Schur-Convex Functions and Inequalities: Volume 2: Applications in Inequalities 9783110607864, 9783110606577

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Huan-nan Shi Schur-Convex Functions and Inequalities

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Huan-nan Shi

Schur-Convex Functions and Inequalities |

Volume 2: Applications in Inequalities

Mathematics Subject Classification 2010 Primary: 26A51, 26B25, 52A41; Secondary: 26D07, 26D10, 26D15, 26D20, 26E60 Author Huan-nan Shi Teacher’s College Beijing Union University Beijing 100011 Beijing People’s Republic of China [email protected]

ISBN 978-3-11-060657-7 e-ISBN (PDF) 978-3-11-060786-4 e-ISBN (EPUB) 978-3-11-060684-3 Library of Congress Control Number: 2019937573 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Harbin Institute of Technology Press Ltd, Harbin, Heilongjiang and Walter de Gruyter GmbH, Berlin/Boston Cover image: eyenigelen / E+ / gettyimages.com Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

|

Dedicated to the memory of my father

Contents Preface | IX Introduction | XI Acknowledgment | XV Notation and symbols | XVII 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3 3.1

Schur-convex functions and sequence inequalities | 1 Definitions and properties of convex sequences | 1 Various convex sequences | 9 An inequality for convex sequences | 15 Majorized proof of several weighted sum properties for convex sequences | 16 Refinement of the discrete Steffensen inequality | 23 Improvement of mean inequalities for convex functions | 24 A kind of jumping factorial inequalities | 33 Convexity and logarithmic convexity of arithmetic and geometric sequences | 37 Convex sequence inequalities associated with Chebyshev inequality | 53 Inequalities for convex sequences and nondecreasing convex functions | 55 Schur-convex functions and integral inequalities | 57 Schur-convex functions related to Hadamard integral inequalities | 57 Schur-convex functions related to Hadamard type integral inequalities | 64 Schur-convex functions related to Schwarz integral inequalities | 76 Schur-convex functions related to Hölder integral inequalities | 79 Schur-convex functions related to Chebyshev integral inequalities | 80 Majorization type integral inequalities | 83 Schur-convex functions and other integral inequalities | 87 Schur-convex functions and gamma functions | 91 Schur-convex functions and mean value inequalities for two variables | 97 Schur-concavity of Stolarsky mean | 97

VIII | Contents 3.2 3.3 3.4 3.5 3.6 3.7 4

Schur-concavity of Gini mean | 103 Comparison of Stolarsky and Gini means | 113 Schur-convexity of generalized Heron mean | 117 Schur-convexity of other two-variable means | 127 Schur-convexity for difference of some means | 158 Two-parameter homogeneous functions | 163

4.3 4.4 4.5

Schur-convex functions and mean value inequalities for multivariables | 169 Schur-convexity of the third k-order symmetric mean | 169 Schur-convexity of weighted generalized logarithmic mean in n variables | 177 On the optimal values for inequalities involving power mean | 178 Schur-p-power convexity of quotient for mean in n variables | 187 Schur-convexity of Bonferroni mean | 189

5 5.1 5.2

Schur-convex functions and geometric inequalities | 197 Schur-convex functions and triangle inequalities | 197 Schur-convex functions and simplex inequalities | 210

4.1 4.2

Bibliography | 221 Index | 233

Preface In 1979, A. M. Marshall and I. Olkin co-published “Inequalities: Theory of Majorization and Its Application.” Since then, majorization theory has become an independent discipline of mathematics. From September 1979 to September 1981, Professor Boying Wang of the Beijing Normal University studied the theory at the University of California (UCSB), as a visiting scholar. In 1984, he took the lead in setting up a course for postgraduate students named “Matrix and Majorization Inequalities” on majorization theory in China. From September 1984 to January 1986, I took this course during my studies at the Mathematics Department of Beijing Normal University. I did not expect this random choice to become my future research direction. So far, I have published 90 papers on majorization theory. In 1990, Professor Boying Wang’s book “Fundamentals of Majorization Inequalities” (in Chinese) was published. In addition to the classical basic theories in Marshall and Olkin’s book, this book also contains a number of wonderful original contents of Professor Boying Wang. In the application part, the book focuses on the application of majorization theory in matrices. As Professor Boying Wang puts it, “The majorization inequalities have almost infiltrated into various fields of mathematics, and played a wonderful role everywhere, because they can always profoundly describe the intrinsic relationship between many mathematical quantities, thus facilitating the derivation of the required conclusions. It can also easily derive many existing inequalities derived by different methods in a uniform way. It is a powerful means to generalize existing inequalities and discover new inequalities, and the theory and application of the majorization inequalities have a bright future.” The publication “Fundamentals of Majorization Inequalities” has greatly promoted the research of majorization theory in China. At present, Chinese scholars have published more than 300 research papers in this field and have formed a research team that has exerted some influence on the world. In 2011, A. M. Marshall, I. Olkin, and B. C. Arnold published “Inequalities: Theory of Majorization and Its Application” (The second edition),” which quoted a number of papers from Chinese scholars (including the five articles from the author). In 2012, China’s Harbin Institute of Technology Press published my monograph titled ““Majorization Theory and Analytic Inequality” (in Chinese),” which raised concerns from my peers. Over the past five years, almost all problems put forward in the book have been pursued and concluded by the following research. This English monograph “Schur-convex functions and inequalities” is a revision and supplement of the monograph “Majorization Theory and Analytical Inequalities.” More than 160 papers published in the past five years have been added, 97 of which were published by Chinese scholars (including 30 works by authors and collaborators).

https://doi.org/10.1515/9783110607864-201

X | Preface This book is divided into nine chapters. Chapters 1 and 2 of the first volume introduce the basic concepts and main theorems of Schur-convex function theory. In order to save space, the book does not include the detailed proofs of some basic theorems (which can be found in monographs [200] and [109]). We introduce the new developments of Schur-convex functions by Chinese scholars. Chapters 3 and 4 of the first volume introduce the wide applications of Schurconvex functions in symmetric function inequalities. Chapters 1 and 2 of the second volume introduce the application of Schur-convex functions in sequence inequalities and integral inequalities, respectively. Chapters 3 and 4 of the second volume introduce the applications of Schur-convex functions to mean inequalities. Chapter 5 of the second volume introduces the applications of Schur-convex functions in geometric inequalities.

Introduction There are many kinds of inequalities, and the techniques to solve them are colorful and numerous, so there is no general theory to deal with all inequalities. In 1923, I. Schur summed up some common and useful elementary and advanced inequalities and deduced a complete set of theories to deal with the inequalities with certain characteristics, which is the theory of majorization. In the theory of majorization, there are two key concepts: majorizing relations (see Definition 1.3.1 (Vol. 1)) and Schur-convex functions (see Definition 2.1.1 (Vol. 1)). Majorizing relations are weaker ordered relations among vectors, and Schur-convex functions are a kind of more extensive functions than classical convex functions. Combining these two objects is an effective method of constructing inequalities. In research of majorization theory, there are two important and fundamental objects: establishing majorizing relations among vectors and finding various Schurconvex functions. The judgement theorem of Schur-convex functions (that is, Theorem 2.1.3 (Vol. 1)) is the main method to determine Schur-convex functions. This theorem only depends on the first derivative of the function, so it is convenient to use. Majorizing relations deeply characterize intrinsic connections among vectors and combining a new majorizing relation with suitable Schur-convex functions can lead to various interesting inequalities. Below we introduce an elementary question that I came across when I first studied majorization theory, hoping to arouse the readers’ interest. Problem 0.0.1 (IMO, 1984). Let x, y, z ≥ 0, x + y + z + 1. Then 0 ≤ xy + yz + zx − 2xyz ≤

7 . 27

(0.1)

This problem has many kinds of proofs. The author found the following equivalent form of inequality (0.1): 3

3

1 1 0 ≤ (1 − x)(1 − y)(1 − z) − xyz ≤ (1 − ) − ( ) . 3 3

(0.2)

Thus, the high-dimensional extension is taken into account, i. e., n

n

i=1

i=1

0 ≤ ∏(1 − xi ) − ∏ xi ≤ (1 −

n

n

1 1 ) −( ) . n n

(0.3)

The author uses a gradual adjustment method to prove (0.3) (see [152]), and then extends (0.3) to the case of elementary symmetric functions. Let x ∈ ℝn+ and n1 ∑ni=1 xi = 1. Then k

k

n 1 1 0 ≤ Ek (1 − x) − Ek (x) ≤ ( )[(1 − ) − ( ) ], k n n https://doi.org/10.1515/9783110607864-202

(0.4)

XII | Introduction where Ek (x) = Ek (x1 , . . . , xn ) =

∑

1≤i1 0 for all i}, ℤn+ = {(p1 , . . . , pn ) | pi is a nonnegative integer, i = 1, . . . , n, n ≥ 2}. Throughout this book, increasing means nondecreasing and decreasing means nonincreasing. Thus if f : ℝ → ℝ, f is increasing if x ≤ y ⇒ f (x) ≤ f (y), strictly increasing if x < y ⇒ f (x) < f (y), decreasing if x ≤ y ⇒ f (x) ≥ f (y), strictly decreasing if x < y ⇒ f (x) > f (y). For any x = (x1 , . . . , xn ) ∈ ℝ, let x[1] ≥ ⋅ ⋅ ⋅ ≥ x[n] denote the components of x in decreasing order, and let x ↓= (x[1] , . . . , x[n] ) denote the decreasing rearrangement of x. Similarly, let x(1) ≥ ⋅ ⋅ ⋅ ≥ x(n) denote the components of x in increasing order, and let x ↑= (x(1) , . . . , x(n) ) denote the increasing rearrangement of x. The elementwise vector ordering xi ≤ yi , i = 1, . . . , n, is denoted by x ≤ y. For any x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ ℝ, we denote x + y = (x1 + y1 , . . . , xn + yn ), xy = (x1 y1 , . . . , xn yn ), αx = (αx1 , . . . , αxn ).

https://doi.org/10.1515/9783110607864-204

XVIII | Notation and symbols Generally, for f : ℝ → ℝ, f (x) = (f (x1 ), . . . , f (xn )), and for f : ℝ2 → ℝ, f (x, y) = (f (x1 , y1 ), . . . , f (xn , yn )). For f : Ω ⊂ ℝn → ℝ, we denote R(f ) = {f (x) | x ∈ Ω}, n

⨁ Ik = {(x1 , . . . , xn ) | xk ∈ Ik ⊂ ℝ++ }; k=1

n! (nk ) = k!(n−k)! is the number of combinations of n elements taken k at a time, defined n (0 ) = 1 and (nk ) = 0 for k > n.

1 Schur-convex functions and sequence inequalities 1.1 Definitions and properties of convex sequences Definition 1.1.1. A sequence {ak } (finite sequence {ak }nk=1 or infinite sequence {ak }∞ k=1 ) of real numbers is said to be a convex sequence if 2ak ≤ ak−1 + ak+1

for all k = 2, . . . , n − 1 (for all k ≥ 2).

(1.1.1)

A real sequence {ak } is said to be a concave sequence if the reverse inequality in (1.1.1) holds. Definition 1.1.2. A sequence {ak } (finite sequence {ak }nk=1 or infinite sequence {ak }∞ k=1 ) of nonnegative real numbers is said to be a logarithmically convex sequence (or logconvex sequence) if a2k ≤ ak−1 ak+1

for all k = 2, . . . , n − 1 (for all k ≥ 2).

(1.1.2)

A nonnegative real sequence {ak } is said to be a logarithmically concave sequence if the reverse inequality in (1.1.2) holds. Theorem 1.1.1. If {an } is a log-convex sequence, then {an } is a convex sequence. Proof. Since {an } is a logarithmically convex column, namely ai−1 ai+1 ≥ a2i , then ai−1 + ai+1 ≥ 2√ai−1 ai+1 ≥ 2ai , so {an } is a convex sequence. Theorem 1.1.2. Let {an } be an increasing nonnegative sequence. If {an } is a convex sequence, {ian } is also a convex sequence; if {an } is a log-concave sequence, {nan } is also a log-concave sequence. Proof. If {an } is a convex sequence, then ai−1 + ai+1 ≥ 2ai ; and because {an } is increasing, we have ai+1 − ai−1 ≥ 0, so (i + 1)ai+1 + (i − 1)ai−1 = i(ai−1 + ai+1 ) + (ai+1 − ai−1 ) ≥ 2iai . Therefore, {nan } is a convex sequence. If {an } is a log-concave sequence, then ai−1 ai+1 ≤ a2i , and then (i + 1)ai+1 ⋅ (i − 1)ai−1 = (i2 − 1)ai−1 ai+1 ≤ i2 a2i . Therefore, {nan } is a logarithmically concave sequence. Theorem 1.1.3. If {an } is both a convex sequence and a log-concave sequence, then { a1 } n is a convex sequence. Proof. If {an } is both a convex sequence and a log-concave sequence, then (ai−1 + ai+1 )ai ≥ 2a2i ≥ 2ai−1 ai+1 , namely, a1 + a1 ≥ a2 , hence { a1 } is a convex sequence. i+1

https://doi.org/10.1515/9783110607864-001

i−1

i

n

2 | 1 Schur-convex functions and sequence inequalities Theorem 1.1.4 ([76, pp. 208–209]). If {ak } is a convex sequence, then {Ak } is also a convex sequence, where Ak = k1 ∑ki=1 ai . The convex sequence is a discrete form of the convex function. The following three theorems reflect the relationship between the two. Theorem 1.1.5. Let {ak } be a convex sequence, f is an increasing convex function. Then {f (ak )} is also a convex sequence. Proof. Since f (ai−1 ) + f (ai+1 ) ≥ 2f (

ai−1 + ai+1 2a ) ≥ 2f ( i ) = 2f (ai ), 2 2

{f (ak )} is also a convex sequence. Theorem 1.1.6 ([78, p. 465]). If φ is a convex function on ℝ++ , then {φ(k)} is a convex sequence. The monograph [128, p. 6] gives the following the remark, but no proof has been given. Remark 1.1.1. If the sequence {ak } is convex, then the function f whose graph is the polygonal line with corner points (k, ak )(k ∈ ℕ) is also convex on [1, ∞). Wu and Lokenath [222] gave the following proofs. Theorem 1.1.7. Let {ak } be a convex sequence, let ψ be a continuous convex function and increasing on Ω, and let the function φ : [1, n] → I(I ⊂ Ω) defined by a1 + (a2 − a1 )(x − 1), { { { { { { a2 + (a3 − a2 )(x − 2), { { { { { {⋅ ⋅ ⋅ φ(x) = { { ai + (ai+1 − ai )(x − i), { { { { { { ⋅ ⋅⋅ { { { { {an−1 + (an − an−1 )(x − n + 1),

1 ≤ x < 2, 2 ≤ x < 3, i ≤ x < i + 1,

(1.1.3)

n − 1 ≤ x < n.

Then ψ(φ(x)) is a continuous and convex function on [1, n]. If ψ is a continuous concave function and decreasing on Ω, then ψ(φ(x)) is a continuous and concave function on [1, n]. Theorem 1.1.8 ([240]). A sequence {ak } is a convex sequence if and only if for any four nonnegative integers m, n, p, q, when p < m < q, p < n < q, and m + n = p + q, the inequality ap + aq ≥ am + an holds.

(1.1.4)

1.1 Definitions and properties of convex sequences | 3

Remark 1.1.2. According to the proof of [240], the condition p < m < q, p < n < q can be relaxed to p ≤ m ≤ q, p ≤ n ≤ q. From the standpoint of majorization, the condition p ≤ m ≤ q, p ≤ n ≤ q, and m + n = p + q means (m, n) ≺ (p, q). It is natural to think of whether the above results can be generalized in n dimensions. In 2001, Shi and Li [157] established the following results. Theorem 1.1.9. Let n ≥ 2. A necessary and sufficient condition that {ak } is a convex sequence is that the inequality ap1 + ⋅ ⋅ ⋅ + apn ≤ aq1 + ⋅ ⋅ ⋅ + aqn

(1.1.5)

holds for any p = (p1 , . . . , pn ), q = (q1 , . . . , qn ) ∈ ℤn+ , and p ≺ q. Proof. Necessity: Use mathematical induction for n. When n = 2, the proposition is established, assuming that the proposition is true when n = m (m ≥ 2). Consider the case of n = m + 1. Let p, q ∈ ℤm+1 + , p = (p1 , . . . , pm+1 ) ≺ (q1 , . . . , qm+1 ) = q. Without loss of generality, we may assume that p1 ≥ p2 ≥ ⋅ ⋅ ⋅ ≥ pm+1 , q1 ≥ q2 ≥ ⋅ ⋅ ⋅ ≥ qm+1 , satisfying (i) ∑ki=1 pi ≤ ∑ki=1 qi , k = 1, . . . , m; m+1 (ii) ∑m+1 i=1 pi = ∑i=1 qi . We divide the proof into two cases. The first case: there is r, 1 ≤ r ≤ m + 1, so that pr = qr . Removing pr and qr , obviously we still have (p1 , . . . , pr−1 , pr+1 , . . . , pm+1 ) ≺ (q1 , . . . , qr−1 , qr+1 , . . . , pm+1 ). By the inductive hypothesis, it follows that aq1 + ⋅ ⋅ ⋅ + aqr−1 + aqr+1 + ⋅ ⋅ ⋅ + apm+1 ≥ ap1 + ⋅ ⋅ ⋅ + apr−1 + apr+1 + ⋅ ⋅ ⋅ + apm+1 . Adding ap on both sides of the above inequality yields aq1 + ⋅ ⋅ ⋅ + apm+1 ≥ ap1 + ⋅ ⋅ ⋅ + apm+1 . The second case: pi ≠ qi , i = 1, 2, . . . , m + 1. From (i) we have p1 < q1 , and by (ii), it is not possible that pi < qi for all i, so there must be r, 2 ≤ r ≤ m + 1, such that pr > qr . Without loss of generality, we can assume that this r is the minimum subscript that satisfies pr > qr . Then p1 < q1 , p2 < q2 , . . ., pr−1 < qr−1 , pr > qr . The smaller of the two 󸀠 natural numbers qr−1 −pr−1 and pr −qr is denoted as h, h > 0, and we write qr−1 = qr−1 −h, 󸀠 qr = qr + h. Then 󸀠 qr−1 ≥ pr−1

and qr󸀠 ≤ pr .

(1.1.6)

󸀠 At least one of the two inequalities is true (when h = qr−1 − pr − 1, qr−1 = qr−1 − h = pr−1 ; 󸀠 󸀠 when h = pr −qr , qr = qr +h = pr ). Also because of pr−1 ≥ pr , by (1.1.5) we have qr−1 ≥ qr󸀠 .

4 | 1 Schur-convex functions and sequence inequalities 󸀠 󸀠 Then qr−1 > qr−1 ≥ qr󸀠 > qr and qr−1 + qr󸀠 = qr−1 + qr , by Theorem 1.1.8 and Remark 1.1.2, we have 󸀠 aqr−1 + aqr󸀠 ≤ aqr−1 + aqr .

(1.1.7)

aqi , we obtain When i ≠ r − 1, r, we agree qi󸀠 = qi , on both sides of (1.1.7) plus ∑i=r−1,r ̸ m+1

m+1

∑ aq󸀠 ≤ ∑ aqi . i=1

i

i=1

(1.1.8)

󸀠 󸀠 󸀠 Look again at (q1󸀠 , . . . , qm+1 ) and (p1 , . . . , pm+1 ). From qr−2 = qr−2 ≥ qr−1 > qr−1 ≥ 󸀠 󸀠 󸀠 󸀠 󸀠 > qr ≥ qr+1 = qr+1 , we know that (q1 , . . . , qn ) = (q[1] , . . . , q[n] ). It is easy to verify that (p1 , . . . , pn ) ≺ (q1󸀠 , . . . , qn󸀠 ). It has been proved that at least one of the two inequalities holds, that is, (p1 , . . . , pn ) and (q1󸀠 , . . . , qn󸀠 ) have at least one component with the same subscripts being equal. Therefore, from the first case proved, we have ∑m+1 i=1 aqi ≤ m+1 m+1 󸀠 . a a ≤ a , and combining with (1.1.8) we obtain ∑ ∑m+1 ∑ qi qi qi i=1 i=1 i=1 To sum up the above two cases, the inductive hypothesis holds when n = m (m ≥ 2). It can be proved that the proposition is correct when n = m + 1 and is correct for all propositions of n ≥ 2. Sufficiency: For any positive integer i, in (1.1.5), take q1 = i − 1, q2 = i + 1, q3 = ⋅ ⋅ ⋅ = qn = 0, p1 = p2 = i, p3 = ⋅ ⋅ ⋅ = pn = 0. Then (p1 , . . . , pn ) ≺ (q1 , . . . , qn ), and (1.1.5) is reduced to ai−1 + ai+1 ≥ 2ai (i = 1, 2, 3, . . .), so {an } is a convex sequence. The proof of Theorem 1.1.9 is completed.

qr󸀠

In 2007, Wu and Lokenath [222] gave a new proof. Corollary 1.1.1. Let f : I ⊂ ℝ → ℝ be an increasing convex function, and let {ak } be a convex sequence on convex set I, ∀p, q ∈ ℤn+ . If p ≺ q, then n

n

i=1

i=1

∑ f (aqi ) ≥ ∑ f (api ).

(1.1.9)

Proof. From Theorem 1.1.5 we know that {f (ak )} is a convex sequence, and using Theorem 1.1.9, we can prove Corollary 1.1.1. Corollary 1.1.2. The necessary and sufficient conditions for the nonnegative sequence {ak } to be a log-convex sequence are ∀p, q ∈ ℤn+ , if p ≺ q, then n

n

i=1

i=1

∏ api ≤ ∏ aqi .

(1.1.10)

Proof. If a log-convex sequence has a item equal to zero, it is easy to prove that each item of this sequence is zero. At this time, it is obvious that Corollary 1.1.2 holds. In the following we assume that {ak } is a positive sequence. Since {log ak } is a convex sequence, according to Theorem 1.1.9, we have ∑ni=1 log aqi ≥ ∑ni=1 log api , and inequality (1.1.10) holds.

1.1 Definitions and properties of convex sequences | 5

The author [158] generalized Theorem 1.1.9 as the following theorem. Theorem 1.1.10. If the sequence {ak } is an increasing convex sequence, for any p, q ∈ ℤn+ , if p ≺w q, then the inequality (1.1.11)

ap1 + ⋅ ⋅ ⋅ + apn ≤ aq1 + ⋅ ⋅ ⋅ + aqn holds.

Corollary 1.1.3. Let the nonnegative sequence {ak } be an increasing log-convex sequence. For any p, q ∈ ℤn+ , if p ≺w q, then n

n

i=1

i=1

(1.1.12)

∏ api ≤ ∏ aqi .

Theorem 1.1.11 ([184]). Let {ak } be a sequence of positive numbers. Then {an } is logconvex if and only if for each x ≥ 0, the sequence Pn (x) = ∑nk=0 ak (nk )x n−k (n ∈ ℕ) is log-convex. Wu [219] extended Theorem 1.1.9 to the case of operator convex sequences. For more properties of convex sequences, see the monograph [128]. Theorem 1.1.12 ([223]). Let {an } be a convex sequence and m, k be nonnegative integers. Then (n − 2m)(ak+1 + ak+3 + ⋅ ⋅ ⋅ + ak+2n+1 ) + (2m − n − 1)(ak+2 + ak+4 + ⋅ ⋅ ⋅ + ak+2n )

(1.1.13)

+ 2m(ak+1 + ak+2n+1 ) − m(ak+2 + ak+2n ) ≥ 0.

Proof. According to Definition 1.3.1 (Vol. 1) and (1.4.42) (Vol. 1), we have k + 2n, . . . , k + 2n) 2k + 4, . . . , k + 4, . . . , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + 2, . . . , k + 2, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1

n+1

n+1

≺ (k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + 1, . . . , k + 1, ⏟⏟ 2k + 3,⏟.⏟⏟. ⏟.⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ , k + ⏟3⏟, . . . , k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + (2n + 1), . . . , k + (2n + 1)). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

n

n

Therefore, from Theorem 1.1.9, it follows that (n + 1)(ak+2 + ak+4 + ⋅ ⋅ ⋅ + ak+2n ) ≤ n(ak+1 + ak+3 + ⋅ ⋅ ⋅ + ak+(2n+1) ). By Theorem 1.3.12(a) (Vol. 1), it is easy to prove that k + 2, . . . , k + 2, k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + 1, . . . , k + 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + 3, . . . , k + 3, 2m

m

2m

+ 2n − 1, . . . , k + 2n − 1, k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + 5, . . . , k + 5, . . . , k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2m

2m

k + 2n, . . . , k + 2n, k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + 2n + 1, . . . , k + 2n + 1) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m

2m

k + 2, . . . , k + 2, ≺ (k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + 1, . . . , k + 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2m

m

(1.1.14)

6 | 1 Schur-convex functions and sequence inequalities k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + (2n − 2), .⏟⏟⏟ . .⏟,⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k + (2n − 2) k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + 4, . . . , k + 4, . . . , ⏟⏟ ⏟⏟, 2m

2m

+ 2n + 1, . . . , k + 2n + 1). k + 2n, . . . , k + 2n, k⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m

2m

Thus by Theorem 1.1.9 we have 2m(ak+1 + ak+3 + ⋅ ⋅ ⋅ + ak+2n+1 ) + m(ak+2 + ak+2n )

(1.1.15)

≤ 2m(ak+2 + ak+4 + ⋅ ⋅ ⋅ + ak+2n ) + 2m(ak+1 + ak+2n+1 ). Adding both sides in inequality (1.1.14) and inequality (1.1.15) correspondingly and making a little deformation yields inequality (1.1.13). When m = 0, the following corollary can be obtained from inequality (1.1.13). Corollary 1.1.4. Let {an } be a convex sequence and k be a nonnegative integer. Then ak+1 + ak+3 + ⋅ ⋅ ⋅ + ak+2n+1 ak+2 + ak+4 + ⋅ ⋅ ⋅ + ak+2n ≥ . n+1 n

(1.1.16)

When k = 0, (1.1.16) is the famous Nanson inequality [76, p. 509] n

n

i=0

i=1

n ∑ a2i+1 ≥ (n + 1) ∑ a2i .

(1.1.17)

Example 1.1.1. If {an } is a convex sequence, then 2n+1

∑ (−1)i+1 ak+i ≥ i=1

1 2n+1 1 n 1 n ∑ ak+2i+1 ≥ ∑ ak+i ≥ ∑ ak+2i . n + 1 i=0 2n + 1 i=1 n i=1

(1.1.18)

Proof. It is not difficult to verify that the three inequalities in (1.1.18) are all equivalent to inequality (1.1.16). Example 1.1.2. Let {an } be a convex sequence, m ∈ ℕ. Then a2 + a4 + ⋅ ⋅ ⋅ + a2m ≥ a3 + a5 + ⋅ ⋅ ⋅ + a2m−1 + am+1

(1.1.19)

a1 + a3 + a5 + ⋅ ⋅ ⋅ + a2m+1 ≥ a2 + a4 + ⋅ ⋅ ⋅ + a2m + am+1 .

(1.1.20)

and

Proof. Let xi = 2m − (i − 1), i = 1, 2, . . . , m. Then x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ x2m ≥ 0, and 2m−1

∑ (−1)i−1 xi = m + 1, i=1

2m

∑(−1)i−1 xi = m. i=1

The following can be obtained from (1.4.22) (Vol. 1) and (1.4.23) (Vol. 1), respectively: (m + 1, 2m − 1, 2m − 3, . . . , 5, 3) ≺ (2m, 2m − 2, . . . , 4, 2)

(1.1.21)

1.1 Definitions and properties of convex sequences | 7

and (m, 2m − 1, 2m − 3, . . . , 3, 1) ≺ (2m, 2m − 2, . . . , 4, 2, 0).

(1.1.22)

From (1.1.22), it is clear that (m + 1, 2m, 2m − 3, 2m − 2, . . . , 4, 2) ≺ (2m + 1, 2m − 1, . . . , 5, 3, 1).

(1.1.23)

Thus by Theorem 1.1.9 and combining with (1.1.21) and (1.1.23), respectively, we can see that inequality (1.1.19) and inequality (1.1.20) hold. Example 1.1.3. Let {an } be a convex sequence, h, m, n ∈ N. Then n−m m n 1 n 1 1 ( ∑ ak + ∑ ak ) (n ≥ 2m), ∑ ak ≤ ∑ ak ≤ n − 2m k=m+1 n k=1 2m k=1 k=n−m+1

(1.1.24)

h−n m m−h n n−m h ∑ ak + ∑ ak + ∑ a ≥ 0 (h < m < n), h k=1 m k=1 n k=1 k

(1.1.25)

m m+n n+m n ( ∑ ak − ∑ ak ) ≤ ∑ ak n − m k=1 k=1 k=1

(m ≠ n).

(1.1.26)

Proof. Since {an } is a convex sequence, by Theorem 1.1.4, we know that {An } is also a convex sequence, where An = n1 ∑ni=1 ai . For n > 2m, it is not difficult to verify that (n − m, . . . , n − m, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n, . . . , n). . . . , m, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n, . . . , n) ≺ (m, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n−m

m

2m

n

By Theorem 1.1.9, the inequality (n − m)An−m + 2mAn ≤ mAm + nAn holds. However, it is easy to see that the two inequalities in (1.1.24) are equivalent to the above inequality. For h < m < n, we have h, . . . , h). . . . , n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (m, . . . , m) ≺ (n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m−h

n−h

n−m

By Theorem 1.1.9, the inequality (n − h)Am ≤ (m − h)An + (n − m)Ah holds. Inequality (1.1.25) is equivalent to the above inequality, so inequality (1.1.25) holds. Since inequality (1.1.26) is about m, n symmetry, without loss of generality, we may assume that m < n. Note that (n, . . . , n) ≺ (m, . . . , m, n + m, . . . , n + m). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

m

n−m

8 | 1 Schur-convex functions and sequence inequalities By Theorem 1.1.9, the inequality nAn ≤ (n − m)An+m + mAm holds. This inequality is equivalent to (1.1.25), so inequality (1.1.26) holds. Example 1.1.4. Let {an } be a convex sequence, n ∈ N, n > 1. Then n

∑ ak ≥

k=0

n + 1 n−1 ∑a n − 1 k=1 k

(1.1.27)

and n 1 n 1 1 n ∑ ai ( ) ≤ ∑ a ≤ (a + an ). n 2 i=0 n + 1 i=0 i 2 0 i

(1.1.28)

Proof. According to Theorem 1.3.12(a) (Vol. 1), we have n, . . . , n) 1, . . . , 1, . . . , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ . . . , 0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ − 1, . . . , n − 1) ≺ (0, 2, . . . , 2, . . . , n (1, . . . , 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1

n−1

n−1

n−1

n+1

n+1

and 1, . . . , 1, ). (0, 0, 1, 1, . . . , n, n) ≺ (0, . . . , 0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1

n+1

Thus from Theorem 1.1.9, it follows that inequalities (1.1.27) and (1.1.28) hold. Example 1.1.5. If {ak } is a convex sequence, then for 0 ≤ i ≤ k, we have ai ≤ (1 −

i i )a + a . k 0 k k

(1.1.29)

Proof. Note that . . . , k , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (i, . . . , i) ≺ (k, 0, . . . , 0). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k

i

k−i

From Theorem 1.1.9, we have kai ≤ (k − i)a0 + iak , i. e., inequality (1.1.29) holds. Example 1.1.6. Let x > 0, n ∈ N. Then 2

1 1 + x2 + ⋅ ⋅ ⋅ + x2n n+1 ≥ ). + (√x − 3 2n−1 √x n x + x + ⋅⋅⋅ + x Proof. Since 2

2

1 1 n+1 n+1 nx2 − (n − 1)x + n + (√x − + (√x − , ) = ) = √x √x n n nx

(1.1.30)

1.2 Various convex sequences | 9

inequality (1.1.30) ⇔ nx(1 + x2 + ⋅ ⋅ ⋅ + x2n ) ≥ (x + x3 + ⋅ ⋅ ⋅ + x2n−1 )[nx 2 − (n − 1)x + n] ⇔ n(x + x3 + ⋅ ⋅ ⋅ + x2n+1 ) + 2n(x2 + x4 + ⋅ ⋅ ⋅ + x2n ) ≥ n(x2n+1 + x) + (n + 1)(x2 + x4 + ⋅ ⋅ ⋅ + x2n ) + 2n(x3 + x5 + ⋅ ⋅ ⋅ + x2n−1 ) ⇔ x[(n − 1)(x + x3 + ⋅ ⋅ ⋅ + x2n−1 ) − n(x 2 + x4 + ⋅ ⋅ ⋅ + x2n−2 )] ≥ 0. Write an = xn , n ∈ N. Then (n − 1)(x + x3 + ⋅ ⋅ ⋅ + x2n−1 ) − n(x2 + x4 + ⋅ ⋅ ⋅ + x2n−2 ) = (n − 1)(a1 + a3 + ⋅ ⋅ ⋅ + a2n−1 ) − n(a2 + a4 + ⋅ ⋅ ⋅ + a2n−2 ). By Theorem 1.1.6, we know that {an } is a convex sequence, and from (1.1.17), it follows that (n − 1)(a1 + a3 + ⋅ ⋅ ⋅ + a2n−1 ) − n(a2 + a4 + ⋅ ⋅ ⋅ + a2n−2 ) ≥ 0. So inequality (1.1.30) holds. Inequality (1.1.30) is a strengthened form of the well-known Wilson inequality [76, p. 125] n+1 1 + x2 + ⋅ ⋅ ⋅ + x2n ≥ . n x + x3 + ⋅ ⋅ ⋅ + x2n−1

1.2 Various convex sequences There are various types of convex and log-convex sequences. This section introduces several examples. Example 1.2.1. Let x ∈ ℝn++ , k ∈ ℕ, and Sk = ∑ni=1 xik . Then {Sk }k∈ℕ is a convex sequence. Proof. We have n

n

n

i=1 n

i=1

i=1 n

Sk−1 + Sk+1 = ∑ xik−1 + ∑ xik+1 = ∑(xik−1 + xik+1 ) ≥ ∑ 2√xik−1 xik+1 = 2 ∑ xik = 2Sk . i=1

i=1

10 | 1 Schur-convex functions and sequence inequalities Example 1.2.2. Let x ∈ ℝn++ , k ∈ ℕ, and Sk = ∑ni=1 xik . Then {Sk }k∈ℕ is a log-convex sequence. Proof. By induction on k, we prove the following inequality holds: Sk2 ≤ Sk−1 Sk+1 ,

k ≥ 2.

