

2.2.3 Applications to Numerical Integration Let In : a = ζ0 < ζ1 < . . . < ζn = b be a partition of the interval [a, b] and ζk = ζk+1 − ζk , k ∈ {0, 1, 2, · · · , n − 1}. Then 

b

f (t)dt = Qn (f, In ) + Rn (f, In ),

(2.2.17)

a

where Rn (f, In ) be the remainder and Qn (f, In ) is defined as Qn (f, In ) :=

n−1 

f (ξk ) −

k=0

  f (ζk+1 ) − f (ζk ) ζk+1 + ζk ξk − ζk . (2.2.18) ζk 2

Theorem 2.2.13 Let all the assumptions of Theorem 2.2.2 be valid. Equation (2.2.17) holds in which Qn (f, In ) is given in the form of (2.2.18) and Rn (f, In ) becomes  |Rn (f, In )| ≤

1 r +1

1 r

(  r+1    )1 n−1 

ζk r+1 r ζk + ζk+1 ζk + ζk+1 r+1 − ξk × + ξk − +2 f q , 2 2 2 k=0

(2.2.19) ∀ ξk ∈ [ζk , ζk+1 ]. Proof If inequality (2.2.4) is applied on [ζk , ζk+1 ], we get 

ζk+1

Rk (f, Ik ) =

ζk

   f (ζk+1 ) − f (ζk ) ζk + ζk+1 f (t)dt − f (ξk ) − ζk , ξk − ζk 2

we sum up Rk (f, Ik ) from 0 to n − 1 over k. This produces 

b

Rn (f, In ) =

f (t)dt− a

  n−1 

f (ζk+1 ) − f (ζk ) ζk + ζk+1 ζk . f (ξk ) − ξk − ζk 2 k=0

It follows from (2.2.4) that     n−1   b 

f (ζk+1 ) − f (ζk ) ζk + ζk+1   |Rn (f, In )| =  ζk  f (ξk ) − ξk − f (t)dt −  a  ζk 2 k=0

2.2 Generalized Ostrowski Type Inequalities for Functions of Lp Spaces and. . .

 ≤

1 r +1 

127

( 1

 r+1  n−1  r ζk + ζk+1 r+1 ζk + ζk+1 − ξk + ξk − 2 2

ζk +2 2

k=0

r+1 ) 1r

f q .

 Corollary 2.2.14 Let r = 1 (and q → ∞) in (2.2.19). Then (2.2.17) is valid in which Qn (In , f ) is given in the form of (2.2.18) and Rn (In , f ) becomes (   ) n−1

ζk + ζk+1 2 [ζk ]2 |Rn (f, In )| ≤ f ∞ , + ξk − 4 2

(2.2.20)

k=0

∀ ξk ∈ [ζk , ζk+1 ]. Theorem 2.2.15 Let all the assumptions of Theorem 2.2.10 be valid. Then (2.2.17) is valid in which Qn (f, In ) is given in the form of (2.2.18) and Rn (f, In ) becomes |Rn (f, In )| ≤

   b  ζk + ζk+1 − 2ξk  ,1 max  (f ),  2 ζk a

n−1

ζk k=0

(2.2.21)

∀ ξk ∈ [ζk , ζk+1 ]. Proof If inequality (2.2.13) is applied on [ζk , ζk+1 ], we get 

ζk+1

Rk (f, Ik ) =

ζk



  f (ζk+1 ) − f (ζk ) ζk + ζk+1 f (t)dt − f (ξk ) − ξk − ζk . ζk 2

We sum up Rk (f, Ik ) from 0 to n − 1 over k. This produces  Rn (f, In ) = a

b

  n−1 

f (ζk+1 ) − f (ζk ) ζk + ζk+1 ζk . f (ξk ) − ξk − f (t)dt − ζk 2 k=0

It follows from (2.2.13) that     n−1   b 

f (ζk+1 ) − f (ζk ) ζk + ζk+1   |Rn (f, In )| =  ζk  f (ξk ) + ξk − f (t)dt −  a  ζk 2 k=0

   n−1 b

 ζk + ζk+1 − 2ξk  ζk ,1 max  (f ). ≤  2 ζk a k=0



128

2 Ostrowski Inequality

If we choose, ξk =

ζk +ζk+1 2

in (2.2.18), then Qn (f, In ) could be defined as

Qn (f, In ) :=

 n−1  

ζk + ζk+1 ζk . f 2

(2.2.22)

k=0

Remark 2.2.10 If (2.2.17) is valid and Qn (f, In ) is given in the form of (2.2.22), then in the following result the remainder Rn (f, In ) from (2.2.6), (2.2.10) and (2.2.15) becomes respectively. Corollary 2.2.16 Let all the assumptions of Theorem 2.2.2 and Theorem 2.2.10 be valid. Then |Rn (f, In )| ≤

1 2



1 r +1

1

n−1 r

[ζk ]

r+1 r

f q ,

k=0

1

[ζk ]2 f ∞ 4 n−1

|Rn (f, In )| ≤

k=0

and  1

ζk (f ). 2 a n−1

|Rn (f, In )| ≤

b

k=0

If we choose, ξk = ζk or ξk = ζk+1 in (2.2.18), then Qn (f, In ) can be defined as Qn (f, In ) : =

 n−1 

f (ζk ) + f (ζk+1 ) k=0

2

ζk .

(2.2.23)

Remark 2.2.11 If (2.2.17) holds and Qn (f, In ) is given by formula (2.2.23), then in the following result the remainder Rn (f, In ) from (2.2.7), (2.2.8), (2.2.9), (2.2.11) and (2.2.16) becomes respectively. Corollary 2.2.17 Let all the assumptions of Theorems 2.2.2 and Theorem 2.2.10 be valid. Then (    )1 n−1 

ζk r+1 r −ζk r+1 |Rn (f, In )| ≤ +3 f q , 1 2 2 r (r + 1) k=0 1

|Rn (f, In )| ≤

1 2



2 r+1

1

n−1 r k=0

(ζk )

r+1 r

f q ,

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter

1 |Rn (f, In )| ≤ 2



1 r+1

1

n−1 r

(ζk )

r+1 r

129

f q ,

k=0

1

[ζk ]2 f ∞ 2 n−1

|Rn (f, In )| ≤

k=0

and  1

ζk (f ). 2 a n−1

|Rn (f, In )| ≤

b

k=0

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter This section introduces weighted Ostrowski type inequality for twice differentiable functions with bounded second derivatives and absolutely continuous first derivatives. It is worth mentioning that throughout this section α = a + b−a 2 and , where

∈ [0, 1]. β = b − b−a 2 In 1976, Milovanovi´c and Peˇcari´c proved a generalization of Ostrowski inequality for n−times differentiable functions from which only the case of twice differentiable functions [134, p. 470] is mentioned. Theorem 2.3.1 Let f : I → R be a twice differentiable function such that f

: (a, b) → R is bounded. Then      b 1   f (x) + (x − a)f (a) + (b − x)f (b) − 1 f (t)dt  2 b−a b−a a ( ) 2 (x − a+b 1 f

∞ 2 2 ) , (b − a) + ≤ 4 12 (b − a)2 ∀ x ∈ [a, b]. In year 1999, Cerone et al. in [36] presented the following inequality. Theorem 2.3.2 Under the assumptions of Theorem 2.3.1, the following inequality holds      b   f (x) − x − a + b f (x) − 1 f (t)dt   2 b−a a

130

2 Ostrowski Inequality

( ≤

  ) 1 a+b 2 (b − a)2 (b − a)2

f

||∞ ≤ + x− f ||∞ , 24 2 2 6

∀ x ∈ [a, b]. In the same year, Dragomir and Barnett in [47] stated the following result. Theorem 2.3.3 Under the assumptions of Theorem 2.3.1, the following inequality holds      b   f (x) − f (b) − f (a) x − a + b − 1  f (t)dt   b−a 2 b−a a ⎧⎡ ⎫ ⎤2 2 ⎪  ⎪ ⎬ (b − a)2 ⎨⎣ x − a+b 1 1 (b − a)2

2 ≤ f

||∞ ≤ f ∞ , + ⎦ + ⎪ 2 b−a 4 12 ⎪ 6 ⎩ ⎭ ∀ x ∈ [a, b]. Cerone et al. established Ostrowski type inequality for functions with bounded second derivatives in [36]. Dragomir and Barnett established a similar inequality in [47]. Dragomir and Sofo in [49] identified Ostrowski type inequality of the same kind in the sense of [36] or [47], as given in the following theorem. Theorem 2.3.4 Let f : [a, b] → R be an absolutely continuous function on [a, b] and f

∈ L∞ [a, b]. Then the following inequality holds    

a

b

      f (a) + f (b) (b − a) a+b 1

f (x) + (b − a) + x− f (x) f (t) − 2 2 2 2    3 a + b  (b − a)3 1  x− + , (2.3.1) ≤ f

∞   3 2 48

∀ x ∈ [a, b]. In 2000, Dragomir et al. established Montgomery identity with parameter [53], stated as follows Theorem 2.3.5 If f : [a, b] → R is differentiable on [a, b] with f integrable on [a, b], where ∈ [0, 1], then generalized integral identity holds (1 − )f (x) +

f (a) + f (b) 1 − 2 b−a



b a

f (t)dt =

1 b−a



b

K (x, t)f (t)dt,

a

(2.3.2)

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter

131

where K (x, t) is defined as

K (x, t) =

 ⎧  b−a ⎪ ⎪ t − a +

, t ∈ [a, x]; ⎪ ⎪ ⎨ 2   ⎪ ⎪ b−a ⎪ ⎪ ⎩t − b − , t ∈ (x, b], 2

(2.3.3)

∀ x ∈ [α, β] . Zafar and Mir established following general form of integral inequality in [193] with the help of (2.3.2), stated as: Theorem 2.3.6 Let all the assumptions of Theorem 2.3.4 be valid. Then the following inequality holds    1  b − a

   1 f (a) + f (b) f (t)dt − (1 − )f (x) + (1 + ) 2 2 a     a+b b−a  f (x) − f (b) − f (a)  −(1 − ) x − 2 4 (  )  0

0 1  1 a + b 3 (b − a)3 0 0 ≤ f ∞ ( ) , + x− (b − a) 3  2  48 b

(2.3.4)

∀ x ∈ [α, β], where ( ) = (1 − )[2(1 − )2 − 1] + 2 , ∈ [0, 1]. Peˇcari´c and Savi´c in [154] first time discussed the weighted version of Ostrowski inequality. Due to the importance of this inequality, in the last few decades, researchers are continuously in an effort for gaining sharp bounds of Ostrowski’s inequality in terms of weight. In the following subsection, we introduce weights on the Ostrowski type inequality (2.3.4) which was proved by Zafar and Mir in [193]. The obtained inequality is then applied to generate certain composite quadrature formulae. The results of current section can be seen in [74].

2.3.1 Weighted Ostrowski Type Inequality with Parameter We need the following lemma from [74] to prove our next main result.

132

2 Ostrowski Inequality

Lemma 2.3.1 Let f : I → R be absolutely continuous function. Further let p : [a, b] → [0, ∞) be a probability density function, where ∈ [0, 1]. Then we have the identity 

β

f (x)



b

p(t)dt =

α

a

 p(t)f (t)dt + f (a)

a



p(t)dt

α



b

+

b

p(t)dt − f (b) β

Kp, (x, t)f (t)dt,

a

where the kernel Kp, (x, t) : [a, b] × [a, b] → R is given by:

Kp, (x, t) =

⎧ t ⎪ ⎪ p(u)du, t ∈ [a, x]; ⎪ ⎪ ⎨ α (2.3.5)

 t ⎪ ⎪ ⎪ ⎪ p(u)du, t ∈ (x, b], ⎩ β

∀ x ∈ [α, β]. Proof Using kernel (2.3.5), after some computation, we obtain 

x



a

t

  w(u)du f (t)dt = f (x)

α

x



a

p(t)dt − f (a)

α



x

p(t)dt −

p(t)f (t)dt

α

a

(2.3.6) and 

b



x

t

  w(u)du f (t)dt = f (x)

β

β x



b

p(t)dt + f (b)



b

p(t)dt −

β

p(t)f (t)dt. x

(2.3.7) By adding (2.3.6) and (2.3.7), we get the required identity (2.3.5).



Theorem 2.3.7 Let f : I → R be a function whose first derivative is absolutely continuous on [a, b] and f

∈ L∞ [a, b]. Also let p : [a, b] → [0, ∞) be a probability density function, where ∈ [0, 1]. Then    

  β 1 b−a p(t)dt + f (x) (f (a)p(a) + f (b)p(b)) 2 2 a α    a  β  b a+b f (x) p(t)dt − f (a) p(t)dt − p(t)dt x − +f (b) 2 β α α b

f (t)p(t)dt −

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter

133

     a  b  a+b b−a

p (t)dt − f (a) − f (t) t − p(t)dt +f (b) p(t)dt  2 2 a α β  x    x 2  x (a + b)x − p(u)du + p(u)du ≤ f

∞  4 4 α β   a+b    2 1  (a + b)t − t 2 p(t)dt  +  2 x   a+b   α  b 2 (a + b)2 ab p(u)du p(u)du + p(u)du − + 8 4 a β α     β  b (a + b)t (a + b)t t2 t2 (2.3.8) −2 − p(t)dt + − p(t)dt , 4 4 4 4 α a 

b

∀ x ∈ [α, β]. Proof We have the following identity from Lemma 2.3.1, 

β

 p(t)dtg(x) =

α

a



b

a

p(t)g(t)dt +



+



α b

b

p(t)dtg(a) −

p(t)dtg(b) β

Kp, (x, t)g (t)dt.

a

Let us consider   a+b f (x). g(x) = x − 2 Equation (2.3.5) implies      b a+b a+b b−a f (x) = f (t)dt − p(t)dt x − p(t) t − 2 2 2 α a   a   b     b a+b f

(t) dt. p(t)dtf (a) + p(t)dtf (b) + Kp, (x, t) f (t) + t − 2 α β a



β

(2.3.9) Integrating by parts, we have  a

b

   b b−a a+b

f (t)p(t)dt p(t)f (t)dt = t− [f (a)p(a) + f (b)p(b)] − 2 2 a    b a+b f (t) t − (2.3.10) − p (t)dt, 2 a

134

2 Ostrowski Inequality

also 

b

Kp, (x, t)f (t)dt =



a

b



a

p(t)dtf (b) −

β



+f (x) α

p(t)dtf (a) α

β



b

p(t)dt −

f (t)p(t)dt. (2.3.11) a

Now using equations (2.3.10) and (2.3.11) in (2.3.9), we get 

β α

  a+b p(t)dt x − f (x) 2

 a  b b−a p(t)dtf (b) − p(t)dtf (a) (f (a)p(a) + f (b)p(b)) + 2 β α   a  b b−a

− p(t)dtf (a) + p(t)dtf (b) 2 α β    β  b  b a+b + p(t)dtf (x) − 2 f (t)p(t)dt − f (t) t − p (t)dt 2 α a a    b a+b

Kp, (x, t) t − + f (t)dt, 2 a

=

or we can write 

b

p(t)f (t)dt =

a

 b−a 1 b p(t)dtf (b) (f (a)p(a) + f (b)p(b)) + 4 2 β   a   b 1 a b−a − p(t)dtf (a) − p(t)dtf (a) + p(t)dtf (b) 2 α 4 α β      β 1 a+b + p(t)dt f (x) − x − f (x) 2 α 2    a+b 1 b f (t) t − − p (t)dt 2 a 2    a+b 1 b Kp, (x, t) t − f

(t)dt, + 2 a 2

∀ x ∈ [α, β], this gives us    



b−a (f (a)p(a) + f (b)p(b)) 2 a α    a  β  b a+b p(t)dtf (b) − p(t)dtf (a) − p(t)dt x − f (x) + 2 β α α b

f (t)p(t)dt −

1 2

β

p(t)dtf (x) +

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter

135

   a  b  a+b b−a

− f (t) t − p(t)dtf (a) + p(t)dtf (b)  p (t)dt − 2 2 a α β   b    1  a+b =  Kp, (x, t) t − f

(t)dt  2 a 2     1 b  a + b  

 ≤ (2.3.12) Kp, (x, t) t − f (t) dt. 2 a 2  



b

It can be easily seen that 

b a

     b        Kp, (x, t) t − a + b  f

(t) dt ≤ f

∞ Kp, (x, t) t − a + b  dt,    2 2  a

(2.3.13) where f

∞ = sup |f

(t)| < ∞. t ∈(a,b)

Also, 

    Kp, (x, t) t − a + b  dt  2 

b

I= a

or 

x

I= a

   t    b  t        p(u)du t − a + b  dt +  p(u)du t − a + b  dt.       2 2  α x β

(2.3.14) Now, we have two cases:   a+b , we obtain (a) For x ∈ a, 2     x  t a+b a+b p(u)du − t dt + − t dt 2 2 a t α α      β  β  a+b  β 2 a+b a+b p(u)du p(u)du t− − t dt + dt + a+b 2 2 x t t 2    b  t a+b + p(u)du t− dt. 2 β β 

I=

α





α

p(u)du

136

2 Ostrowski Inequality

We got after some computations   β (a + b)x x2 (a + b)2 − +2 p(u)du a+b 2 2 8 α x 2    α  b  α (a + b)t ab t2 p(t)dt p(u)du + p(u)du + − − 2 2 2 a β a 

I =−



b

+



β

x



p(u)du −

(a + b)t t2 − 2 2



β

p(u)du





a+b 2

p(t)dt + x



(a + b)t t2 − 2 2

 p(t)dt

   a+b  2 (a + b)t (a + b)t t2 t2 − p(t)dt + − p(t)dt 2 2 2 2 α β  2   x  β  β x (a + b)2 (a + b)x p(u)du − p(u)du p(u)du − +2 =− a+b 2 2 8 α x 2 

−

x



  a+b  2 (a + b)t ab t2 p(t)dt +2 − 2 2 2 a β x    b  β (a + b)t (a + b)t t2 t2 p(t)dt + p(t)dt −2 − − 2 2 2 2 α a (  2   β  a+b  x 2 x (a + b)t (a + b)x =− p(u)du − p(u)du − −2 2 2 2 α x x     α  β  b (a + b)2 ab t2 p(t)dt + 2 p(u)du p(u)du + p(u)du − − a+b 2 8 2 a β 2     β  b (a + b)t (a + b)t t2 t2 (2.3.15) −2 − p(t)dt + − p(t)dt. 2 2 2 2 α a 

−

α



p(u)du +



b

p(u)du

(b) Similarly,     a+b  t 2 a+b a+b p(u)du − t dt + − t dt 2 2 α a t α      β  β  x  t a+b a+b p(u)du t− p(u)du t− dt + dt + a+b 2 2 α x t 2    b  t a+b p(u)du t− + dt. 2 β β 

I=

α





α

p(u)du

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter

137

After some simplification, we get 



  a+b 2 x2 (a + b)2 (a + b)x I= p(u)du − p(u)du p(u)du − +2 2 2 8 α α x    α  b  α ab (a + b)t t2 p(u)du + p(u)du + − p(t)dt − 2 2 2 a β a    a+b   b 2 (a + b)t (a + b)t t2 t2 p(t)dt − p(t)dt − − + 2 2 2 2 x β    β  a+b  2 (a + b)t (a + b)t t2 t2 − p(t)dt − − p(t)dt − 2 2 2 2 x α  2   x  β  α x (a + b)2 (a + b)x p(u)du − p(u)du p(u)du − +2 =− a+b 2 2 8 α x 2 



x

β



  a+b  2 ab (a + b)t t2 −2 − p(t)dt 2 2 2 x a β    β  b (a + b)t (a + b)t t2 t2 −2 − p(t)dt + − p(t)dt 2 2 2 2 α a (  2   β  a+b  x 2 x (a + b)t (a + b)x p(u)du − p(u)du − −2 = 2 2 2 x α x     α  b  α ab (a + b)2 t2 p(u)du p(u)du + p(u)du p(t)dt + 2 − − a+b 2 8 2 a β 2     β  b (a + b)t (a + b)t t2 t2 (2.3.16) −2 − p(t)dt + − p(t)dt, 2 2 2 2 α a −

∀x ∈

α



p(u)du +

 a+b 2

b

p(u)du

 ,b .

Using equations (2.3.13), (2.3.14), (2.3.15) and (2.3.16) we obtain    



b−a (f (a)p(a) + f (b)p(b)) 2 a α    a  β  b a+b p(t)dtf (b) − p(t)dtf (a) − p(t)dt x − f (x) + 2 β α α     a  b  b  a+b b−a

− f (t) t − p(t)dtf (a) + p(t)dtf (b)  p (t)dt − 2 2 a α β 0

0   2   β x 0f 0 x (a + b)x ∞ p(u)du − p(u)du ≤ − 2 2 2 α x b

f (t)p(t)dt −

1 2

β

p(t)dtf (x) +

138

2 Ostrowski Inequality



a+b 2

−2



x

(a + b)t t2 − 2 2

)

 p(t)dt

  α  b ab (a + b)2 p(u)du + p(u)du − 8 2 α a β     β  b (a + b)t (a + b)t t2 t2 −2 − p(t)dt + − p(t)dt 2 2 2 2 α a    x  β 0 0 (a + b)x x2 p(u)du − p(u)du − = 0f

0∞ 4 4 α x )   a+b  2 (a + b)t t2 + − p(t)dt 2 2 x 

+2

a+b 2

p(u)du

  α  b ab (a + b)2 p(u)du + p(u)du − 8 4 a β α    β  b 2 2 (a + b)t (a + b)t t t −2 − p(t)dt + − p(t)dt. 4 4 4 4 α a 

+

a+b 2

p(u)du

 1 in (2.3.8), then we will get (2.3.4). b−a Remark 2.3.1 If we substitute = 0, then α = a and β = b in (2.3.8), we get the following result.

Special Case 1 If we simply put p(t) ≡

Corollary 2.3.8 Let all the assumptions of Theorem 2.3.7 be valid. Then    

 1 b−a f (t)p(t)dt − f (x) + [f (a)p(a) + f (b)p(b)] 2 2 a       b  a+b a+b

− f (t) t − p (t)dt − x − f (x)  2 2 a   x   x 0

0  x2 (a + b)x 0 0  ≤ f ∞  − p(u)du + p(u)du 4 4 a b    a+b   2 t2 (a + b)t  − p(t)dt  +  2 2 x   a+b    b 2 t2 (a + b)t (a + b)2 − − p(t)dt , p(u)du + 8 4 4 a a b

∀ x ∈ [α, β].

(2.3.17)

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter

139

1 in (2.3.8), then we will get (2.3.1). b−a a+b Corollary 2.3.9 If we examine the estimates for the the midpoint x = and 2 end points x = a, x = b in (2.3.8), the midpoint provides us the best estimate

Special Case 2 If we simply put p(t) ≡

   

f (t)p(t)dt −

1 2





 a+b b−a + (f (a)p(a) + f (b)p(b)) 2 2 a α    b  a  b a+b + p(t)dtf (b) − p(t)dtf (a) − f (t) t − p (t)dt 2 β α a  a   b  b−a − p(t)dtf (a) + p(t)dtf (b)  2 α β  ( a+b   a+b  2 2 (a + b)2  

≤ f ∞  p(u)du + p(u)du    α 16 β   a+b  α   b 2 (a + b)2 ab − p(u)du p(u)du + p(u)du + 8 4 a β α     β  b t2 t2 (a + b)t (a + b)t − p(t)dt + − p(t)dt . −2 4 4 4 4 α a b

β

p(t)dtf

Remark 2.3.2 If we substitute = 0, then α = a and β = b in (2.3.8), we get the following result. Corollary 2.3.10 Let all the assumptions of Theorem 2.3.7 be valid. Then    

  b  1 a+b b−a f (t)p(t)dt − p(t)dtf + (f (a)p(a) 2 2 2 a a     b  a+b

f (t) t − p (t)dt  +f (b)p(b)) − 2 a ( a+b    a+b  2 2 (a + b)2  

≤ f ∞  p(u)du + p(u)du   a  16 b b



a+b 2

+ a

 p(u)du

(a + b)2 − 8



b a



t2 (a + b)t − 4 4

Remark 2.3.3 If we substitute = 1, then α = β = following result.



 p(t)dt . (2.3.18)

a+b in (2.3.8), we get the 2

140

2 Ostrowski Inequality

Corollary 2.3.11 Let all the assumptions of Theorem 2.3.7 be valid. Then    

 1 b−a (f (a)p(a) + f (b)p(b)) 2 2 a    b  a  b a+b + p(t)dtf (b) − p(t)dtf (a) − f (t) t − p (t)dt a+b a+b 2 a 2 2  )   a b b−a  − p(t)dtf (a) + p(t)dtf (b)  a+b a+b  2 2 2     b t2 ab (a + b)t ≤ f

∞ − + − p(t)dt . (2.3.19) 4 4 4 a b

f (t)p(t)dt −

17a + 3b 3a + 17b 3 , then α = and β = in Remark 2.3.4 If we substitute = 10 20 20 (2.3.8), we get the following result. Corollary 2.3.12 Let all the assumptions of Theorem 2.3.7 be valid. Then  ( 3a+17b  b 20 b−a 1 a+b  )+ f (t)p(t)dt − p(t)dtf ( (f (a)p(a) f (b)p(b))  17a+3b  a 2 2 2 20    a  b  b a+b p (t)dt p(t)dtf (b) − p(t)dtf (a) − f (t) t − + 3a+17b 17a+3b 2 a 20 20  )  b  a b−a  − p(t)dtf (a) + p(t)dtf (b)  17a+3b 3a+17b  2 20 20   ( a+b  a+b  2  2 2 (a + b)   p(u)du + p(u)du ≤ f

∞   3a+17b  17a+3b  16 20 20  17a+3b    a+b  b 2 20 (a + b)2 ab − p(u)du p(u)du + p(u)du + 3a+17b 17a+3b 8 4 a 20 20 )    3a+17b   b 20 t2 t2 (a + b)t (a + b)t − − −2 p(t)dt + p(t)dt . 17a+3b 4 4 4 4 a 20 (2.3.20) In the next subsection, we will discuss some applications of Composite Quadrature rules.

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter

141

2.3.2 Applications to Numerical Integration In order to get composite quadrature rule estimates with small errors, we can use inequality (2.3.8) to get better results. Let In : a = ζ0 < ζ1 < . . . < ζn−1 < ζn = b be a partition of interval ζk , [a, b], ζk = ζk+1 − ζk , ∈ [0, 1], αk ≤ ξk ≤ βk , where αk = ζk + h 2 ζk βk = ζk+1 − h , k ∈ {0, . . . , n − 1}. Then 2 

b

f (t)p(t)dt = Qn (f, p, In ) + Rn (f, p, In ),

(2.3.21)

a

where Qn (f, p, In ) n−1  βk 1

ζk = p(t)dtf (ξk ) + (f (ζk )p(ζk ) + f (ζk+1 )p(ζk+1 )) 2 2 αk k=0



ζk+1

+



ζk

p(t)dtf (ζk+1 ) −

βk

p(t)dtf (ζk ) αk

     ζk+1 ζk + ζk+1 ζk + ζk+1 f (ξk ) − p (t)dt p(t)dt ξk − f (t) t − 2 2 αk ζk  ζk   ζk+1 ζk p(t)dtf (ζk ) + p(t)dtf (ζk+1 ) ζk , (2.3.22) − 2 αk βk 

−

βk

∀ ξk ∈ [αk , βk ]. Theorem 2.3.13 Let all the assumptions of Theorem 2.3.7 be valid. Equation (2.3.21) holds and Qn (f, p, In ) is given in the form of (2.3.22). Then the remainder becomes |Rn (f, p, In )|   ≤ f

∞  +

1 2



ξk

 −

ζk +ζk+1 2

αk

ζk



 (ζk + ζk+1 )ξk ξk 2 − 4 4 αk βk  ⎛ ⎞   ζk +ζk+1    2 (ζk + ζk+1 )2 2 p(u)du⎠ (ζk + ζk+1 )t − t p(t)dt  + ⎝ 8 αk  ξk



p(u)du +

 p(u)du +

ξk



ζk+1

p(u)du βk

p(u)du

ζk ζk+1 −2 4



βk αk



(ζk + ζk+1 )t t2 − 4 4

 p(t)dt

142

2 Ostrowski Inequality

 +

ζk+1

ζk



(ζk + ζk+1 )t t2 − 4 4





(2.3.23)

p(t)dt ,

∀ ξk ∈ [αk , βk ].

  ζk ζk , ζk+1 − h Proof Applying inequality (2.3.8) on ξk ∈ [αk , βk ]= ζk + h 2 2 and sum up from 0 to n − 1 over k, also applying triangular inequality, we achieve (2.3.22) and (2.3.23).  Special Case 1 If we fixed constant weight in (2.3.22) and (2.3.23) then we will get Theorem 2 of [193]. Corollary 2.3.14 If we substitute = 0, then αk = ζk and βk = ζk+1 in (2.3.22) and (2.3.23), we get Qn (f, p, In ) n−1  ζk+1 1

ζk = p(t)dtf (ξk ) + (f (ζk )p(ζk ) + f (ζk+1 )p(ζk+1 )) 2 2 ζk k=0

  ζk + ζk+1 f (ξk ) p(t)dt ξk − 2 ζk     ζk+1 ζk + ζk+1

p (t)dt ζk f (t) t − − 2 ζk 

−

ζk+1

(2.3.24)

and |Rn (f, p, In )|   ≤ f

∞ 

ξk

 p(u)du +

ζk

1 + 2



ζk +ζk+1 2



⎛  +⎝

ζk +ζk+1 2

ζk

p(u)du ζk+1

(ζk + ζk+1 )t − t 2

ξk



ξk



   p(t)dt  

⎞ p(u)du⎠

(ζk + ζk+1 )2 − 8



ζk+1

ζk

ξk 2 (ζk + ζk+1 )ξk − 4 4



t2 (ζk + ζk+1 )t − 4 4





 p(t)dt . (2.3.25)

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter

143

ζk + ζk+1 in (2.3.22) and (2.3.23) for k ∈ {0, . . . , n − Corollary 2.3.15 For ξk = 2 1}, then Qn (f, p, In ) can be defined as Qn (f, p, In ) n−1  βk 1

ζk + ζk+1 ζk = p(t)dtf ( )+ (f (ζk )p(ζk ) + f (ζk+1 )p(ζk+1 )) 2 2 2 αk k=0









ζk + ζk+1 + p(t)dtf (ζk+1 ) − p(t)dtf (ζk ) − f (t) t − 2 βk αk ζk   ζk  ζk+1 ζk − p(t)dtf (ζk ) + p(t)dtf (ζk+1 ) ζk 2 αk βk ζk+1

ζk

ζk+1



p (t)dt

(2.3.26)

and |Rn (f, p, In )|  ⎞ ⎡⎛ ζ +ζ    k k+1   ζk +ζk+1 2   2 2 + ζ ) (ζ k k+1  p(u)du + p(u)du⎠ ≤ f

∞ ⎣⎝  16 βk   αk ⎞ ⎛ ζ +ζ  αk   ζk+1  k k+1 2 (ζk + ζk+1 )2 ζk ζk+1 ⎠ ⎝ − p(u)du p(u)du + p(u)du + 8 4 αk ζk βk  −2

βk

αk



t2 (ζk + ζk+1 )t − 4 4



 p(t)dt +

ζk+1 ζk



t2 (ζk + ζk+1 )t − 4 4



 p(t)dt . (2.3.27)

Corollary 2.3.16 If we substitute = 0, then αk = ζk and βk = b in (2.3.26) and (2.3.27), k ∈ {0, . . . , n − 1}, we get Qn (f, p, In )   n−1  ζk+1 1

ζk + ζk+1 ζk = p(t)dtf + (f (ζk )p(ζk ) 2 2 2 ζk k=0

+f (ζk+1 )p(ζk+1 )) −



ζk+1 ζk

   ζk + ζk+1 p (t)dt ζk f (t) t − 2 (2.3.28)

144

2 Ostrowski Inequality

and |Rn (f, p, In )|  ⎡⎛ ζ +ζ ⎞   k k+1    ζk +ζk+1 2   2 2 + ζ ) (ζ

k k+1  p(u)du + p(u)du⎠ ≤ g ∞ ⎣⎝  16 ζk+1  ζk  ⎞ ⎤ ⎛ ζ +ζ  ζk+1   k k+1 2 2 2 t + ζ ) + ζ )t (ζ (ζ k k+1 k k+1 − − p(u)du⎠ p(t)dt ⎦ . +⎝ 8 4 4 ζk ζk (2.3.29) Corollary 2.3.17 If we substitute = 1, then αk =βk = (2.3.27), k ∈ {0, . . . , n − 1}, we get

ζk + ζk+1 in (2.3.26) and 2

Qn (f, p, In ) (  ζk+1 n−1 1 ζk = (f (ζk )p(ζk ) + f (ζk+1 )p(b)) + ζ +ζ p(t)dtf (ζk+1 ) k k+1 2 2 2 k=0

  ζk + ζk+1 − ζ +ζ p(t)dtf (ζk ) − p (t)dt f (t) t − k k+1 2 ζ k 2 )   ζk+1  ζk ζk 

− p(t)dtf (ζ ) + p(t)dtf (ζ )  ζk k k+1 ζk +ζk+1 ζk +ζk+1  2 



ζk

ζk+1

2

(2.3.30)

2

and |Rn (f, p, In )|   ≤ f

∞ −

ζk+1

p(u)du ζk

ζk ζk+1 + 4



ζk+1



ζk

t2 (ζk + ζk+1 )t − 4 4



 p(t)dt . (2.3.31)

3 17ζk + 3ζk+1 Corollary 2.3.18 If we substitute = , then αk = and βk = 10 20 3ζk + 17ζk+1 in (2.3.26) and (2.3.27) , k ∈ {0, . . . , n − 1}, we get the inequality 20 Qn (f, p, In ) ⎡ n−1  3ζk +17ζk+1 20 1 ⎣ ζk + ζk+1 ) = p(t)dtf ( 17ζ +3ζ k k+1 2 2 20 k=0

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter

145

 ζk+1 ζk (f (ζk )p(ζk ) + f (ζk+1 )p(ζk+1 )) + 3ζ +17ζ p(t)dtf (ζk+1 ) k k+1 2 20    ζk+1  ζk ζk + ζk+1 p (t)dt f (t) t − − 17ζ +3ζ p(t)dtf (ζk ) − k k+1 2 ζk 20  )  ζk+1  ζk ζk 

− p(t)dtf (ζ ) + p(t)dtf (ζ )  ζk k k+1 17ζk +3ζk+1 3ζk +17ζk+1  2 +

20

20

(2.3.32) and |Rn (f, p, In )|  ⎞ ⎡⎛ ζ +ζ   k k+1    ζk +ζk+1 2  2 2 (ζk + ζk+1 )  ≤ f

∞ ⎣⎝ 17ζ +3ζ p(u)du + 3ζ +17ζ p(u)du⎠  k k k+1 k+1 16   20 20 ⎛ ζ +ζ ⎛ 17ζ +3ζ ⎞  k k+1  k k+1 2 20 (ζk + ζk+1 )2 ⎝ + ⎝ 17ζ +3ζ p(u)du⎠ − p(u)du k k+1 8 ζk 20 

  3ζk +17ζk+1  20 t2 ζk ζk+1 (ζk + ζk+1 )t − 2 17ζ +3ζ − + 3ζ +17ζ p(u)du p(t)dt k k+1 k k+1 4 4 4 20 20    ζk+1  t2 (ζk + ζk+1 )t − + p(t)dt . (2.3.33) 4 4 ζk 

ζk+1

The next section deals with another important inequality that is OstrowskiGrüss. We will discuss its generalization in terms of weight. We will also provide applications related to probability density function and numerical quadrature rules.