(1.2.1)

When n = 1, the equality occurs. Suppose that when n = m, inequality (1.2.1) holds. Now we consider the case n = m + 1. 2

m+1

2

m

2

m

m

2k k k ) = (∑ xik ) + 2xm+1 ( ∑ xik ) = (∑ xik + xm+1 ∑ xik + xm+1 i=1

i=1

i=1

i=1

m

m

m

i=1

i=1

i=1

(1.2.2)

2k k . ≤ (∑ xik−1 )(∑ xik+1 ) + 2xm+1 ∑ xik + xm+1

On the other hand, m+1

m+1

m

m

i=1

i=1

i=1

i=1

k−1 k+1 ( ∑ xik−1 )( ∑ xik+1 ) = (∑ xik−1 + xm+1 )(∑ xik+1 + xm+1 )

(1.2.3)

m

m

m

m

i=1

i=1

i=1

i=1

k+1 k−1 2k = (∑ xik−1 )(∑ xik+1 ) + xm+1 . ∑ xik−1 + xm+1 ∑ xik+1 + xm+1

Comparing the right inequalities in (1.2.2) and (1.2.3), we only need to prove that m

m

m

i=1

i=1

i=1

k−1 k+1 k 2xm+1 ∑ xik+1 . ∑ xik−1 + xm+1 ∑ xik ≤ xm+1

This inequality is equivalent to m

m

i=1 m

i=1

m

m

i=1

i=1

k k+1 k k−1 xm+1 ∑ xik − xm+1 ∑ xik−1 + xm+1 ∑ xik − xm+1 ∑ xik+1 ≤ 0 m

k k−1 ⇔ xm+1 ∑ xik−1 (xi − xm+1 ) + xm+1 ∑ xik (xm+1 − xi ) ≤ 0 i=1 m

i=1

k−1 ⇔ xm+1 ∑ xik−1 (xi − xm+1 )(xm+1 − xi ) ≤ 0

⇔ −

i=1 m k−1 xm+1 ∑ xik−1 (xi i=1

− xm+1 )2 ≤ 0.

The above inequality clearly holds, and inequality (1.2.1) is true. Another proof is given as follows: Sk−1 Sk+1 −

Sk2

=

n

n (∑ xik−1 )(∑ xik+1 ) i=1 i=1

−

2 k (∑ xi ) i=1 n

n

n

n

n

i=1

j=1

i=1

j=1

= (∑ xik−1 )(∑ xjk+1 ) − (∑ xik )(∑ xjk )

1.2 Various convex sequences | 11 n

n

i=1

j=1

n

n

i=1

j=1

= (∑ xik−1 ∑ xjk+1 ) − (∑ xik ∑ xjk ) n

n

n

i=1

j=1

j=1

= ∑(xik−1 ∑ xjk+1 − xik ∑ xjk ) n

n

n

i=1

j=1

j=1

n

n

i=1

j=1

n

n

= ∑(xik−1 (∑ xjk+1 − xi ∑ xjk )) = ∑(xik−1 (∑(xjk+1 − xi xjk ))) = ∑(xik−1 (∑ xjk (xj − xi ))) i=1 n

j=1

n

= ∑ ∑(xjk+1 xik−1 − xjk xik ) i=1 j=1 n

n

= ∑ ∑ xik−1 xjk−1 (xi − xj )2 ≥ 0. i=1 j=1

Example 1.2.3 ([76, p. 585]). Let f , g be a positive continuous function on [a, b]. Write b

n

In = ∫[f (x)] g(x)dx. a

Then {In } is a log-convex sequence. + 1, . . . , m + 1, 0). Then For any natural number m, take p = (m, . . . , m), q = (m ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m+1

m+1 m p ≺ q. From Corollary 1.1.2, it follows that Im ≤ Im+1 ⋅ I0 , namely, m+1

b

(∫ f (x)m g(x)dx)

b

b

m

m

≤ ∫ g(x)dx(∫ f (x)m+1 g(x)dx) .

a

a

(1.2.4)

a

In particular, take m = 1, g(x) = 1, the following well-known inequality can be obtained from inequality (1.2.4): 2

b

(∫ f (x)dx) ≤ a

b

1 (∫ f 2 (x)dx). b−a a

Example 1.2.4 ([76, p. 432]). Let ∞

En (z) = ∫ 1

e−zt dt, tn

Re z > 0, n = 0, 1, 2, . . . .

Then when z > 0, for n = 1, 2, 3, . . . , {En (z)} is a log-convex sequence.

(1.2.5)

12 | 1 Schur-convex functions and sequence inequalities Definition 1.2.1 ([112, p. 365]). A function f said to be an absolutely monotonic function on (0, ∞) if f has derivatives of all orders on (0, ∞) and satisfies f (k) (t) ≥ 0,

t ∈ (0, ∞), k = 0, 1, 2, . . . .

(1.2.6)

Example 1.2.5 ([76, p. 432]). Let b > a > 0, x > 0, and g(x) =

bx − ax . x

Qi [141] proved that g(x) is an absolutely monotonic function on ℝ++ . Definition 1.2.2 ([112, p. 365]). The function f is said to be a completely monotonic function on ℝ++ if f has derivatives of all orders on ℝ++ and satisfies (−1)k f (k) (t) ≥ 0,

t ∈ (0, ∞), k = 0, 1, 2, . . . .

(1.2.7)

Definition 1.2.3 ([136]). The positive value function f is said to be a logarithmically completely monotonic function on interval I ⊂ ℝ if log f has derivatives of all orders on ℝ++ and satisfies (−1)k [log f (t)]

(k)

≥ 0,

t ∈ (0, ∞), k = 0, 1, 2, . . . .

(1.2.8)

Theorem 1.2.1 ([136]). The logarithmically completely monotonic function must be a completely monotonic function. Theorem 1.2.2 ([112, p. 369]). Let f be completely monotonic on ℝ++ , f (k) (t) ≠ 0, k = 0, 1, 2, . . . . Then {(−1)k f (k) (t)} is a log-convex sequence. Remark 1.2.1. If {(−1)k f (k) (t)} is a log-convex sequence, this means that the inequality 2

(−1)k−1 f (k−1) (t) ⋅ (−1)k+1 f (k+1) (t) ≥ [(−1)k f (k) (t)] ,

k = 0, 1, 2, . . . ,

holds, so the inequality 2

f (k−1) (t)f (k+1) (t) ≥ [f (k) (t)] ,

k = 0, 1, 2, . . . ,

(1.2.9)

is true, so {f (k) (t)} is also a log-convex sequence under the conditions of Theorem 1.2.2. It is worth noting that the monograph [112] on page 366 suggests that if the function f is absolutely monotonic on (0, ∞), then (1.2.9) holds, so {f (k) (t)} is also a logconvex sequence. In 2012, Sitnik [185] pointed out that this conclusion is not valid by the following counterexamples. Let f (x) = x2 +1, x ∈ [0, ∞). Then f (x) ≥ 0, f 󸀠 (x) = 2x ≥ 0, f 󸀠󸀠 (x) = 2 ≥ 0, f (k) (x) = 0, k > 2, so that f (x) = x2 +1 is absolutely monotonic on (0, ∞), but f (x)f 󸀠󸀠 (x) ≥ (f 󸀠 (x))2 ⇔ 2(x2 + 1) ≥ 4x2 ⇔ 1 ≥ x2 does not hold on (0, ∞).

1.2 Various convex sequences | 13

Example 1.2.6. For the elementary symmetric functions Ek (x) =

∑

k

∏ xij

1≤i1 ) > ∫ f (x)dx, ∑ f( ∑ f( n i=k+1 n + k n + 1 i=k+1 n + k + 1

(1.6.2)

0

where k ∈ ℤ+ , n ∈ ℕ. Qi and Guo [137] also gave the following result. Theorem 1.6.1. Let f be an increasing convex (or concave) function on [0, 1], and a {an }n∈ℕ is an increasing positive sequence, such that {i( a i − 1)}i∈ℕ is decreasing ({i(

ai+1 ai

i+1

− 1)}i∈ℕ is increasing). Then

1

a a 1 n 1 n+1 ∑f( i ) ≥ ∑ f ( i ) ≥ ∫ f (x)dx. n i=1 an n + 1 i=1 an+1 0

(1.6.3)

1.6 Improvement of mean inequalities for convex functions | 25

Remark 1.6.1. We pointed out that the right inequality in (1.6.3) does not hold, a counterexample is as follows. a Let f (x) = x2 , x ∈ [0, 1]. Taking ai = 2i , then i( a i −1) = − 2i is decreasing. Therefore, i+1 when n ≥ 4, we have 2

1

i

a 1 n 4 1 1 1 ∞ 1 1 n ≤ = ∫ f (x)dx. ∑ f ( i ) = ∑( n−i ) < ∑ ( ) = n i=1 an n i=1 2 4 i=0 4 3n 3 0

Chen et al. [11] further gave the following two theorems. Theorem 1.6.2. Let f be increasing and convex (or concave) on [0, 1]. Then 1

1 n i i 1 n+1 ) ≥ ∫ f (x)dx ∑f( ) > ∑ f( n i=1 n n + 1 i=1 n + 1

(1.6.4)

0

≥

1 n i 1 n−1 i ) ≥ ∑ f ( ). ∑ f( n + 1 i=0 n + 1 n i=0 n

Theorem 1.6.3. Let f be an increasing convex (or concave) function on [0, 1], and a {an }n∈ℕ is an increasing positive sequence, such that {i( a i − 1)}i∈ℕ is decreasing ({i(

ai+1 ai

i+1

− 1)}i∈ℕ is increasing). Then 1

∫ f (x)dx ≥ 0

a 1 n−1 a 1 n ∑ f ( i ) ≥ ∑ f ( i ), n + 1 i=0 an+1 n i=0 an

(1.6.5)

where a0 = 0. Remark 1.6.2. We pointed out that the left inequality in (1.6.5) does not hold, a counterexample is as follows. a Let f (x) = x2 , x ∈ [0, 1]. Taking ai = 1 − 21i , then i( a i − 1) = 2(2ii−1) is decreasing. i+1

Consider the function g(x) = 2xx−1 . It is not difficult to verify that g 󸀠 (x) < 0, that is, g(x) is strictly decreasing on ℝ++ . Therefore, when n ≥ 5, we have 1 2 2 1 n−1 1 − 2i 1 1 n−1 ai 1 n−1 (1 − ) ) > ∑ f ( ) = ∑( ∑ n i=0 an n i=1 1 − 1n n i=1 2i 2

>

1 n−1 n − 1 2 n−1 1 n−1 2 ∞ 1 2 − ∑ i > − ∑ i ∑ (1 − i ) = n i=1 n n i=i 2 n n i=i 2 2 1

n−3 3 2 1 = = 1 − ≥ > = ∫ f (x)dx. n n 5 3 0

In 2007, Xu and Shi [246] improved Theorem 1.6.1 and Theorem 1.6.3, establishing the following two theorems.

26 | 1 Schur-convex functions and sequence inequalities Theorem 1.6.4. Let f be increasing on [0, 1], and {an }n∈ℕ is an increasing positive sequence. a (a) If f is concave and {i( ai+1 − 1)}n∈ℕ is increasing, then i

a a 1 n+1 1 n ∑f( i ) ≥ ∑ f ( i ) ≥ f (0). n i=1 an+1 n + 1 i=1 an+2

(1.6.6)

If f (t) is right continuous at t = 0, the lower bound f (0) is the best. a (b) If f is convex and {i( a i − 1)}n∈ℕ is decreasing, then i+1

a a 1 n 1 n+1 ∑f( i ) ≥ ∑ f ( i ) ≥ f (0). n i=1 an n + 1 i=1 an+1

(1.6.7)

The lower bound f (0) is the best. Theorem 1.6.5. Let f be increasing on [0, 1], and {an }n∈ℕ is an increasing positive sequence, a0 = 0. a (a) If f is concave and {i( ai−1 − 1)}n∈ℕ is increasing, then i

a a 1 n 1 n−1 ∑ f( i ) ≤ ∑ f ( i ) ≤ f (1). n i=1 an−1 n + 1 i=0 an

(1.6.8)

The upper bound f (1) is the best. a (b) If f is convex and {i( ai+1 − 1)}n∈ℕ is decreasing, then i

a 1 n 1 n−1 ai ∑ f( ) ≤ ∑ f ( i ) ≤ f (1). n i=0 an n + 1 i=0 an+1

(1.6.9)

If f (t) is left continuous at t = 1, then the upper bound f (1) is the best. For the natural number n, the Minc–Sathre inequality [110] is √n n! n < 1. < n+1 n+1 √(n + 1)!

(1.6.10)

Using Theorem 1.6.4(a), Xu and Shi [246] strengthened the left inequality in the Minc–Sathre inequality as √n n! n+1 . ≤ n+1 n+2 √(n + 1)!

(1.6.11)

Applying Theorems 1.6.4(a) and 1.6.5(a), Xu and Shi [246] obtained the following inequality: n+1 ≤( n+2 where 0 < r ≤ 1.

1 1 n r r ∑ i n i=1 ) n+1 r 1 ∑ i n+1 i=1

≤

n−1 , n

(1.6.12)

1.6 Improvement of mean inequalities for convex functions | 27

Under the condition 0 < r ≤ 1, the left inequality in (1.6.12) improves the Alzer inequality [2], i. e., n ≤( n+1

1 1 n r r ∑ i n i=1 ) . n+1 r 1 ∑ i n+1 i=1

(1.6.13)

The right inequality in (1.6.13) improves the inequality in Corollary 1 in [11], i. e., (

1 1 n r r ∑ i n i=1 ) 1 ∑n+1 ir n+1 i=1

≤

n . n+1

(1.6.14)

This section uses the majorization method to prove the left inequality in Theorem 1.6.4, improves the left inequality in Theorem 1.6.5, and generalizes inequality (1.6.2). This section is an improvement to [179]. We first give two lemmas. Lemma 1.6.1. Let {an }n∈ℕ be an increasing positive sequence, m ∈ ℕ, and let a a a a x = ( 1 , . . . , 1 , . . . , n+1 , . . . , n+1 ) an+m an+m an+m an+m ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

n

and y=(

an an a1 a1 ,..., ,..., ,..., ). a a a a ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+m−1 n+m−1 n+m−1 n+m−1 n+1

n+1

a

(a) If m = 1 and {i( a i − 1)}i∈ℕ is decreasing, then x ≺w y; i+1

a

(b) if m = 1 or 2 and {i( a i − 1)}i∈ℕ is increasing, then x ≺w y. i+1

Proof. Let ui = (

a ai ,..., i ) an+1 an+1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

and a a a a v i = ( i−1 , . . . , i−1 , i , . . . , i ), a a a a ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n n ⏟⏟⏟⏟⏟⏟⏟⏟⏟ n ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟n⏟⏟ i−1

i = 1, . . . , n + 1.

n−i+1

According to Theorem 1.3.14(a) (Vol. 1), in order to prove x ≺w y, we only need to prove ui ≺w v i . If 1 ≤ k ≤ n − i + 1, then k

∑ ui[j] = k ⋅ j=1

k a ai ≤ k ⋅ i = ∑ vi[j] . an+1 an j=1

28 | 1 Schur-convex functions and sequence inequalities If n − i + 1 < k ≤ n, then k

∑ ui[j] = k ⋅ j=1

k a ai a ≤ (n − i + 1) i + [k − (n − i + 1)] i−1 = ∑ vi[j] an+1 an an j=1

a an ≤ (n − i + 1) + [k − (n − i + 1)] i−1 an+1 ai

⇔k⋅ ⇔ k(

a a an − i−1 ) ≤ (n − i + 1)(1 − i−1 ). an+1 ai ai

(1.6.15)

a

a

If a n − ai−1 ≤ 0, since {an }n∈ℕ is an increasing positive sequence and 1 − n+1 i inequality (1.6.15) obviously holds. a a a If a n − ai−1 > 0, as {i( a i − 1)}i∈ℕ is decreasing, it follows that n+1

k(

i

ai−1 ai

≥ 0,

i+1

a a a a a an − i−1 ) ≤ n( n − i−1 ) = n( n − 1) − n( i−1 − 1) an+1 ai an+1 ai an+1 ai ≤ (i − 1)(

ai−1 a a − 1) − n( i−1 − 1) = (n − i + 1)(1 − i−1 ). ai ai ai

That is, inequality (1.6.15) holds. (b) Let ui = (

a a a ai , . . . , i , i+1 , . . . , i+1 ) a a a a ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+m n+m ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+m n+m n−i+1

i

and ai ai vi = ( ,..., ), a a ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+m−1 n+m−1

i = 1, . . . , n.

n+1

According to Theorem 1.3.14(b) (Vol. 1), in order to prove x ≺w y, we only need to prove v i ≺w ui . If 1 ≤ k ≤ n − i + 1, then k

∑ vi(j) = k ⋅ j=1

ai

an+m−1

≥k⋅

ai

an+m

k

= ∑ ui(j) . j=1

If n − i + 2 < k ≤ n + 1, then k

k

j=1

j=1

∑ vi(j) ≥ ∑ ui(j) ⇔ k ⋅ ⇔k⋅ ⇔ k(

ai

an+m−1

≥ (n − i + 1)

ai

an+m

+ [k − (n − i + 1)]

a an+m ≥ (n − i + 1) + [k − (n − i + 1)] i+1 an+m−1 ai

an+m a a − i+1 ) ≥ (n − i + 1)(1 − i+1 ). an+m−1 ai ai

ai+1 an+m

(1.6.16)

1.6 Improvement of mean inequalities for convex functions | 29

a

a

a

If a n+m − ai+1 ≥ 0, since {an }n∈ℕ is an increasing positive sequence, 1 − ai+1 ≤ 0, so n+m−1 i i inequality (1.6.16) holds. a a a If a n+m − ai+1 < 0, m ≥ 1, implies n + m − 1 ≥ i, and as {i( ai+1 − 1)}i∈ℕ is increasing, n+m−1 i i it follows that (n + m − 1)(

an+m a − 1) ≥ i( i+1 − 1). an+m−1 ai

If m = 1, then k ≤ n + 1 ≤ n + m, and then k(

a a a an+m − i+1 ) ≥ (n + m)( n+m − i+1 ) an+m−1 ai an+m−1 ai = (n + m)(

an+m a − 1) − (n + m)( i+1 − 1) an+m−1 ai

≥ (n + m − 1)( ≥ i(

a an+m − 1) − (n + m)( i+1 − 1) an+m−1 ai

ai+1 a − 1) − (n + m)( i+1 − 1) ai ai

= (n − i + 1)(1 −

ai+1 ), ai

so inequality (1.6.16) holds. If m = 2, then k ≤ n + 1 ≤ n + m − 1, so k(

a a a an+m − i+1 ) ≥ (n + m − 1)( n+m − i+1 ) an+m−1 ai an+m−1 ai = (n + m − 1)( ≥ i(

an+m a − 1) − (n + m − 1)( i+1 − 1) an+m−1 ai

a ai+1 − 1) − (n + 1)( i+1 − 1) ai ai

= (n − i + 1)(1 −

ai+1 ). ai

That is, inequality (1.6.16) holds. This completes the proof of Lemma 1.6.1. Using the same argument as in the proof of Lemma 1.6.1, we can easily carry out the proof of the following lemma. Lemma 1.6.2. Let {an }n∈ℕ be an increasing positive sequence, a0 = 0, m ∈ ℕ, and let x=(

a a a0 a0 , . . . , n−1 , . . . , n−1 ) ,..., a a a a ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+m−1 n+m−1 n+m−1 n+m−1 n+1

n+1

30 | 1 Schur-convex functions and sequence inequalities and y=(

a0 a a a , . . . , 0 , . . . , n , . . . , n ). a a a a ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+m n+m n+m n+m n

n

ai−1 − 1)}i∈ℕ is increasing, ai ai {i( a − 1)}i∈ℕ is decreasing, i−1

(a) If m ≥ 0 and {i(

(b) if m ≥ 1 and

then x ≺w y;

then x ≺w y.

Majorized proof of Theorem 1.6.4. (a) Take m = 2. From Lemma 1.6.1(b), we have (

a a a1 a ,..., 1 ,..., n ,..., n ) a a a a ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1 n+1 n+1 n+1 n+1

n+1

n

n

a a a a ≺w ( 1 , . . . , 1 , . . . , n+1 , . . . , n+1 ). an+2 an+2 an+2 an+2 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ Thus according to Theorem 1.5.2(d) (Vol. 1), we can prove the left inequality in (1.6.6). (b) By Lemma 1.6.1(a) we have a a a a ( 1 , . . . , 1 , . . . , n+1 , . . . , n+1 ) a a a an+1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1 n+1 n+1 n

n

a a a a ≺w ( 1 , . . . , 1 , . . . , n , . . . , n ). an⏟⏟ an⏟⏟ ⏟⏟a⏟⏟⏟⏟⏟⏟⏟ ⏟⏟a⏟⏟⏟⏟⏟⏟⏟ n ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1

n+1

Thus according to Theorem 1.5.2(b) (Vol. 1), we can prove the left inequality in (1.6.7). The following theorem gives a generalization of Theorem 1.6.5. Theorem 1.6.6. Let f be increasing on [0, 1], and {an }n∈ℕ is an increasing positive sequence, a0 = 0. a (a) If f is concave and {i( ai−1 − 1)}n∈ℕ is increasing, then for m ≥ 0, we have i

ai a 1 n−1 1 n ) ≤ ∑ f ( i ) ≤ f (1). ∑ f( n i=0 an+m−1 n i=0 an+m

(1.6.17)

a

(b) If f is convex and {i( a i − 1)}n∈ℕ is decreasing, then for m ≥ 1, we have i−1

ai a 1 n−1 1 n )≤ ∑ f( ∑ f ( i ) ≤ f (1). n i=0 an+m−1 n + 1 i=0 an+m

(1.6.18)

1.6 Improvement of mean inequalities for convex functions | 31

Proof. (a) The right inequality in (1.6.17) clearly holds. According to Theorem 1.5.2(d) (Vol. 1), the left inequality in (1.6.17) can be proved by Lemma 1.6.2(a). (b) The right inequality in (1.6.18) clearly holds. According to Theorem 1.5.2(b) (Vol. 1), it the left inequality of (1.6.18) can be proved by Lemma 1.6.2(b). In Theorem 1.6.6, replacing f by −f gives the following theorem, which is dual to Theorem 1.6.6. Theorem 1.6.7. Let f be decreasing on [0, 1]; {an }n∈ℕ is an increasing positive sequence, a0 = 0. (a) If f is convex and {i(

ai−1 ai

− 1)}n∈ℕ is increasing, then for m ≥ 0, we have

ai a 1 n 1 n−1 ) ≥ ∑ f ( i ) ≥ f (1). ∑ f( n i=0 an+m−1 n i=0 an+m

(1.6.19)

a

(b) If f is concave and {i( a i − 1)}n∈ℕ is decreasing, then for m ≥ 1, we have i−1

ai a 1 n 1 n−1 )≥ ∑ f( ∑ f ( i ) ≥ f (1). n i=0 an+m−1 n + 1 i=0 an+m

(1.6.20)

Theorem 1.6.8. Let f be increasing and convex on [0, 1], k > −1. Then k+i k+i 1 n+1 k+1 1 n )≥ )+( ) − f (0) ∑f( ∑ f( n i=1 n + k n + 1 i=1 n + k + 1 2(n + k)(n + k + 1) ≥ 2f (

n + 2k + 1 ) − f (0) ≥ 0. 4(n + k)

Proof. Taking ai = k + i in Lemma 1.6.1, we have k+n+1 k+1 k+n+1 k+1 ,..., ) ,..., ,..., x=( ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k+n+1 k+n+1 k+n+1 k+n+1 n

n

and y=(

k+1 k+n k+1 k+n ,..., ). ,..., ,..., ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k+n k+n k+n k+n n+1

n+1

a

When k > −1 and {i( a i − 1)}i∈ℕ is decreasing, it is easy to see that x ≺w y. i−1 Let n(n+1)

n(n+1)

n(n+1)

i=1

i=1

i=1

σ = ∑ (yi − xi ) = ∑ yi − ∑ xi

(1.6.21)

32 | 1 Schur-convex functions and sequence inequalities n(n + 1)(n + 2k + 1) (n + 2k + 2)n(n + 1) − 2(n + k) 2(n + k + 1) n(n + 1)(k + 1) = . 2(n + k)(n + k + 1)

=

By (1.3.39) (Vol. 1) and Theorem 1.3.21 (Vol. 1) we have n + 2k + 1 n + 2k + 1 ) ,..., ( 4(n + k) 4(n + k) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(1.6.22)

2n(n+1)

k+n+1 k+1 k+n+1 k+1 ,..., , ,..., ,..., ≺( ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k+n+1 k+n+1 k+n+1 k+n+1 n

n

k+1 k+1 ) ,..., 2(k + n)(n + k + 1) 2(k + n)(n + k + 1) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n(n+1)

k+1 k+n k+1 k+n ≺( ,..., , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 0, . . . , 0). ,..., ,..., ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k+n k+n k+n k + n n(n+1) n+1

n+1

Since f is convex on [0, 1], according to Theorem 1.5.2 (Vol. 1), from (1.6.22) we have n

(n + 1) ∑ f ( n+1

i=1

≥ n ∑ f( i=1

k+i ) + n(n + 1)f (0) n+k

k+i k+1 ) + n(n + 1)f ( ) n+k+1 2(n + k)(n + k + 1)

≥ 2n(n + 1)f (

n + 2k + 1 ). 4(n + k)

This gives inequality (1.6.21). Remark 1.6.3. Note that the left inequality in (1.6.2) can be written k+i k+i 1 n+1 1 n )> ), ∑f( ∑ f( n i=1 n + k n + 1 i=1 n + k + 1 and since f is increasing on [0, 1], f(

k+1 ) − f (0) ≥ 0. 2(n + k)(n + k + 1)

Thus, the left inequality of (1.6.21) is stronger than the left inequality of (1.6.2) if the condition f is increasing on [0, 1] and convex, and k > −1. In 2013, Guan [48] considered the monotonic mean inequality for GA-convex functions (see Section 1.2.5 (Vol. 1)) and obtained the following results.

1.7 A kind of jumping factorial inequalities | 33

Theorem 1.6.9. Let f be increasing on (0, 1) and GA-convex. Then 1 n i i 1 n+1 1 1 )≤ [f (1) − f ( )], ∑f( ) − ∑ f( n i=1 n n + 1 i=1 n + 1 n(n + 1) n+1

n = 1, 2, . . . . (1.6.23)

If f is decreasing and GA-concave, inequality (1.6.23) is reversed. Theorem 1.6.10. Let f be defined on (0, 1), and {an }n∈ℕ is an increasing positive sea quence, such that {( ai+1 )i }i∈ℕ is increasing. i (a) If f is increasing and GA-concave, then a a 1 n+1 1 n ∑f( i ) ≥ ∑ f ( i ). n i=1 an+1 n + 1 i=1 an+2

(1.6.24)

(b) If f is decreasing and GA-convex, inequality (1.6.24) is reversed.

1.7 A kind of jumping factorial inequalities Theorem 1.7.1. Let the positive integer n > 1. We have (2n − 1)!! 1 1 < < √2n (2n)!! 2√2n + 1

(1.7.1)

(2n)!! 1 1 < . < √2n + 1 (2n + 1)!! √n + 1

(1.7.2)

and

Inequality (1.7.1) is the famous Wallis inequality [79, p. 119]. Liu [96] gave a majorized proof that is refreshing. Proof. Let f (x) = log x. It is easy to know that f (x) is a strictly concave function on ℝ++ . From (2, 2) ≺≺ (1, 3), (4, 4) ≺≺ (3, 5), .. . (2n, 2n) ≺≺ (2n − 1, 2n + 1), we have log 2 + log 2 > log 1 + log 3, log 4 + log 4 > log 3 + log 5, .. . log(2n) + log(2n) > log(2n − 1) + log(2n + 1).

34 | 1 Schur-convex functions and sequence inequalities Adding both sides in the above inequalities correspondingly yields 2[log 2 + log 4 + ⋅ ⋅ ⋅ + log(2n)] > 2[log 1 + log 3 + ⋅ ⋅ ⋅ + log(2n − 1) + log(2n + 1)], which implies (2n − 1)!! 1 < . (2n)!! 2√2n + 1

(1.7.3)

Also from (3, 3) ≺≺ (2, 4), (5, 5) ≺≺ (4, 6), .. . (2n − 1, 2n − 1) ≺≺ (2n − 2, 2n), we have log 2 + log 2 > log 1 + log 3, log 4 + log 4 > log 3 + log 5, .. . log(2n) + log(2n) > log(2n − 1) + log(2n + 1). Adding both sides in the above inequalities correspondingly yields 2[log 2 + log 4 + ⋅ ⋅ ⋅ + log(2n)] > 2[log 1 + log 3 + ⋅ ⋅ ⋅ + log(2n − 1) + log(2n + 1)], which implies (2n − 1)!! 1 . < (2n)!! 2√n From inequalities (1.7.3) and (1.7.4) we obtain inequality (1.7.1). Since (3, 3) ≺≺ (2, 4), (5, 5) ≺≺ (4, 6), .. . (2n + 1, 2n + 1) ≺≺ (2n, 2n + 2)

(1.7.4)

1.7 A kind of jumping factorial inequalities | 35

and (2, 2) ≺≺ (1, 3), (4, 4) ≺≺ (3, 5), .. . (2n, 2n) ≺≺ (2n − 1, 2n + 1), in a similar fashion as inequality (1.7.1), we can prove inequality (1.7.2). Theorem 1.7.2. Let the positive integer n > 1. We have 1

1

3 3 1 (3n)!!! 1 ( ) < log(3n) + log(3n) + log(3n + 3). Adding both sides in the above inequalities correspondingly yields 3[log 1+log 4+log 7+⋅ ⋅ ⋅+log(3n+1)] > 3[log 3+log 6+⋅ ⋅ ⋅+log(3n)]−log 3+log(3n+3), which implies 1

3 (3n)!!! 1 log 1 + log 4 + log 4, log 6 + log 6 + log 6 > log 4 + log 7 + log 7, .. . log(3n) + log(3n) + log(3n) > log(3n − 2) + log(3n + 1) + log(3n + 1). Adding both sides in the above inequalities correspondingly yields 3[log 3 + log 6 + ⋅ ⋅ ⋅ + log(3n)] > 3[log 1 + log 4 + ⋅ ⋅ ⋅ + log(3n + 1)] − log(3n + 1), which implies 1

3 1 (3n)!!! ( ) < . 3n + 1 (3n + 1)!!!

From inequalities (1.7.8) and (1.7.9), we obtain inequality (1.7.5). Since (3, 3, 3) ≺≺ (2, 2, 5), (6, 6, 6) ≺≺ (5, 5, 8), .. . (3n, 3n, 3n) ≺≺ (3n − 1, 3n − 1, 3n + 2) and (2, 2, 2) ≺≺ (1, 2, 3), (5, 5, 5) ≺≺ (3, 6, 6), (8, 8, 8) ≺≺ (6, 9, 9), .. . (3n − 1, 3n − 1, 3n − 1) ≺≺ (3n − 3, 3n, 3n), in a similar fashion as inequality (1.7.5), we can prove inequality (1.7.6).

(1.7.9)

1.8 Convexity and logarithmic convexity of arithmetic and geometric sequences | 37

Since (2, 2, 2) ≺≺ (1, 1, 4), (5, 5, 5) ≺≺ (4, 4, 7), .. . (3n − 2, 3n − 2, 3n − 2) ≺≺ (3n − 4, 3n − 1, 3n − 1) and (4, 4, 4) ≺≺ (2, 5, 5), (7, 7, 7) ≺≺ (5, 8, 8), .. . (3n − 2, 3n − 2, 3n − 2) ≺≺ (3n − 4, 3n − 1, 3n − 1), in a similar fashion as inequality (1.7.5), we can prove inequality (1.7.7). Similarly, we can also prove the following theorem. Theorem 1.7.3. Let the positive integer n > 1. We have 1

1

3 3 (3n)!!! 1 1 < , ) ) < ( ( (3n + 2)!!! 2(3n + 1)2 6(n + 1)2 1

(

(1.7.10)

1

3 3 (3n − 1)!!! 1 2 ) < ) n,

n

n

((n + 1)!) ≤ ∏(2k)!,

(1.8.18)

k=1

n

(1.8.17)

(n!)n ≤ ∏(2k − 1)!,

(1.8.19)

2n ⋅ n! ≤ (2n)!,

(1.8.20)

(2n)!! ≤ (n + 1) .

(1.8.21)

k=1

n

Proof. By Theorem 1.8.7, the pre-i item product sequence {i!} of the positive arithmetic sequence {i} is a log-convex sequence. Using Theorem 1.3.12(a) (Vol. 1) it is not difficult to verify that 1, . . . , 1). . . . , m, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (n, . . . , n) ≺ (m, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m−n

n−1

m−1

According to Corollary 1.1.10, inequality (1.8.17) is obtained from the above majorizing relation. From (1.3.39) (Vol. 1), we have (n + 1, . . . , n + 1) ≺ (2, 4, . . . , 2n) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(1.8.22)

(n, . . . , n) ≺ (1, 3, . . . , 2n − 1). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(1.8.23)

(n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2, . . . , 2) ≺ (2n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 1, . . . , 1).

(1.8.24)

n

and n

It is easy to see that n

n

Using Corollary 1.1.10, inequalities (1.8.18), (1.8.19), and (1.8.20) are obtained from (1.8.22), (1.8.23), and (1.8.24) respectively. Using Theorem 1.3.12 (Vol. 1), it is not difficult to verify that 1, . . . , 1). (n, n, n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2, . . . , 2) ≺ (2n, n + 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2n

4n−1

According to Corollary 1.1.10, inequality (1.8.18) can be obtained from the above majorizing relation.