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter This section is intended to generalize weighted Ostrowski-Grüss type inequality for differentiable functions and to deduce explicit error bounds for numerical quadrature formulae using weighted Peano kernel and korkine’s identity. Its applications to the probability density function and numerical quadrature rules are also discussed. The current section’s are from [55] and [78].

146

2 Ostrowski Inequality

Here we need some results. Grüss Inequality Another classical inequality was introduced by Grüss, which depicts a relation between integral of product of two functions and product of integral of two functions which is known as Grüss inequality [65], in fact this inequality establishes a ˇ bounds on Cebyšev functional (see [135, p.297]). This type of inequality has great importance due to their applications in numerical integration, probability theory and integral operator theory etc. Grüss inequality is stated as: Theorem 2.4.1 Let f, g ∈ L[a, b] where γ1 ≤ f (x) ≤ 1 and γ2 ≤ g(x) ≤ 2 ∀ x ∈ [a, b], where γ1 , 1 , γ2 , 2 are real constants. Then |T (f, g)| ≤

1 ( 1 − γ1 )( 2 − γ2 ), 4

(2.4.1)

1 ˇ where Cebyšev functional T (f, g) is defined as in (1.1.57). Here the constant is 4 the best possible. Weighted Grüss Inequality Dragomir established weighted Grüss inequality in [42] which may be stated as Theorem 2.4.2 Let all assumptions of Theorem 2.4.1 be valid. Then |Tp (f, g)| ≤

1 ( 1 − γ1 )( 2 − γ2 ), 4

(2.4.2)

where weighted functional is defined as 



b

Tp (f, g) =

b

p(x)dx a

 f (x)g(x)dx −

a

 

b

f (x)p(x)dx a



b

g(x)p(x)dx a

(2.4.3) 1 and p : [a, b] → [0, ∞) be a probability density function. Here the constant is 4 the best possible. Remark 2.4.1 If we pur f = g in above theorem, then we get another form of weighted Grüss inequality which is stated in [18]. 

b

0≤ a



2

b

p(x)f (x)dx − 2

p(x)f (x)dx a

Interested readers can see [135] for other related results.

≤

1 ( 1 − γ1 )2 . 4

(2.4.4)

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter

147

Ostrowski-Grüss Inequality In 1997, following Ostrowski-Grüss inequality was presented by Dragomir and Wang in [50], that is a linkage between Ostrowski and Grüss inequality. Theorem 2.4.3 Let f : I → R be a differentiable function in I 0 , a, b ∈ I 0 and a < b. If for real constants φ, ; φ ≤ f (x) ≤ , ∀ x ∈ [a, b], then     b  f (b) − f (a) a + b  1 f (x) − 1 f (t)dt − x−  ≤ 4 (b − a)( − φ).  b−a a b−a 2 (2.4.5) In 2000, Ostrowski-Grüss inequality was improved by Mati´c et al. in [117]. Theorem 2.4.4 Let f : I → R be a differentiable function in I 0 and let a, b ∈ I 0 with a < b. If φ ≤ f (x) ≤ , ∀ x ∈ [a, b] for some constants φ,  ∈ R. Then     b  f (b) − f (a) a + b  1 f (x) − 1 f (t)dt − x−   ≤ 4√3 ( − φ)(b − a). b−a a b−a 2

(2.4.6) In the same year in [19], Barnett et al. further improved that inequality (2.4.6), which may be stated as: Theorem 2.4.5 Let f : I → R be an absolutely continuous function such that f ∈ L2 [a, b], if φ ≤ f (x) ≤ , ∀ x ∈ [a, b] for some constants φ,  ∈ R, then     b  a + b  f (b) − f (a) f (x) − 1 x − f (t)dt −   b−a a b−a 2 ( 2 ) 12  (b − a) 1 f (b) − f (a) ≤ √ f 22 − b−a b−a 2 3 1 ≤ √ ( − φ)(b − a). 4 3

(2.4.7)

In 2010, Zafar and Mir in [194] using Montgomery identity with parameter (2.3.2) and generalized the inequality (2.4.7) in the following theorem. Theorem 2.4.6 Let all assumptions of Theorem 2.4.5 be valid. Then       b   f (a) + f (b) 1  (1 − ) f (x) − f (b) − f (a) x − a + b + − f (t)dt   b−a 2 2 b−a a (  )1  a+b 2 2 (b − a)2 2 (3 − 3 + 1) + (1 − ) x − ≤ 12 2

148

2 Ostrowski Inequality

(

1 × f 22 − b−a



f (b) − f (a) b−a

2 ) 12

(  )1  (b − a)2 1 a+b 2 2 2 (3 − 3 + 1) + (1 − ) x − ≤ ( − φ) , 2 12 2

  b−a b−a ∀x ∈ a+ ,b − , 2 2

(2.4.8)

∈ [0, 1].

where

Here we need a lemma, by using the following weighted Korkine’s identity (2.4.9) from [18] and weighted Grüss inequality (2.4.4), in order to prove our results of this section. Lemma 2.4.1 Let p, f, g : [a, b] → R be the measurable function for which the integrals involved in the following identity exist and finite. Then 



b

b

p(t)dt a



a

=

1 2



b

b

b

p(t)f (t)dt

p(t)g(t)dt

a



a



b

p(t)f (t)g(t)dt −

a

p(t)p(s) (f (t) − f (s)) (g(t) − g(s)) dtds.

(2.4.9)

a

In the next subsection, we generalize the inequality (2.4.7) and (2.4.8) for differentiable functions in terms of weight and parameter. The generalization of Ostrowski-Grüss inequality has established by introducing weighted Peano kernel. The parameter and weight could be adjusted to retrieve many established results. b−a b−a and β = b − , where ∈ [0, 1]. Throughout this section α = a + 2 2

2.4.1 Weighted Ostrowski-Grüss Type Inequality with Parameter by Using Korkine’s Identity Theorem 2.4.7 Let all the assumptions of Theorem 2.4.5 be valid. Further let p : [a, b] → (0, ∞) be probability density function. Then we get the following inequality    f (x)    − x

β



α

α

β

α

p(u)du + f (a)

 p(u)du + f (b)

a

 p(u)du + a a

α

b



β

 p(u)du + a β

b

b

p(u)du −

p(t)f (t)dt a



b

p(u)du −

 

b

p(t)sdt a

a

  p(t)f (t)dt 

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter

(

b

≤

2 (x, t)dt Kp,

p(t)

a

(

b



Kp, (x, t)dt a



b

2

p(t)[f (t)] dt −

2 ) 12

p(t)f (t)dt

a

≤

2 ) 12

b

−

149

a

1 ( − φ)Hp, (x, t), 2

(2.4.10)

where 

b

Hp, (x, t) =



2 (x, t)dt Kp,

−

p(t)

a

2

b

Kp, (x, t)dt

,

a

∀ x ∈ [α, β], where ∈ [0, 1] and Kp, (x, t) is defined as in (2.3.5). Proof From (2.4.9), we get the Korkine’s identity in the form of 

b a

Kp, (x, t)f (t)dt −



b

=





b

p(t)p(s) a

a





b

b

Kp, (x, t)dt a

p(t)f (t)dt

a

 Kp, (x, t) Kp, (x, s)  − f (t) − f (s) dtds. p(t) p(s) (2.4.11)

From [74], we have 

b



β

Kp, (x, t)f (t)dt = f (x)

a

 p(u)du + f (a)

α



+f (b)

b

p(u)du a



b

p(u)du −

β

α

p(t)f (t)dt

(2.4.12)

a

and by using simple computation, we have 

b a



β

Kp, (x, t)dt = x α

 p(u)du + a a

α



b

p(u)du + b β



b

p(u)du −

p(t)tdt. a

(2.4.13)

150

2 Ostrowski Inequality

By putting (2.4.12) and (2.4.13) in (2.4.11), we get 



β

α β



=





b

p(t)p(s) a

α

p(u)du + a

α b



a

b

p(u)du + f (b)

a

  − x 

α

p(u)du + f (a)

f (x)



β



b

p(u)du + b

a

p(t)f (t)dt a



 

b

p(u)du −

β

b

p(u)du −

a



b

p(t)tdt

p(t)f (t)dt



a

Kp, (x, t) Kp, (x, s)  − f (t) − f (s) dtds, p(t) p(s)

(2.4.14)

∀ x ∈ [α, β]. Applying Cauchy-Schwartz inequality for double integrals, we get   b b    1  Kp, (x, t) Kp, (x, s) 

 − f (t) − f (s) dtds  p(t)p(s) 2 p(t) p(s) a a    1   2 Kp, (x, t) Kp, (x, s) 2 1 b b ≤ − p(t)p(s) dtds 2 a a p(t) p(s)   b b  12  2 1

× p(t)p(s) f (t) − f (s) dtds . 2 a a

(2.4.15)

By using (2.4.11), we get the following identities 1 2



b b



Kp, (x, t) Kp, (x, s) − p(t) p(s)  2 2 (x, t)dt b Kp, − Kp, (x, t)dt p(t) a

2

p(t)p(s)

a

a



b

= a

dtdt

(2.4.16)

and 1 2

 a

b



b

 2 p(t)p(s) f (t) − f (s) dtds =

a



b

p(t)[f (t)]2 dt −

a



b

p(t)f (t)dt

2 .

a

(2.4.17) Using weighted Grüss inequality (2.4.4), if φ ≤ f (t) ≤  and t ∈ (a, b), we get 

b

0≤ a

  2 p(t) f (t) dt −



b a

p(t)f (t)dt

2

≤

1 ( − φ)2 . 4

(2.4.18)

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter

151

Using (2.4.14) − (2.4.18), we get    f (x)    − x

β



α β

b

≤



b

α

p(u)du + a

2 (x, t)dt Kp,

p(t)



b



b

p(u)du −

p(t)f (t)dt a

 p(u)du −

β

 

b

b

p(t)tdt a

a

  p(t)f (t)dt 

2 ) 12

b

−

Kp, (x, t)dt a

p(t)[f (t)] dt − 2

b

2 ) 12

p(t)f (t)dt

a

a

1 ≤ ( − φ) 2 =

 p(u)du + b



b

β

a

a

(

 p(u)du + f (b)

a

α

(

α

p(u)du + f (a)

(

b

a

2 (x, t)dt Kp,

p(t)



2 )

b

−

Kp, (x, t)dt a

1 ( − φ)Hp, (x, t), 2



which proves our result (2.4.10). We can obtain some special results of (2.4.10) as follows: Remark 2.4.2 If we put p(t) ≡ (2.4.8).

1 in (2.4.10), then we get the special result b−a

Remark 2.4.3 If we substitute = 0 in (2.4.10), then α = a and β = b, then we get following result. (see also Theorem 1 of [78]) Corollary 2.4.8 Let all the assumptions of Theorem 2.4.7 be valid. Then the following inequality holds     f (x) − x −  (

b

≤ a

(

b

 

b

p(t)tdt a

Kp2 (x, t)dt p(t)



p(t)[f (t)]2 dt −

1 ( − φ)Hp (x, t), 2



b a

  p(t)f (t)dt 

2 ) 12

b

−



p(t)f (t)dt −

a

a

≤

b

Kp (x, t)dt a



b

p(t)f (t)dt

2 ) 12

a

(2.4.19)

152

2 Ostrowski Inequality

where Kp (x, t) is defined in [78] as:

Kp (x, t) =

⎧ s ⎪ ⎪ p(u)du, t ∈ [a, x]; ⎪ ⎪ ⎨ a  ⎪ ⎪ ⎪ ⎪ ⎩

(2.4.20) s

p(u)du, t ∈ (x, b].

b

1 in (2.4.19), then we get the inequality (2.4.7) b−a

Remark 2.4.4 If we put p(t) ≡ of [19].

Remark 2.4.5 If we substitute = 1 in (2.4.10), then α = β = following result.

a+b , then we get 2

Corollary 2.4.9 Let all the assumptions of Theorem 2.4.7 be valid. Then the following inequality holds   a+b  a+b  b  2 2  p(u)du + f (b) p(u)du − p(t)f (t)dt f (a)  a a b     a+b   b  b b  2

p(u)du + b p(u)du − p(t)tdt p(t)f (t)dt  − a a+b 2

a

(

b

≤ a

(

b

2 (x, t)dt Kp,1

p(t)



2 )

b

−

a

1 2

Kp,1 (x, t)dt a



b

p(t)[f (t)] dt − 2

a

≤

a

2 ) 12

p(t)f (t)dt a

1 ( − φ)Hp,1 (x, t). 2

(2.4.21)

b−a in (2.4.21), then we get following result. 2 Corollary 2.4.10 Let all the assumptions of Theorem 2.4.7 be valid. Then the trapezoidal inequality (Hermite-Hadamard right bound) holds

Remark 2.4.6 If we put p(t) ≡

   b  f (a) + f (b)  1  − f (t)dt   2 b−a a (  )1  1 f (b) − f (a) 2 2 b−a

2 f 2 − ≤ √ b−a 2 3 b−a

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter

1 ≤ √ ( − φ)(b − a). 4 3

153

(2.4.22)

The above inequality can also be found in [194]. a+b in (2.4.10), then we get following result. 2 Corollary 2.4.11 Let all assumptions of Theorem 2.4.7 be valid. Then the following inequality holds

Remark 2.4.7 If we put x =

   β  α  b  b  f a + b p(u)du + f (a) p(u)du + f (b) p(u)du − p(t)f (t)dt  2 α a β a   b    α  b  b   a+b β p(u)du + a p(u)du + b p(u)du − p(t)tdt p(t)f (t)dt  − 2 α a β a a ( ≤

b

2 Kp,

b

2

, t dt

p(t)

a

(

 a+b



b

−

Kp, a



p(t)[f (t)] dt − 2

a

≤



b

 2 ) 12 a+b , t dt 2

2 ) 12

p(t)f (t)dt a

1 ( − φ)Hp, 2



 a+b ,t . 2

(2.4.23)

1 in (2.4.23), then we get following result. b−a Corollary 2.4.12 Let all the assumptions of Theorem 2.4.7 be valid. Then bound of average midpoint and trapezoidal inequality holds Remark 2.4.8 If we put p(t) ≡

     b   (1 − )f a + b + f (a) + f (b) − 1 f (t)dt   2 2 b−a a (   )1 1 f (b) − f (a) 2 2 b − a 2

f 2 − 3 − 3 + 1 ≤ √ b−a b−a 2 3 ≤

( − φ)(b − a)  2 3 − 3 + 1. √ 4 3

Remark 2.4.9 If we substitute = 1 in (2.4.23), then we get following result.

154

2 Ostrowski Inequality

Corollary 2.4.13 Let all the assumptions of Theorem 2.4.7 be valid. Then the following inequality holds   a+b  b  b  2  p(u)du + f (b) p(u)du − p(t)f (t)dt f (a) a+b  a a 2   a+b     b  b b  2 − a p(u)du + b p(u)du − p(t)tdt p(t)f (t)dt  a+b 2

a

(

b

≤

2 Kp,1

 a+b 2 , t dt p(t)

a

(

b



b

p(t)[f (t)] dt −

1 ≤ ( − φ)Hp,1 2



Kp,1 a

2

a



b

−



a

a

 2 ) 12 a+b , t dt 2

2 ) 12

p(t)f (t)dt a

 a+b ,t . 2

(2.4.24)

1 in (2.4.24), then the trapezoidal inequality b−a (Hermite-Hadamard right bound) holds as we achieved in (2.4.22). Remark 2.4.10 If we put p(t) ≡

Remark 2.4.11 If we substitute = 0, then α = a and β = b in (2.4.23), the we get following result. (see also Corollary 1 of [78]) Corollary 2.4.14 Let all assumptions of Theorem 2.4.7 be valid. Then the weighted midpoint inequality holds        b   b  b  

f a + b − a + b −  p(t)tdt p(t)f (t)dt − p(t)f (t)dt   2 2 a a a (   b  2 ) 12  b K 2 a+b , t dt a + b p 2 ≤ − , t dt Kp p(t) 2 a a (

b



b

p(t)[f (t)] dt −

a

1 ≤ ( − φ)Hp 2

2



2 ) 12

p(t)f (t)dt a

 a+b ,t . 2

Remark 2.4.12 If we put p(t) ≡

(2.4.25) 1 in (2.4.25), then we get following result. b−a

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter

155

Corollary 2.4.15 Let all the assumptions of Theorem 2.4.7 be valid. Then the midpoint inequality (Hermite-Hadamard right bound) holds      b   f a + b − 1  f (t)dt   2 b−a a (   )1 (b − a) f (b) − f (a) 2 2 1 1

2 ≤ √ |f |2 − ≤ √ ( − φ)(b − a). b−a b−a 2 3 4 3 The above inequality can also be found in [78] and [194]. 1 in (2.4.23), then we get following result. 2 Corollary 2.4.16 Let all the assumptions of Theorem 2.4.7 be valid. Then the following inequality holds

Remark 2.4.13 If we substitute =

    a+3b  3a+b  4 4 a+b  p(u)du + f (a) p(u)du f 3a+b  2 a 4  b  b p(u)du − p(t)f (t)dt + f (b)  −

a+3b 4

a+b 2





a a+3b 4

3a+b 4

b

p(t)tdt a

⎡ ≤⎣

a



b

K2

p, 12

(

b

 p(u)du + b

  p(t)f (t)dt 

 a+b 2 , t dt

 a



b

p(t)[f (t)] dt −

1 ≤ ( − φ)Hp, 1 2 2



a+3b 4

p(u)du

Kp, 1 2

⎤1  2 2 a+b , t dt ⎦ 2

2 ) 12

p(t)f (t)dt a

 a+b ,t . 2

Remark 2.4.14 If we put p(t) ≡



b

−

2

a

b

p(t)

a

3a+b 4

a

 

b

−

 p(u)du + a

(2.4.26)

1 in (2.4.26), then we get following result. b−a

156

2 Ostrowski Inequality

Corollary 2.4.17 Let all the assumptions of Theorem 2.4.7 be valid. Then the bound of average midpoint and trapezoidal inequality holds       b  1 f (a) + f (b)  a+b 1   + f f (t)dt − 2  2 2 b−a a (   )1 (b − a) f (b) − f (a) 2 2 1 1

2 ≤ √ f 2 − ≤ √ ( − φ)(b − a). b−a b−a 4 3 8 3 The above inequality can also be found in [194]. 1 in (2.4.23), then we get following result. 3 Corollary 2.4.18 Let all the assumptions of Theorem 2.4.7 be valid. Then the following inequality holds

Remark 2.4.15 If we substitute =

    a+5b  5a+b  6 6 a+b  p(u)du + f (a) p(u)du f 5a+b  2 a 6  b  b + f (b) p(u)du − p(t)f (t)dt  −

a+5b 6

a+b 2





a a+5b 6

5a+b 6

b

p(t)tdt a

⎡ ≤⎣

a



b

K2

p, 13

(

b

 p(u)du + b

  p(t)f (t)dt 

 a+b 2 , t dt

 a



b

p(t)[f (t)] dt −

a



b

−

2

1 ≤ ( − φ)Hp, 1 3 2

b a+5b 6

p(u)du

p(t)

a

5a+b 6

a

 

b

−

 p(u)du + a

Kp, 1 3



a+b , t dt 2

2

⎤ 12 ⎦

2 ) 12

p(t)f (t)dt a



 a+b ,t . 2

Remark 2.4.16 If we put p(t) ≡

(2.4.27)

1 in (2.4.27), then we get following result. b−a

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter

157

Corollary 2.4.19 Let all the assumptions of Theorem 2.4.7 be valid. Then the 1 bound of Simpson’s rule holds 3       b  1 f (a) + f (b)  a+b 1  + 2f − f (t)dt  3 2 2 b−a a (  )1  1 f (b) − f (a) 2 2 1 (b − a)

2 f 2 − ≤ √ ( − φ)(b − a). ≤ 6 b−a b−a 12 3 The above inequality can also be found in [194]. Remark 2.4.17 If we put x = a or x = b and p(t) ≡ get following result.

1 in (2.4.10), then we b−a

Corollary 2.4.20 Let all the assumptions of Theorem 2.4.7 be valid. Then the trapezoidal inequality (Hermite-Hadamard right bound) holds which is independent the value of    b  f (a) + f (b)  1   − f (t)dt   2 b−a a ( 2 ) 12  b−a 1 f (b) − f (a) (b − a)( − φ) ≤ √ f 2 − ≤ . √ b−a 2 3 b−a 4 3 Now, we will discuss some applications for probability density function of our results in the next subsection.

2.4.2 Applications to Probability Theory Suppose X be a continuous random variable with function of probability density f : [a, b] → R+ and the cumulative distribution F : [a, b] → [0, 1], i.e., 

x

F (x) =

x ∈ [α, β] ⊂ [a, b],

f (t)dt, a

 E(X) =

b

tf (t)dt a

158

2 Ostrowski Inequality

and weighted expectation would be 

b

Ep (X) =

p(t)tf (t)dt. a

Theorem 2.4.21 Let all assumptions of Theorem 2.4.7 be valid. In addition we let probability density function belongs to L2 [a, b] space. Then    F (x)    − x

β α



  p(u)du + b

a



b a

b

p(u)du − bp(b) + Ew (X) + α

p(u)du + a

p(b) − ≤

b β

β α



 p(u)du +

  p (t)F (t)dt 

b

a





b

p(u)du −

β

p (t)tF (t)dt

p(t)tdt a

1 ( − φ)Hp, (x, t), 2

(2.4.28)

∀ x ∈ [α, β]. Proof Put f = F in (2.4.10) and by using these two identities mention below, we get (2.4.28), 

b



b

p(t)F (t)dt = bp(b) − Ep (X) −

a

p (t)tF (t)dt

a

and 

b a



b

p(t)F (t)dt = p(b) −

p (t)F (t)dt.

a

 1 Corollary 2.4.22 If we put p(t) ≡ in (2.4.28), then following inequality b−a holds       b − E(X)  a+b

(1 − ) F (x) − 1 x− + −  b−a 2 2 b−a  (   )1 1 1 a+b 2 2  1 2 2 ≤ (b − a)f 22 − 1 (3 − 3 + 1) + (1 − ) x − b − a 12 2 −φ ≤ 2(b − a)

(

 2 ) 12 a + b 1 (3 2 − 3 + 1) + (1 − ) x − . 12 2

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter

159

Corollary 2.4.23 If we substitute = 0 in (2.4.28), then we get the following inequality     b  F (x) − p(b)b + Ep (X) − x − p(t)tdt  a    b   b 

p (t)F (t)dt + p (t)tF (t)dt  p(b) − a

a

1 ≤ ( − φ)Hp (x, t). 2

(2.4.29)

1 in (2.4.29), then b−a      a+b b − E(X)  F (x) − 1 x − −  b−a 2 b−a  ( 2 ) 12  (b − a) 1 f (b) − f (a) 1 ≤ √ f 22 − ≤ √ ( − φ)(b − a). b−a b−a 2 3 4 3

Corollary 2.4.24 If we put p(t) ≡

Next subsection covers some applications to numerical quadrature rules.

2.4.3 Applications to Numerical Integration Let In : a = ζ0 < ζ1 < . . . < ζn = b be a partition of the interval [a, b] and ζk = ζk+1 − ζk , k ∈ {0, 1, 2, . . . , n − 1}. Then for constant weight following identity holds 

b

p(t)f (t)dt = Qn (f, p, In ) + Rn (f, p, In ),

(2.4.30)

a

where Qn (f, p, In ) is defined as Qn (f, p, In )  n−1 

= f (ξ )   − ξk ×

 p(u)du + f (ζk )

αk

k=0



βk

βk

ζk+1 ζk

αk

p(u)du + ζk p(t)f (t)dt

 p(u)du + f (ζk+1 )

ζk



αk

αk



ζk

.

p(u)du βk

 p(u)du + ζk+1

ζk+1

ζk+1 βk

 p(u)du −



ζk+1

p(t)tdt ζk

(2.4.31)

160

2 Ostrowski Inequality

Theorem 2.4.25 Let all assumptions of Theorem 2.4.7 be valid. Then identity (2.4.30) holds where Qn (In , f, p) is given in the form of (2.4.31) and the remainder Rn (f, p, In ) becomes |Rn (f, p, In )| ≤

n−1

( − φ)

2

k=0

(2.4.32)

Hp, (ξk , t).

Proof If inequality (2.4.10) is applied on [ζk , ζk+1 ], we get Rk (f, p, Ik )  ζk+1  = p(t)f (t)dt − f (ξk ) ζk



ζk+1

  p(u)du + ξk

βk



ζk+1

βk



 

ζk+1

p(t)tdt

βk

p(u)du ζk αk

p(u)du + ζk

αk

p(u)du −

αk

p(u)du − f (ζk )

αk

−f (ζk+1 ) +ζk+1



βk



ζk+1

×

ζk

p(u)du ζk

 p(t)f (t)dt .

ζk

Summing Rk (f, p, Ik ) over k from 0 to n − 1. This yields Rn (f, p, In )  b  n−1 

= p(t)f (t)dt − f (ξk ) a

 +f (ζk+1 )

  p(u)du − ξk

βk



ζk+1

+ζk+1

 p(u)du + f (ζk )

αk

k=0

ζk+1

βk

 p(u)du −

βk

βk

 p(u)du + ζk

  p(t)tdt ×

ζk

p(u)du ζk

αk

p(u)du

αk

ζk+1

αk

ζk ζk+1

p(t)f (t)dt



ζk

Applying absolute property on the above identity, we get |Rn (f, p, In )|   βk  αk n−1   b

 f (ξk ) = p(t)f (t)dt − p(u)du + f (ζk ) p(u)du  a αk ζk  +f (ζk+1 )

k=0

ζk+1 βk

  p(u)du − ξk

βk αk

 p(u)du + ζk

αk

p(u)du ζk

.

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter





ζk+1

+ζk+1

p(u)du −

βk

≤



ζk+1

p(t)tdt

 ×

ζk

ζk+1 ζk

161

  p(t)f (t)dt 

1 ( − φ)Hp, (ξk , t). 2 

Corollary 2.4.26 If we fixed constant weight in (2.4.30) and (2.4.31), identity 

b

f (t)dt = Qn (f, In ) + Rn (f, In )

(2.4.33)

a

holds, where Qn (f, In ) is defined as Qn (f, In ) =

   n−1 

f (ζk+1 ) − f (ζk ) ζk + ζk+1 (1 − ) f (ξk ) − ξk − ζk 2 k=0  f (ζk ) + f (ζk+1 ) (2.4.34) + ζk , 2

then remainder Rn (f, In ) satisfies the estimates |Rn (f, In )| ≤

n−1

( − φ) k=0

2

(

 )1  (ζk )2 ζk + ζk+1 2 2 2 (3 − 3 + 1) + (1 − ) ξk − . 12 2

Corollary 2.4.27 If we substitute = 0 in (2.4.31), identity (2.4.30) holds, where Qn (f, p, In ) is defined as Qn (In , f, p) =

n−1 

  f (ξk ) − ξk −

k=0

 

ζk+1

ζk+1

p(t)tdt ζk



p(t)f (t)dt

,

ζk

(2.4.35) then the remainder Rn (f, p, In ) satisfies the estimates |Rn (f, p, In )| ≤

n−1

( − φ) k=0

2

Hp (ξk , t).

(2.4.36)

162

2 Ostrowski Inequality

Corollary 2.4.28 If we fixed constant weight in (2.4.35), identity (2.4.33) holds where the Qn (f, In ) is defined as Qn (f, In ) =

  n−1 

f (ζk+1 ) − f (ζk ) ζk + ζk+1 ζk , f (ξk ) − ξk − ζk 2 k=0

(2.4.37) then remainder Rn (f, In ) satisfies the estimates |Rn (f, In )| ≤

n−1

( − φ)ζk2 √ . 8 3 k=0

In the next section, we would like to present generalized Montgomery type identity using Riemann-Liouville fractional integtral. We will also discuss some generalization of fractional Ostrowski-Grüss type inequality.

2.5 Generalized Fractional Ostrwoski Type Inequality with Parameter In the current section, we use Riemann-Liouville fractional integral to provide generalization of Ostrowski type integral inequality with bounded derivatives. We also get better bounds of the inequality under consideration. Fractional Calculus represents a complex phenomenon in a more accurate and efficient way than classical calculus. It accepts integrals and derivatives of any positive order. This subject has gained importance and popularity during the last four decades because it has various applications in the field of science and engineering. The classical form of fractional calculus is given by Riemann and Liouville. Weyl, Fourier, Abel, Lacrois, Leibniz and Gunwald have also done significant contributions in this field. Fractional integrals play increasingly important role on some mathematical inequalities such as Ostrowski, Grüss, Ostrowski-Grüss and Hadamard inequality etc. We quote from [14], The subject of fractional calculus (that is, calculus of integrals and derivatives of an arbitrary real or complex order) was planted over 300 years ago. Since that time the fractional calculus has drawn the attention of many researchers in. In recent years, the fractional calculus has played a significant role in many areas of science and engineering.

Over the last two decades, fractional integral inequalities have been one of the most important tools for the advancement of many branches of mathematics. Many authors have discussed certain generalizations of fractional integral inequalities. Recent generalizations can be found in the articles [53, 80, 169–172].

2.5 Generalized Fractional Ostrwoski Type Inequality with Parameter

163

For our next result we need here definition of Riemann-Liouville fractional integral from [64]. Definition 2.5.1 The Riemann-Liouville fractional integral operator of order γ ≥ 0 is defined as  x 1 γ Ja f (x) = (x − t)γ −1 f (t)dt

(γ ) a Ja0 f (x) = f (x). where gamma function (γ ) is defined as 

∞

(γ ) =

x γ −1 e−x dx.

0

In [11] by using Riemann-Liouville fractional integrals, the authors obtained fractional Montgomery identity. Theorem 2.5.1 Let f : I → R be differentiable function on I 0 with a, b ∈ I a < b, f ∈ L[a, b] and for γ ≥ 1, then Montgomery fractional identity holds f (x) =

(γ ) γ γ −1 γ (b − x)1−γ Ja f (b) − Ja (K γ (x, b)f (b)) + Ja (K γ (x, b)f (b)), b−a

(2.5.1) where K γ (x, t) is the fractional Peano kernel defined as

K γ (x, t) =

⎧ t −a ⎪ ⎪ (b − x)1−γ (γ ), t ∈ [a, x); ⎪ ⎨b−a ⎪ ⎪ t −b ⎪ ⎩ (b − x)1−γ (γ ), t ∈ [x, b]. b−a

(2.5.2)

In the following subsection, we propose some new results of fractional integral inequalities of Ostrowski type. We begin by establishing Montgomery identity with parameters using Riemann-Liouville fractional integrals and then use this useful identity to generate a lemma, which is further required in our main theorem.

2.5.1 Fractional Ostrowski Type Inequality Involving Parameter We need to proof the following lemmas for our main result.

164

2 Ostrowski Inequality

Lemma 2.5.1 Let f : I → R be an absolutely continuous function on I 0 with a, b ∈ I a < b, f ∈ L[a, b] and for γ ≥ 1, Then the following identity holds (1 − )f (x) = −



(γ ) γ γ −1  γ (b − x)1−γ Ja f (b) − Ja K (x, b)f (b) b−a

(b − x)1−γ 0 γ  Ja f (a) + Ja K γ (x, b)f (b) , 1−γ 2(b − a)

(2.5.3)

  γ b−a ∀ x ∈ a + b−a 2 , b + 2 , where ∈ [0, 1], and K (x, t) is the fractional Peano kernel defined by

K γ (x, t) =

  ⎧ b−a (b − x)1−γ ⎪ ⎪

(γ ), t ∈ [a, x); ⎪ t − a+ ⎪ ⎨ 2 b−a    ⎪ ⎪ ⎪ b−a (b − x)1−γ ⎪ ⎩ t − b−

(γ ), t ∈ [x, b]. 2 b−a

(2.5.4)

Proof Using Riemann-Liouville fractional integral operator twice, we get Ja (K γ (x, b)f (b))  b 1 (b − t)γ −1 K γ (x, t)f (t)dt =

(γ ) a  x    (b − x)1−γ b−a = f (t)dt (b − t)γ −1 t − a + b−a 2 a      b b−a f (t)dt + (b − t)γ −1 t − b − 2 x  b 1−γ (b − x) = (1 − )f (x) − (b − t)γ −1 f (t)dt (b − a) a 

(b − x)1−γ γ −1 b + f (a) + (b − t)γ −2 K γ (x, t)f (t)dt

(γ ) a 2(b − a)1−γ γ

= (1 − )f (x) −

(b − x)1−γ

(b − x)1−γ 0 γ γ −1

(γ )Ja f (b) + J f (a) + Ja (K γ (x, b)). (b − a) 2(b − a)1−γ a

After rearranging the terms, we get (2.5.3).



Remark 2.5.1 If we substitute γ = 1 in (2.5.3), then we get Montgomery identity with parameter as stated in (2.3.2). Remark 2.5.2 If we substitute = 0 in (2.5.3), then we get the identity (2.5.1).

2.5 Generalized Fractional Ostrwoski Type Inequality with Parameter

165

Lemma 2.5.2 Let all the assumptions of Lemma 2.5.1 be valid. Then the generalization of (2.5.3) holds (1 − )f (x) (b − x)2−γ (b − x)1−γ γ γ −1  γ γ −1 K (x, b)f (b) − Ja f (b) − Ja

(γ )Ja f (b) b−a b−a   (b − x)1−γ b−a γ  γ − (2.5.5) Ja0 f (a) + 2Ja K∗ (x, b)f (b) , x − a +

2−γ (b − a) 2 = 2 (γ )

γ

where K∗ (x, t) is the fractional Peano kernel defined by

γ

K∗ (x, t) =

  ⎧ a+x b−a (b − x)1−γ ⎪ ⎪ t − +

(γ ), if t ∈ [a, x); ⎪ ⎪ ⎨ 2 4 b−a    ⎪ ⎪ ⎪ b+x b−a (b − x)1−γ ⎪ ⎩ t− −

(γ ), if t ∈ [x, b]. 2 4 b−a γ

Proof Using Riemann-Liouville fractional integral operator on K∗ (x, t), we get Ja (K∗ (x, b)f (b))  b 1 γ (b − t)γ −1 K∗ (x, t)f (t)dt =

(γ ) a  x    b−a (b − x)1−γ a+x γ −1 + f (t)dt = t− (b − t) b−a 2 4 a      b b−a b+x γ −1

− f (t)dt , t− (b − t) + 2 4 x γ

γ

after some computations, we get Ja (K∗ (x, b)f (b)) γ

γ

(b − x)1−γ 1 γ γ Ja (K (x, b)f (b)) + 2 b−a

=



b

 (b − t)γ −1 (t − x)f (t)dt . (2.5.6)

a

We also have 

b

(b − t)γ −1 (t − x)f (t)dt

a γ −1

= (x − a)(b − a)γ −1 Ja0 f (a) + (b − x) (γ )Ja

γ

f (b) − (γ )Ja f (b). (2.5.7)

166

2 Ostrowski Inequality

Now from Lemma 2.5.1, we have γ  Ja K γ (x, b)f (b) = (1 − )f (x) − +



(γ ) γ γ −1  γ (b − x)1−γ Ja f (b) + Ja K (x, b)f (b) b−a

(b − x)1−γ 0 J f (a). 2(b − a)1−γ a

(2.5.8)

Using (2.5.7) and (2.5.8) in (2.5.6), we get Ja (K∗ (x, b)f (b)) γ

γ

1 1 γ −1 (b − x)1−γ γ (1 − )f (x) − (γ ) Ja f (b) + Ja (K γ (x, b)f (b)) 2 2(b − a) 2    b−a (b − x)1−γ 0 (b − a)1−γ γ −1 J f (a) x − a +

+ + (b − x) (γ )Ja f (b), a 2−γ 2(b − a) 2 2(b − a)

=

which yields the required result.



Remark 2.5.3 If we substitute γ = 1 in (2.5.5), we get the following corollary. Corollary 2.5.2 Let all the assumptions of Lemma 2.5.1 be valid. Then      b b−a 1 1 1 f (b) x − b + f (t)dt + (1 − ) f (x) = 2 b−a a 2(b − a) 2     b b−a 1 −f (a) x − a − K∗ (x, t)f (t)dt, + 2 b−a a (2.5.9) where

K∗ (x, t) =

  ⎧ a+x b−a ⎪ ⎪ t − +

, t ∈ [a, x); ⎪ ⎪ ⎨ 2 4    ⎪ ⎪ b−a b+x ⎪ ⎪ ⎩ t− − , t ∈ [x, b]. 2 4

Remark 2.5.4 If we substitute = 0 in (2.5.5), we get the following corollary.

2.5 Generalized Fractional Ostrwoski Type Inequality with Parameter

167

Corollary 2.5.3 Let all the assumptions of Lemma 2.5.1 be valid. Then f (x) = 2 (γ )

(b − x)1−γ γ γ −1  γ Ja f (b) − Ja K (x, b)f (b) b−a

−

(b − x)2−γ γ −1

(γ )Ja f (b) b−a

−

(b − x)1−γ γ  γ (x − a) Ja0 f (a) + 2Ja K∗ (x, b)f (b) , 2−γ (b − a)

(2.5.10)

where

γ

K∗ (x, t) =

⎧  a + x (b − x)1−γ ⎪ ⎪ t−

(γ ), t ∈ [a, x); ⎪ ⎪ ⎨ 2 b−a   ⎪ ⎪ ⎪ b + x (b − x)1−γ ⎪ ⎩ t−

(γ ), t ∈ [x, b], 2 b−a

and K γ (x, t) is defined in (2.5.2). Remark 2.5.5 If we substitute γ = 1 in (2.5.10), we get the following corollary which can be found in [183] by Tong and Guan. Corollary 2.5.4 Let all the assumptions of Lemma 2.5.1 be valid. Then  b  b 1 (x − b)f (b) − (x − a)f (a) 1 1 f (t)dt + K∗ (x, t)f (t)dt, f (x) = + 2 b−a a 2(b − a) b−a a

where

K∗ (x, t) =

⎧ a+x ⎪ ⎪ t− , t ∈ [a, x); ⎪ ⎨ 2 ⎪ ⎪ b+x ⎪ ⎩t − , t ∈ [x, b], 2

By using Lemma 2.5.2, we obtain generalized Ostrowski fractional integral inequality in the following theorem. Theorem 2.5.5 Let f : I → R be an absolutely continuous function on I o such that a, b ∈ I and a < b. If |f (x)| ≤ M a. e. ∀x ∈ (a, b) where M is positive real constant, then the following inequality holds  1−γ 1 1 γ −1 γ  (1 − )f (x) − (γ ) (b − x) Ja f (b) + Ja (K γ (x, b)f (b)) 2 (b − a) 2   2−γ 1−γ (b − x) b − a  (b − x) γ −1 0

(γ )Ja f (b) + J f (a) x − a +

+  2(b − a) 2(b − a)2−γ a 2

168

2 Ostrowski Inequality

 * M(b − x)1−γ (x − a) 1

(b − a)γ − (b − x)γ + (b − x)γ +1 + (b − a)γ +1 ≤ b−a 2γ 2 "  γ +1 1 a+x b−a + 2 b− + − 2(b − x)γ +1 − (b − a)γ +1 γ (γ + 1) 2 4 2)   b − a γ +1 b−x , (2.5.11) − +2 2 4 for γ ≥ 1, where ∈ [0, 1]. Proof From Lemma 2.5.2, consider  1 (b − x)1−γ γ 1 γ −1  γ Ja f (b) + Ja K (x, b)f (b) I =  (1 − )f (x) − (γ ) 2 (b − a) 2    2−γ 1−γ  (b − x) (b − x) b−a γ −1 0

(γ )Ja f (b) + Ja f (a) + x−a+ 2−γ 2(b − a) 2 2(b − a)  γ γ 

= Ja K∗ (x, b)f (b)     b  1  γ −1 γ

 = (b − t) K∗ (x, t)f (t)dt 

(γ ) a  b    γ 1 ≤ (b − t)γ −1 K∗ (x, t) f (t) dt

(γ ) a  b   γ M ≤ (b − t)γ −1 K∗ (x, t) dt

(γ ) a    x   (b − x)1−γ a+x b − a  γ −1  =M + (b − t) t −  dt b−a 2 4 a      b  b+x b − a  γ −1  − + (b − t) t −  dt 2 4 x =M

(b − x)1−γ [I1 + I2 + I3 + I4 ] , b−a

(2.5.12)

where  I1 = a

= −

a+x b−a 2 + 4



(b − t)γ −1

b−a x−a + 2 4

(b − a)γ +1 , γ (γ + 1)





b−a a+x + 2 4



 − t dt

  (b − a)γ 1 a+x b − a γ +1 + b− − γ γ (γ + 1) 2 4 (2.5.13)

2.5 Generalized Fractional Ostrwoski Type Inequality with Parameter

169

  b−a a+x + dt t− I2 = (b − t) a+x b−a 2 4 2 + 4     a−x b − a (b − x)γ 1 b − a γ +1 a+x = + + − b− 2 4 γ γ (γ + 1) 2 4 

−

x

γ −1

(b − x)γ +1 , γ (γ + 1) 

b+x b−a 2 − 4

I3 =

(b − t)

x

 =

−

(2.5.14)

b−a b−x − 2 4

γ −1





b−a b+x − 2 4



1 (b − x)γ + γ γ (γ + 1)

dt 

b−a b−x + 2 4

(b − x)γ +1 γ (γ + 1)

γ +1

(2.5.15)

and   b−a b+x − dt t− (b − t) I4 = b+x b−a 2 4 2 − 4   1 b − a γ +1 b−x = − . γ (γ + 1) 2 4 

b

γ −1

(2.5.16)

Using (2.5.13)–(2.5.16) in (2.5.12), we get the bound  * M(b − x)1−γ (x − a) 1

(b − a)γ − (b − x)γ + (b − x)γ +1 + (b − a)γ +1 b−a 2γ 2 "  γ +1 a+x b−a 1 − 2(b − x)γ +1 − (b − a)γ +1 2 b− + + γ (γ + 1) 2 4 2)   b−x b − a γ +1 +2 . − 2 4

I≤

 Remark 2.5.6 If we substitute γ = 1 in (2.5.11), we get the following corollary.

170

2 Ostrowski Inequality

Corollary 2.5.6 Let all the assumptions of Theorem 2.5.5 be valid. Then     1 1 b−a  (1 − )f (x) − x − b +

f (b) 2 2(b − a) 2       b  1 b−a f (a) − − x− a− f (t)dt  2 b−a a  

(b − a)2 M (x − a)2 + (b − x)2 + , ≤ 4(b − a) 4

(2.5.17)

where ∈ [0, 1]. Remark 2.5.7 If we substitute = 0 in (2.5.11), we get the following corollary. Corollary 2.5.7 Let all the assumptions of Theorem 2.5.5 be valid. Then  1−γ 1 1 γ −1 γ  f (x) − (γ ) (b − x) Ja f (b) + Ja (P1 (x, b)f (b)) 2 (b − a) 2

  (b − x)1−γ 0 (b − x)2−γ γ −1 

(γ )Ja f (b) + J f (a) − a) (x a  2−γ 2(b − a) 2(b − a)  * M(b − x)1−γ (x − a) 1 1 ≤ (b − a)γ − (b − x)γ + (b − x)γ +1 + b−a 2γ γ (γ + 1) "   γ +1 2)  a + x γ +1 b − x , × 2 b− − 2(b − x)γ +1 − (b − a)γ +1 +2 2 2 (2.5.18) +

for γ ≥ 1. Remark 2.5.8 If we substitute either = 0 in (2.5.17) or γ = 1 in (2.5.18), we get the following corollary. Corollary 2.5.8 Let all the assumptions of Theorem 2.5.5 be valid. Then    b 1  1 1  f (x) − {(x − b) f (b) − (x − a)f (a)} − f (t)dt  2 2(b − a) b−a a   M (2.5.19) (x − a)2 + (b − x)2 . ≤ 4(b − a) Remark 2.5.9 If we put x =

a+b in (2.5.17), we get the following corollary. 2

2.5 Generalized Fractional Ostrwoski Type Inequality with Parameter

171

Corollary 2.5.9 Let all the assumptions of Theorem 2.5.5 be valid. Then      b 1  M f (a) + f (b) a+b 1   + (1 +

) f (t)dt (1 −

)f − (b − a)(2 + ),  ≤ 2  16 2 4 b−a a

(2.5.20) where ∈ [0, 1]. Remark 2.5.10 If we substitute = 0 in (2.5.20), we get the bound for average midpoint and trapezoidal inequality in the following corollary. Corollary 2.5.10 Let all the assumptions of Theorem 2.5.5 be valid. Then      b 1  M  f a + b + f (a) + f (b) − 1 ≤ (b − a). f (t)dt 2  2 4 b−a a 8 1 Remark 2.5.11 If we substitute = in (2.5.20), we get the bound for perturbed 2 trapezoidal inequality in the following corollary. Corollary 2.5.11 Let all the assumptions of Theorem 2.5.5 be valid. Then      b   1   f a + b + 3 f (a) + f (b) − 1  ≤ 5 (b − a)M. f (t)dt 4  32 2 8 b−a a Remark 2.5.12 If we substitute = 13 in (2.5.20), we get the bound for perturbed trapezoidal inequality in the following corollary. Corollary 2.5.12 Let all the assumptions of Theorem 2.5.5 be valid. Then      b   1   f a + b + 1 f (a) + f (b) − 1  ≤ 7 (b − a)M. f (t)dt 3  48 2 3 b−a a Remark 2.5.13 If we substitute = 14 in (2.5.20), we get the bound for perturbed trapezoidal inequality in the following corollary. Corollary 2.5.13 Let all the assumptions of Theorem 2.5.5 be valid. Then      b    3  ≤ 9 (b − a)M.  f a + b + 5 f (a) + f (b) − 1 f (t)dt  64 8 2 16 b−a a Remark 2.5.14 If we put x = a or x = b in (2.5.19), we get the bound for trapezoidal inequality (also Hermite-hadamard right bound) in the following corollary.

172

2 Ostrowski Inequality

Corollary 2.5.14 Let all the assumptions of Theorem 2.5.5 be valid. Then    b  f (a) + f (b)  M 1   ≤ (b − a). − f (t)dt   2 b−a a 4 In the upcoming section, we would like to generalize Montgomery identity and ˇ Ostrowski and Cebyšev type inequality for two variables with parameters.

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery Identity with Parameters Present section deals with the generalization of Montgomery identity for two independent variables using second order differentiable functions with parameters. Some new Ostrowski type inequalities for Lp spaces with better bounds are presented as well. Moreover we will modify Grüss type inequality by using the acquired Montgomery identity. At places we get better bounds of some new obtained inequalities. Montgomery identity is one of the classical results that creates many important inequalities such as Ostrowski, Grüss and Ostrowski-Grüss. Its bivariate form has introduced some new generalization and advancement in different inequalities. These inequalities have many applications in various fields of mathematics such as numerical integration and probability theory. We can also obtain special means with the help of these inequalities. In the last 20 years rapid advancement in generalization and improvement of these types of inequalities has been observed for references see [24, 54, 66, 76, 77, 143, 145, 156]. This section deals with its bivariate form in order to generate our proposed results of Ostrowski and Grüss inequalities in terms of parameters. The idea behind the results based on parameters is to make further generalization of those results of Ostrowski and Grüss which are non parametric based, as parameters extends the region of inequality more wider and provides a family of solutions and the quality of inequality will improve conclusively. If we talk about Lp spaces, this is the first ever combination of Lp space, parameters and bivariate differentiable functions, which some how connects our result with lebesgue measure. In the first subsection we obtain Montgomery identity with parameters of two independent variables, while in the second subsection, we establish Ostrowski type inequality for two variables in terms of parameters for Lp spaces. In the last ˇ subsection, we will achieve Grüss type inequalities with its Cebyšev functional.

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

173

2.6.1 Montgomery Identity for Functions of Two Variables involving Parameters For classical form of Montgomery identity, see the book [134, p. 565], as given in the Theorem 1.1.1. In 2001, Dragomir et al. introduced this identity with parameters in [53] as given in the Theorem 2.3.5. In [16], (see also [52]) authors proved the double integral Montgomery identity for two independent variables stated as follows: Theorem 2.6.1 Let f : I × J = [a, b] × [c, d] → R is differentiable such that ∂ 2 f (t, s) is integrable on interior of I × J . Then ∂t∂s  d 1 f (t, y)dt + f (x, s)ds (d − c) c a  b d 1 − f (t, s)dsdt (b − a)(d − c) a c  b d ∂ 2 f (t, s) dsdt, K(x, t)Q(y, s) + ∂t∂s a c

1 f (x, y) = (b − a)



b

(2.6.1)

where K(x, t) and Q(y, s) are the Peano kernels defined as

K(x, t) =

⎧ t −a ⎪ ⎪ , t ∈ [a, x]; ⎪ ⎨b−a ⎪ ⎪ t −b ⎪ ⎩ , t ∈ (x, b]. b−a

(2.6.2)

and

Q(y, s) =

⎧ s−c ⎪ ⎪ , s ∈ [c, y]; ⎪ ⎨d −c ⎪ ⎪ s−d ⎪ ⎩ , s ∈ (y, d]. d −c

(2.6.3)

Now, we are going to establish new Montgomery identity for two independent variables involving parameters, which will provide generalization of existing Montgomery identities.

174

2 Ostrowski Inequality

Here we state our first main result. Theorem 2.6.2 Let f : I × J → R be absolutely continuous such that is integrable on interior of I × J . Then

 (1 − ) d f (t, y)dt + f (x, s)ds d −c c a  b d 1 1 − f (t, s)dsdt + ψ ,κ (f ) (b − a)(d − c) a c 2  b d 1 ∂ 2 f (t, s) dsdt, + K (x, t)Q1 (y, s) (b − a)(d − c) a c ∂t∂s

(1 − κ) (1 − )(1 − κ)f (x, y) = b−a



∂ 2 f (t, s) ∂t∂s

b

(2.6.4) where ψ ,κ (f ) =

κ (b − a)



b

(f (t, c) + f (t, d)) dt − κ(1 − ) (f (x, c) + f (x, d))

a

 d

(f (a, s) + f (b, s)) ds − (1 − κ) (f (a, y) + f (b, y)) (d − c) c

κ − (f (a, c) + f (a, d) + f (b, c) + f (b, d)) , (2.6.5) 2

+

also K (x, t) is defined in (2.3.3), Q1 (y, s) is defined as

Q1 (y, s) =

 ⎧  d−c ⎪ ⎪ s − c + κ , s ∈ [c, y]; ⎪ ⎪ ⎨ 2   ⎪ ⎪ d−c ⎪ ⎪ ⎩s − d − κ , s ∈ (y, d]. 2

(2.6.6)

where , κ ∈ [0, 1]. Proof By using (2.3.3) and (2.6.6), we have 



∂ 2 f (t, s) dsdt (2.6.7) ∂t∂s a c      2  x y b−a d −c ∂ f (t, s) = dsdt t − a+ s− c+κ 2 2 ∂t∂s a c      2  x d b−a d −c ∂ f (t, s) dsdt + t − a+ s− d −κ 2 2 ∂t∂s a y b

d

K (x, t)Q1 (y, s)

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

175

      2 b−a d −c ∂ f (t, s) dsdt + t − b− s− c+κ 2 2 ∂t∂s x c      2  b d b−a d −c ∂ f (t, s) dsdt + t − b− s− d −κ 2 2 ∂t∂s x y 

b



y

= I1 + I2 + I3 + I4 .

(2.6.8)

After some calculations and simplifications, we have       d −c b−a y− c+κ f (x, y) I1 = x − a + 2 2       x   y d −c b−a − y− c+κ f (t, y)dt − x − a + f (x, s)ds 2 2 a c      y b−a d −c f (a, s)ds + y− c+κ f (a, y) − 2 2 c      x d −c b−a +κ x− a+ f (x, c) − f (t, c)dt 2 2 a  x y (b − a)(d − c) f (a, c) + f (t, s)dsdt, + κ 4 a c       b−a d −c I2 = − x − a + y− d −κ f (x, y) 2 2     x   d   b−a d −c f (t, y)dt − x − a + f (x, s)ds + y− d −κ 2 2 a y       d b−a d−c + f (a, s)ds − y− d −κ f (a, y) − 2 2 y      x d −c b−a +κ f (t, d)dt x− a+ f (x, d) − 2 2 a  x d (b − a)(d − c) + κ f (a, d) + f (t, s)dsdt, 4 a y       b−a d−c I3 = − x − b − y− c+κ f (x, y) 2 2     b   y   b−a d −c f (t, y)dt + x − b − f (x, s)ds − y− c+κ 2 2 x c      y b−a d −c f (b, s)ds + y− c+κ f (b, y) − 2 2 c

176

2 Ostrowski Inequality

      b d −c b−a +κ f (t, c)ds − x− b− f (x, c) − 2 2 x  b y (b − a)(d − c) + κ f (b, c) + f (t, s)dsdt, 4 x c and       b−a d −c I4 = x − b − y− d −κ f (x, y) 2 2     b   d   b−a d −c f (t, y)dt + x − b − f (x, s)ds + y− d −κ 2 2 x y       d d−c b−a − y− d −κ f (b, y) − + f (b, s)ds+ 2 2 y       b d −c b−a f (t, d)dt +κ − x− b− f (x, d) − 2 2 x  b d (b − a)(d − c) + κ f (t, s)dsdt. f (b, d) + 4 x y By substituting the values of I1 , I2 , I3 and I4 in (2.6.8), we get 

b



d

K (x, t)Q1 (y, s) a

c

∂ 2 f (t, s) dsdt ∂t∂s



= (1 − )(1 − κ)(b − a)(d − c)f (x, y) − (1 − κ)(d − c)  − (1 − )(b − a) −κ

d−c 2

d



b

a

× (f (x, c) + f (x, d)) −

b−a 2



d c

(f (t, c) + f (t, d)) dt +

a

f (t, y)dt a

f (t, s)dsdt

c



b

f (x, s)ds −

b



d

κ(1 − )(b − a)(d − c) 2

(f (a, s) + f (b, s)) ds

c

(1 − κ)(b − a)(d − c) (f (a, y) + f (b, y)) 2

κ(b − a)(d − c) + (f (a, c) + f (a, d) + f (b, c) + f (b, d)) , 4

+

which produce the required identity.



2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

177

Remark 2.6.1 1. If we substitute , κ = 0 in (2.6.4), then it gives (2.6.1) of [16] as stated in Theorem 2.6.1. 2. If we substitute f (t, s) = h(t)h(s) and x = y in (2.6.4), then it gives (2.3.2) of [53] as stated in Theorem 2.3.5. Remark 2.6.2 If we substitute = κ, then we get a special type of Montgomery identity as established in [109]. (1 − )2 f (x, y) = − +

(1 − ) b−a



b

f (t, y)dt +

a

1 (b − a)(d − c) 1 (b − a)(d − c)

 

b



b



a

d c

(1 − ) d −c



d

f (x, s)ds c

1 f (t, s)dsdt + ψ (f ) 2

d

K (x, t)Q2 (y, s) a

c

∂ 2 f (t, s) dsdt, ∂t∂s

where  d

(f (t, c) + f (t, d)) dt + (f (a, s) (d − c) c a   +f (b, s)) ds − (1 − ) f (x, c) + f (x, d) + f (a, y) + f (b, y)

ψ (f ) = (b − a)

−



b



2  f (a, c) + f (a, d) + f (b, c) + f (b, d) , 2

also Peano kernels K (x, t) is defined as in (2.3.3), Q2 (y, s) is defined as

Q2 (y, s) =

  ⎧ d −c ⎪ ⎪ s − c +

, s ∈ [c, y]; ⎪ ⎪ ⎨ 2   ⎪ ⎪ d −c ⎪ ⎪ ⎩s − d − , s ∈ (y, d]. 2

where ∈ [0, 1]. Lets recall the concept of Hölder’s inequality from Theorem 1.1.13 that is useful in our results of the coming subsections.

2.6.2 Generalized Ostrowski Type Inequality Now we are going to present Ostrowski type inequality for Lp and L∞ spaces by using the Montgomery identity (2.6.4) as we obtained in the previous subsection.

178

2 Ostrowski Inequality

After the publication of research article [141] by Ostrowski, researchers are in an effort to generalize the Ostrowski inequality and trying to get the sharp bounds. Ostrowski inequality discussed previously in Theorem 2.0.1. In 1997, Dragomir and Wang established the following of Ostrowski type inequality for differentiable functions in [51] where f ∈ Lq space. Theorem 2.6.3 Let f : I → R be a differentiable function on I o where a < b such 1 1 that f ∈ Lq [a, b] where 1 ≤ q ≤ ∞ and + = 1. Then q r    b   1 1 f (x) − 1 f (t)dt  ≤ [B1 (x)] r f q ,  b−a a b−a

(2.6.9)

where, B1 (x) =

(x − a)r+1 + (b − x)r+1 , r +1

(2.6.10)

∀ x ∈ [a, b]. In 2001, Ostrowski type inequality for double integrals was introduced by Barnett and Dragomir in [16]. Theorem 2.6.4 Let f : I × J → R is differentiable such that integrable on interior of I × J and is bounded in L∞ space. Then

∂ 2 f (t, s) is ∂t∂s

  b  d  1 1 f (x, y) − f (t, y)dt − f (x, s)ds  (b − a) a (d − c) c   b d  1 f (t, s)dsdt  + (b − a)(d − c) a c 0   0 0 0 2 1 2 2 2 2 0 ∂ f (t, s) 0 . ≤ (x − a) + (b − x) (y − c) + (d − y) 0 4(b − a)(d − c) ∂t∂s 0∞

(2.6.11) Furthermore in 2000, Dragomir et al. in [54] generalized the results of [16] for Lq space. ∂ 2 f (t, s) is integrable ∂t∂s 1 1 on interior of I × J and is bounded in Lq space where 1 ≤ q ≤ ∞ and + = 1. q r Then   b  d  1 1 f (x, y) − f (t, y)dt − f (x, s)ds  (b − a) a (d − c) c

Theorem 2.6.5 If f : I × J → R is differentiable such that

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

  f (t, s)dsdt  a c 0 2 0 0 ∂ f (t, s) 0 1 0 0 [B1 (x)] 1r [B2 (y)] 1r , ≤ 0 (b − a)(d − c) ∂t∂s 0q 1 + (b − a)(d − c)



b



179

d

(2.6.12)

where B1 (x) is defined as in (2.6.10) and B2 (y) =

(y − c)r+1 + (d − y)r+1 . r+1

(2.6.13)

In 2000, Dragomir et al. in [53] generalized the classical Ostrowski inequality [141] as stated in the following Theorem: Theorem 2.6.6 Let f : I → R be a differentiable functions on I o such that f ∈ L[a, b], where a < b whose derivative f is bounded on (a, b), i.e., f ∞ := sup |f (t)| < ∞. Then

t ∈(a,b)

   b   (1 − )f (x) + f (a) + f (b) − 1  f (t)dt   2 b−a a ( ) * x − a+b 2 11 2 2 2 ≤ (b − a) f (x)∞ ,

+ ( − 1) + 4 (b − a)2

(2.6.14)

where ∈ [0, 1]. In 2003, Yang established Ostrowski inequality for Lp spaces in [190] that is infact a generalization of (2.6.14). Theorem 2.6.7 Let all assumptions of Theorem 2.6.3 be true. Then    b   (1 − )f (x) + f (a) + f (b) − 1  f (t)dt   2 b−a a (  r+1 1 b−a ≤ x− a+ 1 2 (b − a)(r + 1) r    r+1  )1 b−a b − a r+1 r + b− +2 f (x)q , −x 2 2

(2.6.15)

where ∈ [0, 1]. Now we are going to present Ostrowski inequality with parameters of double integrals for Lp and L∞ space with parameters.

180

2 Ostrowski Inequality

∂ 2 f (t, s) is Theorem 2.6.8 Let f : I × J → R is differentiable such that ∂t∂s integrable on interior of I × J and is bounded in Lq space where 1 ≤ q ≤ ∞ 1 1 and + = 1. Then q r   b   (1 − ) d (1 − )(1 − κ)f (x, y) − (1 − κ) f (t, y)dt − f (x, s)ds  (b − a) a (d − c) c   b d  1 1 f (t, s)dsdt − ψ ,κ (f ) + (b − a)(d − c) a c 2 0 2 0 0 ∂ f (t, s) 0 1 0 0 ≤ 2 0 ∂t∂s 0q (b − a)(d − c)(r + 1) r (   r+1   r+1  )1 b−a b−a b − a r+1 r × x− a+ + b− +2 −x 2 2 2 ( ×



d −c y− c+κ 2

r+1

   r+1  )1 d −c d − c r+1 r + d −κ +2 κ , −y 2 2

(2.6.16) where , κ ∈ [0, 1]. Proof From Theorem 2.6.2, we have  1 (1 − κ) b f (t, y)dt − ψ ,κ (f ) (1 − )(1 − κ)f (x, y) − (b − a) a 2  d  b d (1 − ) 1 − f (x, s)ds + f (t, s)dsdt (d − c) c (b − a)(d − c) a c  b d 1 ∂ 2 f (t, s) = K (x, t)Q1 (y, s) dsdt. (2.6.17) (b − a)(d − c) a c ∂t∂s Applying absolute on both sides of (2.6.17) and using Hölder’s inequality, we get   b  1 (1 − )(1 − κ)f (x, y) − (1 − κ) f (t, y)dt − ψ ,κ (f )  (b − a) a 2    b d  1 (1 − ) d f (x, s)ds + f (t, s)dsdt  − (d − c) c (b − a)(d − c) a c    b d   ∂ 2 f (t, s) 1 K (x, t)Q1 (y, s) dsdt  =  (b − a)(d − c) a c ∂t∂s

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

1 ≤ (b − a)(d − c) 1 ≤ (b − a)(d − c)



b a



b a

c

  2  ∂ f (t, s)   dsdt  |K (x, t)Q1 (y, s)|  ∂t∂s 



d



d

|K (x, t)Q1 (y, s)| dsdt

c

r

 1r 

b a



d c

181

1   2 q  ∂ f (t, s) q  dsdt   ∂t∂s 

0 0 2 0 ∂ f (t, s) 0 0 0 2 0 ∂t∂s 0q (b − a)(d − c)(r + 1) r (   r+1   r+1  )1 b−a b−a b − a r+1 r × x− a+ + b− +2 −x 2 2 2 =

1

(      r+1  )1 d −c d − c r+1 r d − c r+1 + d −κ +2 κ . y− c+κ −y 2 2 2 Corollary 2.6.9 Let all the assumptions of Theorem 2.6.6 be valid. If we select r = 1 and q → ∞ in (2.6.16), then we get following result.   b  1 (1 − )(1 − κ)f (x, y) − (1 − κ) f (t, y)dt − ψ ,κ (f )  (b − a) a 2   d  b d  1 (1 − ) f (x, s)ds + f (t, s)dsdt  − (d − c) c (b − a)(d − c) a c   1 ≤ (x − a)2 + (b − x)2 − (1 − )(b − a)2 4(b − a)(d − c) 0  0 0 2 0 2 2 2 0 ∂ f (t, s) 0 × (y − c) + (d − y) − κ(1 − κ)(d − c) 0 ∂t∂s 0 (

) * x − a+b 2 11 2 2

+ ( − 1)2 + = (b − a)(d − c) 4 (b − a)2 ( ) 0 * y − c+d 2 0 0 ∂ 2 f (t, s) 0 11 2 2 2 0 0 × κ + (κ − 1) + 0 ∂t∂s 0 , 4 (d − c)2 ∞

∞

(2.6.18)

where , κ ∈ [0, 1]. Remark 2.6.3 It is to be noted the constant 14 is sharp in (2.6.18) in the first and second bracket in the sense that it cannot be replaced by any smaller values.

182

2 Ostrowski Inequality

To be more specific, if we suppose the inequality (2.6.18) be valid for constants C1 , C2 > 0, i.e.,   b  1 (1 − )(1 − κ)f (x, y) − (1 − κ) f (t, y)dt − ψ ,κ (f )  (b − a) a 2   d  b d  1 (1 − ) f (x, s)ds + f (t, s)dsdt  − (d − c) c (b − a)(d − c) a c ) ( 1 * x − a+b 2 2 2 2 ≤ (b − a)(d − c) C1 + ( − 1) + (b − a)2 ( ) 0 1 * y − c+d 2 0 0 ∂ 2 f (t, s) 0 2 2 2 0 0 × C2 κ + (κ − 1) + 0 ∂t∂s 0 . (d − c)2 ∞ Consider f (s, t) = st, x = a, y = c, and , κ = 0 then above inequality reduces to  1 ≤ C1 + 4  1 1 × ≤ C1 + 2 2

  1 C2 + 4   1 1 C2 + , 4 4 1 4

which gives that C1 ≥ 14 and C2 ≥ 14 . Hence we are true in our claim. In the similar manner one can find out that the improved bounds will be obtained by choosing , κ = 12 . From (2.6.16) and (2.6.18) we can get many results of Ostrowski type inequality. Remark 2.6.4 1. If we substitute = κ = 0 in (2.6.16), the it gives (2.6.12) of [54] as stated in Theorem 2.6.5. 2. If we substitute = κ = 0 in (2.6.18), then it gives (2.6.11) of [16] as stated in Theorem 2.6.4. 3. If we substitute f (t, s) = h(t)h(s), here h be absolutely continuous function, also let h  < ∞ and x = y in (2.6.16), then it gives (2.6.15) of [190] as stated in Theorem 2.6.7. Further if we choose = κ = 0, then we get (2.6.9) of [51] as stated in Theorem 2.6.3. 4. If we substitute f (t, s) = h(t)h(s), here h be absolutely continuous function, also let h  < ∞ and x = y in (2.6.18), then it gives (2.6.14) of [53] as stated in Theorem 2.6.6. Further if we choose = κ = 0, then we get (2.0.1) of [141] as stated in Theorem 2.0.1.