1.8 Convexity and logarithmic convexity of arithmetic and geometric sequences | 43

Example 1.8.7 (Khinchin inequality [79]). Let ni , i = 1, 2, . . . , k be a nonnegative integer and ∑kj=1 nj = n. Then n

k k 1 ∏ nj ! ≤ ( ) ∏(2nj )!. 2 j=1 j=1

(1.8.25)

Proof. From Theorem 1.8.7, we know that the first i-item product sequence {i!} of the positive item sequence {i} is a log-convex sequence. Using Theorem 1.3.12 (Vol. 1), it is not difficult to verify that (n1 , . . . , nk , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2, . . . , 2) ≺ (2n1 , . . . , 2nk , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 1, . . . , 1). n

(1.8.26)

n

According to Corollary 1.1.2, from (1.8.26), it follows that k

k

j=1

j=1

2n ∏ nj ! ≤ ∏(2nj )!. That is, the inequality (1.8.25) holds. Example 1.8.8. Prove that 1 1 1 1 2(n2 − n + 1) + + + ⋅⋅⋅ + 2 ≥ > 1, n n+1 n+2 n(n + 1) n

n > 1,

(1.8.27)

and 1 1 1 3(n + 1) 3 1 + + + ⋅⋅⋅ + ≤ ≤ . n n+1 n+2 2n 4n 2

(1.8.28)

Proof. According to Theorem 1.8.5, we know that the reciprocal sequence { 1i } of the positive sequence {i} is a convex sequence. From (1.3.39) (Vol. 1), it follows that n(n + 1) n(n + 1) ,..., ) ≺ (n, n + 1, . . . , n2 ). ( ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 2 n2 −n+1

According to Corollary 1.1.2, from (1.8.29), it follows that 1 1 1 1 + + + ⋅ ⋅ ⋅ + 2 > (n2 − n + 1) ⋅ n n+1 n+2 n

1

n(n+1) 2

=

2(n2 − n + 1) . n(n + 1)

When n > 2, 2(n2 − n + 1) > 1 ⇔ 2(n2 − n + 1) > n(n + 1) ⇔ (n − 2)(n − 1) > 0. n(n + 1)

(1.8.29)

44 | 1 Schur-convex functions and sequence inequalities And when n = 2, 1 1 1 13 + + = > 1. 2 3 4 12 Therefore, inequality (1.8.27) holds. Using Theorem 1.3.12 (Vol. 1), it is not difficult to verify that . . . , n). (2n, 2n, 2n − 1, 2n − 1, . . . , n + 1, n + 1, n, n) ≺ (2n, . . . , 2n, n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1

n+1

(1.8.30)

Since { 1i } is a convex sequence, from Corollary 1.1.2, it follows that 2(

1 1 1 1 1 3(n + 1) 1 + + + ⋅ ⋅ ⋅ + ) ≤ (n + 1)( + ) = , n n+1 n+2 2n 2n n 2n

and then 1 1 1 3(n + 1) 1 + + + ⋅⋅⋅ + ≤ , n n+1 n+2 2n 4n but 3(n + 1) 3 ≤ ⇔ 2 ≤ 2n. 4n 2 Therefore, inequality (1.8.28) holds. Remark 1.8.1. Inequalities (1.8.27) and (1.8.28) respectively refine two inequalities in [41, p. 185]. 1.8.2 Convexity and logarithmic convexity of geometric sequences Let {bi } be a geometric sequence with common ratio q. Then its general term is bi = b1 qi−1 ,

(1.8.31)

and its sum of the first n terms is n

Sn = ∑ bi = i=1

b1 (1 − qn ) b1 − bn q = . 1−q 1−q

(1.8.32)

The product of the first n terms is n

Tn = ∏ bi = bn1 q i=1

n(n−1) 2

.

(1.8.33)

The author [164] studied the convexity and log-convexity of the geometric sequence and proved some geometric sequence inequalities by using majorization theory.

1.8 Convexity and logarithmic convexity of arithmetic and geometric sequences | 45

Theorem 1.8.8. The geometric sequence {bi } satisfies bi+1 ⋅ bi−1 = b2i ,

i ≥ 2.

(1.8.34)

That is, {bi } is both a log-convex sequence and a log-concave sequence. Proof. Because bi−1 bi+1 = (bi q) ⋅ (

bi ) = b2i , q

according to Definition 1.1.2, {bi } is both log-convex and log-concave. Theorem 1.8.9. If {bi } is a nonnegative geometric sequence, then {bi } is a convex sequence. Proof. It can be directly proved by Theorem 1.8.8 and Theorem 1.1.1. Theorem 1.8.10. If {bi } is a positive geometric sequence, then {bi − bi−1 } is both a logconvex and a log-concave or convex sequence. Proof. By Theorem 1.8.8 and (1.8.31) we have (bi−1 − bi−2 )(bi−1 − bi ) − (bi − bi−1 )2

= bi−1 bi+1 − bi−1 bi − bi−2 bi+1 + bi−2 bi − (b2i − 2bi−1 bi + b2i−1 ) = b2i − bi−1 bi − bi−2 bi+1 + b2i−1 − (b2i − 2bi−1 bi + b2i−1 ) = bi−1 bi − bi−2 bi+1 = b1 q2i−3 − b1 q2i−3 = 0.

Therefore, {bi − bi−1 } is both a log-convex sequence and a log-concave sequence, and then from Theorem 1.1.1, it follows that {bi − bi−1 } is also a convex sequence. Theorem 1.8.11. If {bi } is a positive geometric sequence with a common ratio q ≥ 1, then {ibi } is both a convex and a log-concave sequence. Proof. By Theorem 1.8.8 and Theorem 1.8.9, {bi } is both a convex and a log-concave sequence. Since q ≥ 1 and {bi } is increasing, by Theorem 1.1.2, it follows that {ibi } is both a convex and a log-concave sequence. Theorem 1.8.12. If {bi } is a positive geometric sequence with a common ratio q ≥ 1, then {bii } is both a log-convex and a convex sequence. If the common ratio q ≤ 1, then {bii } is a log-concave sequence. Proof. Since q ≥ (or ≤, respectively)1, bi+1 ≥ (or ≤, respectively) bi , we have i+1 i−1 2 2 Bi−1 i−1 bi+1 = (bi−1 bi+1 ) bi+1 = (bi )

i−1 2 bi+1

i−1 2 bi

≥ (or ≤, respectively)(b2i )

2

= (bii ) .

Therefore, {bii } is a log-convex (or log-concave, respectively) sequence, and then by Theorem 1.1.1, it follows that {bii } is also a convex sequence for q ≥ 1.

46 | 1 Schur-convex functions and sequence inequalities Theorem 1.8.13. If {bi } is a positive geometric sequence with a common ratio q > 0, 1

b1 ≤ q, then {bii } is a log-concave sequence.

Proof. Because 1 i−1

1 i+1

bi−1 bi+1 = (b1 q

i−2

1 i−1

i

) (b1 q )

i2 −i−1 i2 −1

1 i+1

2

i

= (qi−1 b1i −1 )

2

and 1

1

2

1

2

(bii ) = ((b1 qi−1 ) i ) = (b1i q

i−1 i

2

),

to prove 1

1

1

2

i−1 i+1 bi−1 bi+1 ≤ (bii ) ,

we only need to prove i2 −i−1 2

2

i

1

i −1 qi−1 b1i −1 ≤ b1i q

i−1 i

.

This is equivalent to 1

b i(i2 −1) ( 1) ≤ 1. q Since b1 ≤ q, the above inequality is true. b

Theorem 1.8.14. If {bi } is a positive geometric sequence, then { ii } is both a log-convex and a convex sequence. Proof. Since b2 b2 bi+1 bi−1 = 2 i ≥ 2i , i+1i−1 i −1 i b

b

{ ii } is a log-convex sequence, and then by Theorem 1.1.1, it follows that { ii } is also a convex sequence. Theorem 1.8.15. If {bi } is a positive geometric sequence, then the following propositions hold: (a) { b1 } is either a log-convex or a log-concave or convex sequence; i

(b) If c is a nonnegative constant, the sequence { b 1+c } is a log-concave sequence. i

(c) If the common ratio q ≥ 1, c < b1 , then the sequence { b 1−c } is both a log-convex and i a convex sequence.

1.8 Convexity and logarithmic convexity of arithmetic and geometric sequences | 47

1 bi−1

Proof. (a) Because

1 bi+1

⋅

= ( b1 )2 , { b1 } is both a log-convex sequence and a logi

i

concave sequence. By Theorem 1.1.1, { b1 } is also a convex sequence. i (b) If c is a nonnegative constant, by Theorem 1.8.8 and Theorem 1.8.9, we have 1

1

⋅

bi−1 + c bi+1 + c ≤

=

1 bi−1 bi+1 + c(bi−1 + bi+1 ) + c2 2

1 1 ), =( bi + c b2i + 2bi c + c2

and therefore { b 1+c } is a log-concave sequence. i (c) If the ratio is q ≥ 1, c < b1 , by Theorem 1.8.8 and Theorem 1.8.9, we have 1

⋅

1

bi−1 − c bi+1 − c ≥

=

1 bi−1 bi+1 − c(bi−1 + bi+1 ) + c2 2

1 1 ), =( bi − c b2i − 2bi c + c2

and therefore { b 1−c } is a log-convex sequence. i

Theorem 1.8.16. The sum of the first n terms {Si } of the nonnegative geometric sequence {bi } is a log-concave sequence. If the common ratio q ≥ 1, {Si } is also a convex sequence. Proof. Since {bi } is a convex sequence, bi+1 + bi−1 ≥ 2bi , and bi+1 bi−1 = b2i , we have Si+1 Si−1 = = ≤

b1 + bi+1 q b1 + bi−1 q ⋅ 1−q 1−q

b21 − b1 q(bi+1 + bi−1 ) + bi+1 bi−1 q2 (1 − q)2

b21 − 2b1 qbi + b2i q2 (b1 − bi q)2 = = Si2 . (1 − q)2 (1 − q)2

Therefore {Sn } is a log-concave sequence. If the ratio is q ≥ 1, then bi+1 ≥ bi , and then Si+1 + Si−1 = 2Si + bi+1 − bi ≥ 2Si , and therefore {Sn } is also a convex sequence. Theorem 1.8.17. Let {bi } be a nonnegative geometric sequence and c be a positive constant. If the common ratio q ≥ 1 and c < b1 , then {Si − c} is both a convex and a logconcave sequence. Proof. According to Theorem 1.8.16, {Si } is a convex sequence, and we have (Si−1 − c) + (Si+1 − c) = Si−1 + Si+1 − 2c ≥ 2Si − 2c = 2(Si − c)

48 | 1 Schur-convex functions and sequence inequalities and (Si−1 − c) ⋅ (Si+1 − c) = Si−1 Si+1 − c(Si−1 + Si+1 ) + c2 ≤ Si2 − 2cSi + c2 = (Si − c)2 . Therefore, {Si − c} is both a convex sequence and a logarithmically concave sequence. S

Theorem 1.8.18. If {bi } is a nonnegative geometric sequence, then { ii } is a convex sequence. Proof. It is known by Theorem 1.8.9 that {bi } is a convex sequence, and then by TheoS rem 1.1.4, it follows that { ii } is a convex sequence. Theorem 1.8.19. Let {bi } be a nonnegative geometric sequence with a common ratio q ≥ 1, Ti = ∏ij=1 bj be the product of the first i terms. Then {Ti } is both a convex and a logconvex sequence. Proof. For the tolerance q ≥ 1, we have bi+1 ≥ bi , and then i+1

i−1

i

j=1

j=1

j=1

2

Ti+1 Ti−1 = ∏ bj ∏ bj ≥ (∏ bj ) = Ti2 . So {Ti } is a log-convex sequence, and then by Theorem 1.1.1, it follows that {Ti } is also a convex sequence. Example 1.8.9. Let {bi } be a positive geometric sequence. Then when r ≥ 1 or r ≤ 0 we have r r b − bn q 1 n 1 b − bn q ( 1 ) ≤ ∑ bri ≤ ( 1 ). (1.8.35) n(1 − q) n i=1 n 1−q When 0 < r ≤ 1, the above inequalities are reversed. Proof. Without loss of generality, we may assume that the public ratio q ≥ 1. According to (1.3.39) (Vol. 1) and Theorem 1.3.12(a) (Vol. 1), we have b − bn q b − bn q b − bn q ( 1 ) ≺ (bn , bn−1 , . . . , b1 ) ≺ ( 1 ,..., 1 , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 0, . . . , 0). n(1 − q) n(1 − q) 1−q ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n−1

(1.8.36)

n

When r ≥ 1 or r ≤ 0, xr is a convex function. By Theorem 1.5.2(a) (Vol. 1), we have n(

r

r

n b1 − bn q b − bn q ) ≤ ∑ bri ≤ ( 1 ). n(1 − q) 1−q i=1

From this we can get (1.8.35), when 0 < r < 1, xr is a concave function. By Theorem 1.5.2(a) (Vol. 1), the above inequalities are reversed. Example 1.8.10 ([94]). Let {ai } be a positive geometric sequence and let n be a natural number. Then n−1 n−1 n n n (1.8.37) (b2 − b1 )(0 ) (b3 − b2 )( 1 ) ⋅ ⋅ ⋅ (bn+2 − bn+1 )(n ) = (b2 − b1 )2 (bn+2 − bn+1 )2 .

1.8 Convexity and logarithmic convexity of arithmetic and geometric sequences | 49

Proof. By Theorem 1.8.10, {bi − bi−1 } is both a log-convex sequence and a log-concave sequence. By Theorem 1.3.12(a) (Vol. 1), it is not difficult to verify that x = (n + 2, . . . , n + 2, n + 1, . . . , n + 1, . . . , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2, . . . , 2) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (nn ) (n1 ) (0n )

(1.8.38)

2, . . . , 2) = y. ≺ (n + 2, . . . , n + 2, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2n−1

2n−1

In fact, obviously x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ x2n . We have 2n

n n n n n n ∑ xi = ∑ ( )(i + 2) = ∑ i( ) + 2 ∑ ( ) = n2n−1 + 2n+1 i i i i=0 i=0 i=0 i=1

and 2n

n−1

∑ yi = (n + 2)2 i=1

+2⋅2

n−1

= n2

n−1

n+1

+2

2n

= ∑ xi . i=1

It is easy to see that for k = 2n−1 , when i = 1, 2, . . . , k, xi ≤ yi , and when i = k + 1, k + 2, . . . , n, xi ≥ yi . Thus x and y satisfy the condition of Theorem 1.3.12(a) (Vol. 1), so x ≺ y, thus by Corollary 1.1.2, it follows that equality (1.8.37) holds. Example 1.8.11 ([150]). Let {bi } be a positive geometric sequence and let n be a natural number. Then (0n ) b1

+

(n1 ) b2

+ ⋅⋅⋅ +

(nn )

bn+1

≤ 2n−1 (

1 1 + ). b1 bn+1

(1.8.39)

Proof. In a similar fashion as for (1.8.38), we obtain x = (n + 1, . . . , n + 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n, . . . , n, . . . , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 1, . . . , 1) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (nn ) (n1 ) (0n )

(1.8.40)

≺ (n + 1, . . . , n + 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 1, . . . , 1) = y. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2n−1

2n−1

According to Theorem 1.8.15(a), the sequence { b1 } is a convex sequence, so we obi tain (1.8.39) by combining with (1.8.40). Example 1.8.12 ([94]). Let {bi } be a positive geometric sequence. If n < p < m, n < q < m, and n + m = p + q, n, m, p, q, k are positive integers, then (mbm )k + (nbn )k ≥ (pbp )k + (qbq )k ,

(1.8.41)

(mbm ) ⋅ (nbn ) ≤ (pbp ) ⋅ (qbq ) , 1 1 1 1 + ≥ + . (mbm )k (nbn )k (pbp )k (qbq )k

(1.8.42)

k

k

k

k

(1.8.43)

50 | 1 Schur-convex functions and sequence inequalities Proof. The condition n < p < m, n < q < m, and n + m = p + q implies (p, q) ≺ (n, m). By Theorem 1.8.11, {ibi } is a convex sequence and a log-concave sequence. Note that for k ≥ 1, t k is an increasing convex function. By Corollary 1.1.1, it follows that inequality (1.8.41) holds and Corollary 1.1.2 yields (mbm ) ⋅ (nbn ) ≤ (pbp ) ⋅ (qbq ), and then inequality (1.8.42) holds. By Theorem 1.8.15, { ib1 } is a convex sequence. By Corollary 1.1.1, it i follows that inequality (1.8.43) holds. Example 1.8.13 ([94]). Let {bi } be a positive geometric sequence, the ratio q ≥ 1, and Sn is the sum of the first n terms. If n < p < m, n < q < m, and n + m = p + q, then (mSm )k + (nSn )k ≥ (pSp )k + (qSq )k ,

(1.8.44)

(mSm ) ⋅ (nSn ) ≤ (pSp ) ⋅ (qSq ) , 1 1 1 1 + ≥ + . k k k (mSm ) (nSn ) (pSp ) (qSq )k

(1.8.45)

k

k

k

k

(1.8.46)

Proof. By Theorem 1.8.16, {Si } is both a convex sequence and a logarithmically concave sequence. By Theorem 1.1.2, {iSi } is both a convex sequence and a log-concave sequence. The rest is similar to Example 1.8.12, and therefore omitted. Example 1.8.14. Let {bi } be the positive geometric sequence whose ratio is less than 1, and n is a natural number greater than 1. Prove that (2n−1)n . b11 b22 b33 ⋅ ⋅ ⋅ b2n−1 2n−1 ≤ bn

(1.8.47)

Proof. By Theorem 1.8.12, {bii } is a logarithmically concave sequence. By (1.3.39) (Vol. 1), we have (n, . . . , n) ≺ (1, 2, . . . , 2n − 1). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(1.8.48)

2n−1

According to Corollary 1.1.2, (1.8.47) follows from (1.8.48). Example 1.8.15 ([149]). Let {bi } be the positive geometric sequence whose first term is not greater than the public ratio, and n is a positive integer. Prove that 1

1

1

b11 b22 b33 ⋅ ⋅ ⋅ bnn ≥ √bn1 bn .

(1.8.49)

1

Proof. By Theorem 1.8.13, {bii } is a log-concave sequence. Using Theorem 1.3.12(a) (Vol. 1), it is not difficult to verify that (n, n, n − 1, n − 1, . . . , 2, 2, 1, 1) ≺ (n, . . . , n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 0, . . . , 0). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

n

(1.8.50)

According to Corollary 1.1.2, from (1.8.50), we obtain 1

1

1

2

1

n

(b11 b22 b33 ⋅ ⋅ ⋅ bnn ) ≥ bn1 (bnn ) = bn1 bn .

(1.8.51)

Extracting the square root for both sides in inequality (1.8.51), we obtain (1.8.49).

1.8 Convexity and logarithmic convexity of arithmetic and geometric sequences | 51

Example 1.8.16. Let the ratio of the positive geometric sequence {bi } be q ≥ 1, and the product of the first n items is Tn . Prove that T1 T2 ⋅ ⋅ ⋅ Tn−1 ≥ Tn2n−1 (n > 1).

(1.8.52)

Proof. By Theorem 1.8.19, {Ti } is a logarithmically convex sequence. According to (1.3.39) (Vol. 1), we have (n, . . . , n) ≺ (1, 2, . . . , 2n − 1). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(1.8.53)

2n−1

According to Corollary 1.1.2, (1.8.52) follows from (1.8.53). Example 1.8.17 ([111]). If x > 0, proof that 2x 3 + 3x2 − 12x + 7 ≥ 0.

(1.8.54)

Proof. Since x > 0, {xk } is a nonnegative geometric sequence with a public ratio of x. By Theorem 1.8.9, {xk } is a convex sequence, and by (1.3.39) (Vol. 1) we have (1.8.55)

0, . . . , 0). (1, . . . , 1) ≺ (3, 3, 2, 2, 2, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 7

12

According to Corollary 1.1.2, from the above inequality, we have 2x 3 + 3x2 + 7 ≥ 12x, so (1.8.54) is established. Example 1.8.18 ([111]). If x > 0, proof that xn−1 +

1 1 ≤ xn + n . x xn−1

(1.8.56)

Proof. Since x > 0, {x k } is a nonnegative geometric sequence with a public ratio of x. By Theorem 1.8.9, {xk } is a convex sequence, and obviously (2n − 1, 1) ≺ (2n, 0). According to Corollary 1.1.2 we have x2n−1 +x ≤ x2n +1, that is, (1.8.56) is established. Example 1.8.19 ([41]). Let x > 0, p, q being nonnegative integers, p ≤ q. Proof that xp − xq ≥ (q − p)(x q−1 − xq ).

(1.8.57)

Proof. Without loss of generality, we may assume that p < q. It is easy to see that (1.8.57) is equivalent to xq + qxq−1 + px q ≤ xp + qxq + px q−1 .

(1.8.58)

Using Theorem 1.3.12(a) (Vol. 1), it is not difficult to verify that − 1, . . . , q − 1) ≺ (q, . . . , q, q (q, . . . , q, q − 1, . . . , q − 1, p). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ p+1

q

q

p

(1.8.59)

Since x > 0, {xk } is a nonnegative geometric sequence with a public ratio of x. By Theorem 1.8.9, {xk } is a convex sequence, and according to Corollary 1.1.2, (1.8.58) follows from (1.8.59).

52 | 1 Schur-convex functions and sequence inequalities Example 1.8.20. Proof that x + x2 + ⋅ ⋅ ⋅ + x2n ≤ n(x2n+1 + 1) (x ≥ 0).

(1.8.60)

Proof. If x > 0, then {xk } is a nonnegative geometric sequence with a public ratio of x. By Theorem 1.8.9, {xk } is a convex sequence. Using Theorem 1.3.12(a) (Vol. 1), it is not difficult to verify that 0, . . . , 0). (2n, 2n − 1, . . . , 2, 1) ≺ (2n + 1, . . . , 2n + 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

n

(1.8.61)

According to Corollary 1.1.2, (1.8.60) follows from (1.8.61). Example 1.8.21 (Nanson inequality). Let x > 0. For any natural number n, prove that n+1 1 + x2 + x4 + ⋅ ⋅ ⋅ + x2n ≥ . n 1 + x3 + x5 + ⋅ ⋅ ⋅ + x2n−1

(1.8.62)

Proof. Since x > 0, {x k } is a positive geometric sequence with a public ratio of x. By Theorem 1.8.9, {x k } is a convex sequence, and then by Theorem 1.3.12(a) (Vol. 1), from − 1, . . . , 2n − 1) 5, . . . , 5, . . . , 2n 3, . . . , 3, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (1, . . . , 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1

n+1

n+1

n+1

(1.8.63)

≺ (0, . . . , 0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2, . . . , 2, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 4, . . . , 4, . . . , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2n, . . . , 2n) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

n

n

n

(see (1.4.42) (Vol. 1)), we have n(1 + x2 + x4 + ⋅ ⋅ ⋅ + x2n ) ≥ (n + 1)(1 + x3 + x5 + ⋅ ⋅ ⋅ + x2n−1 ). The proof is complete. Example 1.8.22 ([79]). Let Pn (x) = ∑ni=0 xi , n ≥ 2. Then when x > 0, we have Pn (x) n+1 ≥ n Pn (x) − 1 − x n−1

(1.8.64)

Pn (x) ≥ (2n + 1)x n .

(1.8.65)

and

Proof. It is not difficult to verify that (1.8.64) is equivalent to 2Pn (x) ≤ (n + 1)(1 + xn ).

(1.8.66)

Using Theorem 1.3.12 (Vol. 1), it is not difficult to verify that 0, . . . , 0). (n, n, n − 1, n − 1, . . . , 1, 1, 0, 0) ≺ (n, . . . , n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1

n+1

(1.8.67)

According to Corollary 1.1.2, from (1.8.67), it follows that inequality (1.8.66) holds.

1.9 Convex sequence inequalities associated with Chebyshev inequality |

53

By (1.3.39) (Vol. 1) we have (n, . . . , n) ≺ (2n, 2n − 1, . . . , 1, 0). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2n+1

(1.8.68)

According to Corollary 1.1.2, from (1.8.68), it follows that inequality (1.8.65) holds. Example 1.8.23 ([67]). Let x > 0, and x ≠ 1, n ∈ ℕ. Then x−1 . xn − 1

x + x−n ≥ 2n ⋅

(1.8.69)

Proof. It is not difficult to verify that (1.8.69) is equivalent to (xn+1 + 1)(xn − 1) = (xn+1 + 1)(x n−1 + xn−2 + ⋅ ⋅ ⋅ + x + 1) ≥ 2nxn , x−1 namely, x2n + x2n−1 + ⋅ ⋅ ⋅ + xn+1 + xn−1 + xn−2 + ⋅ ⋅ ⋅ + x + 1 ≥ 2nxn .

(1.8.70)

By (1.3.39) (Vol. 1) we have (n, . . . , n) ≺ (2n, 2n − 1, . . . , n + 1, n − 1, . . . , 1, 0). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2n

(1.8.71)

According to Corollary 1.1.2, from (1.8.71), it follows that inequality (1.8.70) holds.

1.9 Convex sequence inequalities associated with Chebyshev inequality A classic result due to Chebyshev (1882–1883) (see [55, 112, 128]) is stated in the following theorem. Theorem 1.9.1. Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) be two sequences of real numbers monotonic in the same direction, and let p = (p1 , . . . , pn ) be a positive sequence. Then n

n

n

n

i=1

i=1

i=1

i=1

(∑ pi )(∑ pi ai bi ) ≥ (∑ pi ai )(∑ pi bi ).

(1.9.1)

If a and b are monotonic in opposite directions, then the reverse of the inequality in (1.9.1) holds. In either case equality occurs if and only if either a1 = ⋅ ⋅ ⋅ = an or b1 = ⋅ ⋅ ⋅ = bn . Note that the sequences a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) are said to be similarly ordered if (ai − aj )(bi − bj ) ≥ 0,

1 ≤ i, j ≤ n

(1.9.2)

holds, and they are said to be oppositely ordered if the reverse inequality holds.

54 | 1 Schur-convex functions and sequence inequalities In 2012, Latreuch and Belaïdi [81] proved a new type of inequalities for convex sequences, and they put a link between these inequalities and Chebyshev’s inequality. Throughout, for any positive integer n we denote the following. If n is odd, N = N1 = N2 =

n+1 n+1 =[ ], 2 2

and if n is even, N = N1 =

n n+1 n =[ ] and N2 = + 1. 2 2 2

We say that a sequence b = (b1 , . . . , bn ) is symmetric about

n+1 2

if

b1 = bn , b2 = bn−1 , . . . , bN1 −1 = bN2 +1 ; bN1 = bN2 . Theorem 1.9.2. Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) be two convex (concave) se](pk = quences, and let p = (p1 , p2 , . . . , pn ) be a positive sequence symmetric about [ n+1 2 pn+1−k , for all k = 1, 2, . . . , n). Then n

n

i=1

i=1

(∑ pi ai bi ) + (∑ pi ai bn+1−i ) ≥

2

∑ni=1 pi

n

n

i=1

i=1

(∑ pi ai )(∑ pi bi ),

(1.9.3)

] is the integer part of n+1 . If a is convex (or concave) and b is concave (or where [ n+1 2 2 convex), then inequality (1.9.3) is reversed. In either case equality occurs if and only if either a1 = ⋅ ⋅ ⋅ = an or b1 = ⋅ ⋅ ⋅ = bn . Corollary 1.9.1. Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) be two convex (concave) sequences. If either a or b is symmetric about [ n+1 ], then 2 n

(∑ ai bi ) ≥ i=1

n 1 n (∑ ai )(∑ bi ). n i=1 i=1

(1.9.4)

If a is convex (or concave) and b is concave (or convex), then inequality (1.9.4) is reversed. In either case equality occurs if and only if either a1 = ⋅ ⋅ ⋅ = an or b1 = ⋅ ⋅ ⋅ = bn . Theorem 1.9.3. Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) be two convex (concave) sequences. (a) If a and b are similarly ordered, then n n n 1 n 1 n (∑ ai bi ) ≥ (∑ ai bi + ∑ an+1−i bi ) ≥ (∑ ai )(∑ bi ); 2 i=1 n i=1 i=1 i=1 i=1

(1.9.5)

(b) if a and b are oppositely ordered, then n

∑ an+1−i bi ≥ i=1

n n 1 n (∑ ai )(∑ bi ) ≥ ∑ ai bi . n i=1 i=1 i=1

(1.9.6)

1.10 Inequalities for convex sequences and nondecreasing convex functions | 55

If a and b are monotonic in opposite directions, then the reverse of the inequality in (1.9.3) holds. In either case equality occurs if and only if either a1 = ⋅ ⋅ ⋅ = an or b1 = ⋅ ⋅ ⋅ = bn . Theorem 1.9.4. Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) be two sequences of real numbers where a is a convex sequence and b is decreasing for all k = 1, . . . , [ n+1 ] and in2 ], . . . , n. Then inequality (1.9.3) holds. creasing for all k = [ n+1 2 Theorem 1.9.5. Let a = (a1 , . . . , an ) be a convex sequence of real numbers and p = ]. Then (p1 , . . . , pn ) be a positive sequence symmetric about [ n+1 2 n

(∑ pi ) i=1

n n aN + an+1−N a + an ≤ (∑ pi ai ) ≤ (∑ pi ) 1 . 2 2 i=1 i=1

(1.9.7)

If a = (a1 , . . . , an ) is a concave sequence, then inequality (1.9.7) is reversed.

1.10 Inequalities for convex sequences and nondecreasing convex functions In 2017, Niezgodaa [123] proved some inequalities for convex sequences and nondecreasing convex functions. Theorem 1.10.1. Let ψ : I → ℝ be a nondecreasing convex function defined on an interval I ⊂ ℝ. Let z = (z1 , . . . , zn ) ∈ I n be a convex sequence. Let pj , qi ∈ {1, 2, . . . n}, j = 1, 2, . . . , k, i = 1, 2, . . . , m. Assume that p = (p1 , . . . , pk ), q = (q1 , . . . , qm ), a = (a1 , . . . , am ) ∈ ℝm ++ , and b = (b1 , . . . , bk ) ∈ ℝk++ . If p = qS

and a = bST

(1.10.1)

for some m × k column stochastic matrix S, then the following inequality holds: k

m

j=1

i=1

∑ bj ψ(zpj ) ≤ ∑ ai ψ(zqi ).

(1.10.2)

In particular, if ψ(t) = t for t ∈ I = ℝ, then (1.10.2) takes the form for some m × k column stochastic matrix S. Then the following inequality holds: k

m

j=1

i=1

∑ bj zpj ≤ ∑ ai zqi .

(1.10.3)

Theorem 1.10.2. Let ψ : I → ℝ be a nondecreasing convex function defined on an interval I ⊂ ℝ. Let z = (z1 , . . . , zn ) ∈ I n be a convex sequence.

56 | 1 Schur-convex functions and sequence inequalities Assume that b = (b1 , . . . , bk ) ∈ ℝk++ . Then the following inequality holds: n

∑ bj ψ(zj ) ≤ a1 ψ(z1 ) + a2 ψ(zn ), j=1

(1.10.4)

where a1 =

1 [(n − 1)b1 + (n − 2)b2 + ⋅ ⋅ ⋅ + 1 ⋅ bn−1 + 0 ⋅ bn ] n−1

(1.10.5)

a2 =

1 [0 ⋅ b1 + 1 ⋅ b2 + ⋅ ⋅ ⋅ + (n − 2)bn−1 + (n − 1)bn ]. n−1

(1.10.6)

and

In particular, if ψ(t) = t for t ∈ I = ℝ, then (1.10.4) reduces to n

∑ bj zj ≤ a1 z1 + a2 zn , j=1

(1.10.7)

where a1 and a2 are given by (1.10.5) and (1.10.6), respectively. Theorem 1.10.3. Let ψ : I → ℝ be a nondecreasing convex function defined on an interval I ⊂ ℝ. Let z = (z1 , . . . , zn ) ∈ I n be a convex sequence. . Then the following Assume that b = (b1 , . . . , bn ) ∈ ℝn++ is symmetric about n+1 2 inequality holds: n

∑ bj ψ(zj ) ≤ j=1

ψ(z1 ) + ψ(zn ) n ∑ bi . 2 i=1

(1.10.8)

Theorem 1.10.4. Let ψ : I → ℝ be a nondecreasing convex function defined on an interval I ⊂ ℝ. Let z = (z1 , . . . , zn ) ∈ I n be a convex sequence. Let b1 , b2 ≥ 0 and p1 , p2 ∈ {1, 2, . . . , n}, where p1 = 1 ⋅ λ1 + 2 ⋅ λ2 + ⋅ ⋅ ⋅ + n ⋅ λn

(1.10.9)

p2 = 1 ⋅ μ1 + 2 ⋅ μ2 + ⋅ ⋅ ⋅ + n ⋅ μn ;

(1.10.10)

and

for some λi ≥ 0, μi ≥ 0, i = 1, 2, . . . , n, ∑ni=1 λi = 1 = ∑ni=1 μi . Then the following inequality holds: n

b1 ψ(zp1 ) + b2 ψ(zp2 ) ≤ ∑ ai ψ(zi ), i=1

(1.10.11)

where ai = b1 ⋅ λi + b2 ⋅ μi

for i = 1, 2, . . . , n.

(1.10.12)

2 Schur-convex functions and integral inequalities 2.1 Schur-convex functions related to Hadamard integral inequalities Theorem 2.1.1. Let f : I ⊂ ℝ → ℝ be a convex (or concave, respectively) function defined in the real interval, and let x, y ∈ I and x < y. Then we have the following double inequality: y

x+y 1 f (x) + f (y) f( ) ≤ (or ≥, respectively) , ∫ f (t) d t ≤ (or ≥, respectively) 2 y−x 2 x

which is known as the Hadamard inequality for convex functions [79, p. 430].