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

Corollary 2.6.10 If we take = κ = 0, x = then we get

183

c+d a+b and y = in (2.6.16), 2 2

     b   1 c+d f a + b , c + d − dt f t,  2 2 (b − a) a 2    d   b d  1 1 a+b , s ds + − f f (t, s)dsdt  (d − c) c 2 (b − a)(d − c) a c 0  1 0 2 1 (b − a)(d − c) r 0 ∂ f (t, s) 0 0 0 ≤ 0 ∂t∂s 0 . 4 (r + 1)2 q The above inequality is Corollary 5 of [54]. Corollary 2.6.11 If we take = κ = 0, x = then we get

c+d a+b and y = in (2.6.18), 2 2

     b   c+d 1 f a + b , c + d − f t, dt  2 2 (b − a) a 2    d   b d  1 a+b 1 , s ds + f f (t, s)dsdt  − (d − c) c 2 (b − a)(d − c) a c 0 2 0 (b − a)(d − c) 0 ∂ f (t, s) 0 0 0 ≤ 0 ∂t∂s 0 . 16 ∞ The above inequality is Corollary 2.2 of [16]. Remark 2.6.5 It is easy to see that in all our results, we get better bounds for a+b c+d 1 substituting x = ,y = and = κ = . 2 2 2 Remark 2.6.6 We can also get many interesting results by varying the values of q and r in our main result (2.6.16). The case q = r = 2 is of special interest.

2.6.3 Generalized Grüss Type Inequalities ˇ Grüss type inequalities usually provide the estimation of Cebyšev bounded functional. In the current subsection, we would like to generalize parametric based ˇ ˇ Cebyšev type inequalities of [66]. Cebyšev introduced the following inequality in his article [30] for two absolutely continuous functions, in the literature this inequality is named as Grüss inequality which is obtained by classical Montgomery identity defined previously in the Theorem 1.1.1. This inequality gives the estimation of

184

2 Ostrowski Inequality

bounded functional for two absolutely continuous functions. Here is the inequality as given in the Theorem stated below: Theorem 2.6.12 Let f, g : I → R be two absolutely continuous function such that f , g ∈ L∞ spaces, for x ∈ [a, b]. Then we have |T (f, g)| ≤

1 (b − a)2 f ∞ g ∞ , 12

(2.6.19)

ˇ where Cebyšev functional T (f, g) defined as in (1.1.57). Pachpatte [143] obtained the another generalized of (2.6.19), which states that: Theorem 2.6.13 Let f, g : I → R be two absolutely continuous function such that 1 1 ˇ f , g ∈ Lq [a, b] spaces where 1 ≤ q ≤ ∞, + = 1 and T (f, g) is a Cebyšev q r functional defined in (1.1.57). Then 1 |T (f, g)| ≤ f q g q (b − a)3



b

2

(B1 (x)) r dx

(2.6.20)

a

and |T (f, g)| ≤

1 2(b − a)2



b



 1 |g(x)|f q + |f (x)|g q (B1 (x)) r dx, (2.6.21)

a

where B1 (x) is defined as in (2.6.10), ∀ x ∈ [a, b]. In 2011, Gauezane-Lakoud and Aissaoui in [66] extended this inequality for two independent variable as can be seen in the following Theorem: Theorem 2.6.14 Let f, g : I × J → R be differentiable functions such that their ∂ 2 f (t, s) ∂ 2 g(t, s) second order partial derivatives and are integrable on I × J . ∂t∂s ∂t∂s Then |T∗ (f, g)| ≤

0 ∂ 2 f (t, s) 0 0 ∂ 2 g(t, s) 0 49 0 0 0 0 (b − a)2 (d − c)2 0 0 0 0 (2.6.22) ∞ ∞ 3600 ∂t∂s ∂t∂s

and 1 8(b − a)2 (d − c)2  b d 0 ∂ 2 f (t, s) 0 0 ∂ 2 g(t, s) 0  0 0 0 0 |g(x, y)| 0 0 + |f (x, y)| 0 0 ∞ ∞ ∂t∂s ∂t∂s a c ,+ , + 2 2 2 2 (2.6.23) × (x − a) + (b − x) (y − c) + (d − y) dydx,

|T∗ (f, g)| ≤

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

185

where T∗ (f, g) =

1 (b − a)(d − c)

−

1 (b − a)2 (d − c)

1 − (b − a)(d − c)2





b

d

f (x, y)g(x, y)dydx a

c





b



d

b

g(x, y) 

1 + 2 (b − a) (d − c)2

a

c



b

f (t, y)dtdydx a



d

d

g(x, y) a

c



f (x, s)dsdydx c

b d



b



d

g(x, y)dydx a

c

f (t, s)dsdt. a

c

(2.6.24) In recent years, a number of research papers related to Grüss type inequality and ˇ its Cebyšev functional have been published, we may mention the works [24, 66, 144–146, 156]. Now we would like to generalize Grüss type inequalities of [66] for functions of Lq space and by introducing some parameters. Theorem 2.6.15 Let f, g : I × J → R be differentiable functions such that their ∂ 2 g(t, s) ∂ 2 f (t, s) and are integrable on I o × J o second order partial derivatives ∂t∂s ∂t∂s 1 1 and are bounded in Lq spaces where 1 ≤ q ≤ ∞ and + . Then q r |T1 (f, g; , κ)| ≤ ⎡

1 (b − a)3 (d − c)3 (r +

4 1) r

0 ∂ 2 f (t, s) 0 0 ∂ 2 g(t, s) 0 0 0 0 0 0 0 0 0 q q ∂t∂s ∂t∂s

)2 r+1   r+1     b−a b − a r+1 r ⎣ x − a + b−a −x + b− +2 2 2 2 c

 b d a

(

r+1   r+1  d−c d −c + d −κ −y y− c+κ 2 2

)2   d − c r+1 r +2 κ dydx, 2

(2.6.25)

where T1 (f, g; , κ) =

(1 − )2 (1 − κ)2 (b − a)(d − c) −

b d



(1 − 2 )(1 − κ)2 (b − a)2 (d − c)

f (x, y)g(x, y)dydx a

c



b





d

b

g(x, y) a

c

f (t, y)dtdydx a

186

2 Ostrowski Inequality

−

(1 − )2 (1 − 2κ) (b − a)(d − c)2



b





d

d

g(x, y) a

c

f (x, s)dsdydx c

 b d   2(2 κ − − κ) + 1 b d g(x, y)dydx f (t, s)dsdt (b − a)2 (d − c)2 a c a c    1 b d 1 − F ψ ,κ (g) + Gψ ,κ (f ) + ψ ,κ (f )ψ ,κ (g) dydx, 2 a c 2 +

(2.6.26) where , κ ∈ [0, 1]. ˜ be defined as follows Proof Let F , G, F˜ and G F

F˜ G

 (1 − κ) b = (1 − )(1 − κ)f (x, y) − f (s, y)ds (b − a) a   b d (1 − ) d 1 1 − f (x, t)dt + f (t, s)dsdt − ψ ,κ (f ), (d − c) c (b − a)(d − c) a c 2  b d ∂ 2 f (t, s) 1 dsdt, K (x, t)Q1 (y, s) = (b − a)(d − c) a c ∂t∂s  (1 − κ) b = (1 − )(1 − κ)g(x, y) − g(s, y)ds (b − a) a   b d 1 1 (1 − ) d g(x, t)dt + g(t, s)dsdt − ψ ,κ (g) − (d − c) c (b − a)(d − c) a c 2

and ˜ = G

1 (b − a)(d − c)



b



d

K (x, t)Q1 (y, s) a

c

∂ 2 g(t, s) dsdt. ∂t∂s

Then using the condition, ˜ F G = F˜ G, 1 and integrate from a to b over x and (b − a)(d − c) integrate c to d over y, we get multiplying the resultant by

(1 − )2 (1 − κ)2 T1 (f, g; , κ) = (b − a)(d − c) −

(1 − 2 )(1 − κ)2 (b − a)2 (d − c)



b



d

f (x, y)g(x, y)dydx 

a b



c



d

b

g(x, y) a

c

f (t, y)dtdydx a

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

−

(1 − )2 (1 − 2κ) (b − a)(d − c)2



b





d

d

g(x, y) a

c

187

f (x, s)dsdydx c

   b d 2(2 κ − − κ) + 1 b d g(x, y)dydx f (t, s)dsdt (b − a)2 (d − c)2 a c a c    1 1 b d F ψ ,κ (g) + Gψ ,κ (f ) + ψ ,κ (f )ψ ,κ (g) dydx − 2 a c 2  b  d  b  d  1 ∂ 2 f (t, s) = K (x, t)Q (y, s) dsdt

1 (b − a)3 (d − c)3 a c ∂t∂s a c    b d  ∂ 2 g(t, s) × K (x, t)Q1 (y, s) dsdt dydx. ∂t∂s a c +

Applying absolute, we get  b  d  b  d   1 ∂ 2 f (t, s)   (x, t)Q (y, s) K dsdt

1 (b − a)3 (d − c)3 a c ∂t∂s a c    b d  ∂ 2 g(t, s)   × K (x, t)Q1 (y, s) dsdt dydx. ∂t∂s a c

|T1 (f, g; , κ)| ≤

Using Hölder’s inequality, we get |T1 (f, g; , κ)|

 b  d  b  d 1 1 r r |K (x, t )Q (y, s)| dsdt

1 (b − a)3 (d − c)3 a c a c 0   b  d 1 0 2 r 0 ∂ g(t, s) 0 r 0 0 dydx |K (x, t )Q1 (y, s)| dsdt 0 × ∂t ∂s 0q a c 0 2 0 0 0 0 ∂ f (t, s) 0 0 ∂ 2 g(t, s) 0 1 0 0 0 0 = (b − a)3 (d − c)3 0 ∂t ∂s 0q 0 ∂t ∂s 0q  b  d  b  d 2 r |K (x, t )Q1 (y, s)|r dsdt dydx × ≤

a

=

c

a

0 2 0 0 ∂ f (t, s) 0 0 0 0 ∂t ∂s 0

q

c

0 2 0 0 0  b d 0 ∂ f (t, s) 0 0 ∂ 2 g(t, s) 0 1 0 0 0 0 H (x, y)2 dydx (b − a)3 (d − c)3 0 ∂t ∂s 0q 0 ∂t ∂s 0q a c

0 ∂ 2 f (t, s) 0 0 ∂ 2 g(t, s) 0 0 0 0 0 0 0 0 0 q q ∂t ∂s ∂t ∂s (b − a)3 (d − c)3 (r + 1) ⎡ r+1   r+1  )2    b d  b−a b−a b − a r+1 r ⎣ x− a+ −x × + b− +2 2 2 2 a c ≤

1

4 r

188

2 Ostrowski Inequality

( ×

r+1   r+1  )2   d −c d −c d − c r+1 r −y y− c+κ + d −κ +2 κ dydx. 2 2 2

Remark 2.6.7 If we substitute = κ = 0 in (2.6.25), we get 0 ∂ 2 f (t, s) 0 0 ∂ 2 g(t, s) 0 1 0 0 0 0 0 0 0 0 q q (b − a)3(d − c)3 ∂t∂s ∂t∂s  b d 2 2 × [B1 (x)] r [B2 (y)] r dydx,

|T∗ (f, g)| ≤

a

(2.6.27)

c

where T∗ (f, g) is defined in (2.6.24). The above result is generalized case for Lp spaces of (2.6.22) of [66]. Corollary 2.6.16 If we substitute r = 1 and p → ∞ in (2.6.25), then we get 0 0 2 0 2 (b − a)2 (d − c)2 0 0 ∂ f (t, s) 0 0 ∂ g(t, s) 0 0 0 0 0 ∞ ∞ 144 ∂t∂s ∂t∂s    7 7 + (1 − ){3(1 − ) − 4} + κ(1 − κ){3(1 − κ) − 4} , × 5 5

|T1 (f, g; , κ)| ≤

(2.6.28) where T1 (f, g; , κ) is defined as in (2.6.26). Remark 2.6.8 If we substitute = κ = 0 in (2.6.28), or r = 1, then p → ∞ in (2.6.27), then we get (2.6.22) of [66] as stated in the Theorem 2.6.14. Now we are going to present the second main result of Grüss inequality. Theorem 2.6.17 Let all assumptions of Theorems 2.6.15 be valid. Then |T2 (f, g, , κ)| 1

≤

2

2(b − a)2 (d − c)2 (r + 1) r  b d 0 ∂ 2 g(t, s) 0  0 ∂ 2 f (t, s) 0 0 0 0 0 × |g(x, y)| 0 0 + |f (x, y)| 0 0 q q ∂t∂s ∂t∂s a c (   r+1   r+1  )1 b−a b−a b − a r+1 r × x− a+ + b− +2 −x 2 2 2 (

    r+1  )1  d−c d − c r+1 r d − c r+1 + d−κ +2 κ dydx, −y y− c+κ 2 2 2

(2.6.29)

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

189

where (1 − )(1 − κ) (b − a)(d − c)

T2 (f, g; , κ) = − −

(1 − κ) (b − a)2 (d − c) (1 − ) (b − a)(d − c)2



b



d

f (x, y)g(x, y)dydx a



c

b





d

b

g(x, y) a



b



c

f (t, y)dtdydx 

d

a d

g(x, y) a

f (x, s)dsdydx

c

c

 b d  b d 1 g(x, y)dydx f (t, s)dsdt + (b − a)2 (d − c)2 a c a c  b d  1 f (x, y)ψ ,κ (g) + g(x, y)ψ ,κ (f ) dydx, − 2(b − a)(d − c) a c

(2.6.30) where , κ ∈ [0, 1]. Proof where identity (2.6.11) to the function g, we get (1 − κ) b−a

(1 − )(1 − κ)g(x, y) = − +



b

g(t, y)dt +

a

1 (b − a)(d − c) 1 (b − a)(d − c)

 

b



b



a

d

c

(1 − ) d−c



d

g(x, s)ds c

1 g(t, s)dsdt + ψ ,κ (g) 2

d

K (x, t)Q1 (y, s) a

c

∂ 2 g(t, s) dsdt. ∂t∂s (2.6.31)

1 1 g(x, y), (2.6.31) by f (x, y), (b − a)(d − c) (b − a)(d − c) summing the resultant identities, then integrate from a to b over x and integrate c to d over y, we obtain Multiplying (2.6.4) by

(1 − )(1 − κ) (b − a)(d − c)



b

−

d

f (x, y)g(x, y)dydx a

(1 − κ) = (b − a)2(d − c) +



(1 − ) (b − a)(d − c)2

c



b





d

b

g(x, y) 

1 (b − a)2 (d − c)2

a b



c

f (t, y)dtdydx



d

a

g(x, y) a

c



b d

d

f (x, s)dsdydx c



b



d

g(x, y)dydx a

c

f (t, s)dsdt a

c

190

+

2 Ostrowski Inequality

1 2(b − a)(d − c)

b d

 a

  f (x, y)ψ ,κ (g) + g(x, y)ψ ,κ (f ) dydx,

c

 b d  d ∂ 2 f (t, s) 1 dtds g(x, y) K (x, t)Q (y, s) +

1 ∂t∂s 2(b − a)2 (d − c)2 a c a c   b d ∂ 2 g(t, s) dtds dydx, (2.6.32) + f (x, y) K (x, t)Q1 (y, s) ∂t∂s a c 

b

from that we deduce, 1 2(b − a)2 (d − c)2  b d  × g(x, y)

T2 (f, g; , κ) =



∂ 2 f (t, s) dtds ∂t∂s a c a c   b d ∂ 2 g(t, s) K (x, t)Q1 (y, s) dtds dydx. + f (x, y) ∂t∂s a c b

d

K (x, t)Q1 (y, s)

(2.6.33)

Consequently taking absolute value on it and then applying Hölder’s Inequality, we have 1

|T2 (f, g; , κ)| ≤

2

2(b − a)2 (d − c)2 (r + 1) r  b d 0 ∂ 2 g(t, s) 0  0 ∂ 2 f (t, s) 0 0 0 0 0 × |g(x, y)| 0 0 + |f (x, y)| 0 0 q q ∂t∂s ∂t∂s a c (  r+1   r+1 b−a b−a × x− a+ + b− −x 2 2 

b−a +2 2  +

 r+1 ) 1r (  d − c r+1 y− c+κ 2

d−c d−κ 2



r+1 −y



d −c +2 κ 2

r+1 ) 1r dydx.

(2.6.34)

Remark 2.6.9 If we substitute = κ = 0 in (2.6.29), then we get 1 2(b − a)2 (d − c)2  b d 0 ∂ 2 f (t, s) 0 0 ∂ 2 g(t, s) 0  0 0 0 0 × |g(x, y)| 0 0 + |f (x, y)| 0 0 q q ∂t∂s ∂t∂s a c

|T∗ (f, g)| ≤

1

1

× [B1 (x)] r [B2 (y)] r dydx,

(2.6.35)

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery. . .

191

where T∗ (f, g) is defined as in (2.6.24). Remark 2.6.10 If we substitute r = 1 and q → ∞ in (2.6.29), then we get 1 8(b − a)2 (d − c)2  b d 0 ∂ 2 f (t, s) 0 0 ∂ 2 g(t, s) 0  0 0 0 0 |g(x, y)| 0 0 + |f (x, y)| 0 0 ∞ ∞ ∂t∂s ∂t∂s a c   × (x − a)2 + (b − x)2 − (1 − )(b − a)2   × (y − c)2 + (d − y)2 − κ(1 − κ)(d − c)2 dydx, (2.6.36)

|T2 (f, g, , κ)| ≤

where T2 (f, g; , κ) is defined as in (2.6.30). Remark 2.6.11 If we substitute = κ = 0 in (2.6.36), then we get inequality (2.6.23) of [66] as stated in Theorem 2.6.14. Remark 2.6.12 We can get many interesting inequalities by varying the values of

and κ. It is to be noted that the better bound for (2.6.25) and (2.6.29) is derived 1 from , κ = . 2

Chapter 3

Functions with Nondecreasing Increments

Mathematics has been called the science of tautology; that is to say, mathematicians have been accused of spending their time proving that things are equal to themselves. This statement (appropriately by a philosopher) is rather inaccurate on two counts. In the first place, mathematics, although the language of science, is not a science. Rather, it is a creative art. Secondly, the fundamental results of mathematics are often inequalities rather than equalities. —Edwin F. Beckenbach and Richard Bellman

The main purpose of the present chapter is to establish the relationship of functions with nondecreasing increments (FWNDI) with other functions of high importance. Our special emphasis will be on the role of representation and connection among functions with nondecreasing increments and arithmetic integral mean, Wright convex functions, convex functions, ∇-convex functions, Jensen m-convex functions, m-convex functions, m-∇-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator as different cases of m h f . We will also consider function with nondecreasing increments of order three and get a generalization of the Levinson’s type inequality and JensenMercer’s type inequality by using Jensen-Boas inequality and also deduce some results. Some contents of the current chapter have been published in 2020 see [87]. In the year 1964, Brunk briefly discussed the “functions with nondecreasing increments in his research article [25, pp. 784]. According to him, Ciesielski (1957–1958) was a man who floated and introduced the idea of functions with nondecreasing increments in [38]”. But Brunk has considerable contribution to raising and exposing the importance of FWNDI in a broader way, he introduced an interesting class of multivariate real-valued functions namely functions with nondecreasing increments and established important results, examples and properties using that class of functions. For further details about the historical literature of functions with nondecreasing increments (see [87, 118]). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Irshad et al., New Perspectives on the Theory of Inequalities for Integral and Sum, https://doi.org/10.1007/978-3-030-90563-7_3

193

194

3 Functions with Nondecreasing Increments

Book [135] having detailed discussion on properties of this distinct topic, i.e., functions with nondecreasing increments. In the past few decades, this topic has gained popularity in several branches of Mathematics, which has increased interest in the study of functions with nondecreasing increments by using finite difference operators. There are interesting topics in numerical methods, differential equations, physics, biology and engineering that play an important role where we use finite difference operators (see [106]). There are many applications of these operators in the areas such as; networking, probabilistic, fractal and random media, fractionally dependent components, applications to mechanics, controls theory, transport phenomena, fractional equations and chaos, and future ideas as documented in the book [174]. Finally, these operators are naturally connected to different inequalities; various general inequalities for functions with nondecreasing increments, for present contribution see [85]. Let us recall a few important definitions and significant results extracted from [97, 113, 126, 158, 188]. Throughout the context we will use I to be an interval in R, I and [a, b] both intervals in Rk . We start some notations to recall the definition of function with nondecreasing increments as follows: Let Rk represent k-dimensional vector lattice of elements x = (x1 , x2 , . . . , xk ), xi be real, with partial ordering “ ≤ " on Rk is here stated as (x1 , x2 , . . . , xk ) ≤ (y1 , y2 , . . . , yk ) ⇐⇒ x1 ≤ y1 , . . . , xk ≤ yk , that is, xi ≤ yi ; i ∈ {1, 2, . . . , k}. We denote ax + by = (ax1 + by1 , . . . , axk + byk ), where 0 stands for k-tuple (0, . . . , 0) and a, b ∈ R. While a, b ∈ Rk , b ≥ a, a set {x ∈ Rk : a ≤ x ≤ b} should be in the interval [a, b]. H. D. Brunk stated the important definition below: Definition 3.0.1 A function f : I → R is said to have nondecreasing increments if following inequality f (a + h) − f (a) ≤ f (b + h) − f (b), holds, where I ⊂ Rk and k is a fixed positive integer, 0 Rk , b ≥ a; a, b + h ∈ I.

(3.0.1) ≤

h

∈

Brunk also stated that inequality (3.0.1) does not imply continuity even if k = 1. Further, Wright-convex is the one dimension case of above definition of function with nondecreasing increments. Now we would give some examples and properties of function with nondecreasing increments which were given by Brunk in paper [25]. Also see [96] for more discussion.

3 Functions with Nondecreasing Increments

195

Examples of FWNDI (i) The simplest example of a FWNDI is a constant function. (ii) Lines in the form x = at + b, where (0, . . . , 0) ≤ a ∈ Rk , b ∈ Rk whose direction cosines are nonnegative, also belong to the family of functions with nondecreasing increments. (iii) An important continuous function ϑ : R2 → R stated as ϑ(x, y) = xy is a function with nondecreasing increments.  (iv) A continuous function ν : [0, ∞)k → R stated as ν(x) = ki=1 xi is another useful function with nondecreasing increments. (v) F (x + y) = F (x) + F (y) is the Cauchy functional equation which is an interesting and widely used example of such functions. Properties of FWNDI FWNDI possesses the following properties: (i) A FWNDI need not be continuous. (ii) If function f : I → R has 1st order partial derivatives ∀ x ∈ I. Then f has nondecreasing increments iff every of those partial derivatives is nondecreasing in every arguments. (iii) If f : I → R has partial derivatives of 2nd order ∀ x ∈ I. Then f has nondecreasing increments iff every of those partial derivatives is non-negative. (iv) A function υ : [0, 1] → R is convex, stated as υ(t) = f (ta +b), if f FWNDI is continuous in b ≤ x ≤ b + a; 0 ≤ a ∈ Rk . In many fields of mathematics several types of differences are used such as finite difference, forward difference, backward difference, divided difference etc. From application point of view there are large number of implementation of these types of differences in the fields as numerical analysis, statistics, vector calculus and physics (see books [2, 68, 179]), we have chosen ∇- operator for our book due to its wide range of application. Now we would like to define further generalized convex functions that can be seen in [95, 97] and [158]. Definition 3.0.2 A function f : I → R, is known as ∇-convex of mth order or m-∇-convex, if ∀ (m + 1) different points xi , xi+1 , . . . , xi+m we have ∇(m) f (xi ) = (−1)m (m) f (xi ) ≥ 0 where (m) f (xi ) represents m-th order divided difference of function f as defined in Definition 1.0.4. We refer to the book [158], for further details of other results about the higher order convex functions. Definition 3.0.3 ([158]) The finite difference of a function of order m on I = [a, b] ∈ R, where m is non- negative integers, is defined by 0h f (x) = f (x) m−1 m f (x + h) − m−1 f (x), h f (x) = h h

196

3 Functions with Nondecreasing Increments

where h = 0, x + ih ∈ I ; i ∈ {0, 1, 2, . . . , m}. Then it can easy to write the statement as m h f (x) =

m  

m f (x + ih)(−1)m−i . i i=0

Further, we conclude that if function f : I → R is satisfied m h f (x) ≥ 0, then function is called Jensen m-convex for all x ∈ I and h > 0. We define here a special type of function which belongs to the class of function with nondecreasing increments and themselves connect/contain the class of other functions that are already provided in the starting section of this chapter, by using several cases of m hf. Here we also recall the important Definitions 3.0.3 and 1.0.4 of finite difference and divided difference of function respectively and we will use these definitions to connect with other definitions that will be seen in the next section. We present some remarks about the relationship among finite difference, divided difference and derivative of the function. Remark 3.0.1 Some important remarks are following: (i) Let us denote [xi , . . . , xi+m ; f ] by (m) f (xi ). The value [xi , . . . , xi+m ; f ] is independent of elements order xi , xi+1 , . . . , xi+m . (ii) We may extend Definition 1.0.4 by including case in which few elements or all elements coincide by supposing that xi ≤ · · · ≤ xi+m (see [158]) and letting [xi , . . . , xi+m ; f ] =

f (m) (xi ) , m!

provided that f (m) (xi ) exists. (iii) The following identity is valid (see [164]): m m h f (x) = m!h m f (x),

provided that xi ’s are equally spaced. Using finite difference operator, we would like to state alternative form of functions with nondecreasing increments as; h1 h2 f (a) ≥ 0, Since

(3.0.2)

h1 (h2 f (a)) = h1 (f (a + h2 ) − f (a)) = f (a + h2 + h1 ) − f (a + h2 ) − f (a + h1 ) + f (a) ≥ 0.

3.1 Functions with Nondecreasing Increments in Real Life

197

Setting h = h1 , b = a + h2 in above statement, then f (b + h) − f (b) − f (a + h) + f (a) ≥ 0. By taking h1 = h2 = h, then we can obtain special case of (3.0.2). h f (b) − h f (a) ≥ 0 h (f (b) − f (a)) ≥ 0 2h f (a) ≥ 0, where a ≤ b. We know that h1 f (x) = f (x + h1 ) − f (x) and further, h1 h2 · · · hm f (x) = h1 (h2 · · · hm f (x)) for m ≥ 2, where x, x + h1 + · · · + hm ∈ I, 0 ≤ hi ∈ Rk for i ∈ {1, 2, . . . , m}. Similarly we can extend this definition for mth order as: Definition 3.0.4 A function f : I → R is known as function with nondecreasing increments of mth order if h1 h2 · · · hm f (x) ≥ 0

(3.0.3)

holds, where x, x + h1 + · · · + hm ∈ I, 0 ≤ hi ∈ Rk for i ∈ {1, 2, . . . , m}. Then the special case is given by m h f (x) ≥ 0,

(3.0.4)

where f is called FWNDI of order m with equally spaced h.

3.1 Functions with Nondecreasing Increments in Real Life We can see the applications of functions with nondecreasing increments in the ultramodular function which is the special case of FWNDI but the only difference is that the range of ultramodular function is [0, 1] while the range of FWNDI is R and ultramodular function is stated as: Ultramodular Function A function f : I → [0, 1] is called ultramodular [168], if for all x, y ∈ I ∈ Rk with x ≤ y and for all h ∈ Rk with h ≥ 0 and x + h, y + h ∈ I. f (x + h) − f (x) ≤ f (y + h) − f (y).

198

3 Functions with Nondecreasing Increments

Statistics In Statistics, ultramodular functions play an important role in modelling stochastic orders and positive dependence among random vectors [177]. Economics In Economics, ultramodular functions f : I → [0, 1] are often said to have nondecreasing increments [168]. Copulas Class of absolutely monotonic functions is a subclass of completely monotonic function and this is subclass of FWNDI if differentiability exists. Absolutely monotonic function is used in copulas [136] and there are many applications of copulas in various branches of mathematics, economics, engineering and medicine which have been highlighted after the following definition. Copula is used in probability theory and specially it is general tool to construct multivariate distributions and to investigate dependence structure between random variables [84]. A function C : [0, 1] × [0, 1] → [0, 1] is a bivariate copula if C(0, u) = C(u, 0) = 0, C(1, u) = C(u, 1) = u and C(u2 , v2 ) − C(u2 , v1 ) − C(u1 , v2 ) + C(u1 , v1 ) ≥ 0 for all 0 ≤ u1 ≤ u2 ≤ 1 and 0 ≤ v1 ≤ v2 ≤ 1. Quantitative Finance In quantitative finance, copulas are applied to risk management, to portfolio management and optimization [112]. Civil Engineering Recently, copula functions have been successfully applied to the database formulation for the reliability analysis of highway bridges, and to various multivariate simulation studies in civil [182], reliability of wind and earthquake engineering [191], mechanical and offshore engineering [195]. Reliability Engineering Copulas are being used for reliability analysis of complex systems of machine components with competing failure modes [161]. Warranty Data Analysis Copulas are being used for warranty data analysis in which the tail dependence is analysed [189]. Medicine Copula functions have been successfully applied to the analysis of neuronal dependencies [57] and spike counts in neuroscience [140]. Solar Irradiance Variability Copulas have been used to estimate the solar irradiance variability in spatial networks and temporally for single locations [137]. Hydrology Research Copulas are used for research of hydrology for more information see in [107].

3.2 Relationship Among Functions with Nondecreasing Increments and Many. . .

199

Climate and Weather Research Copulas are used in climate and weather research for more detailed we can see in [173]. The current chapter has an aim to collect the established facts about the FWNDI, together with some other important functions and notions, that can help out for finding whether a given function is FWNDI or not. In the second section of current chapter, we would like to establish the connection among functions with nondecreasing increments and many other functions by using finite difference operator as different cases of m h f with detailed examples. In the third section, we would like to obtain the generalization of the Levinson’s type inequality and JensenMercer’s type inequality by using Jensen-Boas inequality for FWNDI of order 3 and also deduce some results.

3.2 Relationship Among Functions with Nondecreasing Increments and Many Others We wouldfs like to use the Definition 3.0.4 to establish relationship among functions with nondecreasing increments and many other functions, the detailed list of other functions is already mentioned in the initial section. In current section we would like to recall some important definitions which are extracted from the articles [3, 22, 41, 59, 62, 71, 85, 97, 101, 113, 119, 121, 122, 124, 126, 128, 135, 147, 157, 158, 188] and these will relate to functions with nondecreasing increments, by using the finite difference operator as different cases of m h f ≥ 0. In this connection we will use the relationships of finite difference, derivative and some other differences (see [41, 62, 158, 164]). Arithmetic Integral Mean vs FWNDI Let us have A is mean of arithmetic integral of function f on [0, a] (see [85, 105]). Definition 3.2.1 A function A (nondecreasing function) is said to be a Arithmetic integral mean in the interval [0, a], such that 1 A(t) = t



t

f (x)dx, 0

provided that f : [0, a] → R is a non-negative and nondecreasing, where a > 0. Now, we will recall extension of previously mentioned result to FWNDI of higher order, extracted from [85]. Theorem 3.2.1 Let f : [a, b] → R be continuous and with nondecreasing increments of mth order. Then A is stated as A(t) =

 k # i=1

−1  (ti − ai )

t1 a1

 ···

tk

f (v)dv, ak

200

3 Functions with Nondecreasing Increments

is a FWNDI of order m in the interval [a, b], where v = (v1 , . . . ,vk ) and dv = dv1 · · · dvk . The Alternative form of Arithmetic integral mean of order m in the interval [a, b], using Eq. (3.0.3) it is defined by h1 h2 · · · hm A(t) ≥ 0,

hi ≥ 0.

Remark 3.2.1 By taking h1 = h2 = · · · = hm in above inequality, then obtain special case as m h A(t) ≥ 0. Example 3.2.1 Let f : [0, a] → R+ be a function, stated as 2

f (x) = ex−c . m Since m h f (x) ≥ 0, then from the above remark also holds h A(t) ≥ 0 when h is very small, therefore we can say that A is Arithmetic integral mean of mth order in the interval [0, a] for every a > 0, c ∈ R.

Now first of all we would like to start from 1st dimensional case of function with nondecreasing increments which is Wright–convex function and we will give the equivalent form. Wright–Convex Function vs FWNDI Definition 3.2.2 ([158]) A function f : [a, b] → R is known as Wright–convex, if following inequality is valid ∀ y ≥ x; z ≥ 0; x, y + z belong to [a, b]. f (y + z) − f (y) ≥ f (x + z) − f (x) It can also be written as 2z f (x) ≥ 0 x ≤ y; x, y + z ∈ [a, b]. This is computed same as function with nondecreasing increments, i.e., 2h f (x) ≥ 0.

(3.2.1)

If there exists f

, then f is Wright–convex function iff h2 f

(x) ≥ 0.

(3.2.2)

And the equivalent form of (3.2.2) on [a, b] ∈ I ⊂ R, using Eq. (3.0.4) it is also defined by same inequality (3.2.1), when h is very small. Example 3.2.2 Let f : [a, b] → R be a function, stated as f (x) = x(ax − b). Since h2 f

(x) ≥ 0, i.e., 2h f (x) ≥ 0, therefore f is Wright–convex in the interval [a, b] ⊂ R for every a ≥ 0, b ∈ R.

3.2 Relationship Among Functions with Nondecreasing Increments and Many. . .

201

Remark 3.2.2 Wright-convex function is a special case of FWNDI for k = 1. Now we would like to state generalized convex function may be seen in [97, 158]. m-Convex Function vs FWNDI Definition 3.2.3 A function f : I → R is known as m-convex, if the inequality (m) f (x) ≥ 0 holds ∀ (m + 1) different points x0 , x1 , . . . , xm ∈ I . Special case of convex function of order m, using Eq. (3.0.4) and Remark 3.0.1 it is defined by m h f (x) ≥ 0, m!hm

x ∈ I, h > 0.

(3.2.3)

If there exists f (m) , then function is m-convex or mth order convex iff f (m) (x) ≥ 0. m!

(3.2.4)

And the equivalent form of (3.2.4) on I ⊂ R, using Eq. (3.0.4) it is also defined by same inequality (3.2.3), when h is very small. Example 3.2.3 Let f : I → R be a function, stated as f (x) =

xm . m!

m f (x) f (m) (x) Since ≥ 0, then h m ≥ 0, therefore f is m-convex in the interval I m! m!h for q > 0, m ∈ {0, 1, 2, · · · }. m-∇-Convex Function vs FWNDI Definition 3.2.4 A function f : I → R is known as m-∇-convex, if ∀ (m + 1) different points x0 , x1 , . . . , xm ∈ I we have ∇(m) f (x) = (−1)m (m) f (x) ≥ 0. Special case of ∇-convex of order m, using Eq. (3.0.4) and Remark 3.0.1 it is defined by (−1)m m h f (x) ≥ 0, m!hm

x ∈ I, h > 0.

(3.2.5)

If there exists f (m) , then function is m-∇-convex iff (−1)m f (m) (x) ≥ 0. m!

(3.2.6)

And the equivalent form of (3.2.6) on I ⊂ R, using Eq. (3.0.4) it is also defined by same inequality (3.2.5), when h is very small.

202

3 Functions with Nondecreasing Increments

Example 3.2.4 Let f : I → R be a function, stated as f (x) =

e−rx . rm

(−1)m m (−1)m f (m) (x) h f (x) ≥ 0, therefore f is m-∇-convex in ≥ 0, i.e., m! m!hm the interval I ⊂ R∗ , where r ∈ R\{0}, m ∈ {0, 1, . . .}.

Since

Remark 3.2.3 Convex function and ∇-convex function are the special cases of mconvex function and m-∇-convex respectively, if we put m = 2. Jensen m-Convex Function vs FWNDI Definition 3.2.5 A function f : I → R is known as Jensen m-convex or J-convex of order m if holds m h f (x) ≥ 0,

∀ h > 0 and x ∈ I.

(3.2.7)

This is done by finite difference operator and if there exists f (m) , then f is J– convex of order m iff hm f (m) (x) ≥ 0.

(3.2.8)

And the equivalent form of (3.2.8) on I ⊂ R, using Eq. (3.0.4) is also defined by same inequality (3.2.7), when h is very small. Example 3.2.5 Let a function f : I → R which is stated as f (x) =

eqx . qm

Since hm f (m) (x) ≥ 0, then m h f (x) ≥ 0, therefore f is J -convex of mth order in the interval I ⊂ R∗ for q ∈ R\{0}, m ∈ {0, 1, 2, . . .}. Remark 3.2.4 J -convex function is the special case of J -convex function of mth order, if we put m = 2. Now we recall definitions of convex function and ∇-convex function and their connection with FWNDI using Eq. (3.0.4). Convex Function vs FWNDI Definition 3.2.6 ([158]) A continuous function f : I → R is known as convex, if there exists non-negative second order divided difference, such that 2 f (x) ≥ 0,

x ∈ I.

3.2 Relationship Among Functions with Nondecreasing Increments and Many. . .

203

Special case of convex function on I ⊂ R, using Eq. (3.0.4) and Remark 3.0.1 it is defined by 2h f (x) ≥ 0, 2!h2

x ∈ I, h > 0.

(3.2.9)

If there exists f

, then function is convex iff f

(x) ≥ 0. 2!

(3.2.10)

And the equivalent form of (3.2.10) on I ⊂ R, using Eq. (3.0.4) it is also defined by same inequality (3.2.9), when h is very small. Example 3.2.6 Let f : I → R∗ be a function, stated as f (x) = mx 2 + m2 . 2 f (x) f

(x) ≥ 0, i.e., h 2 ≥ 0, therefore f is convex in the interval I for Since 2! 2!h m ∈ {0, 1, 2, · · · }. Remark 3.2.5 If sets, “C” convex function, “W” Wright-convex function and “J” Jensen convex function then C ⊂ W ⊂ J . Moreover, each inclusion is proper (see [135, 158]). ∇-Convex Function vs FWNDI Definition 3.2.7 [158] A continuous function f : I → R is known as ∇-convex, if there is existence of ∇2 f (x) = (−1)2 2 f (x) and satisfy ∇2 f (x) ≥ 0,

x ∈ I.

Special case of ∇-convex function on I ⊂ R, using Eq. (3.0.4) and Remark 3.0.1 it is defined by (−1)2 2h f (x) ≥ 0, 2!h2

x ∈ I, h > 0.

(3.2.11)

If there exists f

, then f is ∇-convex iff (−1)2f

(x) ≥ 0. 2!

(3.2.12)

And the equivalent form of (3.2.12) on I ⊂ R, using Eq. (3.0.4) it is also defined by same inequality (3.2.11), when h is very small.

204

3 Functions with Nondecreasing Increments

Example 3.2.7 Let f : R+ → R be a function, stated as f (x) = − ln x. (−1)2 2h f (x) (−1)2 f

(x) ≥ 0, therefore f is ∇-convex in the ≥ 0, i.e., 2! 2!h2 interval R+ . Since

Completely Monotonic Function vs FWNDI Definition 3.2.8 ([59, 113, 128]) A continuous function f is known as completely monotonic, if there is existence of derivatives of all orders on I ⊂ R, and satisfy (−1)i f (i) (x) ≥ 0,

i ∈ {0, 1, . . .};

x ∈ I.

Equivalent form of completely monotonic function on I ⊂ R, using Eq. (3.0.4) it is defined by (−1)i ih f (x) ≥ 0, hi

i ∈ {0, 1, . . .};

x ∈ I, h > 0,

when h is very small. Example 3.2.8 Some examples of completely monotonic functions are following: α (i) f (x) = 1−α , 0 ≤ α ≤ 1, x > 0. x (ii) f (x) = − ln(1 − 1/x), for every x ∈ R+ . (iii) f (x) = e1/x , for every x ∈ R+ . Completely monotonic function is generalized form of absolutely monotonic function and k-monotonic function, now we will give connection of absolutely monotonic function and k-monotonic function with FWNDI using Eq. (3.0.4). Absolutely Monotonic Function vs FWNDI Definition 3.2.9 ([59, 113]) A continuous function f is known as absolutely monotonic, if there is existence of derivatives of all orders on I ⊂ R, and satisfy f (i) (x) ≥ 0,

i ∈ {0, 1, . . .};

x ∈ I.

Equivalent form of absolutely monotonic function on I ⊂ R, using Eq. (3.0.4) it is defined by ih f (x) ≥ 0, hi when h is very small.

i ∈ {0, 1, 2, . . .};

x ∈ I, h > 0,

3.2 Relationship Among Functions with Nondecreasing Increments and Many. . .

205

Example 3.2.9 Let a function f : [−1, 1] → R which is stated as f (x) = arcsin x. Since f (i) (x) ≥ 0, then interval [0, 1).

ih f (x) ≥ 0, therefore f is absolutely monotonic in the hi

k-Monotonic Function vs FWNDI Definition 3.2.10 ([59, 113]) A function is known as k–monotonic on I ⊂ R, if all its derivatives f (i) (x) exist and satisfy (−1)i f (i) (x) ≥ 0,

i ∈ {0, 1, . . . , k}, where k is fixed;

x ∈ I.

Equivalent form of k–monotonic function on I ⊂ R, using Eq. (3.0.4) it is defined by (−1)i ih f (x) ≥ 0, hi

i ∈ {0, 1, . . . , k};

x ∈ I, h > 0,

when h is very small. Example 3.2.10 Let a function f : I → R which is stated as f (x) = x −r . (−1)i ih f (x) ≥ 0, therefore the function f is khi monotonic in the interval I ⊂ R∗ for r ≥ 0.

Since (−1)i f (i) (x) ≥ 0, then

Laplace Transform of f vs FWNDI Definition 3.2.11 ([188]) Let f be function which satisfies |f (x)| ≤ Meax and piecewise continuous, where a and M are real constants. Then the Laplace transform of f (x), stated as 

∞

F (s) = L{f (x)} =

f (x)e−sx dx,

s > a.

0

Similarly, the Laplace transformation of Borel measure ϕ(t) on R∗ is stated as 

∞

L{ϕ(t)} =

e−xt dϕ(t).

0

At origin the Laplace transformation is continuous iff ϕ is finite.

206

3 Functions with Nondecreasing Increments

Bernstein [22] proved that f in the interval R∗ is completely monotonic , iff there is existence of increasing function ϕ(t) on R∗ . 

∞

f (x) =

e−xt dϕ(t).

0

Remark 3.2.6 Equivalent form of above statement can also be present as a function (−1)i ih f (x) ≥ 0 (completely monotonic) f on R∗ satisfied the condition hi where h is very small and x ∈ I ⊂ R∗ , i ∈ {0, 1, . . .} iff there exists 

∞

f (x) =

e−xt dϕ(t),

0

here ϕ(t) is an increasing function in the interval R∗ . Exponentially Convex Function vs FWNDI Definition 3.2.12 ([97, 158]) A continuous function ω : I → R on open interval I is called exponentially convex, if m

 ρi ρj ω xi + xj ≥ 0,

i,j =1

∀ m ∈ N and ∀ ρi , ρj ∈ R; such that xi + xj ∈ I and i, j ∈ {1, . . . , m}. For detailed discussion on exponentially convex function we refer the readers to Chap. 1 (See Sect. 1.2.4). Theorem 3.2.2 The function ω : I → R in the interval I is exponentially convex iff  ω(x) =

∞ −∞

et x dϕ(t),

x ∈ I,

for some ϕ(t) : R → R (nondecreasing function). Proof See [3], p. 211.



Little less obvious examples can be deduced by applying above integral representation and some results from Laplace transform are following. Example 3.2.11 The following are examples of Exponentially convex functions as well as Laplace transform of f (x) through above theorem on R+ include: (i) (ii) (iii) (iv)

f (x) = x −α√, for every α > 0. f (x) = e−α√ x , for every α > 0. f (x) = e−x s , for every s > 0. f (x) = e−t x , for every t > 0.

3.3 Functions with Nondecreasing Increments of Order 3

207

Remark 3.2.7 The above functions are also satisfying the condition (−1)i ih f (x) ≥ 0 (completely monotonic) by using Eq. (3.0.4) in these examples, hi when h is very small.

3.3 Functions with Nondecreasing Increments of Order 3 Consider a common scenario that a car is stopped at a traffic signal, as light of signal blinks green if the accelerator of the car is pushed then the engine takes pickup, that in turns causes the j erk when the car is accelerating. J erk is an example a function with nondecreasing increments of order three. Before we give the generalization of Levinson’s type inequality and JensenMercer’s type inequality by using Jensen-Boas inequality for function with nondecreasing increments of order three, we must present following theorem about the inequality of Jensen-Steffensen type for a FWNDI which is collected from [148]. Here in current section [c, d] is an interval in R, I in Rk and Y(t) is a vector of functions. Theorem 3.3.1 Let Y : [c, d] → I be nondecreasing continuous map and let G ∈ BV [c, d] such that G(c) ≤ G(y) ≤ G(d),

G(c) < G(d).

If f : I → R is a continuous FWNDI, then  d f

holds, where

d c

Y dG =

c

Y(t) dG(t) d c dG(t)

 d c



d ≤

Y1 dG, . . . ,

c

d c

f (Y(t)) dG(t) d c dG(t)

 Yk dG .

We also give generalized form of above theorem using Jensen-Boas inequality for function with nondecreasing increments, extracted from [94] as below. Theorem 3.3.2 Let Y : [c, d] → I be a monotonic (either non-increasing or nondecreasing) and continuous map in every of j intervals (di−1 , di ). Let G : [c, d] → R be continuous or of BV satisfying G(c) ≤ G(c1 ) ≤ G(d1 ) ≤ G(c2 ) ≤ · · · ≤ G(dj −1 ) ≤ G(cj ) ≤ G(d)

(3.3.1)

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3 Functions with Nondecreasing Increments

for all ci ∈ (di−1 , di ) (d0 = c, dj = d), and G(d) > G(c). If f is continuous having nondecreasing increments in every of j intervals (di−1 , di ), then  d f

c

Y(t)dG(t) d c dG(t)



d ≤

c

f (Y(t))dG(t) . d c dG(t)

(3.3.2)

Remark 3.3.1 d (Y(t)) dG(t) ∈ I and ∀ y ∈ ci ∈ (di−1 , di ) (d0 = c, dj = d) we have (i) If c  d dG(t) c either G(y) ≥ G(d) or G(y) ≤ G(c), then the inequality in (3.3.2) holds for reverse direction. (ii) By putting j = 1, Theorem 3.3.2 gives a special case as Theorem 3.3.1. Now we are able to give our main theorems which are generalizations of the Levinson’s inequality and Jensen-Mercer’s type inequality using Jensen-Boas inequality in the following next sub-sections.

3.3.1 On Levinson Type Inequalities Inequality of Levinson is the generalization of inequality of Ky Fan, i.e., in other word we say inequality of Ky. Fan is the special case of inequality of Levinson (see [21, 69]) and for generalizations of Levinson’s inequality (see [15]). The next theorem is the generalization of Levinson’s type inequality using Jensen-Boas inequality. Theorem 3.3.3 Let G ∈ BV [c, d] such that (3.3.1) valid and Y : [c, d] → [0, q], (q > 0) be a monotonic (either non-increasing or nondecreasing) and continuous map in every of j intervals (di−1 , di ). If f is a continuous FWNDI of third order in the interval J = [0, 2q] ⊂ Rk , then the inequality d c

  d f (Y(t)) dG(t) Y(t) dG(t) c −f d d dG(t) c c dG(t)  d  d c f (2q − Y(t)) dG(t) c (2q − Y(t)) dG(t) ≤ −f d d c dG(t) c dG(t) (3.3.3)

holds. Proof If f is a FWNDI of third order in the interval J, then inequality h1 h2 h3 f (y) ≥ 0

for y, y + h1 + h2 + h3 ∈ J,

h1 , h2 , h3 ∈ Rk∗

3.3 Functions with Nondecreasing Increments of Order 3

209

holds, i.e., h1 h2 (f (y + h3 ) − f (y)) ≥ 0.

(3.3.4)

If y ∈ J and h3 = 2q − 2y, we have h1 h2 (f (2q − y) − f (y)) ≥ 0, i.e., y → f (2q − y) − f (y) is a FWNDI of second order, i.e., it is a FWNDI. Now, applying Theorem 3.3.2 and get desired Theorem 3.3.3.  Remark 3.3.2 d (2q − Y(t)) dG(t) ∈ J and ∀ y ∈ ci ∈ (di−1 , di ) (d0 = c, dj = d) we (i) If c d dG(t) c have either G(y) ≥ G(d) or G(y) ≤ G(c), then the inequality in (3.3.3) holds for reverse direction. (ii) By putting j = 1, Theorem 3.3.3 gives a special case as Theorem 3.1 of [96]. (iii) We can obtain discrete version of Theorem 3.3.3, using technique as given in Corollary 1 of [100]. Corollary 3.3.4 Let Y satisfies the assumptions of the Theorem 3.3.3. Then inequalities 

k−1 

d

0≤

k d#

dG(t) c



c

k−1 

d

≤

k d#

dG(t) c

i=1

c

Yi (t) dG(t) −

k  #

d

Yi (t) dG(t)

i=1 c

(2qi − Yi (t)) dG(t) −

i=1

k  #

d

(2qi − Yi (t)) dG(t)

i=1 c

hold, where all Y components are non-negative. Proof f (y) = y1 · · · yk is a FWNDI of second and third orders for y ∈ Rk∗ . Now, applying Theorems 3.3.2 and 3.3.3 and get desired Corollary 3.3.4.  Remark 3.3.3 By putting j = 1, Corollary 3.3.4 gives a special case as Corollary 3.3 (i) of [96]. Theorem 3.3.5 Let G ∈ BV [c, d] such that (3.3.1) valid and f be a continuous FWNDI of third order in the interval [p, q] ⊂ Rk . Let 0 < c < q−p. If Y : [c, d] →

210

3 Functions with Nondecreasing Increments

[p, q − c] is a monotonic (either non-increasing or nondecreasing) and continuous map in every of j intervals (di−1 , di ), then the inequality d c

 d  f (Y(t)) dG(t) c Y(t) dG(t) −f d d c dG(t) c dG(t)   d d c f (c + Y(t)) dG(t) c (c + Y(t)) dG(t) ≤ −f d d dG(t) c c dG(t) (3.3.5)

holds. Proof Using (3.3.4) for h3 = c = constant ∈ Rk , since y → f (c + y) − f (y) is a FWNDI, now applying Theorem 3.3.2 and get desired Theorem 3.3.5.  Remark 3.3.4 d (c − Y(t)) dG(t) ∈ J and ∀ y ∈ ci ∈ (di−1 , di ) (d0 = c, dj = d) we (i) If c  d c dG(t) have either G(y) ≥ G(d) or G(y) ≤ G(c), then the inequality in (3.3.5) holds for reverse direction. (ii) By putting j = 1, Theorem 3.3.5 gives a special case as Theorem 3.2 of [96]. (iii) We can obtain discrete version of Theorem 3.3.5, using technique as given in Corollary 1 of [100]. Remark 3.3.5 Theorem 3.3.1 and Theorem 3.3.2 both are special cases of the Theorems 3.3.3 and 3.3.5. Corollary 3.3.6 Let Y satisfies the assumptions of the Theorem 3.3.5, then inequalities 

k−1 

d

0≤

k d#

dG(t) c



c

k−1 

d

≤

k d#

dG(t) c

i=1

c

Yi (t) dG(t) −

k  #

d

Yi (t) dG(t)

i=1 c

(ci + Yi (t)) dG(t) −

i=1

k  #

d

(ci + Yi (t)) dG(t)

i=1 c

hold, where all Y components are non-negative. Proof f (y) = y1 · · · yk is a FWNDI of second and third orders for y ∈ Rk∗ . Now, applying Theorems 3.3.2, and 3.3.5 and get desired Corollary 3.3.6.  Remark 3.3.6 By putting j = 1, Corollary 3.3.6 gives a special case as Corollary 3.3 (ii) of [96].

3.3 Functions with Nondecreasing Increments of Order 3

211

3.3.2 On Jensen-Mercer Type Inequalities The next theorem is the generalization of Jensen-Mercer inequality using JensenBoas inequality, for this purpose using Theorem 3.3.2 for proving the following theorems: Theorem 3.3.7 Let G ∈ BV [c, d] such that (3.3.1) valid and Y : [c, d] → [0, q], (q > 0) be a monotonic (either non-increasing or nondecreasing) and continuous map in every of j intervals (di−1 , di ). If f is a continuous FWNDI of d third order in the interval J = [0, 2q] ⊂ Rk and J = c dG(t) > 0, then inequality 1 J



d

c



  1 d f (Y(t)) dG(t) − f Y(t) dG(t) J c   d   1 1 d f (2q − Y(t)) dG(t) − f (2q − Y(t)) dG(t) ≤ J c J c

(3.3.6) holds. Proof Using (3.3.4) for h3 = 2q − 2y, since y → f (2q − y) − f (y) is a FWNDI of second order, i.e., it is a FWNDI. Now, applying Theorem 3.3.2 and get desired Theorem 3.3.7.  Remark 3.3.7  1 d (2q − Y(t)) dG(t) ∈ J and ∀ y ∈ ci ∈ (di−1 , di ) (d0 = c, dj = d) we (i) If J c have either G(y) ≥ G(d) or G(y) ≤ G(c), then the inequality in (3.3.6) holds for reverse direction. (ii) We can obtain discrete version of Theorem 3.3.7, using technique as given in Corollary 1 of [100]. Theorem 3.3.8 Let G ∈ BV [c, d] such that (3.3.1) valid and f be a continuous FWNDI of third order in the interval [p, q] ⊂ Rk . Let 0 < c < q−p. If Y : [c, d] → [p, q − c] is a monotonic (either non-increasing or nondecreasing) and continuous d map in every of j intervals (di−1 , di ) and J = c dG(t) > 0, then the inequality 1 J



d

 f (Y(t)) dG(t) − f

c

≤

1 J

1 J  d c



d

 Y(t) dG(t)

c

f (c − Y(t)) dG(t) − f



1 J



d

 (c − Y(t)) dG(t)

c

(3.3.7) holds.

212

3 Functions with Nondecreasing Increments

Proof Using (3.3.4) for h3 = c = constant ∈ Rk , since y → f (c + y) − f (y) is a FWNDI, now applying Theorem 3.3.2 and get desired Theorem 3.3.8.  Remark 3.3.8  1 d (i) If (c − Y(t)) dG(t) ∈ J and ∀ y ∈ ci ∈ (di−1 , di ) (d0 = c, dj = d) we J c have either G(y) ≥ G(d) or G(y) ≤ G(c), then the inequality in (3.3.7) holds for reverse direction. (ii) We can obtain discrete version of Theorem 3.3.8, using technique as given in Corollary 1 of [100].

Chapter 4

ˇ Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

All analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove. —G. H. Hardy

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely Monotonic Functions The main aim of this section is to extend the definitions of ∇-convex and completely monotonic functions for two variables. We would construct some examples and applications of completely monotonic functions. In present section, some general   N identities of Popoviciu type for discrete case for sums M i=1 j =1 pij f (xi , yj ) and M N i=1 j =1 pij aij have been deduced for function and sequence involving higher order ∇ operator respectively. Then by applying obtained identities, positivity of these expressions will be characterised for higher order   ∇-convex functions. Some general identities of Popoviciu type for integral P (x, y)f (x, y)dx dy for differentiable function of higher order with two variables will be deduced by three different methods, then by applying obtained identities, positivity of these expressions will be characterised for higher order ∇-convex and completely monotonic functions. These identities and inequalities would be a generalization of several established results. Some applications in terms of generalized Cauchy means and exponential convexity will also be provided. Some contents of the current section have been published in years 2020 and 2021 see [88, 125, 126]. We start from some definitions and preliminaries then we would like to discuss discrete identities of two variables function f (xi , yj ) and sequence aij involving higher order ∇ operator. Using obtained identities we derive various significant results. We will also discuss the characterisation of Popoviciu type positivity of these discrete sums involving ∇-convex functions. Over past few decades the notion of completely monotonic functions has gained popularity among researchers in analysis and other related fields due to their © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Irshad et al., New Perspectives on the Theory of Inequalities for Integral and Sum, https://doi.org/10.1007/978-3-030-90563-7_4

213

214

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

interesting properties (see [104]) and higher applicability (see [59]). As it is evident from the following lines taken from a paper with the title “Completely monotone functions: a digest” [128] written by Milan Merkle. He writes “A brief search in MathSciNet reveals total of 286 items that mention this class of functions in the title from 1932 till the end of the year 2011; 98 of them have been published since the beginning of 2006”. After some preliminaries, we would like to construct some examples and applications of completely monotonic functions. We would get general integral identity for one variable of Popoviciu type and its positivity for higher order ∇-convex and completely monotonic functions which is extracted from [126]. We will give general integral identity by three different methods of higher order differentiable functions of two variables of Popoviciu type. After getting these identities, we would also discuss the characterisation of Popoviciu type positivity of these general integrals involving ∇-convex and completely monotonic functions. These identities and inequalities would be generalization of several established results. We would give new generalized mean value theorems of Lagrange and Cauchy type and will also discuss exponential convexity by the support of different examples. Let us recall, few useful definitions and significant results regarding the convex functions extracted from [158] (see also [97]) and also recall a definition from [135]. Throughout the chapter I and J are interval in R and m, n, M, N are natural numbers. Definition 4.1.1 Let E = {x1 , x2 , . . . , xM } ⊂ R. A function f : E → R is known as discrete m-convex function if [xi , . . . , xi+m ; f ] ≥ 0 holds ∀ (m + 1) different points xi , . . . , xi+m ∈ E for i ∈ {0, 1, . . . , m}. Here, we present extension of previously mentioned Definitions 1.0.4 and 3.0.3 for order (m, n). For the sake of this purpose we use interval I × J = [a, b] × [c, d] ⊂ R2 . Definition 4.1.2 Let f : I × J → R, be a function, then (m, n)-divided difference or divided difference of (m, n)th order of f at distinct points xi , . . . , xi+m ∈ I , yj , . . . , yj +n ∈ J for some i, j ∈ N, is stated as (m,n) f (xi , yj ) = [xi , . . . , xi+m ; [yj , . . . , yj +n ; f ]]. Definition 4.1.3 The finite difference of a function f : I × J → R of order (m, n), where h, k ∈ R and x ∈ I , y ∈ J , is stated as m n n m m,n h,k f (x, y) = h (k f (x, y)) = k (h f (x, y))    m

n

m n (−1)m−i+n−j = f (x + ih, y + j k), i j i=0 j =0

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

215

where x +ih ∈ I , y +j k ∈ J for i ∈ {0, 1, . . . , m} and j ∈ {0, 1, . . . , n}. Moreover, a function f : I × J → R is known as (m, n)-convex, if following condition holds m,n h,k f (x, y) ≥ 0 ∀ x ∈ I , y ∈ J . Definition 4.1.4 Finite difference and Divided difference of (m, n)th order, of a sequence (aij ) are stated as m,n aij = m,n 1,1 f (xi , yj ) and (m,n) aij = (m,n) f (xi , yj ) respectively, where i ∈ {1, . . . , m}, j ∈ {1, . . . , n}. If xi = i, yj = j , then function f is stated as f : {1, . . . , m} × {1, . . . , n} → R which is f (i, j ) = aij . Moreover, a sequence (aij ) is called a (m, n)-convex, if following condition holds m,n aij ≥ 0 for m, n ≥ 0 and i, j ∈ {1, 2, 3, . . .}. Definition 4.1.5 A function f : I × J → R, known as convex of (m, n)th order or (m, n)-convex, if ∀ different elements xi , . . . , xi+m ∈ I and yj , . . . , yj +n ∈ J we have (m,n) f (xi , yj ) ≥ 0. Further that the f is convex of (m, n)th order iff f(m,n) ≥ 0, if partial derivative ∂ m+n f ∂x m ∂y n denoted by f(m,n) and exists. Definition 4.1.6 Let E = {x1 , x2 , . . . , xM }, F = {y1 , y2 , . . . , yN } ⊂ R. A function f : E × F → R, is known as discrete (m, n)-convex if inequality [xi , . . . , xi+m ; [yj , . . . , yj +n ; f ]] ≥ 0, holds ∀ (m + 1) different points xi , . . . , xi+m ∈ E and (n + 1) different points yj , . . . , yj +n ∈ F . Further that in this chapter, throughout we would use the following notations, where I × J ⊂ R × R. For some real sequence (am ), m ∈ N and n ∈ {2, 3, . . .} : ∇ (1) am = ∇am = am − am+1 ,

∇ (n) am = ∇(∇ (n−1) am ).

Also for m different real numbers xi , i ∈ {1, 2, . . . , m} and n ≥ 0: (xk − xi ){n+1} = (xk − xi )(xk−1 − xi ) · · · (xk−n − xi ),

(xk − xi ){0} = 1.

Definition 4.1.7 A function f : I × J → R is known as (m, n) − ∇-convex if inequality ∇(m,n) f (xi , yj ) = (−1)m+n (m,n) f (xi , yj ) ≥ 0, holds ∀ different points xi , . . . , xi+m ∈ I , yj , . . . , yj +n ∈ J . Definition 4.1.8 A function f : I → R is known as completely monotonic (or totally monotonic) of order m or m-completely monotonic if all its derivatives f (i) exist and satisfy (−1)i f (i) (x) ≥ 0,

x ∈ (0, ∞);

i ∈ {0, 1, . . . , m}.

Definition 4.1.9 A function f : I × J → R is known as completely monotonic of order (m, n) or (m, n)-completely monotonic if all its f(i,j ) partial derivatives exist and satisfy the condition below: (−1)(i+j ) f(i,j ) (x, y) ≥ 0, x, y, ∈ (0, ∞); j ∈ {0, 1, . . . , n}, i ∈ {0, 1, . . . , m}.

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

216

Remark 4.1.1 It is simple to observe that the notions of completely monotonic function of order m and (m, n) are generalized notions of m-∇-convex function and (m, n)-∇-convex function respectively if there exists differentiability. Examples of Completely Monotonic Functions In present subsection, we would use variety of classes of completely monotonic function F = {fv : v ∈ I ⊂ R} and construct examples of completely monotonic function. Example 4.1.1 Let a family of functions F1 = {ψv : R → R+ |v ∈ R+ } which is stated as ψv (x) =

e−vx vm .

i

d Since (−1)i dx i ψv (x) > 0 for i ∈ {0, 1, . . . , m}, therefore the function ψv is mcompletely monotonic on R, for every v ∈ R+ .

Example 4.1.2 Let a family of functions F2 = {φv : R+ → R|v ∈ R+ } which is stated as " v −x v=

1 m , (ln φv (x) = (−1)v) i xm , v = 1. m! i

d Since (−1)i dx i φv (x) ≥ 0 for i ∈ {0, 1, . . . , m}, therefore the function φv is mcompletely monotonic on R+ for every v ∈ R+ , x ≥ 0.

Remark 4.1.2 Other examples of completely monotonic functions include: (i) f (x) = c (a nonnegative real constant), ∀ x ∈ R. 1 (ii) f (x) = , α ≥ 0, β ≥ 0, x > 0. (x + α 2 )β (iii) f (x) = − ln x ∀ x ∈ R∗ . Let us give brief description of two subsections about discrete case as follow, after introduction, preliminaries in the next two subsections, we will  andexamples M N N get identities for the sums M p f (x , y ) and p aij for ij i j ij i=1 j =1 i=1 j =1 two dimension involving higher order ∇ operator and investigate the inequality M N i=1 j =1 pij f (xi , yj ) ≥ 0 for ∇-convex functions of order (m, n) for two variables.

4.1.1 Discrete Identity for Two Dimensional Sequences Under the given heading, we would consider  a discrete N sequence of two dimension. Firstly, we will get identities for sequence M i=1 j =1 pij aij which involve higher

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

217

order ∇ operator. Further that we can split this sequence into two sequences as a special case by using aij = ai bj . In the paper [130] the following result for a real sequence (aM ) was proved: Theorem 4.1.1 Let pi ∈ R for i ∈ {1, 2, . . . , M}, then the following identity for any real sequence (aM ) holds: M

pi ai =

i=1

m−1

k=0

M−k

1 (k) ∇ aM−k (M − i){k} pi k! i=1

1 + (m − 1)!

M−m

k=1

 k



(m + k − 1 − i){m−1} pi ∇ (m) ak . (4.1.1)

i=1

Now we would like to obtain the following theorem for a real sequence (aMN ). Theorem 4.1.2 Let pij ∈ R and aij be a sequence, where i ∈ {1, 2, . . . , M} and j ∈ {1, 2, . . . , N}, then M

N

pij aij

i=1 j =1

=

n−1 m−1

M−t

N−k

psr

(M − s){t } (N − r){k} ∇(t,k)a(M−t,N−k) t! k!

psr

(m + t − 1 − s){m−1} (N − r){k} ∇(m,k) a(t,N−k) (m − 1)! k!

psr

(M − s){t } (n + k − 1 − r){n−1} ∇(t,n) a(M−t,k) t! (n − 1)!

k=0 t =0 s=1 r=1

+

n−1 M−m t N−k



k=0 t =1 s=1 r=1

+

N−n k

m−1

M−t

k=1 t =0 s=1 r=1

+

t

k N−n

M−m



k=1 t =1 s=1 r=1

psr

(m + t − 1 − s){m−1} (n + k − 1 − r){n−1} ∇(m,n) a(t,k) (m − 1)! (n − 1)! (4.1.2)

holds. Proof We have M

N

i=1 j =1

⎛ ⎞ M N



⎝ pij aij = qj Ai ⎠ , i=1

j =1

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

218

where pij = qj and Ai : j → a(i,j ) . Using (4.1.1) in the inner sum we get M

N

i=1 j =1

⎛ ⎞ M

N−k n−1



1 ∇(k) Ai(N−k) ⎝ pij aij = qj (N − j ){k} ⎠ k! j =1

i=1 k=0

+

⎛ ⎞ k

1 ∇(n) Ai(k) ⎝ qj (n + k − 1 − j ){n−1} ⎠ (n − 1)!

M N−n



j =1

i=1 k=1

⎛ ⎛ ⎞⎞ n−1

M N−k



1 ⎝ ∇(k) Ai(N−k) ⎝ = qj (N − j ){k} ⎠⎠ k! k=0

i=1

⎛ N−n M



⎝ + k=1

=

i=1

M n−1

k=0

j =1

⎛ ⎞⎞ k

1 ∇(n) Ai(k) ⎝ qj (n + k − 1 − j ){n−1} ⎠⎠ (n − 1)! 

wi Bi

+

i=1

M N−n



k=1

j =1



vi Ci ,

i=1

 N−k k {k} = {k} where wi = N−k j =1 qj (n + k − j =1 qj (N − j ) j =1 pij (N − j ) , vi = 1 1 {n−1} 1 − j) , Bi = k! ∇(k) Ai(N−k) , and Ci = (n−1)! ∇(n) Ai(k) . Using again (4.1.1) in inner sums, then we have M

N

i=1 j =1

n−1 m−1

1 ∇(r) B(M−r) pij aij = r! k=0 r=0

+

k=0 r=1

+

k=1 t =0

+

=

1 ∇(t ) C(M−t ) t!