(2.1.1)

There are many ways to prove the Hadamard inequality. Zheng [277] gave the following majorized proof. Proof. Let xn (i) = x + we have

i(y−x) n

∈ [x, y], i = 1, . . . , n. By the definition of definite integral,

y

n 1 1 lim ∑[f (xn (i))Δxn (i)] ∫ f (t) d t = y−x y − x n→∞ i=1 x

=

n ∑n x (i) 1 lim ∑ f (xn (i)) ≥ lim f ( i=1 n ) n→∞ n n→∞ i=1 n

= lim f ( n→∞

∑ni=1 (x + n

i(y−x) ) n

) = f(

x+y ). 2

On the other hand, y

n−1 l 1 lim ∑ [f (xn (i))Δxn (i) + f (xn (n))Δxn (n)] ∫ f (t) d t = y−x y − x n→∞ i=1 x

=

n 1 n 1 lim [∑ f (xn (i)) + f (y)] = lim [∑ f (xn (i))] n→∞ n n n→∞ i=1 i=1 n 1 n [∑ f (xn (i)) + ∑ f (xn (n − i))] n→∞ 2n i=1 i=1

= lim

1 n ∑[f (xn (i)) + f (xn (n − i))]. n→∞ 2n i=1

= lim

Since y ≥ xn (i) and y ≥ xn (n−i), that is, y ≥ max{xn (i), xn (n−i)}, and xn (i)+xn (n−i) = x + y, (xn (i), xn (n − i)) ≺ (x, y), i = 1, . . . , n. Because f is a convex function on I, by https://doi.org/10.1515/9783110607864-002

58 | 2 Schur-convex functions and integral inequalities Theorem 1.5.2(a) (Vol. 1) we have f (xn (i)) + f (xn (n − i)) ≤ f (x) + f (y),

i = 1, . . . , n,

and then y

1 f (x) + f (y) 1 n . ∑[f (x) + f (y)] = ∫ f (t) d t ≤ lim n→∞ 2n y−x 2 i=1 x

The proof of theorem 2.1.1 is complete. In 2000, Elezovic and Pečarić [32] considered the Schur-convexity of the mean of the function with respect to the upper and lower bounds of the integral. Using the Hadamard integral inequality, the following important results were established. Theorem 2.1.2. Let I be an interval with nonempty interior on ℝ and let f be a continuous function on I. Then y

1 ∫ f (t) d t, F(x, y) = { y−x x f (x),

x, y ∈ I, x ≠ y,

x=y

(2.1.2)

is Schur-convex (or Schur-concave, respectively) on I 2 if and if f is convex (or concave, respectively) on I. Remark 2.1.1. Equality occurs in the Hadamard inequality only when f is a linear function. Therefore, if f is a nonlinear convex (or concave, respectively) function on I, then F(x, y) is a strictly Schur-convex (or Schur-concave, respectively) function on I 2 . About in July and August 2001, I went to the National Library to search for information and saw the concise and beautiful papers of Elezovic and Pečarić. I was very excited at the time because I had been trying to apply controlled theory to some integral inequalities, but without success. This article opened the door for me. Soon after reading this article, I sent a copy to a friend, Prof. Feng Qi, to share the article. Feng Qi immediately wrote back, saying: “This article is very important to me.” Soon Feng Qi wrote an article [134], influenced by reference [134], and I also completed several such papers (see [159, 171, 163]). Corollary 2.1.1 ([22]). Let f be a real Lebesgue integrable function defined on the interval I ⊂ ℝ, with range J. Let k be a real continuous strictly monotonic function on J. Then, for the generalized integral quasiarithmetic mean of function f defined as y

1 k −1 ( y−x ∫x (k ∘ f )(t) d t), Mk (f ; x, y) = { f (a),

x ≠ y,

x = y,

(2.1.3)

the following hold: (a) Mk (f ; x, y) is Schur-convex on I 2 if k ∘ f is convex on I and k is increasing on J or if k ∘ f is concave on I and k is decreasing on J;

2.1 Schur-convex functions related to Hadamard integral inequalities | 59

(b) Mk (f ; x, y) is Schur-concave on I 2 if k ∘ f is convex on I and k is decreasing on J or if k ∘ f is concave on I and k is increasing on J. Long et al. [99] and Sun et al. [190] proved the following theorem in different ways. Theorem 2.1.3. Let f be an increasing (or decreasing, respectively) continuous convex (or concave, respectively) function on I ⊂ ℝ++ . Then F(x, y) in (2.1.3) is both a Schur-geometrically convex (or Schur-geometrically concave, respectively) function and a Schur-harmonically convex (or Schur-harmonically concave, respectively) function on I 2 . Here we give a simple proof using the properties of Schur-geometrically convex functions and Schur-harmonically convex functions. Proof. Without loss of generality, we may assume that y > x. Since f is increasing and convex on I, from (2.1.1), it follows that y

𝜕F 1 1 = [ ∫ f (t) d t − f (x)] 𝜕x y − x y − x x

x+y 1 [f ( ) − f (x)] ≥ 0, ≥ y−x 2 y

𝜕F 1 1 = [f (x) − ∫ f (t) d t] 𝜕y y − x y−x x

f (x) + f (y) 1 f (y) − f (x) 1 [f (y) − ]= ⋅ ≥ 0. ≥ y−x 2 2 y−x That is to say, F(x, y) is increasing on I 2 . From Theorem 2.1.2, we know that F(x, y) is Schur-convex on I 2 , thus by Theorem 2.4.6 (Vol. 1) and Theorem 2.4.18 (Vol. 1), it follows that F(x, y) is both Schur-geometrically convex and Schur-harmonically convex on I 2 . We can prove the case of f decreasing and being concave on I similarly. Theorem 2.1.4 ([274]). Let 0 < a < b, f : [a, b] → ℝ++ be a second-order differentiable function. If for any x ∈ [a, b], 3f 󸀠 (x) + xf 󸀠󸀠 (x) ≥ 0 (or ≤ 0, respectively) holds, then F(x, y) defined by (2.1.3) is a Schur-geometrically convex (or Schur-geometrically concave, respectively) function. In 2005, Qi et al. [138] established the weighted form of Theorem 2.1.2. Theorem 2.1.5. Let f be a continuous function on I and let p be a positive continuous function on I. Then the weighted arithmetic mean of f with weight p y

∫ p(t)f (t) d t

, { x y Fp (f ; a, b) = { ∫x p(t) d t {f (x),

x ≠ y, x=y

(2.1.4)

60 | 2 Schur-convex functions and integral inequalities is the Schur-convex (or Schur-concave, respectively) function on I 2 if and only if inequality y

∫x p(t)f (t) d t y ∫x

p(t) d t

≤

p(x)f (x) + p(y)f (y) p(x) + p(y)

(2.1.5)

holds (or is inverted, respectively) for (x, y) ∈ I 2 . In 2013, Long et al. [99] further investigated the Schur-geometric convexity and Schur-harmonic convexity of Fp (x, y) and obtained the following results. Theorem 2.1.6 ([99]). Under the same condition as Theorem 2.1.5, we have: (a) Fp (x, y) is a Schur-geometrically convex function on I 2 if and only if inequality y

∫x p(t)f (t) d t y

∫x p(t) d t

≤

xp(x)f (x) + yp(y)f (y) xp(x) + yp(y)

(2.1.6)

holds for (x, y) ∈ I 2 ; (b) Fp (x, y) is a Schur-harmonically convex function on I 2 if and only if the inequality y

∫x p(t)f (t) d t y

∫x p(t) d t

≤

x2 p(x)f (x) + y2 p(y)f (y) x2 p(x) + y2 p(y)

(2.1.7)

holds for (x, y) ∈ I 2 . Theorem 2.1.7 ([99]). Let p be a positive continuous function on I and let f be a differentiable function on I satisfying for any x, y ∈ I f 󸀠 (y) ≥

y ∫x

p(y)

f (y) − f (x) . y−x p(t) d t ⋅

Then the following propositions are established: (a) if f and p have the same monotonicity on I, then Fp (x, y) is Schur-convex on I 2 ; (b) if f (t) and tp(t) have the same monotonicity on I, then Fp (x, y) is Schur-geometrically convex on I 2 ; (c) if f (t) and t 2 p(t) have the same monotonicity on I, then Fp (x, y) is Schur-harmonically convex on I 2 . In 1992, by using analytical methods, Dragomir [29] established the following theorem, which is a refinement of the first inequality of (2.1.1). Theorem 2.1.8. If f : [a, b] → ℝ is a convex function and H is defined on [0, 1] by b

1 a+b H(t) = ) d x, ∫ f (tx + (1 − t) b−a 2 a

(2.1.8)

2.1 Schur-convex functions related to Hadamard integral inequalities | 61

then H is convex and increasing on [0, 1], and for all t ∈ [0, 1], we have b

f(

a+b 1 ) = H(0) ≤ H(t) ≤ H(1) = ∫ f (x) d x. 2 b−a

(2.1.9)

a

Using Theorem 2.1.2, the author [159] gave a simple proof of Theorem 2.1.8. Zhang and Chu [273, 275] considered convexity of F(x, y) defined by (2.1.2) and obtained the following result. Theorem 2.1.9 ([273, 275]). Let f be a continuous function on I. Then F(x, y) defined by (2.1.2) is a convex (or concave, respectively) function on I 2 if and only if f is a convex (or concave, respectively) function on I. Remark 2.1.2. From Corollary 2.2.1 (Vol. 1), the symmetrically convex (or concave, respectively) function on the symmetrically convex set must be a Schur-convex (or Schur-concave, respectively) function. Therefore, Theorem 2.1.9 is a strengthened version of Theorem 2.1.2. In 2003, Wulbert [228] proved that if f is a convex function on I, then F is a convex function on I 2 . In 2010, Chu et al. [14] proved the following theorem. Theorem 2.1.10. Suppose that I is an open interval and f : I → ℝ is a continuous function. If Ff (x, y) =

y 1 ∫ y−x x {

0,

f (t) d t − f ( x+y ), x, y ∈ I, x ≠ y, 2 x=y

and f (x)+f (y) 2

Gf (x, y) = { 0,

−

y 1 ∫ y−x x

f (t) d t, x, y ∈ I, x ≠ y, x = y,

(2.1.10)

(2.1.11)

then Ff (x, y) and Gf (x, y) are Schur-convex (or Schur-concave, respectively) functions on I 2 if and only if f is a convex (or concave, respectively) function on I. Remark 2.1.3. In December 2010, Xiaoming Zhang stated the following in a personal 1 2 1 2 letter to me: “If (a1 , b1 ) ≺ (a2 , b2 ), then a1 +b = a2 +b , f ( a1 +b ) = f ( a2 +b ), so by the def2 2 2 2 inition of Schur-convex function, it is easy to prove that Theorem 2.1.10 is equivalent to Theorem 2.1.2.”

as

In 2016, Wang and Bai [209] extended Theorem 2.1.10 as follows. Let I be a nonempty open interval in ℝ. A class of functions S : I 2 → ℝ is defined λ[f (x) + f (y)] + (1 − 2λ)f ( x+y )− 2 Gf (x, y) = { 0,

where (x, y) ∈ I 2 , λ ≥ 0.

y 1 ∫ y−x x

f (t) d t,

x, y ∈ I, x ≠ y,

x = y,

(2.1.12)

62 | 2 Schur-convex functions and integral inequalities If λ = 0 or λ = 21 , then S(x, y) is reduced to Ff (x, y) or Gf (x, y) in Theorem 2.1.10, respectively. Theorem 2.1.11. Let I ⊂ ℝ be an open interval, and let f → ℝ be a twice differentiable mapping such that f 󸀠󸀠 is integrable. If λ ≥ 41 and f is convex (or concave, respectively) on I, then function S(x, y) is Schur-convex (or Schur-concave, respectively) on I 2 . Theorem 2.1.12. Let I ⊂ ℝ be an open interval, and let f → ℝ be a twice differentiable mapping such that f 󸀠󸀠 is integrable. If λ ≥ 21 and f is convex (or concave, respectively) on I, then function S(x, y) is Schur-convex (or Schur-concave, respectively) on I 2 . If f : [a, b] → ℝ is twice differentiable, then the following identity [13] holds: b

f (a) + f (b) b − a 󸀠 1 + [f (b) − f 󸀠 (a)] ∫ f (t) d t − b−a 2 8

(2.1.13)

a

b

=

2

1 a+b ) f 󸀠󸀠 (t) d t. ∫(t − 2(b − a) 2 a

With this identity, Čuljak [22] proved the following theorem. Theorem 2.1.13. If f : I ⊂ ℝ → ℝ is a convex (or concave, respectively) function, then the function f (x)+f (y) 4

P(x, y) = { 0,

− 21 f ( x+y )− 2

y 1 ∫ y−x x

f (t) d t,

x, y ∈ I, x ≠ y,

x=y

(2.1.14)

is Schur-convex (or Schur-concave, respectively) on I 2 if and only if f is a convex (concave) function on I. If f ∈ C 2 (I) and P is Schur-convex (or Schur-concave, respectively), then f is convex (or concave, respectively). By ( x+y , x+y ) ≺ (x, y) and combining the Schur-convexity (Schur-concavity) of P 2 2 on I , it is easy to obtain the following corollary. 2

Corollary 2.1.2. Let f : I → ℝ be a convex (or concave, respectively) function, x, y ∈ I and x < y. Then y

x+y 1 ) ∫ f (t) d t − f ( y−x 2 x

(2.1.15) y

1 f (x) + f (y) − ≤ (or ≥, respectively) ∫ f (t) d t. 2 y−x x

(2.1.16)

2.1 Schur-convex functions related to Hadamard integral inequalities | 63

Let the nonempty open interval I ⊂ ℝ and define the function S : I 2 → ℝ as follows: S(x, y) = {

f (x)+f (y) 6

0,

− 32 f ( x+y )− 2

y 1 ∫ y−x x

f (t) d t,

x, y ∈ I, x ≠ y, x = y.

(2.1.17)

Franjić and Pečarić [35] proved the following result. Theorem 2.1.14. If f ∈ C 4 (I), then the following statements are equivalent: (a) the function S defined by (2.1.17) is Schur-convex on I 2 ; (b) for all x, y ∈ I, x < y, we have y

2 x+y 1 1 1 ) + f (y); ∫ f (t) d t ≤ f (x) + f ( y−x 6 2 2 6 x

(c) the function f is 4-convex (namely, f (4) ≥ 0) on I. In [9], the following double integral inequalities were obtained. Theorem 2.1.15. Let f : [a, b] → ℝ be a twice differentiable mapping on (a, b) and suppose that γ ≤ f 󸀠󸀠 (t) ≤ Γ for all t ∈ (a, b). Then we have b

γ(b − a)2 a+b 1 Γ(b − a)2 ≤ )≤ ∫ f (t) d t − f ( 24 b−a 2 24

(2.1.18)

a

and b

γ(b − a)2 f (a) + f (b) 1 Γ(b − a)2 ≤ − . ∫ f (t) d t ≤ 12 2 b−a 12

(2.1.19)

a

Using Theorem 2.1.10, Theorem 2.1.13, and Theorem 2.1.14, respectively, and combining with (1.4.32) (Vol. 1), the author [165] obtained generalizations and refinements of inequalities (2.1.18) and (2.1.19). Theorem 2.1.16. Under the condition of Theorem 2.1.15, if 0 ≤ p ≤ 21 , then for a < b, we have b

γp(1 − p)(b − a)2 1 1 ≤ ∫ f (t) d t − 6 b−a (2p − 1)(b − a) a

≤

Γp(1 − p)(b − a) 6

2

pb+(1−p)a

∫

f (t) d t

(2.1.20)

pa+(1−p)b

and b

γp(1 − p)(b − a)2 f (a) + f (b) 1 ≤( − ∫ f (t) d t) 3 2 b−a a

(2.1.21)

64 | 2 Schur-convex functions and integral inequalities pb+(1−p)a

f (pa + (1 − p)b) + f (pb + (1 − p)a) 1 −( − 2 (2p − 1)(b − a) ≤

∫

f (t)dt)

pa+(1−p)b

Γp(1 − p)(b − a)2 . 3

Theorem 2.1.17. Under the condition of Theorem 2.1.15, we have b

γp(1 − p)(b − a)2 f (a) + f (b) 1 ≤( − ∫ f (t) d t) 12 4 b−a

(2.1.22)

a

1 f (pa + (1 − p)b) + f (pb + (1 − p)a) − −( 4 (b − a)(2p − 1) ≤

Γp(1 − p)(b − a)2 . 12

pb+(1−p)a

∫

f (t) d t)

pa+(1−p)b

Theorem 2.1.18. Let f : [a, b] → ℝ such that f (4) is continuous on [a, b], and suppose that γ ≤ f (4) (t) ≤ Γ for all t ∈ (a, b). If 0 ≤ p ≤ 21 , then for a < b, we have b

γp(1 − p)(p2 + (1 − p)2 )(b − a)4 f (a) + f (b) 1 ≤( − ∫ f (t) d t) 360 6 b−a

(2.1.23)

a

v

f (u) + f (v) 1 −( − ∫ f (t)dt) 6 v−u 2

≤

2

u

Γp(1 − p)(p + (1 − p) )(b − a)4 . 360

2.2 Schur-convex functions related to Hadamard type integral inequalities This section discusses the Schur-convexity of several classes of functions associated with Hadamard type integral inequalities. 2.2.1 Schur-convex functions related to Dragomir’s integral inequalities In 1992, Dragomir [29] established the following theorem. Theorem 2.2.1. If f : [a, b] → ℝ is a convex function and F is defined on [0, 1] by b b

1 F(t) = ∫ ∫ f (tx + (1 − t)y) d x d y, (b − a)2 a a

(2.2.1)

2.2 Schur-convex functions related to Hadamard type integral inequalities | 65

then (a) F is convex on [0, 1], symmetric about 21 , (i. e., F(t) = F(1 − t) for all t ∈ [0, 1]), increasing on [0, 21 ], and increasing on [ 21 , 1], and for all t ∈ [0, 1], we have b

1 ∫ f (x) d x b−a

F(t) ≤ F(1) =

(2.2.2)

a

and b b

1 x+y 1 a+b F(t) ≥ F( ) = )dx dy ≥ f( ). ∫∫f( 2 2 2 (b − a)2

(2.2.3)

a a

(b) For all t ∈ [0, 1], we have F(t) ≥ max{H(t), H(1 − t)},

(2.2.4)

where H(t) is defined in (2.1.8). Dragomir [30] established the following theorem, which is an extension of the relevant conclusion in [248]. Theorem 2.2.2. If f : [a, b] → ℝ is a convex function and G is defined on [0, 1] by b

G(t) =

1 ∫[f (ta + (1 − t)x) + f (tb + (1 − t)x)] d x, 2(b − a)

(2.2.5)

a

then G is convex on [0, 1], and for all t ∈ [0, 1], we have b

1 f (a) + f (b) . ∫ f (x) d x = G(0) ≤ G(t) ≤ G(1) = b−a 2

(2.2.6)

a

Remark 2.2.1. If f is concave, then (2.2.6) is reversed (note that −f is convex). The author [159] established results similar to Theorem 2.1.2 for F(t) and G(t) and gave the application of these results. Theorem 2.2.3 ([159]). Let I be an interval with nonempty interior on ℝ and define a function of two variables as follows: G(t), P(a, b) = { f (a),

a, b ∈ I, a ≠ b, a = b.

(a) For 21 ≤ t ≤ 1, if f is convex on I, then P(a, b) is Schur-convex on I 2 ; (b) for 0 ≤ t ≤ 21 , if f is concave on I, then P(a, b) is Schur-concave on I 2 .

(2.2.7)

66 | 2 Schur-convex functions and integral inequalities Theorem 2.2.4 ([159]). Let I be an interval with nonempty interior on ℝ and let f be a continuous function on I. For any t ∈ [0, 1], we define a function of two variables as follows: F(t), Q(a, b) = { f (a),

a, b ∈ I, a ≠ b, a = b.

(2.2.8)

If f is convex (or concave, respectively) on I, then Q(a, b) is Schur-convex (or Schurconcave, respectively) on I 2 . β

β

Lemma 2.2.1. Let F(α, β) = ∫α ∫α f (x, y) d x d y, where f (x, y) is continuous on the rectangle [a, p; a, q], α = α(b) and β = β(b) are differentiable with b, a ≤ α(b) ≤ p, and a ≤ β(b) ≤ q. Then β

β

α

α

𝜕F = (∫ f (α, y) d y)α󸀠 (b) + (∫ f (x, β) d x)β󸀠 (b). 𝜕b β

(2.2.9)

β

Proof. Since F(α, β) = ∫α ∫α f (x, y) d x d y, by the derivation rule for the composite functions, we have 𝜕F 𝜕F d α 𝜕F d β = + , 𝜕b 𝜕α d b 𝜕β d b which is equation (2.2.9). Proof of Theorem 2.2.3. It is sufficient to prove (a); the proof of (b) is similar to that of (a). We need only consider the case of 21 ≤ t < 1. It is clear that P(a, b) is symmetric. When a ≠ b, let b

P1 (a, b) = ∫ f (ta + (1 − t)x) d x a

and b

P2 (a, b) = ∫ f (tb + (1 − t)x) d x. a

Then P(a, b) =

1 [P (a, b) + P2 (a, b)] = G(t), a ≠ b. 2(b − a) 1

By the transformation s = ta + (1 − t)x, we obtain 1 P1 (a, b) = 1−t

ta+(1−t)b

∫ a

f (s) d s

2.2 Schur-convex functions related to Hadamard type integral inequalities | 67

1 = [ 1−t

ta+(1−t)b

a

∫

f (s) d s − ∫ f (s) d s],

0

0

𝜕P1 (a, b) 1 = [f (ta + (1 − t)b)t − f (a)] 𝜕a 1−t t f (a) = f (ta + (1 − t)b) − , 1−t 1−t 𝜕P1 (a, b) 1 = [(1 − t)f (ta + (1 − t)b)] = f (ta + (1 − t)b). 𝜕b 1−t

(2.2.10) (2.2.11)

Note that P2 (a, b) = −P1 (b, a). From (2.2.11), we obtain 𝜕P2 (a, b) 𝜕P (b, a) =− 1 = −f (tb + (1 − t)a), 𝜕a 𝜕a

(2.2.12)

and from (2.2.10), we obtain 𝜕P (b, a) f (b) 𝜕P t =− 1 = + f (tb + (1 − t)a). 𝜕b 𝜕b 1−t 1−t

(2.2.13)

And then 𝜕P1 (a, b) 𝜕P1 (a, b) t f (a) − = f (ta + (1 − t)b) − f (ta + (1 − t)b) − 𝜕b 𝜕a 1−t 1−t f (a) 1 − 2t f (ta + (1 − t)b) + , = 1−t 1−t 𝜕P2 (a, b) 𝜕P2 (a, b) f (b) t − = − f (tb + (1 − t)a) + f (tb + (1 − t)a) 𝜕b 𝜕a 1−t 1−t f (b) 1 − 2t = + f (tb + (1 − t)a). 1−t 1−t Since 𝜕P (a, b) 𝜕P2 (a, b) 𝜕P(a, b) 1 1 [P (a, b) + P2 (a, b)] + = {− [ 1 + ]} 𝜕b 2(b − a) 𝜕b 𝜕b 2(b − a)2 1 and 𝜕P (a, b) 𝜕P2 (a, b) 1 1 𝜕P(a, b) [P1 (a, b) + P2 (a, b)] + ={ [ 1 + ]}, 2 𝜕a 2(b − a) 𝜕a 𝜕a 2(b − a) we have 𝜕P(a, b) 𝜕P(a, b) − ) 𝜕b 𝜕a 𝜕P (a, b) 𝜕P2 (a, b) 1 𝜕P (a, b) 𝜕P1 (a, b) − )+( 2 − )] − G(t) = [( 1 2 𝜕b 𝜕a 𝜕b 𝜕a 1 = [f (a) + f (b) + (1 − 2t)(f (ta + (1 − t)b) + f (tb + (1 − t)a))] − G(t) 2(1 − t) (b − a)(

68 | 2 Schur-convex functions and integral inequalities

≥

1 [f (a) + f (b) + (1 − 2t)(tf (a) + (1 − t)f (b) + tf (b) + (1 − t)f (a))] − G(t) 2(1 − t)

(note that f is convex and from

= f (a) + f (b) − G(t) ≥ 0

1 2

≤ t < 1, we have 1 − 2t ≤ 0)

(by the right inequality in (2.2.6)).

According to Theorem 2.1.3 (Vol. 1), it follows that P(a, b) is Schur-convex on I 2 . The proof of Theorem 2.2.3 is completed. Proof of Theorem 2.2.4. Taking α = β = b in Lemma 2.2.1, when a ≠ b, we have b b

−2 𝜕Q(a, b) = ∫ ∫ f (tx + (1 − t)y) d x d y 𝜕b (b − a)3 a a

b

b

a

a

1 + [∫ f (tb + (1 − t)y) d y + ∫ f (tx + (1 − t)b) d x], (b − a)2 b b

2 𝜕Q(a, b) = ∫ ∫ f (tx + (1 − t)y) d x d y 𝜕a (b − a)3 a a

b

b

a

a

1 [∫ f (ta + (1 − t)y) d y + ∫ f (tx + (1 − t)a) d x]. + (b − a)2 Now we only consider the case of convexity; the case of concavity is similar. We have (b − a)(

𝜕Q(a, b) 𝜕Q(a, b) − ) 𝜕b 𝜕a

b

=

1 [∫(f (tb + (1 − t)y) + f (ta + (1 − t)y)) d y b−a a

b

+ ∫(f (tx + (1 − t)b) + f (tx + (1 − t)a)) d x] a

−

b b

b

a a

a

4 4 ∫ ∫ f (tx + (1 − t)y) d x d y ≥ ∫ f (x) d x b−a (b − a)2 b b

− ≥0

4 ∫ ∫ f (tx + (1 − t)y) d x d y (b − a)2

(by the left inequality in (2.2.6))

a a

(by (2.2.2)).

According to Lemma 2.1.3 (Vol. 1), it follows that Q(a, b) is Schur-convex on I 2 . The proof of Theorem 2.2.4 is completed.

2.2 Schur-convex functions related to Hadamard type integral inequalities | 69

Theorem 2.2.5 ([159]). Let t ∈ [0, 1), a, b ∈ ℝ+ = [0, +∞), and let 1

(br − ar ) − (ur − vr ) r−1 Lr (a, b; t) = [ ] , r(1 − t)(b − a)

a ≠ b,

Lr (a, a; t) = a,

where u = tb + (1 − t)a, v = ta + (1 − t)b. (a) If r > 2 and 21 ≤ t ≤ 1, then Lr (a, b; t) is Schur-convex on ℝ2+ ; (b) if 1 < r < 2 and 0 ≤ t ≤ 21 , then Lr (a, b; t) is Schur-concave on ℝ2+ ; (c) if r ≤ 1, r ≠ 0, and 21 ≤ t ≤ 1, then Lr (a, b; t) is Schur-concave on ℝ2+ . Corollary 2.2.1. If (r, t) ∈ {r > 2, 21 ≤ t ≤ 1}, then 1

(t r − 1)r + (1 − t)r r−1 a+b ≤ Lr (a, b; t) ≤ (a + b)( ) . 2 2r(t − 1)

(2.2.14)

If (r, t) ∈ {1 < r < 2, 0 ≤ t ≤ 21 } ∪ {r ≤ 1, r ≠ 0, 21 ≤ t ≤ 1}, then the two inequalities in (2.2.14) are reversed. Remark 2.2.2. The term Lr (a, b; 0) is the generalized logarithmic mean (or Stolarsky’s mean), i. e., 1

br − ar r−1 Sr (a, b) = ( ) . r(b − a) Taking t = 0, from (2.2.14), we can obtain the following known inequality [79, p. 67]: a+b a+b , ≤ Sr (a, b) ≤ 1 2 r r−1 Theorem 2.2.6. Let

1 2

r > 2.

(2.2.15)

≤ t < 1, a, b ∈ ℝ+ , and let

L(a, b; t) =

(log b − log a) − (log u − log v) , 2(1 − t)(b − a)

a ≠ b,

(2.2.16)

L(a, a; t) = a−1 ,

where u = tb + (1 − t)a, v = ta + (1 − t)b. Then L(a, b; t) is Schur-convex on ℝ2+ . From Theorem 2.2.6 and ( a+b , a+b ) ≺ (a, b), we obtain the following. 2 2 Corollary 2.2.2. Let

1 2

≤ t < 1, a, b ∈ ℝ+ . Then L(a, b; t) ≤

2 . a+b

(2.2.17)

b−log a . Taking Remark 2.2.3. The term L(a, b; 0) is the logarithmic mean L(a, b) = log b−a t = 0, from (2.2.16), we can obtain the Ostle–Terwilliger inequality [124], i. e.,

log b − log a 2 ≤ . b−a a+b

(2.2.18)

70 | 2 Schur-convex functions and integral inequalities Theorem 2.2.7. Let t ∈ (0, 1), a, b ∈ ℝ+ , and let 1

ar+1 + br+1 − (ur+1 − vr+1 ) r−1 Qr (a, b; t) = [ ] , r(r + 1)t(1 − t)(b − a)2

a ≠ b,

Qr (a, a; t) = a,

where u = tb + (1 − t)a, v = ta + (1 − t)b. If r ≥ 2, then Qr (a, b; t) is Schur-convex on ℝ2+ ; if r ∈ {1 ≤ r < 2} ∪ {r < 1, r ≠ 0, −1}, then Qr (a, b; t) is Schur-concave on ℝ2+ . Corollary 2.2.3. When r ≥ 2, we have 1

a+b 1 − (1 − t)r+1 − r r+1 r−1 ≤ Qr (a, b; t) ≤ (a + b)( ) . 2 r(r + 1)t(1 − t)

(2.2.19)

When r ∈ {1 ≤ r < 2} ∪ {r < 1, r ≠ 0, −1}, the two inequalities in (2.2.19) are reversed. In 2001, Čuljak [22] generalized Theorem 2.2.3 to the following theorem. Theorem 2.2.8. Let I ⊂ ℝ be an interval with a nonempty interior. Let f be a continuous function on I and α a continuous function on [0, 1]. Let Lα : [0, 1] → ℝ be a function defined by b

1 Lα = ∫[f (α(t)a + (1 − α(t))x) + f (α(t)b + (1 − α(t))x)] d x. b−a

(2.2.20)

a

For a function Pα (a, b) defined on I 2 as Pα (a, b) = {

Lα , f (a),

a, b ∈ I, a ≠ b, a = b,

(2.2.21)

the following hold: (a) for α such that mint∈I α(t) = 21 , maxt∈I α(t) = 1, if f is convex on I, then Pα is Schurconvex on I 2 ; (b) for α such that mint∈I α(t) = 0, maxt∈I α(t) = 21 , if f is concave on I, then Pα is Schurconcave on I 2 . Qian and Zheng [144] obtained the following results. Theorem 2.2.9. Under the conditions of Theorem 2.2.3, P(a, b) is a convex (or concave, respectively) function on I 2 if and only if f is a convex (or concave, respectively) function on I. Theorem 2.2.10. Under the conditions of Theorem 2.2.4, Q(a, b) is a convex (or concave, respectively) function on I 2 if and only if f is a convex (or concave, respectively) function on I.

2.2 Schur-convex functions related to Hadamard type integral inequalities | 71

Remark 2.2.4. From Corollary 2.2.1 (Vol. 1) we know that the symmetrically convex (or concave, respectively) function on a symmetrically convex set must be a Schurconvex (or Schur-concave, respectively) function. Therefore, Theorem 2.2.9 and Theorem 2.2.10 not only generalize the conditions of Theorem 2.2.3, but also strengthen the conclusions of Theorem 2.2.3 and Theorem 2.2.4. 2.2.2 Schur-convex functions related to Lan He’s integral inequalities b

When f , −g both are convex functions satisfying ∫a g(x)dx > 0 and f ( a+b ) ≥ 0, Yang 2 [251] generalized (2.1.1) as follows: f ( x+y ) 2

g( x+y ) 2

y 1 ∫ f (x) d x y−x x . y 1 ∫ g(x) d x y−x x

≤

(2.2.22)

To go further in exploring (2.2.22), He [66] defined two mappings L and F by L : [a, b] × [a, b] → ℝ, y

y

L(x, y; f , g) = [∫ f (t) d t − (y − x)f ( x

x+y x+y )][(y − x)g( ) − ∫ g(t) d t] 2 2 x

and F : [a, b] × [a, b] → ℝ, y

y

x

x

x+y x+y F(x, y; f , g) = g( ) ∫ f (t) d t − f ( ) ∫ g(t) d t, 2 2 and established the following two theorems, which are refinements of the inequality of (2.2.22). Theorem 2.2.11. Let f , −g both be convex functions on [a, b]. Then we have: (a) L(a, y; f , g) is nonnegative increasing with y on [a, b], L(x, b; f , g) is nonnegative decreasing with x on [a, b]; a ) ≥ 0, for any x, y ∈ (a, b) and α ≥ 0 and β ≥ 0 such (b) when ∫b g(x) d x > 0 and f ( a+b 2 that α + β = 1, we have ) f ( a+b 2

g( a+b ) 2

≤

≤

≤

(b − a)f ( a+b ) 2 b

2 ∫a g(t) d t (b − a)f ( a+b ) 2

b 2 ∫a g(t) d t b ∫a f (t) d t + b 2 ∫a g(t) d t

b

+

∫a f (t) d t

(2.2.23)

2(b − a)g( a+b ) 2 b

+

∫a f (t) d t

) 2(b − a)g( a+b 2

2f ( a+b ) 2

2g( a+b ) 2

b

≤

+

αL(a, y; f , g) + βL(x, b; f , g)

∫a f (t) d t b

b

2(b − a)g( a+b ) ∫a g(t) d t 2

∫a g(t) d t

.

72 | 2 Schur-convex functions and integral inequalities Theorem 2.2.12. Let f , −g both be nonnegative convex functions on [a, b] satisfying b ∫a g(x)dx > 0. Then we have the following two results: (a) If f and −g both are increasing, then F(a, y; f , g) is nonnegative increasing with y on [a, b], and we have f ( a+b ) 2

) g( a+b 2

≤

f ( a+b ) 2

) g( a+b 2

+

F(a, y; f , g) b

g( a+b ) ∫a g(t) d t 2

b

≤

∫a f (t) d t b

∫a g(t) d t

,

(2.2.24)

where y ∈ (a, b). (b) If f and −g both are decreasing, then F(a, y; f , g) is nonnegative decreasing with y on [a, b], and we have f ( a+b ) 2

) g( a+b 2

≤

f ( a+b ) 2

) g( a+b 2

+

F(x, b; f , g) b

) ∫a g(t) d t g( a+b 2

b

≤

∫a f (t) d t b

∫a g(t) d t

,

(2.2.25)

where x ∈ (a, b). The author [163] studied the Schur-convexity of L(x, y; f , g) and F(x, y; f , g) with variables (x, y) in [a, b] × [a, b] ⊂ ℝ2 and the Schur-geometric and Schur-harmonic convexity of L(x, y; f , g) with variables (x, y) in [a, b] × [a, b] ⊂ ℝ2+ , obtaining the following results. Theorem 2.2.13. Let f and −g both be a convex function on [a, b]. Then: (a) L(x, y; f , g) is Schur-convex on [a, b] × [a, b] ⊂ ℝ2 , and L(x, y; f , g) is Schur-geometrically convex and Schur-harmonic convex in [a, b] × [a, b] ⊂ ℝ2+ ; (b) if 21 ≤ t2 ≤ t1 ≤ 1 or 0 ≤ t2 ≤ t1 ≤ 21 , then for a < b, we have 0 ≤ L(t1 a + (1 − t1 )b, t1 b + (1 − t1 )a; f , g)

(2.2.26)

≤ L(t2 a + (1 − t2 )b, t2 b + (1 − t2 )a; f , g) ≤ L(a, b; f , g),

and for 0 < a < b, we have 0 ≤ L(bt2 a1−t2 , at2 b1−t2 ; f , g) ≤ L(bt1 a1−t1 , at1 b1−t1 ; f , g) ≤ L(a, b; f , g)

(2.2.27)

and 0 ≤ L(1/(t2 b + (1 − t2 )a), 1/(t2 a + (1 − t2 )b); f , g)

(2.2.28)

≤ L(1/(t1 b + (1 − t1 )a), 1/(t1 a + (1 − t1 )b); f , g) ≤ L(1/a, 1/b; f , g).

Proof. (a) It is clear that L(x, y; f , g) is symmetric with x, y. Without loss of generality, we may assume y ≥ x. A direct calculation yields y

𝜕L x+y y−x 󸀠 x+y x+y = [f (y) − f ( )− f ( )][(y − x)g( ) − ∫ g(t) d t] 𝜕y 2 2 2 2 x

2.2 Schur-convex functions related to Hadamard type integral inequalities | 73 y

+ [∫ f (t) d t − (y − x)f ( x

x+y x+y y−x 󸀠 x+y )][g( )+ g( ) − g(y)], 2 2 2 2 y

x+y y−x 󸀠 x+y x+y 𝜕L = [−f (x) + f ( )− f ( )][(y − x)g( ) − ∫ g(t) d t] 𝜕x 2 2 2 2 x

y

+ [∫ f (t) d t − (y − x)f ( x

x+y x+y y−x 󸀠 x+y )][−g( )+ g( ) + g(x)]. 2 2 2 2

By the Lagrange mean value theorem, there exists ξ ∈ ((x + y)/2, y) such that f (y) − f (

x+y 󸀠 y−x 󸀠 x+y ) = (y − )f (ξ ) = f (ξ ). 2 2 2

), so Since f is convex, f 󸀠 is increasing, and we have f 󸀠 (ξ ) ≥ f 󸀠 ( x+y 2 f (y) − f (

y−x 󸀠 x+y x+y )− f ( ) ≥ 0. 2 2 2

By the same arguments, we have −f (x) + f (

x+y y−x 󸀠 x+y )− f ( ) ≤ 0. 2 2 2

Similarly, since −g is convex, we have g(

y−x 󸀠 x+y x+y )+ g( ) − g(y) ≥ 0 2 2 2

and −g(

y−x 󸀠 x+y x+y )+ g( ) + g(x) ≤ 0. 2 2 2 y

And by Hadamard’s inequality (1), it follows that (y − x)g( x+y ) − ∫x g(t) d t ≥ 0 and 2 y ∫x

) ≥ 0. So f (t) d t − (y − x)f ( x+y 2

𝜕L 𝜕y

≥ 0 and

𝜕L 𝜕x

≤ 0, and, moreover, (y − x)( 𝜕L − 𝜕y

−y2 𝜕L ) ≥ 0. Note that from y ≥ x, we have ln x−ln y and (x−y)(x2 𝜕L 𝜕x 𝜕y 𝜕L 𝜕L ln y)(x 𝜕x − y 𝜕y ) ≥ 0. According to Theorem 2.1.3, Theorem 2.4.3

𝜕L ) 𝜕x

≥0

≤ 0, and then (ln x−

and Theorem 2.4.19

(Vol. 1), it follows that L(x, y; f , g) is Schur-convex in [a, b] × [a, b] ⊆ ℝ2 , and L(x, y; f , g) is Schur-geometrical convex and Schur-harmonic convex in [a, b] × [a, b] ⊆ ℝ2+ . (b) From (1.4.32) (Vol. 1), we have (ln √ab, ln √ab) ≺ (ln(bt2 a1−t2 ), ln(at2 b1−t2 ))

≺ (ln(bt1 a1−t1 ), ln(at1 b1−t1 )) ≺ (ln a, ln b).

(2.2.29)

By Theorem 2.2.13(a), from (1.4.32) (Vol. 1) and (2.2.29) it follows that (2.2.26), (2.2.27) and (2.2.28) hold. The proof of Theorem 2.2.13 is completed.

74 | 2 Schur-convex functions and integral inequalities Theorem 2.2.14. Let f and −g both be a nonnegative convex function on [a, b]. Then (a) F(x, y; f , g) is Schur-convex on [a, b] × [a, b] ⊂ ℝ2 ; (b) if 21 ≤ t2 ≤ t1 ≤ 1 or 0 ≤ t2 ≤ t1 ≤ 21 , then for a < b, we have 0 ≤ F(t1 a + (1 − t1 )b, t1 b + (1 − t1 )a; f , g)

≤ F(t2 a + (1 − t2 )b, t2 b + (1 − t2 )a; f , g) ≤ F(a, b; f , g).

(2.2.30)

Proof. (a) It is clear that F(x, y; f , g) is symmetric. Without loss of generality, we may assume y ≥ x. A direct calculation yields y

𝜕F 1 󸀠 x + y x+y = g( ) ∫ f (t) d t + g( )f (y) 𝜕y 2 2 2 x

y

x+y x+y 1 ) ∫ g(t) d t − f ( )g(y), − f 󸀠( 2 2 2 y

x

𝜕F 1 󸀠 x + y x+y = g( ) ∫ f (t) d t − g( )f (x) 𝜕x 2 2 2 x

y

x+y x+y 1 ) ∫ g(t) d t + f ( )g(x), − f 󸀠( 2 2 2 x

and then (y − x)(

𝜕F 𝜕F − ) 𝜕y 𝜕x

= (y − x)[g(

x+y x+y )(f (x) + f (y)) − f ( )(g(x) + g(y))]. 2 2

Since f and −g both are convex functions on [a, b], f (x) + f (y) ≥ 2f ( x+y ), and 2

) ≥ g(x)+g(y) , and then g( x+y )(f (x) + f (y)) − f ( x+y )(g(x) + g(y)) ≥ 0, we have g( x+y 2 2 2 2 𝜕F 𝜕F (y − x)( 𝜕y − 𝜕x ) ≥ 0. From Theorem 2.1.3 (Vol. 1), it follows that F(x, y; f , g) is Schurconvex on [a, b] × [a, b]. (b) By Theorem 2.1.3(a), from (1.4.32) (Vol. 1) it follows that (2.2.30) holds. a

Theorem 2.2.15. Let f and −g both be convex functions on [a, b] ⊂ ℝ. If ∫b g(x) d x > 0 ) ≥ 0, then and f ( a+b 2

f ( a+b ) 2

g( a+b ) 2 where

1 2

≤

b

tb+(1−t)a

b

tb+(1−t)a

∫a f (t) d t − ∫ta+(1−t)b f (t) d t ∫a g(t) d t − ∫ta+(1−t)b g(t) d t

≤ t < 1 or 0 ≤ t ≤ 21 .

b

≤

∫a f (t) d t b

∫a g(t) d t

,

(2.2.31)

2.2 Schur-convex functions related to Hadamard type integral inequalities | 75

Theorem 2.2.16. Let f , −g both be nonnegative convex functions on [a, b] satisfying b ∫a g(x) d x > 0. Then for a < b, we have f ( a+b ) 2

g( a+b ) 2

b

≤

∫a f (t) d t b

∫a g(t) d t

−

L(ta + (1 − t)b, tb + (1 − t)a; f , g) b

2(b − a)g( a+b ) ∫a g(t) d t 2

b

≤

∫a f (t) d t b

∫a g(t) d t

,

(2.2.32)

and for 0 < a < b, we have b

f ( a+b ) 2

≤

) g( a+b 2 where

1 2

∫a f (t) d t b

∫a g(t) d t

−

L(bt a1−t , at b1−t ; f , g) b

) ∫a g(t) d t 2(b − a)g( a+b 2

b

≤

∫a f (t) d t b

∫a g(t) d t

,

(2.2.33)

≤ t ≤ 1 or 0 ≤ t ≤ 21 .

Theorem 2.2.17. Let a, b ∈ ℝ+ with a < b, and let u = tb + (1 − t)a, v = ta + (1 − t)b, 1 ≤ t < 1 or 0 ≤ t ≤ 21 . Then for 1 ≤ r ≤ 2, we have 2 r

(

r[(log b − log a) − (log u − log v)] r(log b − log a) 2 ) ≤ ≤ . a+b 2(b − a)(1 − t) b−a

(2.2.34)

Theorem 2.2.18. Let a, b ∈ ℝ+ with a < b, and let u = tb + (1 − t)a, v = ta + (1 − t)b, 1 ≤ t < 1 or 0 ≤ t ≤ 21 . Then for 1 ≤ r ≤ 2, we have 2 1

1

ar + br r a+b (b2r − a2r ) − (u2r − v2r ) r ≤[ ] ≤( ) . r r r r 2 2(b − a ) − 2(u − v ) 2

(2.2.35)

2.2.3 Schur-convex functions related to generalized integral quasiarithmetic mean Wang et al. [191] compared (2.1.3) and (2.2.22) and studied the Schur-convexity of the following two functions, involving the quasiarithmetic mean of generalized integrals: M (f ;a,b)

{ p(g;a,b) , a ≠ b, q Hp,q (f , g; a, b) = { M f (a) , a=b { g(a)

(2.2.36)

and [Mp (f ; a, b) − f ( a+b )] ⋅ [g( a+b ) − Mq (g; a, b)], 2 2 Lp,q (f ; g; a, b) = { 0,

a ≠ b,

a = b.

(2.2.37)

They obtained the following results. Theorem 2.2.19. Let f and g be real Lebesgue integrable functions defined on the interval I ⊆ ℝ, with range J1 and J2 , respectively, let p and q be real continuous strictly increasing functions on J1 and J2 , respectively, and let Mp (f ; a, b) ≥ 0, Mq (g; a, b) > 0, and g( a+b ) ≠ 0. 2

76 | 2 Schur-convex functions and integral inequalities (a) If p ∘ f is convex on I and q ∘ g is concave on I, then Hp,q (f , g; a, b) is Schur-convex on I 2 . And then for a < b, we have Mp (f ; a, b)

≥

Mq (g; a, b)

Mp (f ; ta + (1 − t)b, tb + (1 − t)a)

Mq (g; ta + (1 − t)b, tb + (1 − t)a)

≥

) f ( a+b 2

g( a+b ) 2

(2.2.38)

,

where 21 ≤ t ≤ 1 or 0 ≤ t ≤ 21 . (b) If p ∘ f is concave on I and q ∘ g is convex on I, then Hp,q (f , g; a, b) is Schur-concave on I 2 . Then the inequality chain (2.2.38) is reversed. Theorem 2.2.20. Let f and g be a real Lebesgue integrable nonnegative function defined on the interval I ⊂ ℝ, with range J1 and J2 , respectively, and let Mp (f ; a, b) ≥ 0, Mq (g; a, b) > 0, and g( a+b ) ≠ 0. If p, q are real continuous strictly increasing func2 tions on J1 and J2 , respectively, and p ∘ f is convex on I and q ∘ g is concave on I, then Lp,q (f , g; a, b) is Schur-convex on I 2 . Then the following inequality chain holds: Mp (f ; a, b)

Mq (g; a, b)

≥

Mp (f ; a, b)

2Mq (g; a, b)

+

) f ( a+b 2

2g( a+b ) 2

≥

) f ( a+b 2

2Mq (g; a, b)

+

Mp (f ; a, b) 2g( a+b ) 2

≥

) f ( a+b 2

g( a+b ) 2

. (2.2.39)

Corollary 2.2.4. Let f and g be positive integrable functions on I = [a, b] ⊂ ℝ+ , satisfy) > 0. If f (x) is a log-convex function and g 󸀠󸀠 (x) ≤ 0, x ∈ I, then ing g( a+b 2 ta+(1−t)b

b

1 exp{ b−a ∫a log f (t) d t} b

1 exp{ b−a ∫a log g(t) d t}

where

1 2

≥

1 exp{ b−a ∫tb+(1−t)a log f (t) d t} ta+(1−t)b

1 exp{ b−a ∫tb+(1−t)a log g(t) d t}

≥

f ( a+b ) 2

) g( a+b 2

,

(2.2.40)

≤ t < 1 or 0 ≤ t ≤ 21 .

Remark 2.2.5. Taking g(x) = e, x ∈ [a, b], from Corollary 2.2.4, we have b

1 1 exp{ ∫ log f (t) d t} ≥ exp{ b−a b−a a

ta+(1−t)b

∫

log f (t) d t} ≥ f (

tb+(1−t)a

a+b ). 2

(2.2.41)

This is a refinement of an inequality in [31].

2.3 Schur-convex functions related to Schwarz integral inequalities Let f and g be integrable functions on the interval [a, b](a < b). Then b

2

b

2

b

(∫ f (x)g(x) d x) ≤ ∫ f (x) d x ∫ g 2 (x) d x. a

a

This is the famous Schwarz integral inequality.

a

(2.3.1)

2.3 Schur-convex functions related to Schwarz integral inequalities | 77

Zhang et al. [268] used (2.3.1) to define a function of two variables W : [a, b] × [a, b] → ℝ as y

y

2

2

2

y

W(x, y) = W(x, y; f , g) = ∫ f (t) d t ∫ g (t) d t − (∫ f (t)g(t) d t) . x

x

(2.3.2)

x

In this section, we study the Schur-convexity of W(x, y). The contents are derived from the reference [160], which not only greatly simplifies the original proof, but also extends the original conclusion to the case of Schur-harmonically convex and Schurpower convex functions. Theorem 2.3.1. Let f and g be integrable in the interval [a, b]. Then: (a) W(x, b) is decreasing with respect to x on [a, b]; W(a, y) is increasing with respect to y on [a, b]; (b) W(x, y) is Schur-convex on [a, b]×[a, b] ⊂ R2 and Schur-geometrically convex, Schurharmonically convex, and Schur-power convex on [a, b] × [a, b] ⊂ R2++ . Proof. We have y

y

y

x

x

x

𝜕W = −(f 2 (x) ∫ g 2 (t) d t + g 2 (x) ∫ f 2 (t) d t − 2f (x)g(x) ∫ f (t)g(t) d t) 𝜕x y

2

= − ∫(f (x)g(t) − g(x)f (t)) d t ≤ 0 x

and y

y

y

x

x

x

𝜕W = (f 2 (y) ∫ g 2 (t) d t + g 2 (y) ∫ xf 2 (t) d t − 2f (y)g(y) ∫ f (t)g(t) d t) 𝜕y y

2

= ∫(f (y)g(t) − g(y)f (t)) d t ≤ 0. x

Thus (a) is proved. From 𝜕W ≤ 0, 𝜕W ≤ 0, and y ≥ x, it follows that 𝜕x 𝜕y (x − y)(

𝜕W 𝜕W − ) ≥ 0, 𝜕x 𝜕y

(log x − log y)(x (x − y)(x 2

𝜕W 𝜕W −y ) ≥ 0, 𝜕x 𝜕y

𝜕W 𝜕W − y2 )≥0 𝜕x 𝜕y

78 | 2 Schur-convex functions and integral inequalities and xm − ym 1−m 𝜕W 𝜕W (x − y1−m ) ≥ 0, m ≠ 0. m 𝜕x 𝜕y Thus (b) is proved. Theorem 2.3.2. Let a < b. If

1 2

≤ t2 ≤ t1 ≤ 1 or 0 ≤ t1 ≤ t2 ≤ 21 , then

0 ≤ W(t2 b + (1 − t2 )a, t2 a + (1 − t2 )b)

(2.3.3)

≤ W(t1 b + (1 − t1 )a, t1 a + (1 − t1 )b) ≤ W(a, b).

Proof. This theorem can be proved by combining (1.4.32) (Vol. 1) and Theorem 2.3.1(b). Theorem 2.3.3. Let 0 < a ≤ b. If

≤ t2 ≤ t1 ≤ 1 or 0 ≤ t1 ≤ t2 ≤ 21 , then

1 2

0 ≤ W(bt2 a1−t2 , at2 b1−t2 ) ≤ W(bt1 a1−t1 , at1 b1−t1 ) ≤ W(a, b).

(2.3.4)

Proof. By (1.4.32) (Vol. 1) we have (log √ab, log √ab) ≺ (log(bt2 a1−t2 ), log(at2 b1−t2 ))

≺ (log(bt1 a1−t1 ), log(at1 b1−t1 )) ≺ (log a, log b).

Thus, the inequalities in (2.3.4) hold by the Schur-geometric convexity of W(x, y) on [a, b] × [a, b] ⊂ ℝ2++ . Theorem 2.3.4. Let a < b. If

1 2

≤ t2 ≤ t1 ≤ 1 or 0 ≤ t1 ≤ t2 ≤ 21 , then

0 ≤ W((t2 b + (1 − t2 )a) , (t2 a + (1 − t2 )b) ) −1

(2.3.5)

−1

≤ W((t1 b + (1 − t1 )a) , (t1 a + (1 − t1 )b) ) ≤ W(a , b ). −1

−1

−1

−1

Proof. This is proved by Theorem 2.3.1(b) combined with Definition 2.4.1(b) (Vol. 1) and (1.4.32) (Vol. 1). From Theorem 2.3.2, Theorem 2.3.3, and Theorem 2.3.4, the following Corollary 2.3.1, Corollary 2.3.2, and Corollary 2.3.3 can be directly obtained. Corollary 2.3.1. Let f and g be integrable functions on the interval [a, b](a < b), 0 ≤ t ≤ 1. Then 2

b

b

b

a

a

(∫ f (t)g(t) d t) ≤ ∫ f 2 (t) d t ∫ g 2 (t) d t − W(tb + (1 − t)a, ta + (1 − t)b). a

(2.3.6)

Corollary 2.3.2. Let f and g be integrable functions on the interval [a, b](0 < a < b), 0 ≤ t ≤ 1. Then b

2

b

2

b

(∫ f (t)g(t) d t) ≤ ∫ f (t) d t ∫ g 2 (t) d t − W(bt a1−t , at b1−t ). a

a

a

(2.3.7)

2.4 Schur-convex functions related to Hölder integral inequalities | 79

Corollary 2.3.3. Let f and g be integrable functions on the interval [a, b](0 < a < b), 0 ≤ t ≤ 1. Then b

2

b

b

2

(∫ f (t)g(t) d t) ≤ ∫ f (t) d t ∫ g 2 (t) d t − W((tb + (1 − t)a) , (ta + (1 − t)b) ). a

−1

a

−1

a

(2.3.8)

Remark 2.3.1. Because W(tb + (1 − t)a, ta + (1 − t)b) ≥ 0, W(bt a1−t , at b1−t ) ≥ 0 and W((tb + (1 − t)a)−1 , (ta + (1 − t)b)−1 ) ≥ 0, inequalities (2.3.6), (2.3.7), and (2.3.8) all strengthen the Schwarz integral inequality.

2.4 Schur-convex functions related to Hölder integral inequalities The famous Hölder integral inequality [55] is b

b

a

a

1 p

1 q

b

∫ f (x)g(x) d x ≤ (∫ f p (x) d x) (∫ g q (x) d x) ,

(2.4.1)

a

where ak ≥ 0, bk ≥ 0, p > 1, p1 + q1 = 1. In 2018, Dongsheng Wang discussed the Schur-convexity of functions related to Hölder’s inequality and obtained the following results. Theorem 2.4.1. Suppose f (x), g(x) are continuous functions and f (x) ≥ 0, g(x) ≥ 0; b b b ∫a f (x)g(x) d x ≠ 0, ∫a (f (x))p d x ≠ 0, ∫a (g(x))p d x ≠ 0, and p and q are arbitrary real numbers. Let b

p

q

b

p

q

∫ (g(x)) d x ∫ (f (x)) d x { ) ( ba ) , {( ba ∫a f (x)g(x) d x H(a, b) = { ∫a f (x)g(x) d x { pq−p−q , {[f (a)g(a)]

a ≠ b, a = b.

(2.4.2)

Then H(a, b) is Schur-concave (or Schur-convex, respectively) with (a, b) if and only if q(f p (b) + f p (a)) b

∫a f p (x) d x

+

p(g q (b) + g q (a))

(2.4.3)

b

∫a g q (x) d x

≤ (or ≥, respectively)

(f (b)g(b) + f (a)g(a))(p + q) b

∫a f (x)g(x) d x

.

(2.4.4)

Corollary 2.4.1. Suppose f (x), g(x) are two-order differentiable and f (x) ≥ 0, g(x) ≥ 0, b

b

b

∫a f (x)g(x) d x ≠ 0, ∫a (f (x))p d x ≠ 0, ∫a (g(x))q d x ≠ 0. If p, q ≥ 1, f (x), g(x) are convex functions of opposite monotonicity, and f 󸀠󸀠 g + g 󸀠󸀠 f + 2f 󸀠 g 󸀠 ≤ 0, then H(a, b) is Schurconvex with (a, b).

80 | 2 Schur-convex functions and integral inequalities Corollary 2.4.2. Suppose f (x), g(x) are two-order differentiable and f (x) ≥ 0, g(x) ≥ 0, b

b

b

∫a f (x)g(x) d x ≠ 0, ∫a (f (x))p d x ≠ 0, ∫a (g(x))q d x ≠ 0. If p, q < 0 and f (x), g(x) are concave functions of opposite monotonicity, then H(a, b) is Schur-concave with (a, b). Corollary 2.4.3. Suppose f (x), g(x) are two-order differentiable and f (x) ≥ 0, g(x) ≥ 0, b

b

b

∫a f (x)g(x) d x ≠ 0, ∫a (f (x))p d x ≠ 0, ∫a (g(x))q d x ≠ 0. If −1 ≤ p ≤ 0, 0 ≤ q ≤ 1, p + q ≥ 0, and f (x), g(x) are concave functions of opposite monotonicity, then H(a, b) is Schur-convex with (a, b). Theorem 2.4.2. Suppose f (x), g(x) is a continuous function and f (x) ≥ 0, g(x) ≥ 0, b b b ∫a f (x)g(x) d x ≠ 0, ∫a (f (x))p d x ≠ 0, ∫a (g(x))q d x ≠ 0, and f , g are concave functions of opposite monotonicity. If −1 ≤ p < 0, 0 < q ≤ 1, p1 + q1 = 1r ≤ 0, then 1 p

b

b

(∫ f p (x) d x) (∫ g q (x) d x) a

1 q

a

1− 1r

a+b a+b )g( )) ≥ (f ( 2 2

b

(2.4.5) 1 r

(∫ f (x)g(x) d x) . a

If p, q < 0, then this inequality is reversed.

2.5 Schur-convex functions related to Chebyshev integral inequalities For two integrable functions f , g : [a, b] → ℝ, define the functional, which is known in the literature as the Chebyshev functional, by T(p, f , g) := M(p, fg) − M(p, f )M(p, g),

(2.5.1)

where the integral mean is given by M(p, f ) :=

b

1

∫a p(x) d x

b

∫ p(x)f (x) d x,

(2.5.2)

a

and p : [a, b] → ℝ++ is a weight function. In particular, T(1, f , g) is denoted T(f , g). There are two important conclusions for the Chebyshev functional T(p, f , g). Let f , g : [a, b] → ℝ be two Lebesgue integrable functions. If f and g are monotonic in the same sense, we have the well-known Chebyshev inequality T(f , g) ≥ 0.

(2.5.3)

The sign of inequality in (2.5.3) is reversed if f and g are monotonic in the opposite sense.

2.5 Schur-convex functions related to Chebyshev integral inequalities | 81

Let f , g : [a, b] → ℝ be two Lebesgue integrable functions and m1 ≤ f (x) ≤ M1 , m2 ≤ g(x) ≤ M2 , x ∈ [a, b]. Then we have the well-known Chebyshev–Grüss inequality 󵄨 1 󵄨󵄨 󵄨󵄨T(f , g)󵄨󵄨󵄨 ≤ (M1 − m1 )(M2 − m2 ). 4

(2.5.4)

Wang [205] proved the following theorem. Theorem 2.5.1. For a ≤ x ≤ y ≤ b, write y

y

y

y

u(x, y) := ∫ p(t) d t ∫ p(t)f (t)g(t) d t − ∫ p(t)f (t) d t ∫ p(t)g(t) d t. x

x

x

(2.5.5)

x

If the monotonicity of f and g on [a, b] is the same (or opposite, respectively), then u(x, y) is increasing (or decreasing, respectively) with respect to y on [a, b] and decreasing (or increasing, respectively) with respect to x. The functional T(p, f , g) can be regarded as a binary function with integral upper and lower bounds as independent variables T : [a, b]2 → ℝ, denoted as y

T(x, y) = T(p, f , g; x, y) = y

1 ∫ p(t)f (t)g(t) d t y−x x

(2.5.6)

y

1 1 −( ∫ p(t)f (t) d t)( ∫ p(t)g(t) d t). y−x y−x x

x

For the weight function p(t) = 1, Čuljak and Pečarić [23] studied the Schurconvexity of the Chebyshev operator and obtained the following result. Theorem 2.5.2. Let f and g be Lebesgue integrable functions on I = [a, b]. If they are monotonic in the same sense (or in the opposite sense, respectively), then T(x, y) := T(f , g; x, y), (x, y) ∈ [a, b] × [a, b] ⊂ ℝ2 is Schur-convex (or Schur-concave, respectively) on [a, b] × [a, b]. Čuljak [23] further gave the following conclusions on the weighted Chebyshev operator T(P, F, g). Theorem 2.5.3. Let f and g be Lebesgue integrable functions on I = [a, b] and let p be a positive continuous weight on I such that pf and pg are also Lebesgue integrable functions on I = [a, b]. Then T(p; x, y) := T(p; f , g; x, y) is Schur-convex (or Schur-concave, respectively) on I 2 = [a, b] × [a, b] if and only if the inequality T(p, f , g) ≤ p(x)(f p(x, y) − f (x))(gp(x, y) − g(x)) + p(y)(f p(x, y) − f (y)) ⋅ (gp(x, y) − g(y))/(p(x) + p(y))

(2.5.7)

82 | 2 Schur-convex functions and integral inequalities holds (or is reversed, respectively) for all x, y ∈ I, where f p(x, y) =

y

y

1

∫x p(t) d t

∫ p(t)f (t) d t x

and gp(x, y) =

y

y

1

∫x p(t) d t

∫ p(t)g(t) d t. x

In 2013, Long et al. [99] extended Theorem 2.5.3 to the case of Schur-geometrically convex and Schur-harmonically convex functions and obtained the following results. Theorem 2.5.4. Let f and g be Lebesgue integrable functions on the interval [a, b] and let p be a positive continuous weight function on [a, b], making pf and pg Lebesgue integrable in the interval [a, b]. Then: (a) T(p; x, y) is Schur-geometrically convex on [a, b] × [a, b] ⊂ ℝ2 if and only if the inequality T(p, f , g) ≤ xp(x)(f p(x, y) − f (x))(gp(x, y) − g(x))

(2.5.8)

+ yp(y)(f p(x, y) − f (y)) ⋅ (gp(x, y) − g(y))/x(p(x) + yp(y)) holds for all x, y ∈ I; (b) T(p; x, y) is Schur-harmonically convex on [a, b] × [a, b] ⊂ ℝ2 if and only if the inequality T(p, f , g) ≤ x2 p(x)(f p(x, y) − f (x))(gp(x, y) − g(x)) 2

(2.5.9) 2

2

+ y p(y)(f p(x, y) − f (y)) ⋅ (gp(x, y) − g(y))/x (p(x) + y p(y)) holds for all x, y ∈ I. Theorem 2.5.5. Let f and g be Lebesgue integrable functions on the interval [a, b], p be a positive continuous weight function on [a, b], and let Gp (t) = (f p (x, y) − f (t))(g p (x, y) − g(t)) be a differentiable function on I with Gp󸀠 (y) ≥

y ∫x

G(y) − G(x) y−x p(t) d t

p(y)

⋅

for any x, y ∈ I. Then the following statements are true. (a) If Gp (t) and p(t) have the same monotonicity on I, then T(f , g, p) is Schur-convex on I 2 ;

2.6 Majorization type integral inequalities | 83

(b) if Gp (t) and tp(t) have the same monotonicity on I, then T(f , g, p) is Schur-geometrically convex on I 2 ; (c) if Gp (t) and t 2 p(t) have the same monotonicity on I, then T(f , g, p) is Schur-harmonically convex on I 2 . In 2000, the author [180] obtained the following results under stronger conditions. Theorem 2.5.6. Let the function f and g be integrable on the interval [a, b]. (a) If f and g have the same monotonicity and the same convexity, then T(x, y) is Schurconvex on [a, b]2 ⊂ ℝ2 and is Schur-geometrically convex and Schur-harmonically convex on [a, b]2 ⊂ ℝ2++ ; (b) if f and g have the opposite monotonicity and the opposite convexity, then T(x, y) is Schur-concave on [a, b]2 ⊂ ℝ2 and is Schur-geometrically concave and Schurharmonically concave on [a, b]2 ⊂ ℝ2++ .

2.6 Majorization type integral inequalities Definition 2.6.1 ([128, p. 324]). The function y(t) is said to majorize the function x(t), in symbols, x(t) ≺ y(t), for t ∈ [a, b], if they are decreasing in t ∈ [a, b] and s

s

∫ x(t) d t ≤ ∫ y(t) d t 0

for s ∈ [a, b],

(2.6.1)

0

and equality in (2.6.1) occurs for s = b. Theorem 2.6.1 ([128, p. 325]). The functions x(t) and y(t) satisfies that x(t) ≺ y(t), for t ∈ [a, b] if and only if they are decreasing in [a, b] and s

s

∫ h(x(t)) d t ≤ ∫ h(y(t)) d t 0

(2.6.2)

0

holds for every h that is continuous and convex in [a, b] such that the integrals exist. Theorem 2.6.1 is an integral similar to the famous Karamata inequality, namely, Theorem 1.5.2(a) (Vol. 1). In [121], Theorem 2.6.1 has been generalized as follows. Theorem 2.6.2. Let f : [a, b] → [0, ∞) be a continuous function and g : [a, b] → [0, ∞) a continuous, increasing function satisfying b

b

∫ f (t) d t ≥ ∫ g(t) d t, ∀x ∈ [a, b]. x

x

(2.6.3)

84 | 2 Schur-convex functions and integral inequalities Then inequality b

b

∫ h(f (t)) d t ≥ ∫ h(g(t)) d t x

(2.6.4)

x

holds for every convex function h such that h󸀠 > 0 and h󸀠 is integrable on [0, ∞). Corollary 2.6.1. Let f : [a, b] → [0, ∞) be a continuous function and g : [a, b] → [0, ∞) a continuous, nondecreasing function satisfying (2.6.3). Then for every α > 1, we have b

α

b

∫ f (t) d t ≥ ∫ g α (t) d t. a

(2.6.5)

a

Theorem 2.6.3 ([121]). Let f : [a, b] → [0, ∞) be a continuous function and g : [a, b] → [0, ∞) a continuous, nonincreasing function satisfying the reversed inequality of (2.6.3). Then inequality (2.6.4) holds for every convex function h󸀠 ≤ 0 and h󸀠 is integrable on [0, ∞). Theorem 2.6.4 ([187]). Let function f , g : [a, b] → ℝ be integrable such that f ≺ g and let φ : [a, b] → ℝ be an integrable, increasing function. Then b

b

∫ φ(t)f (t) d t ≥ ∫ φ(t)g(t) d t. a

a

Remark 2.6.1. Theorem 2.6.4 is an integral similar to Theorem 1.3.10(a) (Vol. 1). Corollary 2.6.2 (Steffensen inequality). Let function f , g : [0, a] → ℝ be integrable, x 0 ≤ g(x) ≤ 1, let f be decreasing on [0, a], and let F(x) = ∫0 f (t)dt. Then a

a

∫ f (x)g(x) d x ≤ F(∫ g(x) d x). 0

(2.6.6)

0

Corollary 2.6.3 ([106]). Let function f , g : [0, a] → ℝ be integrable, g(x) ≥ 0, and f is decreasing on [0, a]. Then a

c γ

∫ f (x)g(x)dx ≤ γ ∫ f (x) d x, 0

0

where a

γ = sup{g(x); x ∈ [0, a]}, c = ∫ g(x) d x. 0

(2.6.7)

2.6 Majorization type integral inequalities | 85

In 1947, Fuchs [40] gave the following integral majorization theorem for convex functions and two monotonic functions. Theorem 2.6.5. Let κ(τ), ν(τ) : [a, b] → ℝ be continuous and increasing functions, and let μ : [a, b] → ℝ be a function of bounded variation. (a) If b

b

∫ κ(τ)dμ(τ) ≤ ∫ ν(τ)dμ(τ) for x ∈ [a, b] x

x

and b

b

∫ κ(τ)dμ(τ) ≤ ∫ ν(τ)dμ(τ), a

a

then for every continuous convex function φ, we have b

b

∫ ϕ[κ(τ)]dμ(τ) ≤ ∫ ϕ[ν(τ)]dμ(τ). a

(2.6.8)

a

(b) If b

b

∫ κ(τ)dμ(τ) ≤ ∫ ν(τ)dμ(τ) for x ∈ [a, b], x

x

then for every continuous increasing convex function ϕ, we have b

b

∫ ϕ[κ(τ)]dμ(τ) ≤ ∫ ϕ[ν(τ)]dμ(τ). a

(2.6.9)

a

In 1995, Maligranda et al. [104] established the following analogue of the Fuchs inequality. Theorem 2.6.6. Let w be a weight function, and let f and g be positive integrable functions on [a, b]. Suppose that ϕ : [0, ∞) → ℝ is a convex function and that x

x

∫ f (t)w(t) d t ≤ ∫ g(t)w(t) d t a

a

b

b

for x ∈ [a, b]

and ∫ f (t)w(t) d t = ∫ g(t)w(t) d t a

a

for x ∈ [a, b].

86 | 2 Schur-convex functions and integral inequalities (a) If f is a decreasing function on [a, b], then b

b

∫ −aϕ[f (t)]w(t) d t ≤ ∫ ϕ[g(t)]w(t) d t.

(2.6.10)

a

(b) If g is an increasing function on [a, b], then b

b

∫ ϕ[g(t)]w(t) d t ≤ ∫ ϕ[f (t)]w(t) d t. a

(2.6.11)

a

Wu et al. [221] established some majorization integral inequalities for functions defined on rectangles. The following theorem is a generalization of Theorem 2.6.6. Theorem 2.6.7. Let w and p be positive continuous functions on [a, b] and [c, d], respectively, and let f , g and h, k be positive differentiable functions on [a, b] and [c, d], respectively. Suppose that ϕ : [0, ∞) × [0, ∞) → ℝ is a convex function and that x

x

∫ g(t)w(t) d t ≤ ∫ f (t)w(t) d t

for x ∈ [a, b],

∫ k(s)p(s) d s ≤ ∫ h(s)p(s) d s

for y ∈ [c, d],

a y

a y c

c

b

b

∫ g(t)w(t) d t = ∫ f (t)w(t) d t, a

a

d

d

and ∫ k(s)p(s) d s ≤ ∫ h(s)p(s) d s. c

c

(a) If g and k are decreasing functions on [a, b] and [c, d], respectively, then b d

b d

∫ ∫ ϕ[g(t), k(s)]w(t)p(s) d t d s ≤ ∫ ∫ ϕ[f (t), h(s)]w(t)p(s) d t d s. a c

(2.6.12)

a c

(b) If f and h are increasing functions on [a, b] and [c, d] respectively, then b d

b d

∫ ∫ ϕ[f (t), h(s)]w(t)p(s) d t d s ≤ ∫ ∫ ϕ[g(t), k(s)]w(t)p(s) d t d s. a c

a c

(2.6.13)

2.7 Schur-convex functions and other integral inequalities | 87

2.7 Schur-convex functions and other integral inequalities The notation R[a, b] represents the set of all Riemann integrable functions on the b bounded closed interval [a, b], and the Riemann integral ∫a g(x)f n (x) d x is simply b

written as ∫a gf n d x. Chen [12] used double integrals to prove the following conclusion. b

Theorem 2.7.1. Let f , g ∈ R[a, b], m, n > 0, and f , g ≥ 0, m ≤ n, ∫a gf n d x ≠ 0, b

∫a gf m d x ≠ 0. Then

b

∫a gf m+1 d x b

∫a gf m d x

b

≤

∫a gf n+1 d x b

∫a gf n d x

(2.7.1)

.