N−n

M−m

k=1 t =1

wi (M − i){r}

i=1

M−t

i=1



vi (M − i){t }

i=1

 t 

1 {m−1} ∇(m) C(t ) vi (m + t − 1 − i) (m − 1)!

n−1 m−1

M−r

N−k

i=1

pij

(M − i){r} (N − j ){k} ∇(r,k)a(M−r,N−k) r! k!

pij

(m + r − 1 − i){m−1} (N − j ){k} (m − 1)! k!

k=0 r=0 i=1 j =1

+



 r 

1 ∇(m) B(r) wi (m + r − 1 − i){m−1} (m − 1)!

n−1 M−m



N−n

m−1

M−r

r N−k n−1 M−m



k=0 r=1 i=1 j =1

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

219

× ∇(m,k) a(r,N−k) +

k N−n

m−1

M−t



pij

(M − i){t } (n + k − 1 − j ){n−1} t! (n − 1)!

pij

(m + t − 1 − i){m−1} (n + k − 1 − j ){n−1} (m − 1)! (n − 1)!

k=1 t =0 i=1 j =1

× ∇(t,n) a(M−t,k) +

N−n t

k

M−m



k=1 t =1 i=1 j =1

× ∇(m,n) a(t,k). If we change i → s, j → r in all sums and put r → t in first and second sums, then we obtain the required identity (4.1.2).  Remark 4.1.3 If we simply put aij = ai bj in Theorem 4.1.2, then we obtain similar result for two ai and bj sequences as below. Corollary 4.1.3 Let pij ∈ R, b : j → bj and a : i → ai be two sequences, where j ∈ {1, 2, . . . , N} and i ∈ {1, 2, . . . , M}, then M

N

pij ai bj

i=1 j =1

=

n−1 m−1

M−t

N−k

psr

(M − s){t} (N − r){k} ∇(t) a(M−t) ∇(k) b(N−k) t! k!

psr

(m + t − 1 − s){m−1} (N − r){k} ∇(m) a(t) ∇(k) b(N−k) (m − 1)! k!

psr

(M − s){t} (n + k − 1 − r){n−1} ∇(t) a(M−t) ∇(n) b(k) t! (n − 1)!

psr

(m + t − 1 − s){m−1} (n + k − 1 − r){n−1} ∇(m) a(t) ∇(n) b(k) . (m − 1)! (n − 1)!

k=0 t=0 s=1 r=1

+

n−1 M−m t N−k



k=0 t=1 s=1 r=1

+

N−n k

m−1

M−t

k=1 t=0 s=1 r=1

+

N−n t

k

M−m



k=1 t=1 s=1 r=1

4.1.2 Discrete Identity and Inequality for Functions of Two Variables Under present heading, we would consider a discrete function of two variables that is defined  in the Ninterval I × J ⊂ R × R. Firstly, we will get identities for function M i=1 j =1 pij f (xi , yj ) in which involves higher order ∇ operator. Moreover, we can split this function into two functions as a special case by

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

220

f (xi , yj ) = f (xi )g(yj ) and also consider necessary and sufficient conditions of Theorem 4.1.5 for Popoviciu  type Ncharacterisation of positivity of sums for discrete function of two variables M i=1 j =1 pij f (xi , yj ) ≥ 0 holds, for every (m, n)−∇convex function. The following result was proved in [97] for the real function involving ∇ operator and it is a generalization of (4.1.1) which may be stated as: Theorem 4.1.4 Let pi be real numbers for i ∈ {1, 2, . . . , M}, where M ≥ m. Let f be discrete function and xi non mutual elements in the interval I for i ∈ {1, 2, . . . , M}, then following identity holds: M

pi f (xi ) =

i=1

m−1

⎛ ∇(k) f (xM−k ) ⎝

k=0

+

M−k

⎞ pj (xM − xj ){k} ⎠

j =1

M−m

⎛ ⎞ k

∇(m) f (xk )(xm+k − xk ) ⎝ pj (xm+k−1 − xj ){m−1} ⎠ .

k=1

j =1

(4.1.3) Now we are able to give our main general theorem for discrete function in two dimension. Theorem 4.1.5 Let pij ∈ R and f : I × J → R be discrete function, where i ∈ {1, 2, . . . , M} and j ∈ {1, 2, . . . , N}, then M

N

pij f (xi , yj )

i=1 j =1

=

n−1 m−1

M−t

N−k

psr (yN − yr ){k} (xM − xs ){t } ∇(t,k) f (xM−t , yN−k )

k=0 t =0 s=1 r=1

+

n−1 M−m t N−k





psr (yN − yr ){k} (xm+t −1 − xs ){m−1}

k=0 t =1 s=1 r=1

× ∇(m,k) f (xt , yN−k )(xm+t − xt ) +

N−n k

m−1

M−t



psr (yn+k−1 − yr ){n−1} (xM − xs ){t }

k=1 t =0 s=1 r=1

× ∇(t,n) f (xM−t , yk )(yn+k − yk ) +

N−n t

k

M−m



psr (yn+k−1 − yr ){n−1} (xm+t −1 − xs ){m−1} ∇(m,n) f (xt , yk )×

k=1 t =1 s=1 r=1

× (xm+t − xt )(yn+k − yk ).

(4.1.4)

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

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holds, where (xi , yj ) ∈ I × J are distinct points. Proof We have M

N

⎞ ⎛ M N



⎝ pij f (xi , yj ) = qj Gi (yj )⎠ ,

i=1 j =1

j =1

i=1

where pij = qj and Gi : y → f (xi , y). Using (4.1.3) in the inner sum we get M

N

pij f (xi , yj ) =

i=1 j =1

M

n−1

⎛

⎞

∇(k) Gi (yN−k ) ⎝

qj (yN − yj ){k} ⎠

N−k

j =1

i=1 k=0

+

M N−n



∇(n) Gi (yk )(yn+k − yk )

i=1 k=1

⎛ ×⎝

k

⎞

qj (yn+k−1 − yj ){n−1} ⎠

j =1

⎛ ⎛ ⎞⎞ n−1

M N−k



⎝ = ∇(k) Gi (yN−k ) ⎝ qj (yN − yj ){k} ⎠⎠ k=0

+

k=1

⎛ × ⎝

k

⎞⎞

qj (yn+k−1 − yj ){n−1} ⎠⎠

M n−1

k=0

∇(n) Gi (yk )(yn+k − yk )

i=1

j =1

=

j =1

i=1

M N−n



i=1

 wi F (xi ) +

N−n

k=1



M

 vi H (xi ) ,

i=1

N−k N−k {k} = {k} where wi = = j =1 qj (yN − yj ) j =1 pij (yN − yj ) , vi k {n−1} , F (xi ) = ∇(k) Gi (yN−k ), and H (xi ) = ∇(n) Gi (yk ) j =1 qj (yn+k−1 − yj ) (yn+k − yk ).

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Using again (4.1.3) in the inner sums, then we have N M



pij f (xi , yj ) =

i=1 j =1

n−1 m−1



∇(r) F (xM−r )

k=0 r=0

+

n−1 M−m



×

r

∇(m) F (xr )(xm+r − xr ) 

wi (xm+r−1 − xi )

i=1

+

N−n

m−1

∇(t ) H (xM−t )

{m−1}

M−t

k=1 t =0

+

N−n

M−m

×

t

 vi (xM − xi )

{t }

i=1

∇(m) H (xt )(xm+t − xt )

k=1 t =1



wi (xM − xi )

{r}

i=1

k=0 r=1





M−r



vi (xm+t −1 − xi )

{m−1}

i=1

N M



pij f (xi , yj ) =

i=1 j =1

n−1 m−1

M−r

N−k

pij (yN − yj ){k} (xM − xi ){r}

k=0 r=0 i=1 j =1

× ∇(r,k) f (xM−r , yN−k ) +

n−1 M−m r N−k





pij (yN − yj ){k} (xm+r−1 − xi ){m−1}

k=0 r=1 i=1 j =1

× ∇(m,k) f (xr , yN−k )(xm+r − xr ) +

N−n k

m−1

M−t



pij (yn+k−1 − yj ){n−1} (xM − xi ){t}

k=1 t=0 i=1 j =1

× ∇(t,n) f (xM−t , yk )(yn+k − yk ) +

N−n t

k

M−m



pij (yn+k−1 − yj ){n−1}

k=1 t=1 i=1 j =1

× (xm+t−1 − xi ){m−1} ∇(m,n) f (xt , yk )(xm+t − xt )(yn+k − yk ).

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If we change i → s, j → r in all sums and put r → t in first and second sums, then we obtain the required identity (4.1.4).  Corollary 4.1.6 Let pij ∈ R and function f : I 2 → R be discrete, where i, j ∈ {1, 2, . . . , M}, then following identity holds: M

M

pij f (xi , yj )

i=1 j =1

=

n−1 m−1

M−t

M−k

psr (yM − yr ){k} (xM − xs ){t } ∇(t,k) f (xM−t , yM−k )

k=0 t =0 s=1 r=1

+

t M−k n−1 M−m





psr (yM − yr ){k} (xm+t −1 − xs ){m−1}

k=0 t =1 s=1 r=1

× ∇(m,k) f (xt , yM−k )(xm+t − xt ) +

k M−n

m−1

M−t



psr (yn+k−1 − yr ){n−1} (xM − xs ){t }

k=1 t =0 s=1 r=1

× ∇(t,n) f (xM−t , yk )(yn+k − yk ) +

M−n t

k

M−m



psr (yn+k−1 − yr ){n−1} (xm+t −1 − xs ){m−1} ∇(m,n) f (xt , yk )×

k=1 t =1 s=1 r=1

× (xm+t − xt )(yn+k − yk ). Remark 4.1.4 If we simply put f (xi , yj ) = f (xi )g(yj ) in Theorem 4.1.5, then we obtain similar result for both f and g functions as below. Corollary 4.1.7 Let pij ∈ R and functions f : I → R and g : J → R be both discrete, where i ∈ {1, 2, . . . , M} and j ∈ {1, 2, . . . , N}, then M

N

pij f (xi )g(yj )

i=1 j =1

=

n−1 m−1

M−t

N−k

psr (xM − xs ){t } ∇(t ) f (xM−t )(yN − yr ){k} ∇(k) g(yN−k )

k=0 t =0 s=1 r=1

+

n−1 M−m t N−k





psr (yN − yr ){k}

k=0 t =1 s=1 r=1

× ∇(k) g(yN−k )(xm+t −1 − xs ){m−1} ∇(m) f (xt )(xm+t − xt )

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224

+

N−n

m−1

M−t

k

psr (yn+k−1 − yr ){n−1}

k=1 t =0 s=1 r=1

× ∇(n) g(yk )(yn+k − yk )(xM − xs ){t } ∇(t ) f (xM−t ) +

t

k N−n

M−m



psr (yn+k−1 − yr ){n−1} ∇(n) g(yk )(yn+k − yk )×

k=1 t =1 s=1 r=1

× (xm+t −1 − xs ){m−1} ∇(m) f (xt )(xm+t − xt ), holds, where (xi , yj ) ∈ I × J are distinct points. Now its time to present necessary and sufficient conditions of Theorem 4.1.5 for Popoviciu type characterisation of positivity of sums for discrete function of two variables involving (m, n) − ∇-convex functions. Theorem 4.1.8 Let pij ∈ R and f : I × J → R be discrete function, where i ∈ {1, 2, . . . , M}, j ∈ {1, 2, . . . , N} and I = {xM−r , xM−r+1 , . . . , xM }, J = {yN−k , yN−k+1 , . . . , yN } and xM−r < · · · < xM , yN−k < · · · < yN , then the following inequality holds for all (m, n) − ∇-convex function f M

N

pij f (xi , yj ) ≥ 0

(4.1.5)

i=1 j =1

iff M−t

N−k

psr (yN − yr ){k} (xM − xs ){t } = 0,

s=1 r=1

k ∈ {0, 1, · · · , n − 1} t ∈ {0, 1, · · · , m − 1} (4.1.6)

t N−k



psr (yN − yr ){k} (xm+t −1 − xs ){m−1} = 0,

s=1 r=1

k ∈ {0, 1, · · · , n − 1} t ∈ {1, 2, · · · , M − m} (4.1.7)

M−t k



psr (yn+k−1 − yr ){n−1} (xM − xs ){t } = 0,

s=1 r=1

k ∈ {1, 2, · · · , N − n} t ∈ {0, 1, · · · , m − 1} (4.1.8)

t

k

s=1 r=1

psr (xm+t −1 − xs ){m−1} (yn+k−1 − yr ){n−1} ≥ 0,

k ∈ {1, 2, · · · , N − n} t ∈ {1, 2, · · · , M − m}. (4.1.9)

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

225

Proof If (4.1.6), (4.1.7) and (4.1.8) hold, then 1st, 2nd and 3rd terms are zero in (4.1.4), then by using (4.1.9) we obtain the required inequality (4.1.5). Conversely, if substitute the following functions in (4.1.5). Then we obtain the required equality (4.1.6) f1 (xs , yr ) = (yN − yr ){k} (xM − xs ){t }

and f2 = −f1

for 0 ≤ t ≤ m − 1 and 0 ≤ k ≤ n − 1 such that ∇(m,n) fj ≥ 0, M−t

N−k

psr (yN − yr ){k} (xM − xs ){t } = 0,

j ∈ {1, 2}

0 ≤ k ≤ n − 1; 0 ≤ t ≤ m − 1.

s=1 r=1

In the similar manner, if take the following functions in (4.1.5) for 0 ≤ k ≤ n − 1 and 1 ≤ t ≤ M − m f3 (xs , yr ) =

(yN − yr ){k} (xm+t −1 − xs ){m−1} , s < t 0, s ≥ t f4 = −f3

such that ∇(m,n) fj ≥ 0, t

k

j ∈ {3, 4}, we obtain the equality (4.1.7), i.e.,

psr (yN −yr ){k} (xm+t −1 −xs ){m−1} = 0,

0 ≤ k ≤ n−1;

1 ≤ t ≤ M−m.

s=1 r=1

Similarly, if take the following functions in (4.1.5) for 0 ≤ t ≤ m − 1 and 1 ≤ k ≤ N −n f5 (xs , yr ) =

(yn+k−1 − yr ){n−1} (xM − xs ){t } , r < k 0, r ≥ k f6 = −f5

such that ∇(m,n) fj ≥ 0, M−t k

s=1 r=1

j ∈ {5, 6}, we obtain the equality (4.1.8) as above, i.e.,

psr (yn+k−1 −yr ){n−1} (xM −xs ){t } = 0, 0 ≤ t ≤ m−1;

1 ≤ k ≤ N −n.

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226

We get the last inequality (4.1.9) by considering the following function in (4.1.5) for 1 ≤ t ≤ M − m and 1 ≤ k ≤ N − n f7 (xs , yr ) =

r < k; (xm+t −1 − xs ){m−1} (yn+k−1 − yr ){n−1} , s < t, 0, s ≥ t or r ≥ k. 

Remark 4.1.5 Similar remark as given in Remark 4.1.4 and for sequence aij and split this sequence into two sequences as a special case by using aij = ai bj also holds for this result. Now, let us briefly explain the format of the following subsections: Firstly we have discussed the discrete case followed by integral case, then in the third subsection we present identity for the integral P (x)f (x)dx which involves function of higher order derivatives and also discuss necessary  and sufficient conditions of this identity ∀ (m+1)−∇-convex function in which P (x)f (x)dx ≥ 0 holds and only necessary condition holds for (m + 1)-completely monotonic function (see also [126]). In the fourth subsection, we will give general integral identity by three different methods for higher order differentiable function of two variables of Popoviciu type and also deduce positivity of above obtained case for (M + 1, N + 1) − ∇-convex functions and (M + 1, N + 1)-completely monotonic functions for two variables, then consider an identity of linear functional (f ) in double integral. In the fifth subsection, we would state some mean value theorems of Lagrange and Cauchy types and consider the nonnegative functional (f ) and apply this on exponentially convex functions ψ (q) of certain type, and give some properties. At the end of the first section, construct examples of completely monotonic and exponentially convex functions by using various classes of functions.

4.1.3 Integral Identity and Inequality for Functions of One Variable We give an integral identity of one variable that is analogous to the result of Theorem 4.1.4. Theorem 4.1.9 Let f ∈ C (m+1) and both p, f : I → R be integrable functions, then 

b

p(x)f (x)dx =

a

m

i=0



b

+ a



b

(−1)i f (i) (b)

p(x) a



(b − x)i dx i!

s

(−1)m+1 f (m+1) (s)

p(x) a



 (s − x)m dx ds. m!

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

227

The following two theorems are proved and extracted from [126] and the following first theorem is generalized form of result (see [151, pp. 121–122]) and (see also [97]). Theorem 4.1.10 Let assumption of the Theorem 4.1.9 be valid, then following inequality holds 

b

f (x)p(x)dx ≥ 0

a

for all (m + 1) − ∇-convex function f , iff 

(b − x)i dx = 0, i!

i ∈ {1, 2, . . . , m};

(s − x)m dx ≥ 0, m!

∀ s ∈ [a, b].

b

p(x) a  s

p(x) a

Theorem 4.1.11 Let assumption of the Theorem 4.1.9 be valid, then following inequality holds 

b

p(x)f (x)dx ≥ 0

a

for each completely monotonic function f of order m + 1 if 

(b − x)i dx = 0, i!

i ∈ {1, 2, . . . , m};

(s − x)m dx ≥ 0, m!

∀ s ∈ [a, b].

b

p(x) a  s

p(x) a

Remark 4.1.6 Previous result also holds for every exponentially convex function [81]. Moreover, each completely monotonic is log-convex so the stated result also holds for every log-convex function [128].

4.1.4 Integral Identity and Inequality for Functions of Two Variables Under above heading, we can suppose function in x and y variables which is stated in the interval I × J = [a, b] × [c, d]. Moreover, m, n, M, N ∈ N ∪ {0} throughout

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

228

this subsection, and useful notations are: f(0,0) = f (function itself), f(1,0) = f(i,j ) =

∂ 2f ∂f ∂f ∂ 2f , f(0,1) = , f(1,1) = = , ∂x ∂y ∂x∂y ∂y∂x

∂ i+j f ∂ i+j f = . ∂x i ∂y j ∂y j ∂x i

Let f : I × J → R and pi,j be both integrable functions then we also introduce following notations as: 

b

p(1,1)(x, y) = x



d



b

p(s, t)dtds, p(m+1,n+1) (x, y) =

y



and p(m+1,n+1) (x, y) =

y

(y − t)n (x − s)m dtds. n! m!

p(s, t) x

d

p(m,n) (s, t)dtds x

b d



y

To prove our main upcoming theorem, we will have to need Two-dimensional Induction method, for this purpose we required following statement: Definition 4.1.10 Let G(M, N) denotes a statement involving two variables M and N. Suppose (i) G(0, 0) is true. (ii) If G(m, 0) is true for some integer m ≥ 0, then G(m + 1, 0) is also true. (iii) If G(m, n) holds for some integers m, n ≥ 0, then G(m, n + 1) is also true. Therefore, G(M, N) is true ∀ integers M, N ≥ 0. This process is called Twodimensional Induction. Here we need to write following lemma which is a special case of Theorem 4.1.12. Lemma 4.1.1 Let f has continuous partial derivatives f(0,1) , f(1,0) , f(1,1) and f, p : I × J → R be both integrable functions, then 

d c



b

 p(x, y)f (x, y)dxdy =

a

c



a

p(s, t)f (s, t)dsdt d

b

= p(1,1)(b, d)f(0,0)(b, d)  a p(1,1)(s, d)f(1,0) (s, d)ds + b

 +

c

p(1,1)(b, t)f(0,1) (b, t)dt d

 +

c



a

p(1,1)(s, t)f(1,1) (s, t)dsdt. d

b

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

229

Now we recall a result from [97] which would be helpful to prove our upcoming main result: Lemma 4.1.2 Let f has continuous partial derivatives f(i,j ) and p, f : I ×J → R be both integrable functions, where i ∈ {0, 1, . . . , M +1} and j ∈ {0, 1, . . . , N +1}, then 

b d

p(x, y)f (x, y)dydx a

c M

N 

=

b



i=0 j =0 a

+

N 

+

M 

i=0

 +

p(s, t)

(s − x)M (t − c)j f(M+1,j ) (x, c)dtdsdx M! j!

p(s, t)

(s − a)i (t − y)N f(i,N+1) (a, y)dtdsdy i! N!

d c



c

b



x d

(s − a)i (t − c)j f(i,j ) (a, c)dtds i! j!

c

b b

j =0 a

p(s, t)

d

b a

d



d

y b

d

p(s, t) a

c

x

y

(s − x)M (t − y)N f(M+1,N+1) (x, y)dtdsdydx. M! N! (4.1.10)

Remark 4.1.7 This above result can also be proved by using Two-dimensional mathematical induction as given in prove of Theorem 4.1.12. Now we state main integral identity of this subsection using higher order derivatives. We would like to prove following result in three different ways, first by interchanging technique, second by Two-dimensional Induction and third by Taylor expansion. Theorem 4.1.12 Let f has continuous partial derivatives f(i,j ) and p, f : I ×J → R be both integrable functions, where i ∈ {0, 1, . . . , M + 1}, j ∈ {0, 1, . . . , N + 1}, then 

b



d

f (x, y)p(x, y)dy dx a

=

c M

N 

b



i=0 j =0 a

+

N 

j =0 a

b



s a

d

c



d c

(−1)i+j p(x, y)

(b − x)i (d − y)j f(i,j ) (b, d)dy dx i! j!

(−1)M+j +1 p(x, y)

(s − x)M (d − y)j f(M+1,j ) (s, d)dy dx ds M! j!

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

230

+

M 

c

i=0

b

 +



d

a

b



a d

c

t

(−1)i+N+1 p(x, y)

c s

a

t

(−1)M+N p(x, y)

c

(b − x)i (t − y)N f(i,N+1) (b, t)dy dx dt i! N!

(s − x)M (t − y)N f(M+1,N+1) (s, t)dy dx dt ds. M! N!

(4.1.11) Proof (Method I) We restate the identity given in Lemma 4.1.2 as follows 



B

D

p(x, y)f (x, y)dy dx A

=

C M

N 

B



p(s, t)f(i,j ) (A, C)

i=0 j =0 A

+

N 

B



M 

D



D



C B



D

p(s, t)f(i,N+1) (A, y)

B

+

B x

C

i=0



C

(s − A)i (t − C)j dt ds i! j!

p(s, t)f(M+1,j ) (x, C)

j =0 A

+

D

A D

y B

(s − x)M (t − C)j dt ds dx M! j!

(s − A)i (t − y)N dt ds dy i! N!

D

p(s, t)f(M+1,N+1) (x, y) A

C

x

y

(s − x)M (t − y)N dt ds dy dx. M! N! (4.1.12)

Let us substitute [A, B] = [b, a] and [C, D] = [d, c]. Then and we change the variables names x ↔ s, y ↔ t, then 

a



B A

=

a b

=−

b a

etc.

c

p(x, y)f (x, y)dy dx b

=

d N  M



a



f(i,j ) (b, d)p(x, y)

i=0 j =0 b

+

N 

a



M 

i=0





s c

c



d a



c

f(i,N+1) (b, t)p(x, y) d

a

+

a

d

(x − b)i (y − d)j dy dx i! j!

f(M+1,j ) (s, d)p(x, y)

j =0 b

+

c

b c

t a

(x − s)M (y − d)j dy dx ds M! j!

(x − b)i (y − t)N dy dx dt i! N!

c

f(M+1,N+1) (s, t)p(x, y) b

d

s

t

(x − s)M (y − t)N dy dx dt ds. M! N! (4.1.13)

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

Left hand side of (4.1.13) may write   a c p(x, y)f (x, y)dy dx = b

d



b a b

=

 

d

231

(−1)2 p(x, y)f (x, y)dy dx

c d

p(x, y)f (x, y)dy dx. a

c

First summand on right side may also write N  M





a

p(x, y)

i=0 j =0 b

=

c d

M

N 

b



i=0 j =0 a

=

M

N 

d

(x − b)i (y − d)j f(i,j ) (b, d)dy dx i! j! (b − x)i (d − y)j (−1)j f(i,j ) (b, d)dy dx i! j!

(−1)2 p(x, y)(−1)i

c b



i=0 j =0 a

d

(−1)i+j p(x, y)

c

(b − x)i (d − y)j f(i,j ) (b, d)dy dx. i! j!

Second summand on right side may also write N 

a





c

p(x, y)

j =0 b

=

a s

N 

d

b s

j =0 a



a

d

(x − s)M (y − d)j f(M+1,j ) (s, d)dy dx ds M! j! (s − x)M (d − y)j (−1)j M! j!

(−1)3 p(x, y)(−1)M

c

× f(M+1,j ) (s, d)dy dx ds =

N 

b s

j =0 a



a

d

(−1)M+1+j p(x, y)

c

(s − x)M (d − y)j f(M+1,j ) (s, d)dy dx ds. M! j!

Similarly the third summand is rewritten as M 



a



c

p(x, y) d

b

M 

d

i=0

=

c

i=0

t



c

b



a

t

(x − b)i (y − t)N f(i,N+1) (b, t)dy dx dt i! N!

(−1)3 p(x, y)(−1)i

c

(b − x)i (t − y)N (−1)N i! N!

× f(i,N+1) (b, t)dy dx dt =

M 

i=0

d c



b a



t c

(−1)N+1+i p(x, y)

(b − x)i (t − y)N f(i,N+1) (b, t)dy dx dt. i! N!

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

232

Finally, last summand on right side rewritten as 

a



c



a



c

p(x, y) b

d



s

b d

= a

c

t



s a



t

(x − s)M (y − t)N f(M+1,N+1) (s, t)dy dx dt ds M! N!

(−1)4 p(x, y)(−1)N

c

(t − y)N (s − x)M (−1)M N! M!

× f(M+1,N+1) (s, t)dydxdtds  b d s t (s − x)M (t − y)N (−1)M+N p(x, y) = M! N! a c a c × f(M+1,N+1) (s, t)dy dx dt ds. By substituting all these expression in (4.1.13) we would arrive at our required result.  Proof (Method II) First we claim that 

c d



a

p(x, y)f (x, y)dxdy =

b

M

N

f(i,j ) (b, d)p(i+1,j +1) (b, d)

i=0 j =0

+

N 

a

f(M+1,j ) (s, d)p(M+1,j +1) (s, d)ds

j =0 b

+

M 

i=0



c

+

c

f(i,N+1) (b, t)p(i+1,N+1) (b, t)dt d



a

f(M+1,N+1) (s, t)p(M+1,N+1) (s, t)dsdt. d

b

(4.1.14) Now we prove this equality by using the Definition 4.1.10 of two-dimensional induction and considering base case, i.e., M = N = 0 

d c



b



c

p(x, y)f (x, y)dxdy =



a

p(s, t)f (s, t)dsdt

a

d

b

= f(0,0) (b, d)p(1,1)(b, d)  a f(1,0) (s, d)p(1,1) (s, d)ds + 

b



d

c

+

f(0,1) (b, t)p(1,1)(b, t)dt c

+



a

f(1,1)(s, t)p(1,1) (s, t)dsdt, d

b

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

233

which is the result of Lemma 4.1.1 so it is proved. Let us assume that our hypothesis is true for M = m and N = 0, i.e., 



c

a

p(x, y)f (x, y)dxdy d

=

b m



f(m+1,0) (s, d)p(m+1,1) (s, d)ds b

i=0

+

a

f(i,0) (b, d)p(i+1,1)(b, d) +

m 

c

 f(i,1) (b, t)p(i+1,1) (b, t)dt +

a

f(m+1,1) (s, t)p(m+1,1) (s, t)dsdt.

d

i=0



c d

b

(4.1.15) We would show that it is valid for M = m + 1 and N = 0, i.e., following inequality holds  c a p(x, y)f (x, y)dxdy d

=

b m+1



m+1

 c i=0

f(m+2,0) (s, d)p(m+2,1) (s, d)ds b

i=0

+

a

f(i,0) (b, d)p(i+1,1)(b, d) +





c

f(i,1) (b, t)p(i+1,1) (b, t)dt +

a

f(m+2,1) (s, t)p(m+2,1) (s, t)dsdt.

d

d

b

(4.1.16) To prove (4.1.16) we consider 2nd term of (4.1.15) 

a

f(m+1,0) (s, d)p(m+1,1) (s, d)ds b



a

=

  f(m+1,0) (b, d) +

b



= f(m+1,0) (b, d)

s

 f(m+2,0) (θ, d)dθ p(m+1,1) (s, d)ds

b a



a

p(m+1,1) (s, d)ds +



s

f(m+2,0) (θ, d)p(m+1,1)(s, d)dθ ds,

b

b

b

by using Fubini theorem and interchanging θ ↔ s 

a

f(m+1,0) (s, d)p(m+1,1) (s, d)ds b



a

= f(m+1,0) (b, d)p(m+2,1)(b, d) +



a

f(m+2,0) (θ, d)p(m+1,1)(s, d)dsdθ b

θ

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

234





a

= f(m+1,0) (b, d)p(m+2,1)(b, d) +

a

f(m+2,0) (s, d) 

b

p(m+1,1) (θ, d)dθ ds s

a

= f(m+1,0) (b, d)p(m+2,1)(b, d) +

f(m+2,0) (s, d)p(m+2,1) (s, d)ds. (4.1.17) b

Now consider 4th term of (4.1.15)  c a f(m+1,1) (s, t)p(m+1,1) (s, t)dsdt d

b c



= 

d

a

  f(m+1,1) (b, t) +

b



a

f(m+2,1) (θ, t)dθ p(m+1,1) (s, t)dsdt

a

f(m+1,1) (b, t) d  c



s b



c

= +



p(m+1,1) (s, t)dsdt b



s

f(m+2,1) (θ, t)p(m+1,1) (s, t)dθ dsdt d



b c

=

f(m+1,1) (b, t)p(m+2,1) (b, t)dt d  c

+

b



a



a

f(m+2,1) (θ, t)p(m+1,1) (s, t)dsdθ dt d



b c

=

f(m+1,1) (b, t)p(m+2,1) (b, t)dt d  c

+

θ





a

a

f(m+2,1) (s, t) d



b c

=

p(m+1,1) (θ, t)dθ dsdt s



c

f(m+1,1) (b, t)p(m+2,1) (b, t)dt +



a

f(m+2,1) (s, t)p(m+2,1) (s, t)dsdt.

d

d

b

(4.1.18) For obtaining desired equality (4.1.16), we will have to substitute values of (4.1.17) and (4.1.18) in (4.1.15) it means that our 1st hypothesis is true. So, let us set again the hypothesis for M = m and N = n, implies that  c a p(x, y)f (x, y)dxdy d

=

b n m

i=0 j =0

+

m 

i=0



c

+

f(i,j ) (b, d)p(i+1,j +1) (b, d) +

n 

a

j =0 b

c

f(i,n+1) (b, t)p(i+1,n+1) (b, t)dt d



a

f(m+1,n+1) (s, t)p(m+1,n+1) (s, t)dsdt. d

f(m+1,j ) (s, d)p(m+1,j +1) (s, d)ds

b

(4.1.19)

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

235

We would show, further it is valid for M = m and N = n + 1  c a p(x, y)f (x, y)dxdy d

=

b n+1 m



f(i,j ) (b, d)p(i+1,j +1) (b, d) +

i=0 j =0

+

m 

i=0



c

+

n+1 

a

j =0 b

f(m+1,j ) (s, d)p(m+1,j +1) (s, d)ds

c

f(i,n+2) (b, t)p(i+1,n+2) (b, t)dt d



a

f(m+1,n+2) (s, t)p(m+1,n+2) (s, t)dsdt. d

(4.1.20)

b

To prove (4.1.20) we consider 3rd term of (4.1.19) m 

f(i,n+1) (b, t)p(i+1,n+1) (b, t)dt d

i=0

=

c

m 



c

+

c

p(i+1,n+1) (b, t)dt

d





t

f(i,n+2) (b, φ)p(i+1,n+1) (b, t)dφdt d

d

m

 f(i,n+1) (b, d)p(i+1,n+2)(b, d) i=0



c

+





c

f(i,n+2) (b, φ)p(i+1,n+1) (b, t)dtdφ d

=

d

 m 

f(i,n+1) (b, d) i=0

=

   t f(i,n+2) (b, φ)dφ p(i+1,n+1) (b, t)dt f(i,n+1) (b, d) +

d

i=0

=

c

φ

m

 f(i,n+1) (b, d)p(i+1,n+2)(b, d) i=0





c

+

f(i,n+2) (b, t) d



c

p(i+1,n+1) (b, φ)dφdt t

 m 

= f(i,n+1) (b, d)p(i+1,n+2) (b, d) + i=0

c

 f(i,n+2) (b, t)p(i+1,n+2) (b, t)dt .

d

(4.1.21)

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

236

Now consider 4th term of (4.1.19) 

c d



f(m+1,n+1) (s, t)p(m+1,n+1) (s, t)dsdt b a c



=

f(m+1,n+1) (s, t)p(m+1,n+1) (s, t)dtds 

b a

= 



b

d



c



c

p(m+1,n+1) (s, t)dtds d



t

f(m+1,n+2) (s, φ)p(m+1,n+1) (s, t)dφdtds 

d

d

a

f(m+1,n+1) (s, d)p(m+1,n+2) (s, d)ds b  a



c



c

f(m+1,n+2) (s, φ)p(m+1,n+1) (s, t)dtdφds b



d

φ

a

=

f(m+1,n+1) (s, d)p(m+1,n+2) (s, d)ds b  a





c

c

f(m+1,n+2) (s, t) b



d

p(m+1,n+1) (s, φ)dφdtds t

a

= +

   t f(m+1,n+2) (s, φ)dφ p(m+1,n+1) (s, t)dtds f(m+1,n+1) (s, d) +

f(m+1,n+1) (s, d) b  a

=

+

c d

b

+

d

a

= +

a

f(m+1,n+1) (s, d)p(m+1,n+2) (s, d)ds b  a



c

f(m+1,n+2) (s, t)p(m+1,n+2) (s, t)dtds. b

(4.1.22)

d

For obtaining desired equality (4.1.20), we will have to substitute values of (4.1.21) and (4.1.22) in (4.1.19) that shows by applying method of induction we have proved our required result. Here, we need to apply notations that are introduced in starting of subsection for proving further. Considering terms from (4.1.14) separately and find their values such that N M



f(i,j ) (b, d)p(i+1,j +1)(b, d)

i=0 j =0

=

M

N 

i=0 j =0 b

a



c

f(i,j ) (b, d)p(x, y) d

(x − b)i (y − d)j dydx i! j!