Shi et al. [183] used majorization theory to prove and generalize (2.7.1) and established a series of such integral inequalities to show the distinct characteristics of the “batch production inequality” of the theory of majorization. b

Lemma 2.7.1. Let f , g ∈ R[a, b] and f , g ≥ 0. Note that I(r) = ∫a gf r d x, r ≥ 0. Then log I(r) is a convex function on ℝ+ . Proof. Let α, β ≥ 0, 0 < t < 1. By the integral Hölder inequality [79, p. 6], we have b

I(tα + (1 − t)β) = ∫ gf

tα+(1−t)β

a

b

α

b

t

α

a

t

b

β

1−t

≤ (∫ gf d x) (∫ gf d x) a

1−t

d x = (∫ gf ) (gf β )

a

dx

= I t (α)I 1−t (β),

which is log I[(tα + (1 − t)β)] ≤ t log I(α) + (1 − t) log I(β). This shows that log I(r) is a convex function on ℝ+ . b

Lemma 2.7.2. Let f , g ∈ R[a, b] and f , g ≥ 0, n ≥ 2. Note that I(r) = ∫a gf r d x, r ≥ 0; ∀p, q ∈ ℝn+ , if p ≺ q, then n

n

i=1

i=1

∏ I(pi ) ≤ ∏ I(qi ).

(2.7.2)

Proof. From Lemma 2.7.2 we know that log I(r) is a convex function on ℝ+ , so by Theorem 1.5.2(a) (Vol. 1) we have n

n

i=1

i=1

∑ log I(pi ) ≤ ∑ log I(qi ), that is, inequality (2.7.2) holds.

88 | 2 Schur-convex functions and integral inequalities Proof of Theorem 2.7.1. Taking p = (m + 1, n), q = (m, n + 1), it is easy to see p ≺ q, so by inequality (2.7.2), it follows that I(n)I(m + 1) ≤ I(m)I(n + 1), i. e., b

b

n

∫ gf d x ⋅ ∫ gf a

m+1

a

b

b

m

d x ≤ ∫ gf d x ⋅ ∫ gf n+1 d x. a

a

This inequality is equivalent to inequality (2.7.1). Now, we establish a series of such inequalities by constructing various majorizing relations. Theorem 2.7.2. Let f , g ∈ R[a, b] and f ≥ 0, g > 0, m, n ∈ ℕ, m ≤ n. Then b

(

∫a f m d x b

∫a g d x

b

n

) (

∫a gf n d x b

∫a g d x

m

(2.7.3)

) .

Proof. It is easy to see that p := (m, . . . , m) ≺ (n, . . . , n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 0, . . . , 0) =: q. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

m

n−m

Thus from inequality (2.7.2) it follows that [I(m)]n ≤ [I(n)]m ⋅ [I(0)]n−m , namely, b

n

m

b

n

m

n−m

b

(∫ gf d x) ≤ (∫ gf d x) ⋅ (∫ g d x) a

a

.

a

This inequality is equivalent to inequality (2.7.3). In particular, taking n = m + 1, inequality (2.7.3) is reduced to Theorem 4 of [95], that is, b

m+1

m

(∫ gf d x)

b

b

≤ ∫ g d x ⋅ (∫ gf

a

a

m+1

a

m

d x) .

(2.7.4)

Taking g(x) = 1, inequality (2.7.3) is reduced to Theorem 4 of [95], that is, n

b

m

b

1 1 (∫ f m d x) ≤ (∫ f n d x) , m ≤ n. b−a b−a a

i. e.,

(2.7.5)

a

Further taking n = 2, m = 1, inequality (2.7.5) is reduced to a familiar inequality, b

2

b

(∫ f d x) ≤ (b − a) ∫ f 2 d x, a

a

(2.7.6)

2.7 Schur-convex functions and other integral inequalities | 89

and taking n = 3, m = 2, (2.7.5) is reduced to 3

b

2

b

1 1 (∫ f 2 d x) ≤ (∫ f 3 d x) . b−a b−a a

(2.7.7)

a

Similarly, using (m, n − m) ≺ (n, 0), we have the following. Theorem 2.7.3. Let f , g ∈ R[a, b] and f , g ≥ 0, m, n ∈ ℕ, m ≤ n. Then b

b

b

b

∫ gf m d x ⋅ ∫ gf n−m d x ≤ ∫ g d x ⋅ ∫ gf n d x. a

a

a

(2.7.8)

a

If f > 0, taking n = 2, m = 1, and g = f1 , the following corollary is obtained from inequality (2.7.8). Corollary 2.7.1. Let f ∈ R[a, b] and f > 0. Then b

b

(∫ f d x)(∫ a

a

1 d x) ≥ (b − a)2 . f

(2.7.9)

This is also a well-known result. By using (n − m, . . . , n − m, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n, . . . , n), . . . , m, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n, . . . , n) ≺ (m, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n−m

n

m

2m

we can obtain the following theorem. Theorem 2.7.4. Let f , g ∈ R[a, b] and f , g ≥ 0, m, n ∈ ℕ, 2m < n. Then b

(∫ gf

n−m

n−m

d x)

b

2m

n

⋅ (∫ gf d x)

a

a

b

m

m

b

n

n

≤ (∫ gf d x) ⋅ (∫ gf d x) . a

(2.7.10)

a

By using h, . . . , h), h < m < n, . . . , n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (m, . . . , m) ≺ (n, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n−m

m−h

n−h

we obtain the following theorem. Theorem 2.7.5. Let f , g ∈ R[a, b] and f , g ≥ 0, h, m, n ∈ ℕ, h < m < n. Then b

m

n−h

(∫ gf d x) a

b

n

m−h

≤ (∫ gf d x) a

b

h

n−m

⋅ (∫ gf d x) a

.

(2.7.11)

90 | 2 Schur-convex functions and integral inequalities By Theorem 1.3.12(a) (Vol. 1), it is easy to verify that n, . . . , n). (0, 0, 1, 1, . . . , n, n) ≺ (0, . . . , 0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1

n+1

Based on this, the following theorems can be obtained. Theorem 2.7.6. Let f , g ∈ R[a, b] and f , g ≥ 0, n ∈ ℕ. Then n

b

i=0

a

2

i

n+1

b

b

∏(∫ gf d x) ≤ (∫ g d x)

n+1

n

⋅ (∫ gf d x)

a

.

a

(2.7.12)

Taking g(x) = 1, from inequality (2.7.12), it follows that n

b

b

a

a

1 1 ∏( ∫ f i d x) ≤ ( ∫ f n d x) b − a b − a i=0

n+1 2

(2.7.13)

.

In particular, take n = 3; inequality (2.7.13) is then reduced to (

b

b

b

a

a

a

1 1 1 ∫ f d x) ⋅ ( ∫ f 2 d x) ≤ ∫ f 3 d x. b−a b−a b−a

(2.7.14)

By Theorem 1.3.12(a) (Vol. 1), it is easy to verify that n, . . . , n). 1, . . . , 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ . . . , 0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ − 1, . . . , n − 1) ≺ (0, 2, . . . , 2, n (1, . . . , 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n−1

n−1

n−1

n+1

n+1

n+1

Based on this, the following theorems can be obtained. Theorem 2.7.7. Let f , g ∈ R[a, b] and f , g ≥ 0, n ∈ ℕ. Then n−1

b

i=1

a

i

n+1

∏(∫ gf d x)

n

b

i=0

a

n−1

i

≤ ∏(∫ gf d x)

(2.7.15)

,

that is, n−1

b

i=1

a

i

2

n−1

b

∏(∫ gf dx) ≤ (∫ g d x) a

b

n

n−1

⋅ (∫ gf d x)

.

a

Using (1.4.42) (Vol. 1), namely, + 1, . . . , 2n + 1), 3, . . . , 3, 2n . . . , 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2n, . . . , 2n) ≺ (1, 4, . . . , 4, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (2, . . . , 2, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n+1

n+1

n+1

the following theorem can be obtained.

n

n

n

(2.7.16)

2.8 Schur-convex functions and gamma functions | 91

Theorem 2.7.8. Let f , g ∈ R[a, b] and f , g ≥ 0, n ∈ ℕ. Then n

b

∏(∫ gf 2i d x) i=1

n+1

a

n

b

i=0

a

n

≤ ∏(∫ gf 2i+1 d x) .

(2.7.17)

From (1.3.39) (Vol. 1) we have (n, . . . , n) ≺ (0, 1, . . . , 2n). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2n+1

Based on this, the following theorem can be obtained. Theorem 2.7.9. Let f , g ∈ R[a, b] and f , g ≥ 0, n ∈ ℕ. Then b

2n+1

n

(∫ gf d x) a

2n

b

i=0

a

≤ ∏(∫ gf i d x).

(2.7.18)

It should be noted that since the Hölder inequality holds for the Lebesgue integrals on the measurable set, Lemma 2.7.1 and Lemma 2.7.2 also hold. Therefore, the above conclusion is not limited to Riemann integrable functions. For example, the following is an example of infinite generalized integrals. 2 For g(t) = exp(− t2 ), f (t) = t, noting that (1, 1) ≺ (0, 2), by Lemma 2.7.2, we have the following. Theorem 2.7.10. For x > 0 we have 2

t2 t2 t2 [ ∫ t exp(− ) d t] ≤ ∫ t exp(− ) d t ⋅ ∫ t 2 exp(− ) d t. 2 2 2 ∞

x

∞

∞

x

x

(2.7.19)

2.8 Schur-convex functions and gamma functions For x > 0, the Euler gamma function Γ(x) is defined by ∞

Γ(x) = ∫ e−t t x−1 dt.

(2.8.1)

0

Marshall and Olkin [108] applied Corollary 2.1.3 (Vol. 1) to obtain the following Theorem 2.8.1. They studied the Schur-convexity of some gamma function products and, using Theorem 2.8.1, obtained Theorem 2.8.2. Theorem 2.8.1. If γ is a measure on [0, ∞) such that g(x) = ∫0 z x d γ(z) exists for all x in an interval I, then log g is convex on I. Unless γ concentrates its mass on a set of the form {0, z0 }, log g is strictly convex on I. ∞

92 | 2 Schur-convex functions and integral inequalities When γ is a probability measure [g(0) = 1], the log-convexity of g(x) ≡ μx . μ is equivalent to the Lyapunov inequality [79, p. 787] r−s r−t μr−t s ≤ μt μr , r ≥ s ≥ t.

The following is a consequence. If μr is the rth moment of a nonnegative variable, that is, ∞

μr = ∫ z r dγ(z) 0

for some probability measure γ, and if μr exists for all r in the interval I ⊂ ℝ, then n

(2.8.2)

φ(x) = φ(x1 , . . . , xn ) = ∏ μxi i=1

is Schur-convex in x = (x1 , . . . , xn ) ∈ I n . Theorem 2.8.2 ([108]). Let functions f and g be integrable on the interval [a, b]. Then: (a) φ1 (x) = ∏ni=1 Γ(xi + a) is Schur-convex on (−a, ∞)n , where a ≥ 0; Γ(xi +a) (b) φ2 (x) = ∏ni=1 Γ(x +a+b) is Schur-convex on (−a, ∞)n , where a, b > 0; (d) φ4 (x) =

i

Γ(mxi +a) is Schur-convex on (− ma , ∞)n , Γ(xi +a) xi +1 x ∏ni=1 Γ(xi +1) is Schur-concave on ℝn+ . i

(c) φ3 (x) = ∏ni=1

where a > 1, m ≥ 2, s ≤ m;

The following theorem summarizes the relevant conclusions in the monograph [107]. Theorem 2.8.3. Let functions f and g be integrable on the interval [a, b]. Then: (a) ϕ1 (x) = ∏ni=1 Γ(xi ) is Schur-convex on ℝn++ ; Γ(x +a) (b) ϕ2 (x) = ∏ni=1 Γ(xi +1) is Schur-convex on (−a, ∞)n , where 0 < a ≤ 1; i

Γ(mx +a)

i is Schur-convex (or Schur-concave, respectively) on ℝn+ , (c) ϕ3 (x) = ∏ni=1 Γ(x +a+b)Γ(x i i +1) where a, b ≤ 1 (or a, b ≥ 1, respectively); x

xi i is Γ(xi +1) n Γ(mxi +a) ∏i=1 Γ(x +a) i

(d) ϕ4 (x) = ∏ni=1

Schur-convex on ℝn+ ;

(e) ϕ5 (x) = is Schur-convex on (− ma , ∞)n , where a ≥ m > 1, the convexity is strict; (f) Γ(x)−Γ(y)

x−y ϕ6 (x) = { Γ󸀠 (x) , Γ(x)

is strictly Schur-concave on ℝn++ .

,

x ≠ y, x=y

m−1 , k−1

k > 1; when

(2.8.3)

2.8 Schur-convex functions and gamma functions | 93

In 2005, by using a geometrical method, Alsina and Toms [1] proved the following double inequality: 1 Γ(1 + x)n ≤ ≤ 1. n! Γ(1 + nx)

(2.8.4)

In 2009, Nguyen and Ngo [122] obtained the following generalization of (2.8.4): ∏ni=1 Γ(1 + αi x) ∏ni=1 Γ(1 + αi ) 1 , ≤ ≤ n Γ(β + ∑i=1 αi ) Γ(β + (∑ni=1 αi )x) Γ(β)

(2.8.5)

where x ∈ [0, 1], β ≥ 1, αi > 0, n ∈ N. For k > 0, the Γk function is defined [27] by x

n!k n (nk) k −1 , n→∞ (x)n,k

Γk (x) = lim

x ∈ C\kZ − ,

(2.8.6)

where (x)n,k = x(x + k(x + 2k) ⋅ ⋅ ⋅ (x + (n − 1)k). The above definition is a generalization of the definition of Γ(x) functions. For x ∈ ℂ with R(x) > 0, the function Γk (x) is given by the integral [27] ∞

tk

Γk (x) = ∫ e− k t x−1 d t.

(2.8.7)

0

As a generalization of the Riemann zeta function ζ (x), the following k-Riemannian zeta function ζk (x) is defined in [74]: ζk (x) =

1 t x−k d t, ∫ t Γk (x) e − 1 ∞

x > k.

(2.8.8)

0

Zhang and Shi [267], by using methods from the theory of majorization, extended inequality (2.8.5) to Γk (x) and ζk (x) and obtained the following two theorems. Theorem 2.8.4. We have ∏ni=1 Γk (1 + αi x) ∏ni=1 Γk (1 + αi ) 1 ≤ ≤ , n n Γk (β + ∑i=1 αi ) Γk (β + (∑i=1 αi )x) Γk (β)

(2.8.9)

where x ∈ [0, 1], β ≥ 1, αi > 0, n ∈ N. tk

Proof. Taking g(t) = e− k , f (t) = t, a = 0, b = +∞, we have b

x

+∞

tk

I(x) = ∫ g(t)(f (t)) dt = ∫ e− k t x dt = Γk (x + 1). a

0

(2.8.10)

94 | 2 Schur-convex functions and integral inequalities By Lemma 2.7.1, I(x) is log-convex on [0, +∞), and then from Corollary 2.1.3(a) (Vol. 1), n+1 . Combining with (1.4.47) (Vol. 1) and φ(x) = ∏n+1 i=1 I(xi ) is Schur-convex on [0, +∞) (1.4.48) (Vol. 1), respectively, we have φ(u) ≤ φ(v) and φ(w) ≤ φ(z), i. e., n

n

n

n

i=1

i=1

i=1

i=1

Γk (β + (∑ αi )x) ∏ Γk (1 + αi ) ≤ Γk (β + ∑ αi ) ∏ Γk (1 + αi x)

(2.8.11)

and n

n

i=1

i=1

Γk (β) ∏ Γk (1 + αi x) ≤ Γk (β + (∑ αi )x).

(2.8.12)

Thus, we have proved the double inequality (2.8.9). The proof of Theorem 2.8.4 is completed. Theorem 2.8.5. We have ∏ni=1 ζk (k + 1 + αi )Γk (k + 1 + αi ) ζk (β + k + ∑ni=1 αi )Γk (β + k + ∑ni=1 αi ) ≤ ≤

(2.8.13)

∏ni=1 ζk (k + 1 + αi )Γk (k + 1 + αi x) ζk (β + k + (∑ni=1 αi )x)Γk (β + k + (∑ni=1 αi )x) (π 2 /6) , ζk (β + k)Γk (β + k)

where x ∈ [0, 1], β ≥ 1, αi > 0, i = 1, . . . , n, n ∈ N. Proof. Let ∞

ξk (x) = ∫ 0

t x−k dt, et − 1

x > k,

i. e., ξk (x) = ζk (x)Γk (x). Taking g(t) =

t , et −1

f (t) = t, a = 0, b = +∞, we have b

x

+∞

J(x) = ∫ g(t)(f (t)) dt = ∫ a

0

t x+1 dt = ξk (x + k + 1). et − 1

(2.8.14)

2.8 Schur-convex functions and gamma functions | 95

By Lemma 2.7.1, J(x) is log-convex on [0, +∞), and then by Corollary 2.1.3(a) (Vol. 1), n+1 . Combining with (1.4.47) (Vol. 1) and φ(x) = ∏n+1 i=1 J(xi ) is Schur-convex on [0, +∞) (1.4.48) (Vol. 1), respectively, we have ψ(u) ≤ ψ(v) and ψ(w) ≤ ψ(z), i. e., n

n

ξk (β + k + (∑ αi )x) ∏ ξk (k + 1 + αi ) i=1

i=1

n

n

i=1

i=1

≤ ξk (β + k + ∑ αi ) ∏ ξk (k + 1 + αi x) and n

ξk (β + k) ∏ ξk (k + 1 + αi x) i=1

n

≤ ξk (β + k + (∑ αi )x)( i=1

n

π2 ) . 6

2

Note that ξk (k + 1) = π6 . Further, we have n

n

n

ζk (β + k + (∑ αi )x)Γk (β + k + (∑ αi )x) ∏ ζk (k + 1 + αi )Γk (k + 1 + αi ) i=1

i=1

n

n

n

i=1

i=1

i=1

i=1

≤ ζk (β + k + ∑ αi )Γk (β + k + ∑ αi ) ∏ ζk (k + 1 + αi x)Γk (k + 1 + αi x)

(2.8.15)

and n

ζk (β + k)Γk (β + k) ∏ ζk (k + 1 + αi x)Γk (k + 1 + αi x) i=1

n

n

i=1

i=1

≤ ζk (β + k + (∑ αi )x)Γk (β + k + (∑ αi )x)(

n

π2 ) . 6

(2.8.16)

Rearranging (2.8.15) and (2.8.16) gives the double inequality (2.8.5). The proof of Theorem 2.8.5 is completed. Substituting k = 1 and αi = 1 (i = 1, . . . , n) into (2.8.13) and taking into account that 2 Γ(3) = 2 and ζ (2) = π6 , we obtain the following corollary.

96 | 2 Schur-convex functions and integral inequalities Corollary 2.8.1. We have (2ζ (3))n ζ (1 + β + n)Γ(1 + β + n) (ζ (2 + x))(Γ(2 + x))n ≤ ζ (1 + β + nx)Γ(1 + β + nx) (ζ (2))n ≤ , ζ (1 + β)Γ(1 + β)

(2.8.17)

where x ∈ [0, 1], β ≥ 1, n ∈ N. Theorem 2.8.6 ([3, 74]). Let a = (a1 , . . . , an ), b = (b1 , . . . , bn ) ∈ ℝn++ satisfy a ≺w b. Then the function n

x→∏ i=1

Γk (x + ai ) Γk (x + bi )

(2.8.18)

is completely monotonic on (0, ∞) (see Definition 1.2.2). Theorem 2.8.7 ([42]). Let a = (a1 , . . . , an ), b = (b1 , . . . , bn ) ∈ ℝn++ satisfy a ≺w b. If f 󸀠󸀠 (x) is completely monotonic on (0, +∞), then the function n

exp{∑(f (x + ai ) − f (x + bi ))} i=1

(2.8.19)

is logarithmically completely monotonic on (0, +∞). Catalan numbers Cn are an important class of natural numbers in combinatorial mathematics. Their formula can be expressed as a gamma function, i. e., Cn =

4n Γ(n + 21 )

√πΓ(n + 2)

.

(2.8.20)

Qi et al. [139] gave a generalization of (2.8.20), i. e., C(a, b; x) =

Γ(b) b Γ(x + a) , Γ(a) a Γ(x + b)

(2.8.21)

where a, b > 0, x ≥ 0. Because C( 21 , 2, x) = Cn , we call C(a, b; x) the Catalan–Qi function (or the generalized Catalan function). Qi et al. [140] studied the Schur-convexity of this function and obtained the following results. Theorem 2.8.8. For a, b > 0, x ≥ 0, let Fx (a, b) = log C(a, b, x). Then for x ≥ 0, Fx (a, b) with regard to (a, b) is Schur-convex on ℝ2++ .

3 Schur-convex functions and mean value inequalities for two variables The mean inequality plays a central role in inequality theory. This chapter discusses the applications of majorization theory to the mean value inequalities for two variables. The mean value inequalities for two variables have been favored by many researchers because of their delicacy and changeableness. For the binary mean, early attention was paid to monotony, log-convexity, geometric convexity, and the comparison inequalities of binary averages. In recent years, attention has been paid to the Schur-convexity, Schur-geometric convexity, and binary convexity of means for two variables.

3.1 Schur-concavity of Stolarsky mean Let (r, s) ∈ ℝ2 , (x, y) ∈ ℝ2++ . The generalized mean (or Stolarsky mean) of (x, y) is defined in [188] and [189] as s

s

−x 1/(s−r) ( sr ⋅ yyr −x , rs(r − s)(x − y) ≠ 0, { r ) { { { { yr −x r 1 1/r { ( r ⋅ log y−log x ) , r(x − y) ≠ 0, { { { { 1 xxr 1/(xr −yr ) E(r, s; x, y) = { 1/r ( yr ) , r(x − y) ≠ 0, e { y { { { { x ≠ y, {√xy, { { { x = y. {x,

(3.1.1)

The Stolarsky means are sometimes called the “difference means,” or the “generalized means” (see [125]). Many binary means are special cases of the Stolarsky mean. For example, x+y = A(x, y) is the arithmetic mean; 2 E(0, 0; x, y) = √xy = G(x, y) is the geometric mean; 2xy = H(x, y) is the harmonic mean; E(−2, −1; x, y) = x+y x−y E(1, 0; x, y) = = L(x, y) is the logarithmic mean; log x − log y 1 x 1− x E(1, 1; x, y) = x x−y y x−y = I(x, y) is the exponential mean; e 2 x2 + xy + y2 ) = g(x, y) is the centroid mean; E(2, 3; x, y) = ( 3 x+y x + √xy + y 1 3 E( , ; x, y) = = h(x, y) is the Heron mean; 2 2 3 E(1, 2; x, y) =

https://doi.org/10.1515/9783110607864-003

98 | 3 Schur-convex functions and mean value inequalities for two variables 1

xp + yp p E(p, 2p; x, y) = ( ) = Mp (x, y) is a power mean (Hölder mean); 2 1

xp − yp p−1 E(1, p; x, y) = [ ] = Sp (x, y) is a generalized logarithmic mean. p(x − y) Stolarsky’s mean is a type of binary mean which is rich in connotation, not only because it contains many important means, but it also has many good properties. Proposition 3.1.1 (Symmetry). We have E(r, s; x, y) = E(s, r; x, y) and E(r, s; x, y) = E(r, s; y, x). Proposition 3.1.2 (Homogeneity). We have E(r, s; λx, λy) = λE(r, s; x, y), λ > 0. Proposition 3.1.3. We have E(r, s; x, y) = [E(−r, −s; x−1 , y−1 )]−1 . Proposition 3.1.4 (Monotonicity). The mean E(r, s; x, y) is increasing with respect to (x, y) on ℝ2++ and (s, r) on ℝ2++ . Proposition 3.1.5 (Logarithmic convexity [51]). For fixed x, y ∈ ℝ++ , (a) if (r, s) ∈ ℝ2++ , E(r, s; x, y) is log-concave for either r or s; (b) if (r, s) ∈ ℝ2−− , E(r, s; x, y) is log-convex for either r or s. Proposition 3.1.6 (Geometric convexity [21]). We have the following: (a) E(r, s; x, y) is geometrically convex with respect to (x, y) on ℝ2++ if and only if s+r ≥ 0; (b) E(r, s; x, y) is geometrically concave with respect to (x, y) on ℝ2++ if and only if s + r ≤ 0. Theorem 3.1.1 (Minkowski type inequality [145]). If s ≠ r, then r

1 ∫ log It d t, s−r

(3.1.2)

1 xt log x + yt log y ). It = exp(− + t xt − yt

(3.1.3)

log E(r, s; x, y) =

s

where

Remark 3.1.1. It is not difficult to verify that E(r, s; x, y) can also be expressed as the following integral form: yr

s 1 E(r, s; x, y) = ( r ∫ t r −1 dt) r y −x

xr

1 s−r

,

r(s − r) ≠ 0

(3.1.4)

3.1 Schur-concavity of Stolarsky mean

| 99

Theorem 3.1.2 (Stolarsky mean comparison theorem [125]). Let x, y ∈ ℝ++ , p, q, r, s ∈ ℝ, (p − q)(r − s) ≠ 0. Then p + q ≤ r + s, E(p, q; x, y) ≤ E(r, s; x, y) ⇔ { m(p, q) ≤ m(r, s),

(3.1.5)

where { u−vu , if min{p, q, r, s} ≥ 0 or max{p, q, r, s} ≤ 0, m(u, v) = { log( v ) |u|−|v| { u−v , if min{p, q, r, s} < 0 < max{p, q, r, s}.

(3.1.6)

Theorem 3.1.3 (Minkowski type inequality [131]). The inequality E(r, s; x1 + x2 , y1 + y2 ) ≤ E(r, s; x1 , y1 ) + E(r, s; x2 , y2 )

(3.1.7)

holds if and only if r + s ≥ 3 and min r, s ≥ 1. When (r, s) ≠ (1, 2), (r, s) ≠ (2, 1), equality occurs if and only if xx1 = yy1 . 2

2

Theorem 3.1.4 ([135]). For a fixed (x, y) ∈ ℝ2++ and x ≠ y, E(r, s; x, y) for (r, s) are Schurconcave on ℝ2+ and Schur-convex on ℝ2− . Sándor [146] gave a simple proof of Theorem 3.1.4. Later, Guo and Qi [52] also gave a simple proof of Theorem 3.1.4. By combining Theorem 3.1.2 and Theorem 3.1.4, Li et al. [84] proved a set of mean single-parameter inequalities. Theorem 3.1.5. For a fixed (x, y) ∈ ℝ2++ and x ≠ y, if p ≤ 0, then I p ≤ M p ≤ h p ≤ Lp ≤ Sp−1 ;

(3.1.8)

Sp−1 ≤ Lp ≤ h p ≤ M p ≤ I p ;

(3.1.9)

Lp ≤ Sp−1 ≤ h p ≤ M p ≤ I p ;

(3.1.10)

Lp ≤ h p ≤ Sp−1 ≤ M p ≤ I p ;

(3.1.11)

Lp ≤ h p ≤ M p ≤ Sp−1 ≤ I p ,

(3.1.12)

2

3

2

if 0 < p ≤ 1, then 2

if 1 < p ≤

4 3

4 3

4, then 2

if

3

3

2

< p ≤ 3, then 2

3

2

100 | 3 Schur-convex functions and mean value inequalities for two variables where 1

xp + yp p Mp = Mp (x, y) = ( ) , 2 Lp = Lp (x, y) = [

(3.1.13) 1

p xp − yp ] , p(log x − log y) p

(3.1.14)

1

p

xp + x 2 y 2 + yp p ) , hp = hp (x, y) = ( 3 1 xp log x − yp log y Ip = Ip (x, y) = exp(− + ), p xp − yp

(3.1.15) (3.1.16)

and 1

xp − yp p−1 ] . Sp = Sp (x, y) = [ p(x − y)

(3.1.17)

Proof. Note that Lp = E(p, 0; x, y), hp = E( 3p , q ; x, y), Mp = E(2p, p; x, y), Ip = 2 2 E(p, p; x, y), Sp = E(p, 1; x, y); in particular √ab = E(0, 0; x, y). We only prove (3.1.8) and the rest can be similarly proved. It is not difficult to prove that when p ≤ 1, we have p p 2p p 3p p ( , ) ≺ ( , ) ≺ ( , ) ≺ (p, 0) ≺ (p − 1, 1), 2 2 3 3 4 4

(3.1.18)

and then for a fixed (x, y) ∈ ℝ2++ and p ∈ ℝ, when p ≤ 0, from Theorem 3.1.4 and (3.1.18), we have p p 2p p 3p p E( , ; x, y) ≤ E( , ; x, y) ≤ E( , ; x, y) ≤ E(p, 0; x, y), 2 2 3 3 4 4 that is, I p ≤ M p ≤ h p ≤ Lp . 2

3

2

On the other hand, because min{p − 1, 1, p, 0} = p − 1 < 0 < max{p − 1, 1, p, 0} = 1, p + 0 ≤ (p − 1) + 1,

and m(p, 0) =

2 |p − 1| − |1| p |p| − |0| = −1 < −1 + = = = m(p − 1, 1), p−0 2−p p−1−1 2−p

by Theorem 3.1.2, it follows that Lp = E(p, 0; x, y) ≤ E(p − 1, 1; x, y) = Sp−1 . At this point, (3.1.8) has been proved.

3.1 Schur-concavity of Stolarsky mean

| 101

For a fixed (r, s), Shi et al. [178] discussed the Schur-convexity of E(r, s; x, y) with respect to (x, y). They pointed out the omissions in [138] and used (3.1.4) to obtain the following theorem. Theorem 3.1.6. For fixed (r, s) ∈ ℝ2 , (1) if 2 < 2r < s or 2 ≤ 2s ≤ r, then E(r, s; x, y) is Schur-convex with (x, y) ∈ (0, ∞) × (0, ∞); (2) if (r, s) ∈ {r < s ≤ 2r, 0 < r ≤ 1} ∪ {s < r ≤ 2s, 0 < s ≤ 1} ∪ {0 < s < r ≤ 1} ∪ {0 < r < s ≤ 1} ∪ {s ≤ 2r < 0} ∪ {r ≤ 2s < 0}, then E(r, s; x, y) is Schur-concave with (x, y) ∈ (0, ∞) × (0, ∞). The following theorems of Chu and Zhang [19] improve the result of Theorem 3.1.6. Theorem 3.1.7. For fixed (r, s) ∈ ℝ2 , (a) E(r; s; x; y) are Schur-convex with respect to (x; y) ∈ ℝ2++ if and only if (r; s) ∈ {s ≥ 1; r ≥ 1; s + r ≥ 3}; (b) E(r; s; x; y) are Schur-concave with respect to (x; y) ∈ ℝ2++ if and only if (r; s) ∈ {r ≤ 1; s + r ≤ 3} ∪ {s ≤ 1; s + r ≤ 3}. Chu et al. [20] investigated the Schur-geometric convexity of E(r, s; x, y) and obtained the following. Theorem 3.1.8. For fixed (r, s) ∈ ℝ2 , (a) E(r; s; x; y) are Schur-geometrically convex with respect to (x; y) ∈ ℝ2++ if and only if s + r ≥ 0; (b) E(r; s; x; y) are Schur-geometrically concave with respect to (x; y) ∈ ℝ2++ if and only if s + r ≤ 0. Xia et al. [237] investigated the Schur-harmonic convexity of E(r, s; x, y) and obtained the following. Theorem 3.1.9. For fixed (r, s) ∈ ℝ2 , (a) E(r; s; x; y) are Schur-harmonically convex with respect to (x; y) ∈ ℝ2++ if and only if (r; s) ∈ {s ≥ −1, s ≥ r, r + s + 3 ≥ 0} ∪ {r ≥ −1, r ≥ s, s + r + 3 ≥ 0}; (b) E(r; s; x; y) are Schur-harmonically concave with respect to (x; y) ∈ ℝ2++ if and only if (r, s) ∈ {s ≤ −1, r ≤ −1, r + s + 3 ≤ 0}. In 2012, Yang [260] investigated the Schur-power convexity of E(r, s; x, y) and obtained the following. Theorem 3.1.10. For fixed (r, s) ∈ ℝ2 , (a) if m > 0, E(r, s; x, y) are Schur-m-power convex (or Schur-m-power concave, respectively) with respect to (x, y) on ℝ2++ if and only if r+s ≥ 3m (or r+s ≤ 3m, respectively) and min{r, s} ≥ m (or min{r, s} ≤ m, respectively);

102 | 3 Schur-convex functions and mean value inequalities for two variables (b) if m < 0, E(r, s; x, y) are Schur-m-power convex (or Schur-m-power concave, respectively) with respect to (x, y) on ℝ2++ if and only if r+s ≥ 3m (or r+s ≤ 3m, respectively) and max{r, s} ≥ m (or max{r, s} ≤ m, respectively); (c) if m = 0, E(r, s; x, y) are Schur-m-power convex (or Schur-m-power concave, respectively) with respect to (x, y) on ℝ2++ if and only if r + s ≥ 0 (or r + s ≤ 0, respectively). Two applications are given at the end of this section. Example 3.1.1. For (x, y) ∈ ℝ2++ , x ≠ y, Kuang’s interpolation inequality [76] is as follows: H(x, y) < G(x, y) < Q 1 (x, y) < L(x, y) < M 1 (x, y) < M 1 (x, y) 3

3

2

(3.1.19)

< h(x, y) < M 2 (x, y) < I(x, y) < A(x, y) < g(x, y) < M2 (x, y), 3

where Q 1 = ar bs + as br , 3

1 1 ), r = (1 + √3 2

1 1 s = (1 − ). √3 2

If removing Q 1 (x, y) in (3.1.19), we can use the symbol of the generalized mean 3 E(r, s; x, y) to write 1 2 E(−2, −1; x, y) < E(0, 0; x, y) < E(1, 0; x, y) < E( , ; x, y) 3 3 1 3 2 4 1 < E( , 1; x, y) < E( , ; x, y) < E( , ; x, y) 2 2 2 3 3

(3.1.20)

< E(1, 1; x, y) < E(1, 2; x, y) < E(2, 3; x, y) < E(2, 4; x, y).

Note that 1 2 1 (−2, −1) < (0, 0) < (1, 0) ≻≻ ( , ) < ( , 1) 3 3 2 1 3 2 4 < ( , ) ≻≻ ( , ) ≻≻ (1, 1) < (1, 2) < (2, 3) < (2, 4) 2 2 3 3

(3.1.21)

and combining the strict monotonicity of E(r, s; x, y) on ℝ2 and the strict Schurconvexity on ℝ2++ with respect to the parameters (r, s) yields (3.1.20). This example shows the characteristics of the “mass production” inequality of the majorization method, that is, “to easily derive many existing inequalities from different methods in a unified way.” Example 3.1.2. For x ≠ y, Guo and Qi [53] used analytical methods to verify inequalities log

ex + ey ex − ey (x − 1)ex − (y − 1)ey < . < log x y x−y e −e 2

(3.1.22)

3.2 Schur-concavity of Gini mean |

103

Shi and Wu [177] gave a simple proof based on the strict monotonicity of E(r, s; x, y) with respect to the parameters (r, s) (see [269]). Let ex = u and ey = v. Then inequality (3.1.22) becomes log

u−v (log u − 1)u − (log v − 1)v u+v < < log , log u − log v u−v 2

(3.1.23)

that is, E(1, 0; u, v) < E(1, 1; u, v) < E(1, 2; u, v).

(3.1.24)

By the strict monotonicity of E(r, s; x, y) with respect to the parameters (r, s), the inequalities in (3.1.24) hold. This example can also be considered by considering the Schur-convexity of the following two difference functions: log

ex − ey (x − 1)ex − (y − 1)ey − x−y ex − ey

log

ex + ey (x − 1)ex − (y − 1)ey − . 2 ex − ey

and

This is omitted here.

3.2 Schur-concavity of Gini mean Let (r, s) ∈ ℝ2 , (x, y) ∈ ℝ2++ . This section describes another type of binary mean: the Gini mean, which is rich in connotation and reads as follows: s

s

{( x r +yr )1/(s−r) , G(r, s; x, y) = { x +y xs log x+ys log y exp( ), xr +yr {

r ≠ s,

r = s.

(3.2.1)

The Gini means are also called the “sum means.” The Gini mean also contains many important means. For example, G(0, −1; x, y) is the harmonic mean, G(0, 0; x, y) is the geometric mean, G(1, 0; x, y) is the arithmetic mean, and G(p − 1, p; x, y) is the Lehmer mean, i. e., Lp (a, b) =

ap + bp , + bp−1

ap−1

−∞ ≤ p ≤ +∞.

The Gini mean has similar properties to the Stolarsky mean.

(3.2.2)

104 | 3 Schur-convex functions and mean value inequalities for two variables Theorem 3.2.1 ([145]). If s ≠ r, then s

log G(r, s; x, y) =

1 ∫ log Jt d t, s−r

(3.2.3)

r

where Jt = exp(

xt log x + yt log y ). xt + yt

Theorem 3.2.2 (Gini mean comparison theorem [126]). Let x, y ∈ ℝ++ , p, q, r, s ∈ ℝ, (p − q)(r − s) ≠ 0. Then p + q ≤ r + s, G(p, q; x, y) ≤ G(r, s; x, y) ⇔ { m(p, q) ≤ m(r, s),

(3.2.4)

where min{u, v}, if min{p, q, r, s} ≥ 0, { { { |u|−|v| m(u, v) = { u−v , if min{p, q, r, s} < 0 < max{p, q, r, s}, { { {max{u, v}, if max{p, q, r, s} ≤ 0.

(3.2.5)

Example 3.2.1. Yang [252] Conjecture: Let x, y ≥ 0, n ∈ ℕ, n > 1. Then 2(xn + yn )

n+1

≥ (x n+1 + yn+1 )(xn−1 + yn−1 )

n+1

,

(3.2.6)

where equality occurs if and only if x = y. He [58] proved this conjecture using a certain skill and combined with software calculations, but the proof process has a large amount of calculations and it is a semimanual proof. Using the derivative, Li [87] proved that when x, y ≥ 0, n ∈ ℝ, n = 2 or n ≥ 3, inequality (3.2.6) holds, and when 2 < n < 3, it does not. The author [162] used Pales’ Gini mean comparison theorem to give a simple proof of this conjecture for the generalized case of x, y ≥ 0, n ∈ ℝ, n ≥ 2. Proof. If xy = 0, it is easy to see that inequality (3.2.6) holds. Now suppose that x > 0, y > 0. Then inequality (3.2.6) is equivalent to 1

G(n, n − 1; x, y) =

xn + yn xn+1 + yn+1 n+1 ≥( ) = G(n + 1, 0; x, y). n−1 n−1 2 x +y

(3.2.7)

Since when n ∈ ℝ, n ≥ 2, n + 1 + 0 ≤ n + n − 1, we have min{n + 1, 0, n, n − 1} ≥ 0 and min{n + 1, 0} ≤ min{n, n − 1}, inequality (3.2.7) holds by the Gini mean comparison theorem, thus inequality (3.2.6) is proved.

3.2 Schur-concavity of Gini mean |

105

Theorem 3.2.3 (Minkowski type inequality [100]). We have G(r, s; x1 + x2 , y1 + y2 ) ≤ G(r, s; x1 , y1 ) + G(r, s; x2 , y2 )

(3.2.8)

if and only if r + s ≥ 1 and 0 ≤ min{r, s} ≤ 1. Theorem 3.2.4 ([146]). For a fixed (x, y) ∈ ℝ2++ and x ≠ y, G(r, s; x, y) with respect to (r, s) are Schur-concave on ℝ2+ and Schur-convex on ℝ2− . Proof. Jt is log-concave for t > 0 and log-convex for t < 0 (see [146]). This conclusion is combined with Theorem 3.2.1 and Theorem 2.1.2, and Theorem 3.2.4 is verified. Using Theorem 3.2.4 and Theorem 3.2.2, Li et al. [85] proved the following theorem. Theorem 3.2.5. For a fixed (x, y) ∈ ℝ2++ and x ≠ y, if p ≥ 1, then Mp ≤ L p−1 ≤ J p ;

(3.2.9)

J p ≤ L p−1 ≤ Mp ;

(3.2.10)

J p ≤ Mp ≤ L p−1 ;

(3.2.11)

L p−1 ≤ Mp ≤ J p ;

(3.2.12)

M0 = L− 1 = J0 = √xy,

(3.2.13)

2

2

if p ≤ −1, then 2

2

if −1 < p < 0, then 2

2

if 0 < p < 1, then 2

2

if p = 0, then 2

where 1

p xp − yp Lp = Lp (x, y) = [ ] p(log x − log y)

(3.2.14)

and 1

xp + yp p Mp = Mp (x, y) = ( ) . 2

(3.2.15)

We write Jp = Jp (x, y) = exp(

xp log x + yp log y ). xp + yp

(3.2.16)

106 | 3 Schur-convex functions and mean value inequalities for two variables Proof. Note that Mp = G(p, 0; x, y), Lp = (p+1, p; x, y), Jp = (p, p; x, y), √xy = G(0, 0; x, y). We only prove (3.2.11); the rest can be similarly proved. It is not difficult to prove that when −1 < p < 1, we have p p p−1 p+1 ( , ) ≺ (p, 0) ≺ ( , ). 2 2 2 2

(3.2.17)

By Theorem 3.2.4, from (3.2.17), it follows that p p J p = G( , ; x, y) ≤ G(p, 0; x, y) = Mp . 2 2 2 On the other hand, because m(p, 0) =

p−1 2

1, (a) g(t, z) is increasing on (−∞, 0) with t; (b) g(t, z) is increasing on (0, ξz ) with t; (c) g(t) is decreasing on (ξz , 1) or (1, +∞) with t, where ξz is a zero of the function g1 (t, z) = t(z t + z t−1 ) ln z + (z t + 1)(z t−1 − 1) with 0 < ξz < 1/2. Proof. Differentiate g(t, z) with respect to t to obtain 𝜕g(t, z) tz t (z t−1 − 1) log z − (z t + 1)(z t−1 − 1) − tz t−1 (z t + 1) log z = 𝜕t t 2 (z t−1 − 1)2 g (t, z) = − 2 1t−1 . t (z − 1)2 For fixed z > 1, g1 (t, z) < 0, and

𝜕g(t,z) 𝜕t

> 0, g(t, z) increases on (−∞, 0) with t, and

< 0, g(t, z) decreases on (1, +∞) with t. for g1 (t, z) > 0 and Differentiate g1 (t, z) with respect to t to obtain 𝜕g(t,z) 𝜕t

𝜕g1 (t, z) = [2z 2t−1 + 2z t−1 + t(z t + z t−1 ) log z] log z. 𝜕t Since 𝜕g1𝜕t(t,z) > 0 on (0, 1), g1 (t, z) increases on (0, 1). It follows that g1 (0, z) ≤ g1 (t, z) ≤ g1 (1, z). Furthermore, g1 (0, z) = 2(z −1 − 1) < 0 and g1 (1, z) = (z + 1) log z > 0, hence there ≥ 0 for 0 < t ≤ ξz , and exist ξz ∈ (0, 1) such that g1 (ξz , z) = 0, and g1 (t, z) ≤ 0 and 𝜕g(t,z) 𝜕t

g1 (t, z) > 0 and 𝜕g(t,z) < 0 for ξz < t < 1, so g(t, z) increases on (0, ξz ) and decreases on 𝜕t (ξz , 1). Differentiate g1 (t, z) with respect to z to obtain 𝜕g1 (t, z) = tz t−1 (z t−1 − 1) + (t − 1)z t−2 (z t + 1) 𝜕z + t(z t−1 + z t−2 ) + t[tz t−1 + (t − 1)z t−2 ] log z

= (2t − 1)z 2t−2 + t 2 z t−1 log z + (2t − 1)z t−2 + (t 2 − t)z t−2 log z.

For 1 > t ≥ 1/2, we have 𝜕g1 (t, z) ≥ t 2 z t−1 log z + (2t − 1)z t−2 + (t 2 − t)z t−2 log z 𝜕z = (t 2 z + t 2 − t)z t−2 log z > (2t 2 − t)z t−2 log z = t(2t − 1)z t−2 log z ≥ 0.

108 | 3 Schur-convex functions and mean value inequalities for two variables Hence, for 1 > t ≥ 1/2, g1 (t, z) increases on (1, +∞) with z, and then g1 (t, z) > lim+ g1 (t, z) = g1 (t, 1) = 0. z→1

Thus we conclude that 0 < ξz < 1/2. Lemma 3.2.3. For fixed (x, y) with x > y > 0, if (r, s) ∈ {r > 1, s < 0, r + s ≤ 1} ∪ {1 < r ≤ s} ∪ {0 < r ≤ 1 − r ≤ s < 1} ∪ {1/2 ≤ r ≤ s < 1}, then s(xr + yr )(xs−1 − ys−1 ) ≥ r(x s + ys )(xr−1 − yr−1 ).

(3.2.18)

If (r, s) ∈ {s > 1, r < 0, r + s ≤ 1} ∪ {r ≤ s < 0}, then (3.2.18) is reversed. t

Proof. Let g(t) = t(zzt−1+1−1) with z = x/y > 1. Note that y > 0. It is easy to see that (3.2.18) is equivalent to g(r) ≥ g(s). For r > 1, we first prove that g(r) ≥ g(1 − r), i. e., y(z r + 1) y(z 1−r + 1) y(z r + z) ≥ = . −r r−1 r(z − 1) (1 − r)(z − 1) (r − 1)(z r − 1) It is sufficient prove that h(z) := (r − 1)(z r − 1)(z r + 1) − r(z r−1 − 1)(z r + z) ≥ 0. A direct calculation yields h(z) = (r − 1)z 2r − rx 2r−1 + rx − r + 1,

h󸀠 (z) = 2r(r − 1)z 2r−1 − r(2r − 1)z 2r−2 + r, h󸀠󸀠 (z) = 2r(r − 1)(2r − 1)z 2r−3 (z − 1).

By r > 1 and z > 1, it follows that h󸀠󸀠 (z) > 0. Therefore, h󸀠 (z) > h󸀠 (1) = 0; moreover, h(z) > h(1) = 0, that is, g(r) ≥ g(1 − r). If r > 1, s < 0, r + s ≤ 1, then s ≤ 1 − r < 0, and from Lemma 3.2.2(a), we have g(r) ≥ g(s), that is, (3.2.18) holds. If s > 1, r < 0, r + s ≤ 1, replacing r by s and replacing s by r in the above case, it follows that g(r) ≤ g(s), i. e., (3.2.18) is reversed. If 0 < r ≤ 1/2 ≤ 1 − r ≤ s < 1, then h󸀠󸀠 (z) > 0. It follows that h󸀠 (z) > h󸀠 (1) = 0; moreover, h(z) > h(1) = 0, i. e., g(r) ≥ g(1 − r). From Lemma 3.2.2(c), we have g(r) ≥ g(1 − r) ≥ g(s), i. e., (3.2.18) holds. If 1/2 ≤ r ≤ s < 1 or 1 < r ≤ s, from Lemma 3.2.2(c), we have g(r) ≥ g(s), so (3.2.18) holds. If r ≤ s < 0, from Lemma 3.2.2(a), we have g(r) ≤ g(s), i. e., (3.2.18) is reversed. Proof of Theorem 3.2.6. Let φ(x, y) =

xs +ys . xr +yr

When r ≠ s, for fixed (x, y) ∈ ℝ2 , we have

𝜕φ sx s−1 (xr + yr ) − rx r−1 (xs + ys ) , = 𝜕x (xr + yr )2

3.2 Schur-concavity of Gini mean |

109

𝜕φ sys−1 (xr + yr ) − ryr−1 (xs + ys ) = , 𝜕y (xr + yr )2 and 𝜕φ 𝜕φ s(xr + yr )(xs−1 − ys−1 ) − r(xs + ys )(xr−1 − yr−1 ) − = 𝜕x 𝜕y (xr + yr )2 = =

s(xr−1 − yr−1 ) s − 1 (r − 1)(xs−1 − ys−1 ) r xs + ys − ⋅ [ ⋅ ] (xr + yr ) r − 1 (s − 1)(xr−1 − yr−1 ) s xr + yr

r s(xr−1 − yr−1 ) s − 1 s−r [ ⋅ E (r − 1, s − 1; x, y) − ⋅ Gs−r (r, s; x, y)], (xr + yr ) r−1 s

and then 𝜕G 𝜕G − ) 𝜕x 𝜕y 1 x − y 𝜕φ 𝜕φ = ( − )φ s−r −1 (x, y) s − r 𝜕x 𝜕y

Δ := (x − y)(

=

s(x − y)(xr−1 − yr−1 ) (s − r)(xr + yr )

×[

1 s − 1 s−r r ⋅ E (r − 1, s − 1; x, y) − ⋅ Gs−r (r, s; x, y)]φ s−r −1 (x, y). r−1 s

In Lemma 3.2.1, taking l = r, t = s, p = r − 1, q = s − 1, we have r > 1, s > 1 { { { { { {r > s ⇔{ { {r + s ≥ −1 { { { {s/3 ≤ r ≤ 3s

l > 0, t > 0, p > 0, q > 0 { { { { { {p > q { { {p + q ≤ 3(l + t) { { { {1/3 ≤ l/t ≤ 3

⇔ 3s ≥ r > s > 1

and l > 0, t > 0, p > 0, q > 0 { { { { { {p > q { { p + q ≤ 3(l + t) { { { { {q ≤ l + t

r > 1, s > 1 { { { { { {r > s ⇔{ {r + s ≥ −1 { { { { {r ≥ −1

⇔ r > s > 1.

Hence, when r > s > 1, we have 1

r − 1 r−s ) E(r − 1, s − 1; x, y), G(r, s; x, y) ≤ ( s−1 i. e., Gs−r (r, s; x, y) ≥

s − 1 s−r ⋅ E (r − 1, s − 1; x, y). r−1

(3.2.19)

110 | 3 Schur-convex functions and mean value inequalities for two variables When r > s > 1, we have s − r < 0 and (x − y)(xr−1 − yr−1 ) ≥ 0. It follows that Δ ≥ 0. By Theorem 2.1.3 (Vol. 1), G(r, s; x, y) is Schur-convex with (x, y) ∈ ℝ2++ . Now we consider other cases. Note that (x − y)( =

𝜕φ 𝜕φ − ) 𝜕x 𝜕y

s(x r + yr )(x − y)(xs−1 − ys−1 ) − r(xs + ys )(x − y)(xr−1 − yr−1 ) , (xr + yr )2

where r ≥ 1, 0 ≤ s ≤ 1. Since t r−1 and t s−1 are increasing and decreasing in ℝ+ , respectively, it follows that (x − y)(xs−1 − ys−1 ) ≥ 0 and (x − y)(xr−1 − yr−1 ) ≤ 0. Moreover, (x − y)( 𝜕x − 𝜕φ

𝜕φ ) 𝜕y

≤ 0 and

Δ=

1 x − y 𝜕φ 𝜕φ ( − )φ s−r −1 (x, y) ≥ 0. s − r 𝜕x 𝜕y

That is, when r ≥ 1, 0 ≤ s ≤ 1, G(r, s; x, y) is Schur-convex with (x, y) ∈ ℝ2+ . When r < 0, 0 < s ≤ 1, since t r−1 and t s−1 are decreasing in ℝ++ , it follows that 𝜕φ 𝜕φ (x − y)(xs−1 − ys−1 ) ≤ 0 and (x − y)(xr−1 − yr−1 ) ≤ 0. Moreover, (x − y)( 𝜕x − 𝜕y ) ≤ 0 and

Δ ≤ 0, that is, when r < 0, 0 < s ≤ 1, G(r, s; x, y) is Schur-concave with (x, y) ∈ ℝ2+ . Without loss of generality, we may assume x > y > 0. Note that Δ=

x − y s(x r + yr )(xs−1 − ys−1 ) − r(xs + ys )(xr−1 − yr−1 ) s−r1 −1 φ ⋅ (x, y). s−r (xr + yr )2

When r > 1, s < 0, r + s ≤ 1, from Lemma 3.2.3, it follows that Δ ≤ 0, so G(r, s; x, y) is Schur-concave with (x, y) ∈ ℝ2+ . Similarly, we can prove that when r ≤ s < 0, G(r, s; x, y) is Schur-concave with (x, y) ∈ ℝ2+ , and when 0 < r ≤ 1 − r ≤ s or 1/2 ≤ r ≤ s < 1, G(r, s; x, y) is Schur-convex with (x, y) ∈ ℝ2+ . When r = s ≥ 1, let ψ(x, y) =

xs log x + ys log y xs log x + ys log y = . xr + yr xs + ys

Then 𝜕ψ xs−1 h(x, y) , = s 𝜕x (x + ys )2

𝜕ψ ys−1 k(x, y) , = s 𝜕y (x + ys )2

3.2 Schur-concavity of Gini mean

| 111

where h(x, y) = (s log x + 1)(x s + ys ) − s(xs log x + ys log y), k(x, y) = (s log y + 1)(x s + ys ) − s(xs log x + ys log y).

By computation, xs−1 h(x, y) − ys−1 k(x, y)

= (x s + ys )[x s−1 (s log x + 1) − ys−1 (s log y + 1)] − s(xs log x + ys log y)(xs−1 − ys−1 )

= ss−1 ys−1 (x + y)(log x − log y) + (x s−1 − ys−1 )(xs + ys ), and then, (x − y)( =

𝜕ψ 𝜕ψ ψ(x,y) 𝜕G 𝜕G − ) = (x − y)( − )e 𝜕x 𝜕y 𝜕x 𝜕y

sxs−1 ys−1 (x + y)(x − y)(log x − log y) + (x − y)(xs−1 − ys−1 )(xs + ys ) ψ(x,y) e . (xs + ys )2

Since log t and t s−1 are increasing in ℝ+ with t for s ≥ 1, we have (x − y)(log x − log y) ≥ 0 and (x − y)(xs−1 − ys−1 ) ≥ 0. Moreover, (x − y)( 𝜕G − 𝜕G ) ≥ 0. That is, when 𝜕x 𝜕y

r = s ≥ 1, G(r, s; x, y) is Schur-convex with (x, y) ∈ ℝ2+ . In conclusion, if (r, s) ∈ {r > s > 1} ∪ {r = s ≥ 1} ∪ {r ≥ 1, 0 ≤ s ≤ 1} ∪ {0 < r ≤ 1 − r ≤ s} ∪ {1/2 ≤ r ≤ s < 1}, then G(r, s; x, y) is Schur-convex with (x, y) ∈ ℝ2+ , and if (r, s) ∈ {r < 0, 0 < s ≤ 1} ∪ {r > 1, s < 0, r + s ≤ 1} ∪ {r ≤ s < 0}, then G(r, s; x, y) is Schur-concave with (x, y) ∈ ℝ2+ . Since G(r, s; x, y) is symmetric with (r, s), if (r, s) ∈ {s > r > 1} ∪ {s ≥ 1, 0 ≤ r ≤ 1} ∪ {0 < s ≤ 1 − s ≤ r} ∪ {1/2 ≤ s ≤ r < 1}, then G(r, s; x, y) is also Schur-convex with (x, y) ∈ ℝ2+ , and if (r, s) ∈ {s < 0, 0 < r ≤ 1} ∪ {s > 1, r < 0, r + s ≤ 1} ∪ {s ≤ r < 0}, then G(r, s; x, y) is also Schur-concave with (x, y) ∈ ℝ2+ . The proof is complete.

Remark 3.2.1. The Schur-convexity of the function G(r, s; x, y) on the set {s < 0, r + s > 1} or {r < 0, r + s > 1} or {r > 0, s > 0, r + s < 1} with (x, y) is uncertain. Example 1. Let (r, s) = (2.5, −1.2). It is clear that (2.5, −1.2) ∈ {s < 0, r + s > 1}. For (3, 3) ≺ (5, 1), a direct calculation yields G(2.5, −1.2; 3, 3) = 3.000000000 > G(2.5, −1.2; 5, 1) = 2.873884533. But, for (1.25, 1.25) ≺ (1.5, 1), a direct calculation yields G(2.5, −1.2; 1.25, 1.25) = 1.25.0000000 < G(2.5, −1.2; 1.5, 1) = 1.256253447.

112 | 3 Schur-convex functions and mean value inequalities for two variables Example 2. Let (r, s) = (−0.2, 1.5). It is clear that (−0.2, 1.5) ∈ {r < 0, r + s > 1}. For (8, 8) ≺ (15, 1), a direct calculation yields G(−0.2, 1.5; 8, 8) = 8.000000000 < G(−0.2, 1.5; 15, 1) = 8.412747770. But, for (25.5, 25.5) ≺ (50, 1), a direct calculation yields G(−0.2, 1.5; 25.5, 25.5) = 25.5.0000000 > G(−0.2, 1.5; 50, 1) = 25.32833093. Example 3. Let (r, s) = (0.6, 0.2). It is clear that (0.6, 0.2) ∈ {r > 0, s > 0, r + s < 1}. For (10.5, 10.5) ≺ (20.9, 0.1), a direct calculation yields G(0.6, 0.2; 10.5, 10.5) = 10.5.0000000 < G(0.6, 0.2; 20.9, 0.1) = 11.03249418. But, for (10.5, 10.5) ≺ (18, 3), a direct calculation yields G(0.6, 0.2; 10.5, 10.5) = 10.50000000 > G(0.6, 0.2; 18, 3) = 9.970045812. Witkowski [215] proved this fact using Theorem 2.1.6 (Vol. 1). Wang [213] investigated the Schur-geometric convexity of G(r, s; x, y) and obtained the following results. Theorem 3.2.7. For fixed (r, s) ∈ ℝ2 , (a) G(r, s; x, y) is Schur-geometrically convex for (x, y) on ℝ2++ if and only if r + s ≥ 0; (b) G(r, s; x, y) is Schur-geometrically concave for (x, y) on ℝ2++ if and only if r + s ≤ 0. Remark 3.2.2. The Schur-geometric convexity of the Gini mean and the Stolarsky mean are exactly the same. Xia and Chu [235] discussed the Schur-harmonic convexity of the Gini mean and obtained the following conclusions. Theorem 3.2.8. For fixed (r, s) ∈ ℝ2 , (a) G(r, s; x, y) is Schur-harmonically convex for (x, y) on ℝ2++ if and only if (r, s) ∈ {(r, s) | r ≥ 0, r ≥ s, r + s + 1 ≥ 0} ∪ {(r, s) | s ≥ 0, s ≥ r, r + s + 1 ≥ 0}; (b) G(r, s; x, y) is Schur-harmonically concave for (x, y) on ℝ2++ if and only if (r, s) ∈ {(r, s) | r ≤ 0, s ≤ 0, r + s + 1 ≤ 0}. Note the Lehmer mean Lp (x, y) = G(p − 1, p; x, y); the following corollaries can be obtained from Theorem 3.2.4, Theorem 3.2.7, and Theorem 3.2.8. Corollary 3.2.1. For fixed (r, s) ∈ ℝ2 , (a) [44] when p ≥ 1 (or p ≤ 1, respectively), Lp (x, y) is Schur-convex (or Schur-concave, respectively) with respect to (x, y) on ℝ2++ ;

3.3 Comparison of Stolarsky and Gini means | 113

(b) [44] when p ≥ 21 (or p ≤ 21 , respectively), Lp (x, y) is Schur-geometrically convex (or Schur-geometrically concave, respectively) with respect to (x, y) on ℝ2++ ; (c) [231] when p ≥ 0 (or p ≤ 0, respectively), Lp (x, y) is Schur-harmonically convex (or Schur-harmonically concave, respectively) with respect to (x, y) on ℝ2++ . Remark 3.2.3. Yang [259] gave another proof for Schur-harmonic convexity of the Gini mean. In 2013, Yang [261] studied the Schur-power convexity of the Gini mean and obtained the following theorem. Theorem 3.2.9. For fixed (r, s) ∈ ℝ2 , (a) if m > 0, then G(r, s; x, y) is m-order Schur-power convex (or Schur-power concave, respectively) with respect to (x, y) on ℝ2++ if and only if r + s ≥ m (or r + s ≤ m, respectively) and min{r, s} ≥ 0 (or min{r, s} ≤ 0, respectively); (b) if m < 0, then G(r, s; x, y) is m-order Schur-power convex (or Schur-power concave, respectively) with respect to (x, y) on ℝ2++ if and only if r + s ≥ m (or r + s ≤ m, respectively) and max{r, s} ≥ 0 (or max{r, s} ≤ 0, respectively); (c) if m = 0, then G(r, s; x, y) is m-order Schur-power convex (or Schur-power concave, respectively) with respect to (x, y) on ℝ2++ if and only if r + s ≥ 0 (or r + s ≤ 0, respectively).

3.3 Comparison of Stolarsky and Gini means The Stolarsky mean and the Gini mean are two relatively independent and important binary two-parameter means. This section discusses the relationship between the two. First, we introduce the comparison theorems between the two. Theorem 3.3.1 ([24]). Let x, y ∈ ℝ2++ and x ≠ y, r, s ∈ ℝ. Then E(r, s; x, y) ≤ G(r, s; x, y),

(3.3.1)

provided r + s > 0. Inequality (3.3.1) is reversed if r + s < 0 and it becomes an equality if and only if r + s = 0. Theorem 3.3.2 ([119]). Let r, s ∈ ℝ, r ≠ s. In order for the inequality E(r, s; x, y) ≤ G(r − 1, s − 1; x, y)

(3.3.2)

to be valid for all x, y > 0, it is necessary that r + s ≥ 3 and

min{r, s} ≥ 1.

(3.3.3)

Conversely, if max{3, log 2 ⋅ L(r, s) + 2} ≤ r + s

and

min{r, s} ≥ 1,

(3.3.4)

114 | 3 Schur-convex functions and mean value inequalities for two variables then (3.3.4) holds for all positive x, y. Inequality (3.3.4) becomes an equality if and only if either (a, b) ∈ {(2, 1), (1, 2)} or (a, b) ∉ {(2, 1), (1, 2)} and x = y, where r−s

L(r, s) = { log r−log s s,

, r ≠ s, r=s

is the logarithmic mean of r and s. Theorem 3.3.3 ([119]). Let r, s ∈ ℝ, r ≠ s. In order for the inequality E(r, s; x, y) ≥ G(r − 1, s − 1; x, y)

(3.3.5)

to be valid for all x, y > 0, it is necessary that r + s ≤ 3 and

min{r, s} ≤ 1.

(3.3.6)

Conversely, if r+s≤2

(3.3.7)

or and 0 ≤ min{r, s} ≤ 1,

r + s ≤ min{3, log 2 ⋅ L(r, s) + 2}

(3.3.8)

then (3.3.5) holds for all positive x, y. Inequality (3.3.5) becomes an equality if and only if either (a, b) ∈ {(2, 1), (1, 2)} or (a, b) ∉ {(2, 1), (1, 2)} and x = y. Theorem 3.3.4 ([119]). Let r ∈ ℝ. Then E(r, r; x, y) < G(r − 1, r − 1; x, y)

(3.3.9)

holds for all distinct positive x and y if and only if r ≥ 32 . The reversed inequality is valid if and only if r ≤ 1. Jiang and Shi [73] established a new result about the comparison of E(r − 1, s − 1; x, y) and G(r, s; x, y), namely, the following theorem. Theorem 3.3.5. Let x > y > 0. (a) If (r, s) ∈ {r > 1, s < 0, r + s ≤ 1} ∪ {0 < r ≤ 1 − r ≤ s < 1} ∪ {s ≤ r < 0} ∪ { 21 ≤ r ≤ s < 1}, then 1

r(r − 1) r−s G(r, s; x, y) ≥ [ ] ⋅ E(r − 1, s − 1; x, y). s(s − 1)

(3.3.10)

(b) If (r, s) ∈ {s > 1, r < 0, r + s ≤ 1} ∪ {0 < s ≤ 1 − s ≤ r < 1} ∪ { 31 ≤ s ≤ r < 1} ∪ {r ≤ s < 0} ∪ {r > 1, s > 1}, then inequality (3.3.10) is reversed. (c) If (r, s) ∈ {sr < 0, r + s > 1} ∪ {s > 0, r > 0, r + s < 1}, then the result is undefined.

3.3 Comparison of Stolarsky and Gini means | 115

To prove Theorem 3.3.5, the following lemma is used. Lemma 3.3.1. Let x > 1, r, k ∈ ℝ, (r − 1)(r + k) ≠ 0, m = min{−k, 1}, M = max{−k, 1}. Then for the function g(r, x) =

xr + 1 , (r + k)(xr−1 − 1)

(a) when r < m or m < r ≤ ξ , g(r, x) is increasing with respect to r, (b) when r > M or ξ < r < M, g(r, x) is decreasing with respect to r, where ξ is the unique real root in the interval (m, M)(k ≠ −1) of the equation with respect to r (xr + 1)(xr−1 − 1) + (r + k)(xr + xr−1 ) log x = 0. ), when k > 1, ξ ∈ (−k, 0), and when k < −1, ξ ∈ (1, 1−k ). And when −1 < k ≤ 1, ξ ∈ (−k, 1−k 2 2 For (r, s), (u, v) ∈ ℝ2 , (x, y) ∈ ℝ2++ , Witkowski [216] defined the following fourparameter binary mean: u

u

1

u ≠ v, [ E(r,s;x v ,yv ) ] u−v , R(u, v; r, s; x, y) = { E(r,s;xd ,y ) u u exp{ du log E(r, s; x , y )}, u = v.

(3.3.11)

The R-mean contains many important means. In particular, it includes the Stolarsky mean and the Gini mean. It is easy to see that R(1, 0; r, s; x, y) = E(r, s; x, y) is the Stolarsky mean, R(2, 1; r, s; x, y) = G(r, s; x, y) is Gini mean, R( 32 , 21 ; r, s; x, y) is Heron mean, and 1

xn + xn−1 y + ⋅ ⋅ ⋅ + xyn−1 + yn n ) . R(1, n + 1; 0, 1; x, y) = ( n+1 For the R-mean, the following comparison theorem is established in [216]. Theorem 3.3.6 (Comparison theorem for R-means). In the case u ≠ v, the inequality R(u, v; r, s; x, y) ≤ R(u, v; p, q; x, y) holds for all x, y > 0 if and only if either u+v =0 or u + v > 0, r + s ≤ p + q

and

w(r, s) ≤ w(p, q)

(3.3.12)

116 | 3 Schur-convex functions and mean value inequalities for two variables or u + v < 0, r + s ≥ p + q

and

w(r, s) ≥ w(p, q),

where e(r, s), if uv = 0, w(r, s) = { m(r, s), if uv ≠ 0,

(3.3.13)

{ u−vu , if uv = 0, e(u, v) = { log( v ) {m(r, s), if uv ≠ 0.

(3.3.14)

The expression for e(u, v) and m(r, s) is shown in (3.1.6) and (3.2.5), respectively. Theorem 3.3.7 ([217]). The function R(u, v; r, s; x, y) is Schur-geometrically convex (or Schur-geometrically concave, respectively) with respect to (x, y) on ℝ2++ if and only if (u + v)(r + s) ≥ 0 (or (u + v)(r + s) ≤ 0, respectively). Remark 3.3.1. According to Theorem 3.3.7, R(1, 0; r, s; x, y) = E(r, s; x, y) and R(2, 1; r, s; x, y) = G(r, s; x, y) give Theorem 3.1.10 and Theorem 3.2.7, respectively. Problem 3.3.1. How Schur-convexity and Schur-harmonic convexity of R(u, v; r, s, x, y) with respect to (x, y) on ℝ2++ ? Problem 3.3.2. How Schur-convexity and Schur-harmonic convexity of R(u, v; r, s, x, y) with respect to (u, v) or (r, s) on ℝ2++ ? In 2011, Witkowski [217] proved the following theorem. Theorem 3.3.8. For any positive number x, y, if r + s ≥ 0 (or r + s ≤ 0, respectively), then R(u, v, r, s; x, y) is Schur-concave (or Schur-convex, respectively) with respect to (u, v) on ℝ2+ , and it is Schur-convex (or Schur-concave, respectively) with respect to (u, v) on ℝ2− . In 2016, for (x, y) ∈ ℝ2++ , (r, s), (p, q) ∈ ℝ2 , and r + s = 1, Murali and Nagaraja [115] defined a Stolarsky extended type mean value, 1

p2 rx p + syp xq − yq q−p Np,q (a, b; r, s) = [ 2 ( q )( )] , xp − yp q rx + syq

(3.3.15)

and obtained the following results. Theorem 3.3.9. For fixed (p, q) ∈ ℝ2 and r = s, if p + q − 3 ≤ 0 (or p + q − 3 ≥ 0, respectively), then Np,q (x, y; r, s) is Schur-convex (or Schur-concave, respectively) related to (x, y) on ℝ2++ . Problem 3.3.3. In the case of r ≠ s, how Schur-convexity of Np,q (x, y; r, s)?

3.4 Schur-convexity of generalized Heron mean |

117

The Schur-harmonic convexity of Np,q (x, y; r, s) is studied in [117]. Theorem 3.3.10. For a fixed (p, q) ∈ ℝ2 , if p + q + 3 ≥ 0 (or p + q + 3 ≤ 0, respectively), then Np,q (x, y; r, s) is Schur-harmonically convex (or Schur-harmonically concave, respectively) with respect to (x, y) on ℝ2++ . The Schur-geometric convexity of Np,q (x, y; r, s) is studied in [68]. Theorem 3.3.11. For a fixed (p, q) ∈ ℝ2 and r = s, if −8(p+q) ≤ 0 (or −8(p+q) ≥ 0, 3 3 respectively), then Np,q (x, y; r, s) is Schur-geometrically convex (or Schur-geometrically concave, respectively) with respect to (x, y) on ℝ2++ .