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . . M

N 

=

b

i=0 j =0 a N 

d

(−1)i+j p(x, y)

c

f(M+1,j ) (s, d)p(M+1,j +1) (s, d)ds

N 



a

N 

b



p(x, y)

a

d

(−1)M+1+j p(x, y)

c

(s − x)M (d − y)j f(M+1,j ) (s, d)dydxds, M! j!

f(i,N+1) (b, t)p(i+1,N+1) (b, t)dt

M 



c

a



c

f(i,N+1) (b, t) d

M 





d

 (x − s)M (y − d)j dydx ds M! j!

c

p(x, y) b

b



a

i=0



s a

i=0

=

c

d

i=0

=



s

j =0 a M 

a

f(M+1,j ) (s, d)

j =0 b

=

(b − x)i (d − y)j f(i,j ) (b, d)dydx, i! j!

a

j =0 b

=



237

d



c

t

t

(−1)i+N+1 p(x, y)

c

 (x − b)i (y − t)N dydx dt i! N!

(b − x)i (t − y)N f(i,N+1) (b, t)dydxdt, i! N!

c

f(M+1,N+1) (s, t)p(M+1,N+1) (s, t)dsdt b

d



a

=





c

a



c

f(M+1,N+1) (s, t) 

b b

= a



d d c



s a

p(x, y) s



t

t

(−1)M+N p(x, y)

c

 (x − s)M (y − t)N dydx dsdt M! N!

(s − x)M (t − y)N f(M+1,N+1) (s, t)dydxdtds. M! N!

Put all these values in (4.1.14) we will get the desired identity (4.1.11)



Proof (Method III) Let H (y) = f (x, y), i.e., considering f (x, y) as a function of y, where x is fixed. Then H may be written as Taylor expansion f (x, y) = H (y) =

N

j =0

=

H (j ) (d)

(y − d)j + j!



y d

H (N+1) (t)

(y − t)N dt N!

 d N

(d − y)j (t − y)N f(0,j ) (x, d) + f(0,N+1) (x, t)dt, (−1)j (−1)N+1 j! N! y j =0

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

238

where use H (j )(d) = f(0,j ) (x, d) and H (N+1) (t) = f(0,N+1) (x, t). Multiplying above equation by p(x, y) and integrate it by y over the limit c to d, then 

d

p(x, y)f (x, y)dy =

c

 N

(−1)j f(0,j ) (x, d) j =0



d

+

d

p(x, y)

c



d

p(x, y)(−1) c

N+1

y

(d − y)j dy j!

 (t − y)N dt dy. f(0,N+1) (x, t) N! (4.1.23)

Now we may write functions x → f(0,j ) (x, d), x → f(0,N+1) (x, t) by applying Taylor expansions: f(0,j ) (x, d) =

M

(b − x)i f(i,j ) (b, d) (−1)i i! i=0



b

+

(−1)M+1

x

f(0,N+1) (x, t) =

(s − x)M f(M+1,j ) (s, d)ds, M!

M

(b − x)i f(i,N+1) (b, t) (−1)i i! i=0



b

+

(−1)M+1

x

(s − x)M f(M+1,N+1) (s, t)ds. M!

Putting these equations in (4.1.23), then 

d

p(x, y)f (x, y)dy c

=

N

 (−1)

j

j =0

M

(b − x)i f(i,j ) (b, d) (−1)i i! i=0

 d (d − y)j − x)M + f(M+1,j ) (s, d)ds dy (−1) p(x, y) M! j! x c M  d  d

(b − x)i + f(i,N+1) (b, t) (−1)N+1 p(x, y) (−1)i i! c y 

b



b

M+1 (s

i=0

+

(−1) x

M+1 (s

− x)M M!

f(M+1,N+1) (s, t)ds



 (t − y)N dt dy N!

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

=

M N



j =0

+

i=0

N 



d

+ c



(−1)

d y

M+1+j

c



d

p(x, y) c

(s − x)M f(M+1,j ) (s, d)ds M!

(d − y)j dy j!



d

p(x, y) c

(d − y)j dy j!

M  i

(t − y)N i+N+1 (b − x) f(i,N+1) (b, t) dt dy p(x, y) (−1) i! N! i=0

d 

d

+

b

(b − x)i f(i,j ) (b, d) i!

x

j =0



(−1)

i+j

239

y

b

(−1)M+N p(x, y)

x

(s − x)M f(M+1,N+1) (s, t)ds M!



(t − y)N dt dy. N!

Now integrate f (x, y)p(x, y) by x over limit a to b and obtain: 

b



d

p(x, y)f (x, y)dy dx a

c



b

= a

⎤ ⎡  M N d i j



(b − x) (d − y) ⎣ f(i,j ) (b, d) dy ⎦ dx (−1)i+j p(x, y) i! j! c ⎡



b

+ a

d

× c

b

+

i=0

N 

b

(−1)M+1+j

x

j =0





⎣

j =0

(

d



d

c

y

⎤ (t − y)N × dt dy ⎦ dx N! b  d

 + a

×



⎤ (d − y)j ⎦ dy dx p(x, y) j! p(x, y)

a

(s − x)M f(M+1,j ) (s, d)ds M!

c



d y



b

M

(−1)

i+N+1 (b

i=0

(−1)M+N p(x, y)

x

⎤

(t − y)N dt dy ⎦ dx. N!

− x)i f(i,N+1) (b, t) i!



(s − x)M f(M+1,N+1) (s, t)ds M!



ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

240

Now changing the order of summation in first summand, and using integral’s linearity we obtain: M

N 

b d

i=0 j =0 a

(−1)i+j p(x, y)

c

(b − x)i (d − y)j f(i,j ) (b, d)dy dx. i! j!

The second summand is rewritten as: ⎡   b

N  b (s − x)M ⎣ f(M+1,j ) (s, d)ds (−1)M+1+j M! a x j =0

 (d − y)j dy dx × p(x, y) j! c ⎡  b

N  b  d (d − y)j ⎣ = (−1)M+1+j p(x, y) j! a x c 

d

j =0

 (s − x)M f(M+1,j ) (s, d)dy ds dx × M! N  b b d

(s − x)M (d − y)j f(M+1,j ) (s, d)dy ds dx = (−1)M+1+j p(x, y) M! j! a x c j =0

=

N 

b



j =0 a

s



a

d

(−1)M+1+j p(x, y)

c

(s − x)M (d − y)j f(M+1,j ) (s, d)dy dx ds. M! j!

Here, Fubini theorem for variables s and x is applied in the last step. The variable x has been changed from a → b and s from x → b. The order of integration is then changed followed by changing s from a → b and x from a → s. Similarly the third summand can be rewritten as: ) M   b ( d  d i

(t − y)N i+N+1 (b − x) f(i,N+1) (b, t) dt dy dx p(x, y) (−1) i! N! a c y i=0

=

M 

i=0

=

a

M 

i=0

=

b

i=0

c b

a

M 



d



(−1)i+N+1 p(x, y)



t



t c

(b − x)i (t − y)N f(i,N+1) (b, t)dt dy dx i! N!

(−1)i+N+1 p(x, y)

(b − x)i (t − y)N f(i,N+1) (b, t)dy dt dx i! N!

(−1)i+N+1 p(x, y)

(b − x)i (t − y)N f(i,N+1) (b, t)dy dx dt. i! N!

c b

a

d y

c d

c

d

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

241

In the above step Fubini theorem has been applied twice. First, t and y have been changed and then t and x. Therefore, last summand is rewritten as: 



b

a

 =

c

y d

c b

=



a



d

b

a





d



c

(−1)M+N p(x, y)

x d

y d

b

b

(−1)M+N p(x, y)

x



s

a

t

(−1)M+N p(x, y)

c

(s − x)M f(M+1,N +1) (s, t)ds M!



 (t − y)N dt dy dx N!

(s − x)M (t − y)N f(M+1,N +1) (s, t)ds dt dy dx M! N!

(s − x)M (t − y)N f(M+1,N +1) (s, t)dy dx dt ds. M! N!

The above steps involve application of Fubini theorem various times. The variables are changed in the following order, first t and y followed by y and s, after that s and t, then s and x and finally t and x. The required identity by using all these results. 



b

d

p(x, y)f (x, y)dy dx a

=

c M

N 

b



i=0 j =0 a

+

N 

b



j =0 a

+

M 

i=0



b

+ a

c



s





d

b



c

(b − x)i (d − y)j f(i,j ) (b, d)dy dx i! j!

(−1)M+1+j p(x, y)

c



a d

(−1)i+j p(x, y)

c

a d

d

t

(−1)i+N+1 p(x, y)

c s



a

t

(−1)M+N p(x, y)

c

(s − x)M (d − y)j f(M+1,j ) (s, d)dy dx ds M! j!

(b − x)i (t − y)N f(i,N+1) (b, t)dy dx dt i! N!

(s − x)M (t − y)N f(M+1,N+1) (s, t)dy dx dt ds. M! N!

 Remark 4.1.8 We may also obtain corollary of Theorem 4.1.12 for the variables names on right side x ↔ s, y ↔ t.

I2

by changing

Corollary 4.1.13 Let both p, f : I 2 → R be functions, f ∈ C (M+1,N+1) (I 2 ) and p is an integrable, where j ∈ {0, 1, . . . , N + 1} and i ∈ {0, 1, . . . , M + 1}, then 

b



b

p(x, y)f (x, y)dy dx a

=

a M

N 

i=0 j =0 a

b



b a

(−1)i+j p(s, t)

(b − s)i (b − t)j f(i,j ) (b, b)dt ds i! j!

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

242 N 

+

b x

j =0 a M 

+

a



a b

a

b

(−1)M+1+j p(s, t)

a

b b

b

+

a

a

i=0





y

(−1)i+N+1 p(s, t)

a x

a

y

(−1)M+N p(s, t)

a

(x − s)M (b − t)j f(M+1,j ) (x, b)dt ds dx M! j!

(b − s)i (y − t)N f(i,N+1) (b, y)dt ds dy i! N!

(x − s)M (y − t)N f(M+1,N+1) (x, y)dt ds dy dx M! N!

holds. Remark 4.1.9 If we replace f (x, y) by f (x)g(y) into Theorem 4.1.12, then we obtain the below statement. Corollary 4.1.14 Let f ∈ C (M+1) (I ), g ∈ C (N+1) (J ), be two different functions and function p : I × J → R be integrable, then 



b

d

f (x, y)p(x, y)dy dx a

=

c M

N 

b



i=0 j =0 a

+

N 

b

j =0 a

+

M 

i=0

 +

a

b

c



d

b

 a

(d − y)j (j ) (b − x)i (i) g (d) f (b)dy dx j! i!

(−1)M+1+j p(x, y)

c



a d

c

s



(−1)i+j p(x, y)

c

a d





d

t

(−1)N +1+i p(x, y)

c s



t

(−1)N +M p(x, y)

c

(d − y)j (j ) (s − x)M (M+1) g (d) f (x)dy dx ds j! M!

(t − y)N (N +1) (b − x)i (i) g f (b)dy dx dt (y) N! i!

(t − y)N (N +1) (s − x)M (M+1) g f (y) (x)dy dx dt ds. N! M!

We obtain necessary and sufficient conditions by using results of previous theorem that (f ) ≥ 0 holds ∀ (M + 1, N + 1) − ∇-convex function and only necessary condition ∀ (M + 1, N + 1)-completely monotonic function for two variables function. Theorem 4.1.15 Let all the assumptions of Theorem 4.1.12 be valid, then following inequality holds 

b

(f ) = a



d c

p(x, y)f (x, y)dy dx ≥ 0

(4.1.24)

4.1 Linear Inequalities for Higher Order ∇-Convex and Completely. . .

243

∀ (M + 1, N + 1) − ∇-convex function f on I × J , iff 

b



d

p(x, y) a

c

(b − x)i (d − y)j dy dx = 0, i ∈ {0, 1, . . . , M}; j ∈ {0, 1, . . . , N}; i! j!

(4.1.25) 

s

d

p(x, y) a

c



b

(s − x)M (d − y)j dy dx = 0, j ∈ {0, 1, . . . , N}; ∀ s ∈ [a, b]; M! j!

(4.1.26) t

p(x, y) a

c

(b − x)i (t − y)N dy dx = 0, i ∈ {0, 1, . . . , M}; ∀ t ∈ [c, d]; i! N!

(4.1.27) 



s

t

p(x, y) a

(s

− x)M

(t

M!

c

− y)N N!

dy dx ≥ 0, ∀ s ∈ [a, b]; ∀ t ∈ [c, d].

(4.1.28)

Proof If (4.1.25), (4.1.26) and (4.1.27) hold, then first, second and third sums are zero in (4.1.11), then using (4.1.28) we get desired inequality (4.1.24). Conversely, if substitute the following functions in (4.1.24). Then f 1 (x, y) =

(b − x)m (d − y)n m! n!

and f 2 = −f 1 j

for 0 ≤ n ≤ N; 0 ≤ m ≤ M such that (−1)M+N f(M+1,N+1) ≥ 0, then obtain the desired equation (4.1.25), i.e., 

b



d

p(x, y) a

c

j ∈ {1, 2},

(b − x)m (d − y)n dy dx = 0, 0 ≤ m ≤ M; 0 ≤ n ≤ N. m! n!

In the similar manner, if take the following functions in (4.1.24) ∀ s ∈ [a, b] and 0≤n≤N " (s−x)M (d−y)n 3 M! n! , x < s f (x, y) = and f 4 = −f 3 0, x ≥ s j

such that (−1)N+M f(M+1,N+1) ≥ 0, (4.1.26), i.e., 

s



d

p(x, y) a

c

j ∈ {3, 4}, we obtain desired equation

(s − x)M (d − y)n dy dx = 0, 0 ≤ n ≤ N; ∀ s ∈ [a, b]. M! n!

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

244

Similarly, if take the following functions in (4.1.24) ∀ t ∈ [c, d] and 0 ≤ m ≤ M "

(b−x)m (t −y)N m! N! ,

y 0, the function ζq (x, y) is (M + 1, M + 1)-completely monotonic on R+ × R+ since (−1)2i ζq(i+1,i+1) (x, y) ≥ 0 for i ∈ {0, 1, . . . , M}, ∀ q ∈ R and q → ζq(M+1,M+1)(x, y) is exponentially convex by definition. Example 4.1.5 Let F2 = {φq : R+ × R+ → R+ |q ∈ R+ } be family of functions which is stated as ⎧ (xy)q ⎨ , q∈

{0, 1, . . . , M}; [q(q−1)···(q−M)]2 φq (x, y) = (xy)q ln(xy)2 ⎩ , q ∈ {0, 1, . . . , M}. 2[q!(M−q)!]2 Clearly φq(M+1,M+1)(x, y) = e(q−M−1) ln(xy) > 0, the function φq (x, y) is (M + 1, M +1)-completely monotonic on R+ ×R+ since (−1)2i φq(i+1,i+1) (x, y) ≥ 0 for i ∈ {0, 1, . . . , M}, ∀ q ∈ R and q → φq(M+1,M+1) (x, y) is exponentially convex by definition.

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex Functions In the current section we would establish two generalizations, first generalization of ˇ discrete Cebyšev identity for function of higher order ∇ operator of two variables and give its special case as sequence of higher order ∇ operator and would also ˇ deduce results of discrete inequality of Cebyšev involving higher order ∇-convex ˇ function. We have plan to give generalization for integral Cebyšev and integral Ky Fan identities for function of higher order derivative and will discuss its inequalities using ∇-convex function. Some contents of the present section are extracted from [127]. Over past few decades, there were some reviewers by Mitrinovi´c and Vasi´c [133] and Mitrinovi´c and Peˇcari´c [132] respectively, which traced completely the ˇ chronological and historical development of Cebyšev identities and its connected inequalities. These research works are remarkable due to many instances incorrect quotations of consequences, sometimes by change of several mathematical scholars–have been uncritically transferred paper to paper and book to book. It is ˇ well known that the famous Cebyšev functional is applied in many fields such as

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

252

numerical quadrature, probability, transform theory, special functions and statistical problems (see [33]). This current section is divided into four parts. In the first part, we introduce ˇ inequality of Cebyšev and give its some related results and will define some notations as well. In the second part, we will discuss the generalization of discrete ˇ identity and inequality of Chebyšev type. In third and fourth parts, we would deduce ˇ the generalization of integral identities and inequalities of Cebyšev and Ky Fan type respectively. ˇ We start this section from a significant result of Cebyšev [31, 32] may be stated as (see also [158, p. 197]): Theorem 4.2.1 Let functions f, h : [a, b] → R be both integrable and p : [a, b] → R+ , where p is also integrable. If f and h are monotone in the same direction, then the inequality 



b

b

p(x)dx a





b

f (x)h(x)p(x)dx ≥

b

h(x)p(x)dx

a

a

f (x)p(x)dx

(4.2.1)

a

holds, there exists integrability. If f (x) and h(x) are monotone in the opposite directions, then (4.2.1) is also valid for reverse inequality. Equality holds in (4.2.1) in the both cases, iff either h or f is constant function almost everywhere. A discrete form of above theorem can be present as following (see [158]). Theorem 4.2.2 Let p be a nonnegative m-tuple and a and b be both real m-tuples monotone in the same direction. Then M

i=1

pi ai bi

M

pi ≥

i=1

M

pi bi

i=1

M

(4.2.2)

pi ai

i=1

holds. If a and b are monotone in the opposite directions, then (4.2.2) is also valid for reverse inequality. Equality holds in (4.2.2) in the both cases, iff either a1 = a2 = · · · = am or b1 = b2 = · · · = bm . ˇ For more discussion about the Cebyšev inequality, we suggest [120, 135, 158]. ˇ In [142] Ostrowski obtained the result which is related to inequality of Cebyšev as follows: Theorem 4.2.3 Let p : I → R+ be an integrable function and f, h ∈ C (1) (I ) be both monotone functions. Then ∃ ν, ζ ∈ I, such that T (f, h, p) = f (ν)h (ζ )T (x − a, x − a, p),

(4.2.3)

where 



b

T (f, h, p) =

p(x)f (x)h(x)dx a



b

p(x)dx− a



b

p(x)h(x)dx a

b

p(x)f (x)dx. a

(4.2.4)

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

253

For other generalizations of Theorem 4.2.3, (see [149]). By using the functional, Peˇcari´c has given the main generalization of Theorem 4.2.3 in [150] which is as follows: 

b

C(f, p) = a



b



b

p(x, y)f (x, x)dy dx −

a



b

p(x, y)f (x, y)dy dx, a

a

(4.2.5)

where the functions f and p are integrable. Theorem 4.2.4 Let p be integrable function and stated as p : I 2 → R, such that Y (x, x) = Y (x, x) f or every x belongs to I and let either Y (x, y) ≥ 0,

Y (x, y) ≥ 0,

a ≤ y ≤ x ≤ b;

a≤x≤y≤b

or their reverse inequalities be valid, where 

b

Y (x, y) =



y

p(s, t) dt ds x

a

and 

x

Y (x, y) =



b

p(s, t) dt ds. a

y

If function f : I 2 → R has continuous partial derivatives, i.e., f(0,1) = ∂ f(1,0) = ∂x f (x, y) and f(1,1) = [a, b], such that

∂2 ∂x∂y f (x, y),

∂ ∂y f (x, y),

then there is existence of ν, ζ ∈

C(f, p) = C((x − a)(y − a), p)f(1,1) (ν, ζ ).

(4.2.6)

ˇ Now its time to describe discrete Cebyšev identity and inequality which are stated as following [150]. Let C  (a, p) =

M

M

pij aii −

i=1 j =1

M

M

pij aij ,

(4.2.7)

i=1 j =1

where pij , aij ∈ R; i, j ∈ {1, . . . , M}. Theorem 4.2.5 The following given inequality C  (a, p) ≥ 0

(4.2.8)

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

254

holds, ∀ real numbers aij , for i, j ∈ {1, 2, . . . , M} such that (1,1)aij ≥ 0 for i, j ∈ {1, 2, . . . , M − 1}, iff Yj +1,j = Y j,j +1

j ∈ {1, 2, . . . , M − 1}

and Yij ≥ 0,

i ∈ {j + 1, . . . , m} f or

Y ij ≥ 0,

i ∈ {1, 2, . . . , j − 1}

j ∈ {1, 2, . . . , M − 1} f or

j ∈ {2, 3, . . . , M}

hold. The reverse inequality of above (4.2.8) is also valid for i, j ∈ {1, 2, . . . , M − 1}, if (1,1)aij ≤ 0, where Yij =

j M



prs

Y ij =

and

r=i s=1

i

M

prs .

r=1 s=j

Ky Fan [58] proposed the following result in 1952, as a problem (see also [133]): Theorem 4.2.6 Let (x, y) → v(x, y) be a function of non-negative Lebesgue integrable over square {(x, y) : a ≤ x ≤ b ; a ≤ y ≤ b} and let D be b b positive constant such that a v(x, y)dx ≤ D and a v(x, y)dy ≤ D for almost all y and x ∈ [a, b] respectively. If f and h finite valued functions and both are non-increasing and non-negative in the interval [a, b], then inequality 

b

D



b

f (x)h(x)dx ≥

a



b

v(x, y)f (x)h(y)dx dy a

(4.2.9)

a

holds. Remark 4.2.1 If put v(x, y) = constant, then (4.2.9) becomes special case of inequality (4.2.1). In [150] J. Peˇcari´c considered the following expression for f, p and q integrable functions for generalization of result of K. Fan 

b

R(f, p, q) =



b

q(x)f (x, x)dx −

a



b

p(x, y)f (x, y)dx dy a

(4.2.10)

a

and gave the result as follows. Theorem 4.2.7 Let q : I → R and p : I 2 → R be both integrable functions, such that P1 (x, y) ≤ S1 (max{x, y}); P1 (x, a) = S1 (x), P1 (a, y) = S1 (y), ∀ x, y ∈ [a, b],

where S1 (x) =

b x

q(t)dt,

P1 (x, y) =

bb x

y

p(s, t)dt ds.

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

255

If f : I 2 → R has f(0,1) , f(1,0), and f(1,1) continuous partial derivatives on I 2 . Then there is existence of (ν, ζ ) ∈ I 2 , such that R(f, p, q) = f(1,1) (ν, ζ )R((x − a)(y − a), p, q)

for ν, ζ ∈ [a, b].

We give following theorem for our chapter from above theorem. Theorem 4.2.8 Let q : I → R and p : I 2 → R be both integrable functions, such that P (x, y) ≤ S(max{x, y});

P (x, b) = S(x),

P (b, y) = S(y),

∀ x, y ∈ [a, b],

x x y where S(x) = a q(t)dt, P (x, y) = a a p(s, t)dt ds. If f ∈ C

(I 2 ), then there exists (ν, ζ ) ∈ I 2 such that R(f, p, q) = f(1,1) (ν, ζ )R((b − x)(b − y), p, q).

(4.2.11)

Under the assumptions of Theorem 4.2.8, we would like to use some notations for easy to present the statements for following upcoming theorems:  P P

(i,j )

(i,j )

(x, y) = 

a x

(x, y) = 

S

x

(i,j )

(x) =

(x, y) = − (x, y) = −

a

 

p(s, t)

(x − s)i (y − t)j dtds, i! j!

(4.2.12)

p(s, t)

(x − s)i (y − s)j dtds, i! j!

(4.2.13)

y a y a

(x − s)i (b − s)j ds, i! j! a  max{x,y}  b (x − s)M (y − s)N dt ds p(s, t) M! N! a a  x y (x − s)M (y − t)N dt ds, p(s, t) M! N! a a  max{x,y} (x − s)M (y − s)N q(s) ds M! N! a  x y (x − s)M (y − t)N dt ds. p(s, t) M! N! a a x

q(s)

(4.2.14)

(4.2.15)

(4.2.16)

Now at this stage we require useful definition which is taken from [180] and for this purpose we also require few notations that are follow: let B = [a, b] × [c, d] represents rectangle in two dimension, S(B) represents the all rectangles system [x1 , x2 ] × [y1 , y2 ] contained in B and provided that a function ω : B → R, we put Eω ([x1 , x2 ] × [y1 , y2 ]) = ω(x2 , y2 ) − ω(x2 , y1 ) − ω(x1 , y2 ) + ω(x1 , y1 ) for [x1 , x2 ] × [y1 , y2 ] ∈ S(B). The function of rectangles Eω : S(B) → R is just stated as a function of rectangles associated with ω.

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

256

Definition 4.2.1 A function ω : B → R is known as absolutely continuous in the Carathéodory’s sense on B, if the following statements hold: (a) The rectangles function Eω associated with ω is absolutely continuous, i.e. ∀

> 0, ∃ δ > 0 such that, if P1 , P2 , . . . , Pj ∈ S(B) are rectangles (mutually  non-overlapped) with property ij =1 |Pj | ≤ δ, where | · | denotes the rectangle  area, then ij =1 Eω (Pj ) ≤ . (b) The functions ω(·, c) : [c, d] → R and ω(a, ·) : [a, b] → R are absolutely continuous. If function ω : B → R is absolutely continuous in the Carathéodory’s sense on B, then it admits the representation of integral ∀ (x, y) ∈ B ω(x, y) = ω(a, c) +

 x a

ω(1,0) (s, c) ds +

 y c

ω(0,1) (a, t) dt +

 x y a

c

ω(1,1) (s, t) dtds,

(4.2.17) provided that the existence of partial derivatives almost everywhere in the above equation (see more details in [133]). Let q : I → R and f, p : I 2 → R be functions and in which p, q are integrable and there should be existence of f(M,N) with absolutely continuous in the Carathéodory’s sense, then for this chapter R(f, p, q) and C(f, p) are defined as: C(f, p) = C(f, p) −

M

N 

P

(i,j )

 (b, b) − P (i,j ) (b, b) f(i,j ) (b, b)

i=0 j =0

−

N 

b

 P

(M,j )

 (x, b) − P (M,j ) (x, b) f(M+1,j ) (x, b) dx

j =0 a

−

M 

i=0

b

 P

(i,N)

 (b, y) − P (i,N) (b, y) f(i,N+1) (b, y) dy, (4.2.18)

a

where C(f, p) is stated in (4.2.5). R(f, p, q) = R(f, p, q) −

N

M 

 S (i,j ) (b) − P (i,j ) (b, b) f(i,j ) (b, b)

j =0 i=0

−

N 

b



 S (M,j ) (x) − P (M,j ) (x, b) f(M+1,j ) (x, b) dx

b



 S (i,N) (y) − P (i,N) (b, y) f(i,N+1) (b, y) dy,

j =0 a

−

M 

i=0

a

where R(f, p, q) is stated in (4.2.10).

(4.2.19)

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

257

ˇ 4.2.1 Generalized Discrete Cebyšev Identity and Inequality ˇ In present subsection we will get discrete identity and inequality of Cebyšev in sequential manner. In this subsection we present some important identities followed by some inequalities. For that purpose we would require Theorem 4.1.4 and Corollary 4.1.6 from Sect. 4.1.2. ˇ Now we would obtain our main theorem about discrete Cebyšev’s identity in the following. Theorem 4.2.9 Let f : I 2 → R be function and (xi , yj ) ∈ I 2 = [a, b] × [a, b] be mutually different points, where i, j ∈ {1, 2, . . . , M}. Let pij be real numbers. Then M

M

C ∇ (f, p) =

pij f (xi , yi ) −

i=1 j =1

=

n−1 m−1



M

M

pij f (xi , yj )

i=1 j =1

∇(t,k) f (xM−t , yM−k )×

k=0 t =0

⎡

M−max{t,k} M−k



×⎣

s=1

−

psr (yM − ys ){k} (xM − xs ){t }

r=1

M−t

M−k

) psr (yM − yr ){k} (xM − xs ){t }

s=1 r=1

+

n−1 M−m



∇(m,k) f (xt , yM−k )(xm+t − xt )×

k=0 t =1

⎡

max{t,k}

M−k

×⎣

s=1

−

t M−k



psr (yM − ys ){k} (xm+t −1 − xs ){m−1}

r=1

) {k}

psr (yM − yr ) (xm+t −1 − xs )

{m−1}

s=1 r=1

+

M−n

m−1

∇(t,n) f (xM−t , yk )(yn+k − yk )×

k=1 t =0

⎡

M−max{t,k} k



×⎣

s=1

r=1

psr (yn+k−1 − ys ){n−1} (xM − xs ){t }

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

258

−

M−t

k

) psr (yn+k−1 − yr )

{n−1}

(xM − xs )

{t }

s=1 r=1

+

M−n

M−m

∇(m,n) f (xt , yk )(xm+t − xt )(yn+k − yk )×

k=1 t =1

×

 max{t,k} k



s=1

−

psr (yn+k−1 − ys ){n−1} (xm+t −1 − xs ){m−1}

r=1

t

k

psr (yn+k−1 − yr ){n−1} (xm+t −1 − xs ){m−1}

 (4.2.20)

s=1 r=1

holds, where ∇(m,n) f (x, y) = (m,n) f (x, y)(−1)m+n . Proof By considering the following expression we begin the proof of this theorem M

M

p˜ij f (xi , yi ),

i=1 j =1

where p˜ij is defined as p˜ ij = M M



M

p˜ ij f (xi , yi ) =

i=1 j =1

i = j, i = j.

r=1 pir ,

0,

M M



pij f (xi , yi ).

i=1 j =1

We have M

M

⎛ ⎞ M M



⎝ pij f (xi , yi ) = qj Gi (yi )⎠ ,

i=1 j =1

i=1

j =1

where pij = qj and Gi : y → f (xi , y). Using (4.1.3) in the inner sum we get M M



pij f (xi , yi ) =

i=1 j =1

n−1 M



⎛

⎞

∇(k) Gi (yM−k ) ⎝

qj (yM − yi ){k} ⎠

i=1 k=0

+

M M−n



i=1 k=1

M−k

j =1

∇(n) Gi (yk )(yn+k

⎞ ⎛ k

− yk ) ⎝ qj (yn+k−1 − yi ){n−1} ⎠ j =1

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

259

⎛ ⎞⎞ ⎛ n−1

M M−k



⎝ = ∇(k) Gi (yM−k ) ⎝ qj (yM − yi ){k} ⎠⎠ k=0

j =1

i=1

⎛ ⎞⎞ ⎛ M−n M k



⎝ + ∇(n) Gi (yk )(yn+k − yk ) ⎝ qj (yn+k−1 − yi ){n−1} ⎠⎠ k=1

=

n−1

j =1

i=1



M

k=0

 wi F (xi ) +

i=1

M−n



M

k=1

 vi H (xi ) ,

i=1

M−k M−k {k} = {k} where wi = j =1 qj (yM − yi ) j =1 pij (yM − yi ) , vi = k {n−1} , F (x ) = ∇ G (y i (k) i M−k ), and H (xi ) = ∇(n) Gi (yk ) j =1 qj (yn+k−1 − yi ) (yn+k − yk ). Using again (4.1.3) in the inner sums, then we have M

M

pij f (xi , yi )

i=1 j =1

=

n−1 m−1



∇(r) F (xM−r )

M−r

k=0 r=0

+

n−1 M−m



+

∇(m) F (xr )(xm+r − xr )

∇(t )H (xM−t )

M−t

k=1 t =0

+

M−n

M−m

+

wi (xm+r−1 − xi ) 

vi (xM − xi )  t

{m−1}

{t }

 vi (xm+t −1 − xi )

{m−1}

i=1

i=1

i=1

pij (yM − yi ){k} (xM − xi ){r} ∇(r,k)f (xM−r , yM−k )

j =1

n−1 M−m

max{r,k}

M−k

k=0 r=1



i=1

∇(m) H (xt )(xm+t − xt )

n−1 m−1 M−k

M−max{r,k}



k=0 r=0

 r

i=1

k=1 t =1

=

wi (xM − xi )

i=1

k=0 r=1 M−n

m−1

 {r}

pij (yM − yi ){k} (xm+r−1 − xi ){m−1} ∇(m,k) f (xr , yM−k )×

j =1

× (xm+r − xr ) +

M−n k

m−1

M−max{t,k}



k=1 t =0

i=1

j =1

pij (yn+k−1 − yi ){n−1} (xM − xi ){t } ∇(t,n) f (xM−t , yk )×

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

260

× (yn+k − yk ) +

M−n k

M−m

max{t,k}



k=1 t =1

i=1

pij (yn+k−1 − yi ){n−1} (xm+t −1 − xi ){m−1} ×

j =1

× ∇(m,n) f (xt , yk )(xm+t − xt )(yn+k − yk ). If we change i → s, j → r in all sums and put r → t in and second sums, first M then we obtain the required result by putting values of M i=1 j =1 pij f (xi , yi )  M M  M M ∇ and Corollary 4.1.6 in C (f, p) = i=1 j =1 pij f (xi , yi ) − i=1 j =1 pij f (xi , yj ).  Remark 4.2.2 If we put xi = i, yj = j and f (xi , yj ) = f (i, j ) = aij in Theorem 4.2.9, then get following corollary. Corollary 4.2.10 Let pij and aij ∈ R for i, j ∈ {1, 2, . . . , M}. Then following identity holds M

M

C ∇ (a, p) =

pij aii −

i=1 j =1

=

n−1 m−1



M

M

⎡

∇(t,k) a(M−t,M−k) ⎣

M−max{t,k} M−k



k=0 t=0

−

s=1

M−t

M−k

psr

s=1 r=1

+

pij aij

i=1 j =1

n−1 M−m



(M − s){t} (M − r){k} t! k!

∇(m,k) a(t,M−k)

k=0 t=1

r=1

(M − s){t} (M − s){k} t! k!

)

 max{t,k}

M−k

s=1

psr

psr

r=1

(M − s){k} (m + t − 1 − s){m−1} k! (m − 1)!

 (M − r){k} (m + t − 1 − s){m−1} k! (m − 1)! s=1 r=1 ⎡ M−max{t,k} M−n k

m−1



(M − s){t} (n + k − 1 − s){n−1} ∇(t,n) a(M−t,k) ⎣ psr + t! (n − 1)! −

t M−k



k=1 t=0

−

M−t k

s=1 r=1

psr

s=1

r=1

(M − s){t} (n + k − 1 − r){n−1} psr t! (n − 1)!

)

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

+

M−n

M−m

⎡ ∇(m,n) a(t,k) ⎣

k=1 t=1

−

t

k

s=1 r=1

max{t,k} k



s=1

psr

r=1

261

(n + k − 1 − s){n−1} (m + t − 1 − s){m−1} (n − 1)! (m − 1)!

(n + k − 1 − r){n−1} (m + t − 1 − s){m−1} psr (n − 1)! (m − 1)!

)

Before starting the next theorem, we would like to state few notations, under assumptions of Theorem 4.2.9: ∇

C (f, p) = C ∇ (f, p) − ⎡ ×⎣

∇(t,k)f (xM−t , yM−k )×

k=0 t =0 M−max{t,k} M−k



s=1

−

n−1 m−1



psr (yM − ys ){k} (xM − xs ){t }

r=1

M−t

M−k

) psr (yM − yr )

{k}

(xM − xs )

{t }

s=1 r=1

−

n−1 M−m



∇(m,k) f (xt , yM−k )(xm+t − xt )×

k=0 t =1

⎡ ×⎣

max{t,k}

M−k

s=1

−

psr (yM − ys ){k} (xm+t −1 − xs ){m−1}

r=1

t M−k



) {k}

psr (yM − yr ) (xm+t −1 − xs )

{m−1}

s=1 r=1

−

M−n

m−1

∇(t,n) f (xM−t , yk )(yn+k − yk )×

k=1 t =0

×

 M−max{t,k} k



s=1

−

M−t k

s=1 r=1

psr (yn+k−1 − ys ){n−1} (xM − xs ){t }

r=1

psr (yn+k−1 − yr )

{n−1}

(xM

 − xs ) , {t }

(4.2.21)

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

262

⎡ ∇ (t, k) = ⎣

max{t,k} k



s=1

−

psr (yn+k−1 − ys ){n−1} (xm+t −1 − xs ){m−1}

r=1

t

k

) psr (yn+k−1 − yr )

{n−1}

(xm+t −1 − xs )

{m−1}

.

(4.2.22)

s=1 r=1

Theorem 4.2.11 Let (xi ) and (yj ) for i, j ∈ {1, 2, . . . , M} be real sequences and monotonic in the same sense and f is ∇-convex function of order (m, n) and pij ∈ R for i, j ∈ {1, 2, . . . , M}. Then ∇

C (f, p) ≥ 0

if ∇ (t, k) ≥ 0;

t ∈ {m + 1, . . . , M},

k ∈ {n + 1, . . . , M}.

∇

Where C (f, p) and ∇ (t, k) are stated respectively in (4.2.21) and (4.2.22). 

Proof This result can easily obtain using (4.2.20).

Remark 4.2.3 If we put xi = i, yj = j and f (xi , yj ) = f (i, j ) = aij in last theorem for m = n = 1 then can obtain similar result for ∇-convex function of Theorem 3 of paper [150] and therefore in this result for aij = f (ai , bj ) we can also get similar result for ∇-convex function of Corollary 2 of paper [150]. Theorem 4.2.12 Let (xi , yj ) ∈ I 2 = [a, b] × [a, b] where i, j ∈ {1, 2, . . . , M}, be mutually different elements, pij ∈ R for i, j ∈ {1, 2, . . . , M} and suppose that f, h : I 2 → R be (m, n) − ∇-convex functions, such that inequalities ∇ (t, k) ≥ 0;

t ∈ {m + 1, . . . , M}

,

k ∈ {n + 1, . . . , M}

(4.2.23)

and L∇(m,n) h(xi , yj ) ≤ ∇(m,n) f (xi , yj ) ≤ U ∇(m,n) h(xi , yj )

(4.2.24)

hold, then below are valid ∇

∇

∇

LC (h, p) ≤ C (f, p) ≤ U C (h, p),

(4.2.25)

where ∇ (t, k) is stated in (4.2.22) and U , L are some real constants. Proof Let functions F1 (xi , yj ) = f (xi , yj ) − Lh(xi , yj ) and F2 (xi , yj ) = U h(xi , yj ) − f (xi , yj ), then ∇(m,n) F1 (xi , yj ) ≥ 0 and ∇(m,n) F2 (xi , yj ) ≥ 0, now using Theorem 4.2.11 we get our required result.  Remark 4.2.4 If reverse inequalities hold in (4.2.23) and (4.2.24), then inequalities in (4.2.25) remain hold. Further that the reverse inequalities in (4.2.25) are also valid, if reverse of inequality holds in (4.2.23).

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

263

Remark 4.2.5 If put xi = i, yj = j and f (xi , yj ) = f (i, j ) = aij and h(i, j ) = bij in previous theorem then we get similar result for ∇-convex function of Theorem 4 of paper [150].

ˇ 4.2.2 Generalized Integral Cebyšev Identity and Inequality ˇ Now we would obtain our main theorem about integral Cebyšev’s identity in the following, for the sake of this purpose we apply the Corollary 4.1.13 from Chap. 4 of this book. Theorem 4.2.13 Let p, f : I 2 → R be both functions, where p is an integrable and there should be existence of partial derivatives f(M+1,N) and f(M,N+1) those are absolutely continuous, then 

b

C(f, p) =



a



b

b

p(x, y)f (x, x)dy dx −

a



b

p(x, y)f (x, y)dy dx a

a

M

N  

(i,j ) = (−1)i+j P (b, b) − P (i,j ) (b, b) f(i,j ) (b, b) i=0 j =0

+

N 

b

 (M,j )  (−1)M+1+j P (x, b) − P (M,j ) (x, b) f(M+1,j ) (x, b) dx

j =0 a

+

M 

i=0



b

+ a

where P (i,j ) , P respectively.

b

 (i,N)  (−1)i+N+1 P (b, y) − P (i,N) (b, y) f(i,N+1) (b, y) dy

a



b

(−1)M+N  (x, y)f(M+1,N+1) (x, y) dy dx,

(4.2.26)

a (i,j )

, and (x, y) are stated in (4.2.12), (4.2.13), and (4.2.15)

Proof For fixed x we define a function f (x, y) = Fx (y). Now we write Taylor expansion of Fx (y) as follows: f (x, y) = Fx (y) =

N

j =0

=

F (j ) (b)

(y − b)j + j!



y b

F (N+1) (t)

(y − t)N dt N!

 b N

(b − y)j (t − y)N f(0,j ) (x, b) + f(0,N+1) (x, t) dt, (−1)j (−1)N+1 j! N! y j =0

where F (j ) (b) = f(0,j ) (x, b) and F (N+1) (t) = f(0,N+1) (x, t).

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

264

Now, for y = x we have f (x, x) =

 b N

(b − x)j (t − x)N f(0,j ) (x, b) + f(0,N+1) (x, t) dt, (−1)j (−1)N+1 j! N! x j =0

Multiplying above equation by p(x, y) and integrate it by y over the limit a to b, then 

b

p(x, y)f (x, x)dy =

a

N

j =0



b

+

 j

(−1) f(0,j ) (x, b)

b

p(x, y) a



a

b

(−1)N+1 p(x, y)

x

(b − x)j dy j!

 (t − x)N f(0,N+1) (x, t)dt dy. N! (4.2.27)

Now we use further representation of functions x → f(0,j ) (x, b) and x → f(0,N+1) (x, t) by Taylor expansions: f(0,j ) (x, b) =

M

(b − x)i f(i,j ) (b, b) (−1)i i! i=0



b

+

(−1)M+1

x

f(0,N+1) (x, t) =

M

(−1)i

i=0



b

+

(b − x)i f(i,N+1) (b, t) i!

(−1)M+1

x

(s − x)M f(M+1,j ) (s, b)ds, M!

(s − x)M f(M+1,N+1) (s, t)ds. M!

Putting these above formulae in (4.2.27), then 

b

p(x, y)f (x, x)dy a

M N



(b − x)i j = f(i,j ) (b, b) (−1) (−1)i i! j =0

i=0

 b (s − x)M (b − x)j f(M+1,j ) (s, b)ds dy p(x, y) M! j! x a M  b  b

(b − x)i N+1 f(i,N+1) (b, t) + (−1) p(x, y) (−1)i i! a x 

+

b

(−1)M+1

i=0

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

265

  − x)M (t − x)N f(M+1,N+1) (s, t)ds dt dy + (−1) M! N! x M  N b i



(b − x)j i+j (b − x) = f(i,j ) (b, b) dy (−1) p(x, y) i! j! a 

b

M+1 (s

j =0

+

i=0

N 

(−1)

b b

+

p(x, y) a



M

x b b

+

(s − x)M f(M+1,j ) (s, b)ds M!

M+1+j

x

j =0



b

a

i=0



x

(−1)

i+N+1 (b

b

(−1)M+N p(x, y)

x



b

p(x, y) a

(b − x)j dy j!

 (t − x)N − x)i f(i,N+1) (b, t) dt dy i! N!

(s − x)M f(M+1,N+1) (s, t)ds M!



(t − x)N dt dy. N!

Now integrate p(x, y)f (x, x) by x over the limit a to b and obtain: 



b

b

p(x, y)f (x, x)dy dx a

a



b

= a

⎤ ⎡  M N i b j



(b − x) (b − x) i+j ⎣ f(i,j ) (b, b) dy ⎦ dx (−1) p(x, y) i! j! a j =0

i=0

⎡ ⎤  b  b

N  b M j (b − x) (s − x) ⎣ + (−1)M+1+j p(x, y) f(M+1,j ) (s, b)ds dy ⎦ dx M! j! a x a j =0

+

 b (

b





b

p(x, y) a



a b

+ a

 a

x b



b

i+N +1 (b

(−1)

i=0



x

M

b

(−1)M+N p(x, y)

x

 ) (t − x)N − x)i f(i,N +1) (b, t) dt dy dx i! N!

(s − x)M f(M+1,N +1) (s, t)ds M!



 (t − x)N dt dy dx. N!

Now changing the order of summation in first summand, and use integral linearity property and obtain: N  M

i=0 j =0 a

b



b a

(−1)i+j p(x, y)

(b − x)i (b − x)j f(i,j ) (b, b)dy dx. i! j!

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

266

The second summand is rewritten as: 

⎡ ⎤  b N  b M j

(b − x) (s − x) ⎣ f(M+1,j ) (s, b)ds dy ⎦ dx (−1)M+1+j p(x, y) M! j! x a

b a

j =0



b

=

⎡ N 

⎣

a

=

b



j =0 a

=

x

j =0

N 

N 

b



j =0 a

a



x b



b

b

b

⎤  j (s − x)M (b − x) f(M+1,j ) (s, b)dy ds ⎦ dx (−1)M+1+j p(x, y) j! M!

(−1)M+1+j p(x, y)

(s − x)M (b − x)j f(M+1,j ) (s, b) dy ds dx M! j!

(−1)M+1+j p(x, y)

(s − x)M (b − x)j f(M+1,j ) (s, b) dy dx ds. M! j!

a s



a

b

a

Applying Fubini theorem for variables s and x in the last step. Let us first, the change of variable x from a → b while changing of variable s from x → b. After the change of order of integration, s is changed a → b while x is changed a → s. In the similar manner the third summand may be rewritten as: 

b

(



b



b

p(x, y) a

a

x

M  b

=

a

i=0

b



b

)  i N (t − x) (b − x) f(i,N+1) (b, t) dt dy dx (−1)i+N+1 i! N!

(−1)i+N+1 p(x, y)

(b − x)i (t − x)N f(i,N+1) (b, t) dt dy dx i! N!

(−1)i+N+1 p(x, y)

(b − x)i (t − x)N f(i,N+1) (b, t) dy dt dx i! N!

(−1)i+N+1 p(x, y)

(b − x)i (t − x)N f(i,N+1) (b, t) dy dx dt. i! N!

b

x



a

M 

=



b a

i=0

b a

M 

=

i=0



a

i=0

M

t

a b



a

t

a

In above using Fubini theorem twice. First, changing t and y, then changing t and x. Therefore, the last summand may be rewritten as: 

b



a

×

b a



b

x

(t

− x)N



b

= a

N!  b a



b

(−1)M+N p(x, y)

x

 dt dy dx b x



b x

(−1)M+N p(x, y)

(s − x)M f(M+1,N+1) (s, t) ds M!



(s − x)M (t − x)N f(M+1,N+1) (s, t) ds dt dy dx M! N!

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .





b

= a

b



a

max{s,t}  b a

(−1)M+N p(x, y)

a

267

(s − x)M (t − x)N M! N!

× f(M+1,N+1) (s, t) dy dx dt ds.

Now, adding up all these summand results to obtain: 



b

b

p(x, y)f (x, x)dy dx a

=

a M

N 

b



i=0 j =0 a

+

N 

b



j =0 a

+

M 

a

i=0



b

+ a

s





(b − x)i (b − x)j f(i,j ) (b, b) dy dx i! j!

(−1)M+1+j p(x, y)

a

a

a

b

b t

b

(−1)i+j p(x, y)

a

a b

b

(−1)i+N+1 p(x, y)

a max{s,t }  b

a

(s − x)M (b − x)j f(M+1,j ) (s, b) dy dx ds M! j!

(b − x)i (t − x)N f(i,N+1) (b, t) dy dx dt i! N!

(−1)M+N p(x, y)

a

(s − x)M (t − x)N M! N!

× f(M+1,N+1) (s, t) dy dx dt ds. After changing x ↔ s, y ↔ t on right side, then get 



b

b

p(x, y)f (x, x)dy dx a

=

a N  M



b



i=0 j =0 a

+

N 

+

M 

i=0

 +

a

a

a b

a



a

b

(−1)M+1+j p(s, t)



y

(−1)i+N+1 p(s, t)

a max{x,y}  b

a

(b − s)i+j f(i,j ) (b, b) dt ds i!j !

a

b b

b

(−1)i+j p(s, t)

a

b x

j =0 a

b

(x − s)M (b − s)j f(M+1,j ) (x, b) dt ds dx M! j!

(b − s)i (y − s)N f(i,N+1) (b, y) dt ds dy i! N!

(−1)M+N p(s, t)

a

×f(M+1,N+1) (x, y) dt ds dy dx.

(x − s)M (y − s)N M! N!

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

268

Now by defined notations, finally we obtain b b



p(x, y)f (x, x)dy dx a

=

a M

N

(−1)i+j P

(i,j )

(b, b)f(i,j ) (b, b)

i=0 j =0

+

N 

b

(−1)M+1+j P

(M,j )

(x, b)f(M+1,j )(x, b) dx

j =0 a

+

M 

+ ×

(−1)i+N+1 P

(i,N)

(b, y)f(i,N+1) (b, y) dy

a

i=0



b



b



b

(−1) a

a

(y

− s)N

N+M

max{x,y}  b

f(M+1,N+1) (x, y)

p(s, t) a

N!

a

(x − s)M M!

dt ds dy dx,

(i,j )

where P is stated in (4.2.13). Use above expression for f (x, x)dy dx and Corollary 4.1.13 in the following 

b

C(f, p) = a



b



b

p(x, y)f (x, x)dy dx −

a



bb a

a

p(x, y)

b

p(x, y)f (x, y)dy dx, a

a



we get our required identity.

Remark 4.2.6 If put f (x, y) = f (x)h(y) and p(x, y) = p(x)p(y) in Theorem 4.2.13, then we can give corollary as: Corollary 4.2.14 Let p, h, f : I → R be three functions such that p is an integrable and there should be existence of derivatives f (M) and h(N) with absolutely continuous, then T (f, h, p) = T (PM (f ), PN (h), p) + T (M (f ), PN (h), p) + T (PM (f ), N (h), p)  b  b  b  max{x,y} (x − s)M f (M+1) (x) (y − s)N h(N+1) (y) + p(x) dx (−1)M+N M! N! a a a a  b  b × p(s) ds dy dx − (−1)M+N p(x) M (f )(x) dx p(x) N (h)(x) dx, a

a

(4.2.28) x  i (i) where Pk (g)(x) = ki=0 (b−x)i!g (b) , k (g)(x) = a and g is a function and T (f, h, p) is stated in (4.2.4).

(x−s)M g (M+1) (s) M!

ds, k ∈ N,

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

269

Proof Applying Taylor formula for f and h, then we can easily obtain (4.2.28).  Corollary 4.2.15 Let p, f : I 2 → R be both functions, p is an integrable and there should be existence of partial derivatives f(M+1,N) and f(M,N+1) with absolutely continuous, then for 1s + 1t = 1; s, t > 1; we have 

b b

| C(f, p) |≤ a



b b

× a

|  (x, y)| dy dx t

 1t

a

|(−1)M+N f(M+1,N+1) (x, y)|s dy dx

 1s ,

(4.2.29)

a

where C(f, p) and (x, y) are stated in (4.2.18) and (4.2.15) respectively. Proof Applying Hölder inequality for integrals on Theorem 4.2.13, we may obtain (4.2.29).  Theorem 4.2.16 Let p, f : I 2 → R be functions, p is integrable and f is (M + 1, N + 1) − ∇-convex, then C(f, p) ≥ 0

if (x, y) ≥ 0

∀ x , y ∈ [a, b],

where (x, y) and C(f, p) are stated in (4.2.15) and (4.2.18) respectively. Proof If function f is ∇-convex function of (M +1, N +1)th order and on domain, it can be approximated uniformly by polynomials containing non-negative (M + 1, N + 1)th order partial derivatives. From polynomials of Bernstein B

m,n

n    m

n m f (ai , bj )(x − a)i (y − a)j (b − y)n−j (b − x)m−i , (x, y) = j i i=0 j =0

(where k = (b − a)/n and h = (b − a)/m) converge to function f uniformly in the domain I 2 limits as n → ∞, m → ∞ provided that function is continuous. Furthermore, if function f (M + 1, N + 1)th order ∇-convex function, where polynomial containing non-negative (M + 1, N + 1)th order partial derivatives, m,n that is, (−1)M+N B(M+1,N+1) ≥ 0, applying following formula it may be proved by using method of induction: m,n (−1)M+N B(M+1,N+1) (x, y)

   m−M−1

n−N−1

n m = (N + 1)!(M + 1)! N +1 M+1    n−N −1 m−M −1 × × j i

i=0

j =0

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

270

  × (M+1,N+1) f (a + ih, a + j k) (y − a)j (b − y)n−N−1−j h,k × (x − a)i (b − x)m−M−1−i  = ((N + 1)!) ((M + 1)!) h 2

2 M+1 N+1

k

  m−M−1

n m N +1 M+1 i=0

n−N−1

m − M − 1 × i j =0

  n−N −1  × ∇(M+1,N+1) f (ai , bj ) (y − a)j (b − y)n−N−1−j j × (x − a)i (b − x)m−M−1−i , where ai = a + ih, bj = a + j k and as (ai ) and (bj ) increasing sequences. Since f is (M+1, N +1)th order ∇-convex, so 0 ≤ ∇(M+1,N+1) f . Since (x, y) m,n is continuous and (−1)M+N B(M+1,N+1) ≥ 0 in the domain I 2 so by (4.2.18), we get C(B m,n , p)  b b m,n (−1)N+M B(M+1,N+1) (x, y) = a



a

max{x,y}  b

(y − s)N (x − s)M dt ds N! M! a a   x y (x − s)M (y − t)N dt ds dy dx ≥ 0 − p(s, t) M! N! a a ×

p(s, t)

or we can write 

b

C(B m,n , p) = a



b a

m,n (x, y)(−1)M+N B(M+1,N+1) (x, y)dy dx ≥ 0.

(4.2.30)

m,n Now convergence of B(M+1,N+1) uniformly to f(M+1,N+1) by supposing n, m → ∞ through an appropriate sequence, provides required result. 

Theorem 4.2.17 Let p, f : I 2 → R be two functions, where f ∈ C (M+1,N+1) is (M + 1, N + 1)th order ∇-convex function in the interval I 2 and p is integrable. If (x, y) ≥ 0

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

271

holds, ∀ x, y belong to [a, b], ∃ ν, ζ belong to [a, b], such that C(f, p) = (−1)M+N C (G0 , p) f(M+1,N+1) (ν, ζ ),

(4.2.31)

where G0 (x, y) = (−1)M+N

(b − x)M+1 (b − y)N+1 (M + 1)!(N + 1)!

(4.2.32)

and C(f, p), (x, y) are stated in (4.2.18) and (4.2.15) respectively. Proof Since 

b

C(f, p) = a



b

(−1)N+M  (x, y)f(M+1,N+1) (x, y)dy dx,

a

by applying mean value theorem for the purpose of double integrals, then obtain  C(f, p) = (−1)

N+M

b



b

f(M+1,N+1) (ν, ζ ) a

(x, y)dy dx.

a

In above equation, if put f (x, y) = G0 (x, y) then we can write as: 

b b

C (G0 , p) = C (G0 , p) = a

(x, y) dy dx

a



and hence we get what we wanted.

Remark 4.2.7 By putting f (x, y) = f (x)h(y) and p(x, y) = p(x)p(y) in Theorem 4.2.17 with M = N = 0, then can obtain similar result for ∇-convex function of (4.2.3). Theorem 4.2.18 Let h, f : I 2 → R be both functions and p : I 2 → R is integrable, such that f is (M + 1, N + 1) − ∇-convex function and h ∈ C (M+1,N+1) (I 2 ) with h(M+1,N+1) = 0 on I 2 . If (x, y) ≥ 0

∀ x, y belong to [a, b]

holds, ∃ ν, ζ belong to [a, b], such that C(f, p) =

f(M+1,N+1) (ν, ζ ) C(h, p), h(M+1,N+1) (ν, ζ )

where (x, y) and C(f, p) are stated in (4.2.15) and (4.2.18) respectively.

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

272

Proof (Method I) Applying mean value theorem of integral and (4.2.31), then 

b



b

f(M+1,N+1) (x, y) h(M+1,N+1) (x, y)  (x, y)dy dx h (M+1,N+1) (x, y) a a   f(M+1,N+1) (ν, ζ ) b b = (−1)N+M  (x, y)h(M+1,N+1) (x, y)dy dx h(M+1,N+1) (ν, ζ ) a a

C(f, p) =

=

(−1)M+N

f(M+1,N+1) (ν, ζ ) C(h, p). h(M+1,N+1) (ν, ζ ) 

be (M + 1, N + 1)th order ∇-convex Proof (Method II) Let u ∈ function in the interval I × J , is stated as: C (M+1,N+1)

u = C(h, p)f − C(f, p)h, by applying Theorem 4.2.17, ∃ ν, ζ ∈ I , such that 0 = C(u, p) = (−1)N+M C(G0 , p)u(M+1,N+1) (ν, ζ ) or [C(h, p)f(M+1,N+1) (ν, ζ ) − C(f, p)h(M+1,N+1) (ν, ζ )]C(G0 , p) = 0. 

This gives required result.

Remark 4.2.8 If we put M = N = 0 in Theorem 4.2.18, then we get similar result for ∇-convex function of Theorem 2 of [150]. Theorem 4.2.19 Let p, f : I 2 → R be two functions, f is (M + 1, N + 1)th order ∇-convex and p is integrable. ∃ ν, ζ ∈ [a, b], such that  C(f, p) = (−1)M+N  (ν, ζ ) f(M,N) (b, b) − f(M,N) (b, a) −f(M,N) (a, b) + f(M,N) (a, a) , where (x, y) and C(f, p) are stated in (4.2.15) and (4.2.18). m,n Proof Since (−1)M+N B(M+1,N+1) ≥ 0 in the interval I 2 and (x, y) is continum,n ous, here B is polynomial of Bernstein, using same statement that was applied in the proof of the Theorem 4.2.17, we start from (4.2.30), we obtain

 C(B

m,n

b

, p) = a



b a

m,n (−1)M+N  (x, y)B(M+1,N+1) (x, y) dy dx



b b

= (−1)M+N  (νm,n , ζm,n ) a

a

m,n B(M+1,N+1) (x, y) dy dx

ˇ 4.2 Generalized Cebyšev and Ky Fan Identities and Inequalities for ∇-Convex. . .

273

 m,n m,n = (−1)N+M  (νm,n , ζm,n ) B(M,N) (b, b) − B(M,N) (a, b)  m,n m,n (b, a) + B(M,N) (a, a) . −B(M,N) The points xm,n = (νm,n , ζm,n ) have a limit point (ν, ζ ) on I 2 as m, n → ∞, so m,n the uniform convergence of B(M,N) to f(M,N) by letting m, n → ∞ through an appropriate sequence, gives our desired result.  Remark 4.2.9 For case M = N = 0 in Theorem 4.2.19, we may get similar result for ∇-convex function of Theorem 6 of [150].

4.2.3 Generalized Integral Ky Fan Identity and Inequality According to MathSciNet, Ky Fan (1914–2010) published 126 research papers and books. The contributions of Ky Fan in mathematics, have provided many of influence in development of convex analysis, nonlinear analysis, linear algebra, operator theory, mathematical economics, approximation theory and mathematical programming (see [108]). In literature, there are different kinds of inequalities due to Ky Fan worked in several fields; cf. [26]. Now in this subsection we have to obtain some important identities and inequalities as below: Theorem 4.2.20 Let q : I → R and f, p : I 2 → R be functions, such that q and p are integrable and there should be existence of partial derivatives f(M+1,N) and f(M,N+1) with absolutely continuous, then R(f, p, q) =

N

M

  (−1)i+j S (i,j ) (b) − P (i,j ) (b, b) f(i,j ) (b, b)

j =0 i=0

+

N 

b

  (−1)M+1+j S (M,j ) (x) − P (M,j ) (x, b) f(M+1,j ) (x, b) dx

j =0 a

+

M 

i=0



b

+ a

b

  (−1)i+N+1 S (i,N) (y) − P (i,N) (b, y) f(i,N+1) (b, y) dy

a



b

(−1)M+N (x, y)f(M+1,N+1) (x, y) dy dx,

a

where S (i,j ) , P (i,j ) , and (x, y) are stated in (4.2.14), (4.2.12), and (4.2.16) respectively.

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ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

Proof By applying the substitution 

b

p(x, y)dy = q(x),

a

we may prove this theorem in similar manner as done in Theorem’s 4.2.13 proof.  Remark 4.2.10 By putting f (x, y) = f (x)h(y) and p(x, y) = q(x)q(y) in b a q(t ) dt b Theorem 4.2.20, here q is integrable and a q(t) dt = 0, then we can give corollary as below. Corollary 4.2.21 Let f, h, q : I → R be functions, such that q is integrable, where b (M) and g (N) with a q(t) dt = 0 and there should be existence of derivatives f absolutely continuous. Then T (f, h, q) = T (PM (f ), PN (h), q) + T (M (f ), PN (h), q) + T (PM (f ), N (h), q)  b  b  max{x,y} (x − s)M f (M+1) (x) (y − s)N h(N+1) (y) + (−1)M+N M! N! a a a  b  b × q(s) ds dy dx − (−1)M+N M (f )(x)q(x) dx N (h)(x)q(x) dx, a

a

x i (i) M g (M+1) (s)  where Pk (g)(x) = ki=0 (b−x)i!g (b) , k (g)(x) = a (x−s) M! ds, k ∈ N, and here g is a function and T (f, h, p) is defined in (4.2.4). Corollary 4.2.22 Let q : I → R and f, p : I 2 → R be functions and also p and q are integrable and there should be existence of partial derivatives f(M+1,N) and f(M,N+1) with absolutely continuous. Then for 1s + 1t = 1; s, t > 1; we have 

b

| R(f, p, q) | ≤



a



|(−1)

M+N

s

 1s

f(M+1,N+1) (x, y)| dy dx

a b

×

b

a



b

|(x, y)| dy dx t

 1t ,

a

where R(f, p, q) and (x, y) are stated in (4.2.19) and (4.2.16) respectively. Theorem 4.2.23 Let q : I → R and f, p : I 2 → R be three functions and also p and q are integrable and function f is (M + 1, N + 1)th order ∇-convex. Then R(f, p, q) ≥ 0

if (x, y) ≥ 0,

∀ x, y ∈ [a, b],

where R(f, p, q) and (x, y) are stated in (4.2.19) and (4.2.16) respectively.

4.3 Weighted Montgomery Identities for Higher Order Differentiable Function. . .

275

Remark 4.2.11 We can prove Theorem 4.2.23 in similar manner as done in Theorem’s 4.2.16 proof. Theorem 4.2.24 Let q : I → R and f, p : I 2 → R be functions and also p and q are integrable and function f is (M + 1, N + 1)th order ∇-convex and assuming ∀ x, y belong to [a, b] (x, y) ≥ 0. ∃ ν, ζ ∈ [a, b], such that R(f, p, q) = (−1)N+M f(M+1,N+1) (ν, ζ )R (G0 , p, q) , where R(f, p, q) and G0 are stated in (4.2.19) and (4.2.32) respectively. Remark 4.2.12 We can give proof of Theorem 4.2.24 in similar way as done in Theorem’s 4.2.17 proof. Further that we obtain Theorem 4.2.8 from Theorem 4.2.24 by putting M = N = 0. Theorem 4.2.25 Let q : I → R and f, h, p : I 2 → R be four functions and also q and p are integrable and function f is (M + 1, N + 1) − ∇-convex and h ∈ C (M+1,N+1) (I 2 ) with h(M+1,N+1) = 0 in the interval I 2 and assuming ∀ x, y belong to [a, b] (x, y) ≥ 0. Then ∃ ν, ζ ∈ [a, b], such that R(f, p, q) =

f(M+1,N+1) (ν, ζ ) R(h, p, q), h(M+1,N+1) (ν, ζ )

where R and  are stated in (4.2.19) and (4.2.16) respectively. Remark 4.2.13 We can give proof of Theorem 4.2.25 in similar ways as done in Theorem’s 4.2.18 proof by two different methods. Further, if put M = N = 0 in Theorem 4.2.25, then we get similar result for ∇-convex function of Theorem 16 of [150].

4.3 Weighted Montgomery Identities for Higher Order Differentiable Function of Two Variables and Related Inequalities We would provide double weighted integrals identities of Montgomery for differentiable function of higher order for two variables and by help of those identities

276

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

we would get generalization of Ostrowski and Grüss type inequalities for weighted integrals for differentiable functions of higher order for two variables. The some contents of the current section have published in 2019 see [86]. In this section we would deduce the generalizations of weighted integral identities of Montgomery for differentiable function of higher order for two variables and their consequences. Above said identities recapture other known and important ˇ identities and inequalities like Ostrowski type, Cebyšev type and Grüss type inequalities. The topic of Montgomery’s identities have many applications and cover other important known identities and inequalities in which involve Ostrowski and Grüss type inequalities (also see in present section). There are many applications of Ostrowski inequalities in the field of numerical integrations and probability theory (see [23, 48, 56, 110, 123, 138, 176]). We can also get special means using such ˇ inequalities (see [9, 10]). The special case of Ostrowski type inequalities is Cebyšev inequality which is very popular (see [149, 150]). Also there are many applications of Grüss type inequalities in the numerical integrations and other different fields (see [28, 34, 45, 175]). Now-a-days, due to rapid advancement of these types of inequalities are very popular. In current section, several generalizations of identities of Montgomery and inequalities of Ostrowski type, Grüss type also proposed by us for differentiable function of higher order. All these identities and inequalities generalize several consequences that we can see in [16, 52, 54, 66, 141, 156] etc. In this section, we use interval I × J = [a, b] × [c, d] ⊂ R2 . We recall Montgomery identity (1.1.1) from Chap. 1. We present here the generalization of Montgomery identity which is collected from [146]. Theorem 4.3.1 Let f be function and provided f is continuous in the interval I . Then 

b

f (x) =



b

v(r)f (r)dr +

a

pv (x, r)f (r)dr,

a

holds for weighted Peano kernel pv , defined as pv (x, r) =

V (r) , V (r) − 1 ,

a ≤ r ≤ x; x < r ≤ b;

where v : I → R∗ is some probability density function, i.e., it is a function that b satisfies a v(r)dr = 1 and V (r) =

⎧ ⎨ ⎩

r a

0, v(ν)dν , 1,

r < a; r ∈ [a, b]; r > b.

4.3 Weighted Montgomery Identities for Higher Order Differentiable Function. . .

277

The following generalized identities are obtained from [16] and [54] for functions with 2 independent variables. Theorem 4.3.2 Let f be function and provided f(1,1) is continuous in the interval I × J . Then  

d

+(b − a)



c



b

a

b



b

d

b

f (r, s) ds dr + (d − c)

c



f (r, y) dr a

d

c



a





d

p(x, r)q(y, s)f(1,1) (r, s) ds dr a

and (d − c)(b − a)f (x, y) = +

a

f (x, s) ds +



b

(d − c)(b − a)f (x, y) = −

d



b

f (r, s) ds dr +

c



b

q(y, s)f(0,1) (r, s) ds dr +

d

p(x, r)f(1,0) (r, s) ds dr a



 c

d

p(x, r) q(y, s)f(1,1) (r, s) ds dr

c

a

c

hold, here p and q are Peano kernels as defined above. In [156] authors gave the identities of weighted Montgomery for two variables functions. Theorem 4.3.3 Let function p : I × J → R be integrable and P is stated as 

b

P (x, y) =



d

(4.3.1)

p(ν, ζ ) dζ dν. x

y

If f is a function and provided that f(1,1) is continuous in the interval I × J . Then 

b

P (a, c)f (x, y) =  +

a d



d



b

p(r, s)f (r, s)dsdr +

c

P˜ (y, s)f(0,1)(x, s)ds −



c

b



a

Pˆ (x, r)f(1,0)(r, y) dr

a d

P¯ (x, r, y, s)f(1,1)(r, s)dsdr

c

holds, where

Pˆ (x, r) =

P˜ (y, s) =

"  r d a

"  b t a

c

c

p(ν, ζ )dζ dν , −P (r, c) ,

p(ν, ζ )dζ dν , −P (a, s) ,

a ≤ r ≤ x; x < r ≤ b;

c ≤ s ≤ y; y < s ≤ d;

(4.3.2)

ˇ 4 Popoviciu and Cebyšev-Popoviciu Type Identities and Inequalities

278

⎧ ⎪ ⎪ ⎪ ⎨

r s p(ν, ζ )dζ dν, ab cs − r c p(ν, ζ )dζ dν, r d and P¯ (x, r, y, s) = ⎪ − ⎪ a s p(ν, ζ )dζ dν, ⎪ ⎩ P (r, s),

a x a x

≤r