3.4 Schur-convexity of generalized Heron mean 3.4.1 Generalized Heron mean The aforementioned Stolarsky mean and Gini mean are the two types of binary means that are most used in recent years, followed by the Heron mean. Let (x, y) ∈ ℝ2+ . The classical Heron mean [79, p. 55] is defined as He (x, y) =

x + √xy + y 2A + G = , 3 3

(3.4.1)

where A and G are the arithmetic mean and geometric mean of x and y, respectively. There are well-known double inequalities on such mean [54], i. e., Mα ≤ He (x, y) ≤ Mβ ,

(3.4.2)

log 2 where α = log , β = 32 , and Mp (x, y) is the power mean. 3 Mao [105] defined the dual form of the Heron mean,

̃e (x, y) = x + 4√xy + y = A + 2G , H 6 3

(3.4.3)

and established the following double inequality: ̃e (x, y) ≤ M 1 (x, y). M 1 (x, y) ≤ H 3

2

(3.4.4)

The author [161] used the majorization method to generalize the right inequality in (3.4.4) to obtain the following theorem. Theorem 3.4.1. Let x, y ∈ ℝ++ , n ∈ ℕ, p ∈ ℝ. Then n

Mp (x, y) ≥ [ ∑

i=0

(ni )

2

(2n ) n

ip (n−i)p

x y

1 np

]

(3.4.5)

118 | 3 Schur-convex functions and mean value inequalities for two variables

≥[

2A(xnp , ynp ) + ((2n ) − 2)G(xnp , ynp ) n (2n ) n

1 np

] .

When n = 1 or a = b, equality occurs in (3.4.5). When n = 2, inequality (3.4.5) becomes 1

b2p + 4ap bp + a2p 2p ) Mp (x, y) ≥ ( 6

(3.4.6) 1

A(x 2p , y2p ) + 2G(x2p , y2p ) 2p ) . =( 3 In particular, when p = 21 , inequality (3.4.6) becomes inequality (3.4.4), so it can be said that inequality (3.4.5) gives a generalization with two parameters of the right inequality in (3.4.4). When n = 3, inequality (3.4.6) becomes 1

b3p + 9ap b2p + 9a2p bp + a3p 3p Mp (x, y) ≥ ( ) 20

(3.4.7)

1 3p

≥(

A(x3p , y3p ) + 9G(x3p , y3p ) ) . 10

̃e (x, y), Walther [199] defined a generalized As a unified extension of He (x, y) and H Heron mean, x+w√xy+y

Hw(x, y) = { w+2 √xy,

if 0 ≤ w < ∞,

,

if w = ∞.

(3.4.8)

The comparability between Hw (x, y) and other means is also discussed. For the generalized Heron mean p

p

p

1

{[ x +w(xy) 2 +y ] p , if p ≠ 0, 3 Hp (x, y) = { xy, if p = 0, √ {

(3.4.9)

L(x, y) ≤ Hp (x, y) ≤ Mq (x, y),

(3.4.10)

Jia and Gao [69] obtained the double inequality

, and p = 21 , q = 31 are the best constants. where p ≥ 21 , q ≥ 2p 3 Li et al. [82] discussed the monotonicity and Schur-convexity of Hp (x, y) with respect to (x, y) and obtained the following. Theorem 3.4.2. The function Hp (x, y) is increasing on ℝ2+ with respect to (x, y). When p ≤ 32 , Hp (x, y) is Schur-concave with respect to (x, y) on ℝ2+ ; when p ≥ 2, Hp (x, y) is Schur-convex with respect to (x, y) on ℝ2+ , and when 32 < p < 2, Schur-convexity of Hp (x, y) is uncertain with respect to (x, y) on ℝ2+ .

3.4 Schur-convexity of generalized Heron mean |

119

Shi et al. [167] defined a more generalized Heron mean, i. e., p

p

p

1

{[ x +w(xy) 2 +y ] p , w+2 Hw,p (x, y) = { xy, √ {

if p ≠ 0, w ≥ 0,

if p = 0, w = +∞,

(3.4.11)

and discussed the Schur-convexity and Schur-geometric convexity of Hw,p (x, y). Firstly, the following two lemmas are given. Lemma 3.4.1 ([78, p. 43]). The generalized logarithmic mean (or Stolarsky mean) of two positive numbers a and b is defined as follows: p

p

1

[ b −a ] p−1 , Sp (a, b) = { p(b−a) b,

p ≠ 0, 1, a ≠ b, a = b.

When a ≠ b, Sp (a, b) is a strictly increasing function for p. Lemma 3.4.2 ([97]). Let a, b > 0 and a ≠ b. If x > 0, y ≤ 0, and x + y ≥ 0, then y bx+y − ax+y x + y ≤ (ab) 2 . x x b −a x

Theorem 3.4.3. For fixed (p, w) ∈ ℝ2 , (a) Hp,w (a, b) is increasing for (a, b) ∈ ℝ2+ ; (b) if (p, w) ∈ {p ≤ 1, w ≥ 0} ∪ {1 < p ≤ 32 , w ≥ 1} ∪ { 32 < p ≤ 2, w ≥ 2}, then Hp,w (a, b) is Schur-concave for (a, b) ∈ ℝ2+ ; (c) if p ≥ 2, 0 ≤ w ≤ 2, then Hp,w (a, b) is Schur-convex for (a, b) ∈ ℝ2+ ; (d) if 3/2 < p ≤ 2, w < 2, then the Schur-convexity of Hp,w (a, b) is uncertain with (a, b) ∈ ℝ2+ . Proof. Let φ(a, b) =

ap + w(ab)p/2 + bp . w+2 1

When p ≠ 0 and w ≥ 0, we have Hp,w (a, b) = φ p (a, b). It is clear that Hp,w (a, b) is symmetric with (a, b) ∈ ℝ2+ . Since 𝜕Hp,w (a, b) 𝜕a

=

1 p 1 wb −1 [ap−1 + (ab) 2 −1 ]φ p (a, b) ≥ 0 w+2 2

=

1 p 1 wa −1 [bp−1 + (ab) 2 −1 ]φ p (a, b) ≥ 0, w+2 2

and 𝜕Hp,w (a, b) 𝜕b

from Theorem 1.1.4(a) (Vol. 1), we obtain that Hp,w (a, b) is increasing with (a, b) ∈ ℝ2+ .

120 | 3 Schur-convex functions and mean value inequalities for two variables Let Λ := (b − a)(

𝜕Hp,w (a, b)

When a = b, Λ = 0. When a ≠ b, Λ = Q=

𝜕b

−

1

𝜕Hp,w (a, b) 𝜕a

(b−a)2 p −1 φ (a, b)Q, w+2

).

where

p bp−1 − ap−1 w − (ab) 2 −1 . b−a 2 p−1

p−1

−a If p ≤ 1, w ≥ 0, since xp−1 is decreasing on ℝ++ , we have b b−a Therefore, by Theorem 2.1.3 (Vol. 1), Hp,w (a, b) is Schur-concave. If p ≥ 2, 0 ≤ w ≤ 2, note that p−2

Q = (p − 1)[Sp−1 (a, b)]

−

≤ 0 and Λ ≤ 0.

w p−2 [S (a, b)] . 2 −1

By Lemma 3.4.1, Sp (a, b) is increasing for p and p−2 ≥ 0. It follows that [Sp−1 (a, b)]p−2 ≥ [S−1 (a, b)]p−2 . And then, since p − 1 ≥ w2 , we have Λ ≥ 0. Therefore, Hp,w (a, b) is Schurconvex. If 1 < p ≤ 32 , w ≥ 1, then p − 1 ≤ 21 ≤ w2 ; taking x = 1, y = p − 2 in Lemma 3.4.2, we have x > 0, y ≤ 0, x + y ≥ 0. It follows that p−2 p w bp−1 − ap−1 ≤ (p − 1)(ab) 2 ≤ (ab) 2 −1 , b−a 2

and then Λ ≤ 0; therefore, Hp,w (a, b) is Schur-concave. If 32 < p ≤ 2, w ≥ 2, then x > 0, y ≤ 0, x + y ≥ 0; taking x = 1, y = p − 2 in Lemma 3.4.2, we have p − 1 ≤ 1 ≤ w2 . It follows that p−2 p w bp−1 − ap−1 ≤ (p − 1)(ab) 2 ≤ (ab) 2 −1 , b−a 2

and then Λ ≤ 0. Therefore, Hp,w (a, b) is Schur-concave. If 3/2 < p ≤ 2, w < 2, then the Schur-convexity of Hp,w (a, b) is uncertain. For example, taking w = 1, a = 4, b = 2, p = 1.6, then Q = 0.0609630938 and Λ > 0; taking w = 1, a = 21, b = 12221, p = 1.6, then Q = −0.01869701065 and Λ < 0 (the two examples are due to Yong-ming Jiang). Theorem 3.4.4. For fixed (p, w) ∈ ℝ2 , (a) if p < 0, w ≥ 0, then Hp,w (a, b) is Schur-geometrically concave for (a, b) ∈ ℝ2++ ; (b) if p > 0, w ≥ 0, then Hp,w (a, b) is Schur-geometrically convex for (a, b) ∈ ℝ2++ . Proof. We have a

𝜕Hp,w (a, b) 𝜕a

=

1 p 1 wb −1 [ap + (ab) 2 ]φ p (a, b) w+2 2

3.4 Schur-convexity of generalized Heron mean |

121

and 𝜕Hp,w (a, b)

1 p 1 wa −1 [bp + (ab) 2 ]φ p (a, b), w+2 2 𝜕Hp,w (a, b) 𝜕Hp,w (a, b) −b ) Δ := (log b − log a)(a 𝜕b 𝜕a (log b − log a)(bp − ap ) p1 −1 = φ (a, b). w+2

b

𝜕b

=

When p < 0, w ≥ 0, since log x is increasing and xp is decreasing on ℝ++ , we have (log b − log a)(bp − ap ) ≤ 0 and Δ ≤ 0. Therefore, Hp,w (a, b) is Schur-geometrically concave. When p > 0, w ≥ 0, since log x and xp are both increasing on ℝ++ , we have (log b − log a)(bp − ap ) ≥ 0 and Δ ≥ 0. Therefore, Hp,w (a, b) is Schur-geometrically convex. Fu et al. [39] improved Theorem 3.4.3 and obtained the following theorem. Theorem 3.4.5. The generalized Heron mean Hw,p (x, y) is Schur-convex if and only if (p, w) ∈ E1 , and it is Schur-concave if and only if (p, w) ∈ E2 , where E1 = {(p, w) | p ≥ 2, 0 ≤ w ≤ 2(p − 1)} ∪ {(p, w) | 1 < p ≤ 2, w = 0}

(3.4.12)

and E2 = {(p, w) | p ≤ 2, max{2(p − 1), 0} ≤ w}.

(3.4.13)

Remark 3.4.1. Let E3 = {(p, w) | 1 +

w < p < 2, 0 < w < 2} 2

(3.4.14)

and E4 = {(p, w) | 2 < p < 1 +

w , w > 2}. 2

(3.4.15)

Then, when (p, w) ∈ E3 ∪ E4 , the Schur-convexity of Hw,p (x, y) is uncertain. Theorem 3.4.6 ([38]). Let (x, y) ∈ ℝ2++ , w ∈ ℝ+ , and p ∈ ℝ. If (p, w) ∈ F1 , then Hw,p (x, y) is Schur-harmonically convex with respect to (x, y) on ∈ ℝ2++ . If (p, w) ∈ F2 , then Hw,p (x, y) is Schur-harmonically concave with respect to (x, y) on ∈ ℝ2++ , where F1 = {(p, w) | p ≥ −2, max{0, −2(p + 1)} ≤ w}

(3.4.16)

and F2 = {(p, w) | p ≤ −2, 0 ≤ w ≤ −2(p + 1)} ∪ {(p, w) | p ≤ −1, w = 0}.

(3.4.17)

122 | 3 Schur-convex functions and mean value inequalities for two variables Remark 3.4.2. Let F3 = {(p, w) | −2 < p < −1, 0
0, x ≠ y, w ≥ 0, and p ∈ ℝ. Then Hp,w (x, y) are strictly increasing with respect to p. Theorem 3.4.11. For 0 ≤ w ≤ 4, (a) if p ≥ 0, then Hw,p (x, y) is log-concave with respect to p; (b) if p ≤ 0, then Hw,p (x, y) is log-convex with respect to p.

3.4 Schur-convexity of generalized Heron mean

| 123

Theorem 3.4.12. For 0 ≤ w ≤ 4, (a) if p ≥ 0, then Hw,p (x, y) is concave with respect to p; (b) if p ≤ 0, then Hw,p (x, y) is convex with respect to p. 3.4.2 Generalizations of generalized Heron mean Guan and Zhu [50] gave the generalized Heron in n variables. Let x = (x1 , . . . , xn ) ∈ ℝ2+ . The generalized Heron mean in n variables x1 , . . . , xn is defined as Hw (x) = {

nAn (x)+wGn (x) , w+n

Gn (x),

if 0 ≤ w < ∞, if w = ∞,

(3.4.20)

where An (x) and Gn (x) are the arithmetic mean and geometric mean of x, respectively. Guan and Zhu [50] obtained the following results. Theorem 3.4.13. The function Hw (x) (w > 0) is an increasing and strictly Schur-concave function on ℝn++ . H (x) (w w−1 (x)

Theorem 3.4.14. The function H w

> 1) is a strictly Schur-concave function on ℝn++ .

For x ∈ ℝn++ , Lv et al. [102] defined a mixed mean of geometric mean and harmonic mean in n variables x1 , . . . , xn , i. e., nHn (x)+wGn (x)

1+w Mw (x) = { Gn (x),

, if 0 ≤ w < ∞, if w = ∞,

(3.4.21)

where Hn (x) is the harmonic mean. They obtained the following results. Theorem 3.4.15. If w ≥ 0, Mw (x) is Schur-concave, Schur-geometrically concave, and Schur-harmonically convex on ℝn++ . Theorem 3.4.16. M (x) (a) If w ≥ 1, then Φw (x) := M w (x) is Schur-geometrically convex and Schur-harmoniw−1 cally convex on ℝn++ ; (b) Φw (x) is Schur-convex on ℝ2++ for all w ≥ 1, but when n ≥ 3, Φw (x) is neither Schurconvex nor Schur-concave on ℝn++ for all w ≥ 1. Theorem 3.4.17. If w ≥ 0, then: (a) for x ∈ ℝn++ and α ≥ 0, we have Mw+1 (x)Mw+α (x) ≥ Mw (x)Mw+1+α (x); (b) for x ∈ (0, 21 ]n , we have

Hn (x) Gn (x) Mw (x) ≤ ≤ . Hn (1 − x) Mw (1 − x) Gn (1 − x)

124 | 3 Schur-convex functions and mean value inequalities for two variables In 2009, Kuang [79, p. 61] defined a binary mean with three parameters, i. e., 1

K(w1 , w2 , p) = (

w1 A(xp , yp ) + w2 G(xp , yp ) p ) , w1 + w2

(3.4.22)

where p ≠ 0, w1 , w2 ≥ 0, w1 + w2 ≠ 0. 2 . In particular, K(1, w22 , p) is the general Heron mean (3.4.11). If w1 ≠ 0, let w = 2w w1 Then K(w1 , w2 , p) also becomes the generalized Heron mean (3.4.11). In 2016, Fu et al. [37] studied Schur-convexity, Schur-geometric convexity, and Schur-harmonic convexity of K(w1 , w2 , p) with respect to (x, y). They obtained the following three theorems. Theorem 3.4.18. (a) When ω1 ω2 ≠ 0, if p ≥ 2 and p(ω1 − ω22 ) − ω1 ≥ 0, then K(ω1 , ω2 , p) is Schurconvex with (x, y) ∈ ℝ2+ ; if 1 ≤ p < 2 and p(ω1 − ω22 ) − ω1 ≤ 0, then K(ω1 , ω2 , p) is Schur-concave with (x, y) ∈ ℝ2+ ; if p < 1, then K(ω1 , wω2 , p) is Schur-concave with (x, y) ∈ ℝ2+ . (b) When ω1 = 0, ω2 ≠ 0, K(ω1 , ω2 , p) is Schur-concave with (x, y) ∈ ℝ2+ . (c) When ω1 ≠ 0, ω2 = 0, if p ≥ 2, then K(ω1 , ω2 , p) is Schur-convex with (x, y) ∈ ℝ2+ ; if p < 2, then K(ω1 , ω2 , p) is Schur-concave with (x, y) ∈ ℝ2+ . Theorem 3.4.19. If p ≥ 0, then K(ω1 , ω2 , p) is Schur-geometrically convex with (x, y) ∈ ℝ2+ . If p < 0, then K(ω1 , ω2 , p) is Schur-geometrically concave with (x, y) ∈ ℝ2+ . Theorem 3.4.20. If p ≥ −1, then K(ω1 , ω2 , p) is Schur-harmonically convex with (x, y) ∈ ℝ2+ . If −2 < p < −1 and ω1 (p + 1) + ω2 ( p2 + 1) ≥ 0, then K(ω1 , ω2 , p) is Schur-harmonically convex with (x, y) ∈ ℝ2+ . If p ≤ −2 and ω1 ( p2 + 1) + ω2 = 0, then K(ω1 , ω2 , p) is Schurharmonically concave with (x, y) ∈ ℝ2+ . Wang et al. [201] studied the Schur-power convexity of K(w1 , w2 , p) with respect to (x, y) and obtained the following results. Theorem 3.4.21. (I) For m > 0, 2 )m, 2m}, then K(ω1 , ω2 , p) is Schur-m-power convex with respect (a) if p ≥ max{(1 + ω ω 1

to (x, y) ∈ ℝ2+ ; (b) if m ≤ p ≤ min{(1 +

ℝ2+ ;

ω2 )m, 2m}, ω1

then K(ω1 , ω2 , p) is Schur-m-power concave with

respect to (x, y) ∈ (c) if 0 ≤ p < m, then K(ω1 , ω2 , p) is Schur-m-power concave with respect to (x, y) ∈ ℝ2+ ; (d) if p < 0, then K(ω1 , ω2 , p) is Schur-m-power concave with respect to (x, y) ∈ ℝ2+ . (II) For m < 0, (a) if p ≥ 0, then K(ω1 , ω2 , p) is Schur-m-power convex with respect to (x, y) ∈ ℝ2+ ; (b) if m ≤ p < 0, then K(ω1 , ω2 , p) is Schur-m-power convex with respect to (x, y) ∈ ℝ2+ ;

3.4 Schur-convexity of generalized Heron mean |

125

ω2 2 )m (0 < ω < 1), then K(ω1 , ω2 , p) is Schur-m-power ω1 ω1 2 convex with respect to (x, y) ∈ ℝ+ ; ω1 2 if p < 2m and p = (1 + ω )m ( ω > 1), then K(ω1 , ω2 , p) is Schur-m-power concave ω1 2 2 with respect to (x, y) ∈ ℝ+ .

(c) if 2m ≤ p < m and p = (1 + (d)

Wang et al. [202] also extended K(ω1 , ω2 , p) to n variables, defining 1

ω1 An (x1p , . . . , xnp ) + ω2 Gn (x1p , . . . , xnp ) p ] , Kn (ω1 , ω2 , p; x) = [ ω1 + ω2

(3.4.23)

where An (x) and Gn (x) are the arithmetic mean and geometric mean of x, respectively, and parameters p ≠ 0, ω1 , ω2 ≥ 0 with ω1 + ω2 ≠ 0. Schur-convexity, Schur-geometric convexity, and Schur-harmonic convexity of Kn (ω1 , ω2 , p; x) are discussed next. Theorem 3.4.22. Let ω1 , ω2 ≥ 0 with ω1 + ω2 ≠ 0. (a) If p ≤ 1, then Kn (ω1 , ω2 , p; x) is Schur-concave with x ∈ ℝn+ . (b) If p ≥ 1, ω2 = 0, then Kn (ω1 , ω2 , p; x) is Schur-convex with x ∈ ℝn+ . (c) If p ≥ 1, ω1 = 0, then Kn (ω1 , ω2 , p; x) is Schur-concave with x ∈ ℝn+ . (d) Let B = {x = (x1 , . . . , xn )|xi ≥ 1, i = 1, . . . , n, and x1 + ⋅ ⋅ ⋅ + xn ≤ s}. If p ≥ 2 and p ω1 ≥ ωn2ps , then Kn (ω1 , ω2 , p; x) is Schur-convex with x ∈ B. (e) Let C = {x = (x1 , . . . , xn )|xi ≤ 1, i = 1, . . . , n, and x1 + ⋅ ⋅ ⋅ + x1 ≥ r}. If 2 ≥ p ≥ 1 and ω1 ≤

ω2 np , rp

1

n

then Kn (ω1 , ω2 , p; x) is Schur-concave with respect to x ∈ C.

Theorem 3.4.23. Let ω1 , ω2 ≥ 0 with ω1 + ω2 ≠ 0. (a) If p > 0, then Kn (ω1 , ω2 , p; x) is Schur-geometrically convex with x ∈ ℝn+ . If p < 0, then Kn (ω1 , ω2 , p; x) is Schur-geometrically concave with x ∈ ℝn+ . (b) If p ≥ −1, then Kn (ω1 , ω2 , p; x) is Schur-harmonically convex with x ∈ ℝn+ . Yang [262] extended the range w ≥ 0 of the generalized Heron mean Hw,p (x, y) parameter w to w > −2, called the generalized Heron mean the Daróczy mean, and obtained the following results. Theorem 3.4.24. For fixed p ∈ ℝ, m > 0, and w > −2, the Daróczy mean Hw,p (x, y) is m-order Schur-power convex with respect to (x, y) on ℝn++ if and only if (p, w) ∈ Ω1 , where Ω1 = {−2 < w ≤ 0, p ≥

w+2 w+2 m} ∪ {w > 0, p ≥ max( m, 2m)}. 2 2

(3.4.24)

Theorem 3.4.25. For fixed p ∈ ℝ, m > 0, and w > −2, the Daróczy mean Hw,p (x, y) is m-order Schur-power concave with respect to (x, y) on ℝn++ if and only if (p, w) ∈ Ω2 , where Ω2 = {−2 < w < 0, p < 0} ∪ {w ≥ 0, p ≤ min(

w+2 m, 2m)}. 2

(3.4.25)

126 | 3 Schur-convex functions and mean value inequalities for two variables Theorem 3.4.26. For fixed p ∈ ℝ, m < 0, and w > −2, the Daróczy mean Hw,p (x, y) is m-order Schur-power convex with respect to (x, y) on ℝn++ if and only if (p, w) ∈ E1 , where E1 = {−2 < w < 0, p > 0} ∪ {w ≥ 0, p ≥ max(

w+2 m, 2m)}. 2

(3.4.26)

Theorem 3.4.27. For fixed p ∈ ℝ, m < 0, and w > −2, the Daróczy mean Hw,p (x, y) is m-order Schur-power concave with respect to (x, y) on ℝn++ if and only if (p, w) ∈ E2 , where E2 = {−2 < w < 0, p ≤

w+2 w+2 m} ∪ {w > 0, p ≤ min( m, 2m)}. 2 2

(3.4.27)

Theorem 3.4.28. For fixed p ∈ ℝ, m = 0 and w > −2, the Daróczy mean Hw,p (x, y) is m-order Schur-power convex (or Schur-power concave, respectively) with respect to (x, y) on ℝn++ if and only if p ≥ 0 (or p ≤ 0, respectively). In 2014, Deng et al. [26] generalized the generalized Heron mean Hw,p (x, y) and Gini mean G(r, s; x, y) to the following generalized Gini–Heron mean with three parameters and studied its Schur-geometric convexity: p

p 2

p

1

x +w(xy) +y p−q if p ≠ q, { {( xq +w(xy) q2 +yq ) , Hp,q,w (x, y) = { p p p {exp{ x log x+w(xy) 2 log xy+y log y }, if p = q, p { xp +w(xy) 2 +yp

(3.4.28)

where (p, q) ∈ ℝ2 , (x, y) ∈ ℝ2++ . Theorem 3.4.29. For fixed (p, q, w) ∈ ℝ3 , (a) if p + q ≥ 0 and w ≥ 0, then Hp,q,w (x, y) is Schur-geometrically convex on ℝ2++ with respect to (x, y), (b) if p + q ≤ 0 and w ≥ 0, then Hp,q,w (x, y) is Schur-geometrically concave on ℝ2++ with respect to (x, y). Definition 3.4.1. Let (x, y) ∈ ℝ2+ , k ∈ ℤ+ , (α, β) ∈ ℝ2 . The Gnan mean G(x, y; k, α, β) and its duality g(x, y; k, α, β) are defined as follows: β

1 β

1 k (k + 1 − i)xα + iyα α ) ] , G(x, y; k, α, β) = [ ∑( k i=1 k+1

(3.4.29)

1 k (k+1−i)β k+1 G(x, y; k, 0, β) = [ ∑ x k+1 b ] , k i=1

(3.4.30)

iβ

k

1 β

1

(k + 1 − i)xα + iyα kα ) , G(x, y; k, α, 0) = ∏( k+1 i=1

(3.4.31)

G(x, y; k, 0, 0) = √xy,

(3.4.32)

3.5 Schur-convexity of other two-variable means | 127

β

1 β

1 k (k − i)xα + iyα α g(x, y; k, α, β) = [ ) ] , ∑( k + 1 i=0 k

(3.4.33)

1 k (k−i)β k g(x, y; k, 0, β) = [ ∑ x k b k+1 ] , k + 1 i=0

(3.4.34)

1 β

iβ

1

k

(k − i)xα + iyα (k+1)α , g(x, y; k, α, 0) = ∏( ) k i=0

(3.4.35)

g(x, y; k, 0, 0) = √xy.

(3.4.36)

β

Theorem 3.4.30 ([98]). If α, α ≥ 1, then the Gnan mean G(x, y; k, α, β) and its dual form are Schur-convex on ℝ2+ . β

Theorem 3.4.31 ([98]). If α, α ≥ 1, then the Gnan mean G(x, y; k, α, β) and its dual form are Schur-geometrically convex (or Schur-geometrically concave, respectively) if and only if α, β ≥ 0 (or α, β ≤ 0, respectively) on ℝ2+ . Theorem 3.4.32 ([116]). Let α, β, x and y be real numbers, with x ≥ y, and let k be a nonnegative integer. Then: (a) for α, β ≠ 0, G(x, y; k, α, β) and g(x, y; k, α, β) are Schur-convex with respect to (x, y) for 1 ≤ α ≤ β; (b) for α = 0, β ≠ 0, G(x, y; k, 0, β) and g(x, y; k, 0, β) are Schur-concave with respect to (x, y) for β ≤ 0; (c) for α ≠ 0, β = 0, G(x, y; k, α, 0) and g(x, y; k, α, 0) are Schur-concave with respect to (x, y) for α ≤ 21 ; (d) for α = 0, β = 0, G(x, y; k, 0, 0) and g(x, y; k, 0, 0) are Schur-concave with respect to (x, y). Problem 3.4.1. How about the harmonic convexity and Schur-power convexity of the Gnan mean and its dual form?

3.5 Schur-convexity of other two-variable means 3.5.1 Generalized Muirhead mean For (x, y) ∈ ℝ2++ , (r, s) ∈ ℝ2 , and r+s ≠ 0, Trif [198] introduced the following generalized Muirhead mean: 1

xr ys + xs yr r+s M(r, s; x, y) = ( ) . 2

(3.5.1)

Theorem 3.5.1 ([43]). For fixed (r, s) ∈ ℝ2 , (a) M(r, s; x, y) is Schur-convex with respect to (x, y) on ℝ2 if and only if (r, s) ∈ {(r, s) ∈ ℝ2+ , (r − s)2 ≥ r + s, and rs ≤ 0};

128 | 3 Schur-convex functions and mean value inequalities for two variables (b) M(r, s; x, y) is Schur-concave with respect to (x, y) on ℝ2 if and only if (r, s) ∈ {(r, s) ∈ ℝ2+ , (r − s)2 ≤ r + s, and (r, s) ≠ (0, 0)} ∪ {(r, s) ∈ ℝ2 , r + s < 0}. Theorem 3.5.2 ([10]). For fixed (x, y) ∈ ℝ2++ with x ≠ y, M(r, s; x, y) is Schur-convex on ℝ2++ and Schur-convex on ℝ2−− with respect to (r, s). Proposition 3.5.1. When (r, s) is in the second and fourth quadrant limits, how do we prove Schur-convexity of M(r, s, x, y) with respect to (r, s)? Theorem 3.5.3 ([232]). For fixed (r, s) ∈ ℝ2 , (a) M(r, s; x, y) is Schur-geometrically convex with respect to (x, y) on ℝ2 if and only if r + s > 0; (b) M(r, s; x, y) is Schur-geometrically concave with respect to (x, y) on ℝ2 if and only if r + s < 0. Theorem 3.5.4 ([232]). For fixed (r, s) ∈ ℝ2 , (a) M(r, s; x, y) is Schur-harmonically convex with respect to (x, y) on ℝ2 if and only if (r, s) ∈ {(r, s) | r + s > 0} ∪ {(r, s) | r ≤ 0, s ≤ 0, (r − s)2 + (r + s) ≤ 0, r 2 + s2 ≠ 0}; (b) M(r, s; x, y) is Schur-harmonically concave with respect to (x, y) on ℝ2 if and only if (r, s) ∈ {(r, s) | r ≥ 0, r + s < 0, (r − s)2 + r + s ≥ 0} ∪ {(r, s) | s ≥ 0, r + s < 0, (r − s)2 + r + s ≥ 0}. Theorem 3.5.5 ([16]). For fixed (r, s) ∈ ℝ2 , write E1 (m) = {(r, s) : r + s > 0, m ≤ 0}

∪ {(r, s) : r + s > 0, rs ≤ 0, (r − s)2 − m(r + s) ≥ 0, m > 0}

∪ {(r, s) : r + s < 0, rs ≥ 0, (r − s)2 − m(r + s) ≤ 0, m < 0} and E2 (m) = {(r, s) : r + s < 0, m ≥ 0}

∪ {(r, s) : r + s < 0, rs ≤ 0, (r − s)2 − m(r + s) ≥ 0, m < 0}

∪ {(r, s) : r + s > 0, rs ≥ 0, (r − s)2 − m(r + s) ≤ 0, m > 0}. (a) When (r, s) ∈ E1 (m), M(r, s; x, y) is m-order Schur-power convex with respect to (x, y) on ℝ2++ ; (b) when (r, s) ∈ E2 (m), M(r, s; x, y) is m-order Schur-power concave with respect to (x, y) on ℝ2++ . In 2014, Deng et al. [25] unified the generalized Heron mean (3.4.11) and the generalized Muirhead mean (3.5.1) and defined the following more general mean Hp,q,w (x, y),

3.5 Schur-convexity of other two-variable means | 129

studying monotonicity and Schur-convexity of Hp,q,w (x, y) with respect to (x, y) on ℝ2++ : p q

p+q

{( x y +w(xy) 2 w+2 Hp,q,w (x, y) = { xy, √ {

+x q yp

1

) p+q ,

if p + q ≠ 0,

if p = q = 0,

(3.5.2)

where (p, q) ∈ ℝ2 , (x, y) ∈ ℝ2++ . On the basis of [25], He et al. [62] further discussed the Schur-geometric convexity and Schur-harmonic convexity of Hp,q,w (x, y) with respect to (x, y) on ℝ2++ (for details, see the original text). Theorem 3.5.6 ([62]). Let p, q ∈ ℝ. Write S1 = {(p, q, w) | p = q ≤ 0, w ≥ 0} ∪ {(p, q, w) | p + q > 0, w ≥ 0} and S2 = {(p, q, w) | p = q ≥ 0, w ≥ 0} ∪ {(p, q, w) | p + q < 0, w ≥ 0}. Then Hp,q,w (x, y) is Schur-geometrically convex with respect to (x, y) on ℝ2++ if and only if (p, q, w) ∈ S1 , and Hp,q,w (x, y) is Schur-geometrically concave with respect to (x, y) on ℝ2++ if and only if (p, q, w) ∈ S2 . Theorem 3.5.7 ([62]). Let p, q ∈ ℝ. Write A1 = {(p, q, w) | p − q − 2 ≥ 0}

∪ {(p, q, w) | p > q, p − q − 2 < 0, 8p(p − q + 1) − (p + q)(p − q − 2)w > 0},

A2 = {(p, q, w) | q − p − 2 ≥ 0}

∪ {(p, q, w) | q > p, q − p − 2 < 0, 8q(q − p + 1) − (p + q)(q − p − 2)w > 0},

A3 = {(p, q, w) | p − q − 2 ≥ 0, p < 0, 8p(p − q + 1) − (p + q)(p − q − 2)w ≤ 0} ∪ {(p, q, w)p > q, p − q − 2 < 0, p < 0},

A4 = {(p, q, w) | q − p − 2 ≥ 0, q < 0, 8q(q − p + 1) − (p + q)(q − p − 2)w ≤ 0} ∪ {(p, q, w) | q > p, q − p − 2 < 0, q < 0},

A5 = {(p, q, w) | p − q − 2 ≥ 0, p > 0}

∪ {(p, q, w) | p > q, p − q − 2 < 0, p > 0, 8p(p − q + 1) − (p + q)(p − q − 2)w ≥ 0},

A6 = {(p, q, w) | q − p − 2 ≥ 0, q > 0}

∪ {(p, q, w) | q > p, q − p − 2 < 0, q > 0, 8q(q − p + 1) − (p + q)(q − p − 2)w ≥ 0}

and S3 = {(p, q, w) | p = q, w ≥ 0}

∪ {(p, q, w) | p = 0, max{−(w + 2)/2, −2} ≤ q < 0, w ≥ 0}

130 | 3 Schur-convex functions and mean value inequalities for two variables ∪ {(p, q, w) | q = 0, max{−(w + 2)/2, −2} ≤ p < 0, w ≥ 0} ∪ {(p, q, w) | p + q > 0, w ≥ 0} ∩ (A1 ∪ A2 )

∪ {(p, q, w) | p + q < 0, 2(p − q)2 + (2 + w)(p + q) ≤ 0, w ≥ 0} ∩ (A3 ∪ A4 ),

S4 = {(p, q, w) | p = 0, q ≤ −1, w = 0}

∪ {(p, q, w) | p = 0, q ≤ min{−(w + 2)/2, −2}, w > 0} ∪ {(p, q, w) | q = 0, p ≤ −1, w = 0}

∪ {(p, q, w) | q = 0, p ≤ min{−(w + 2)/2, −2}, w > 0}

∪ {(p, q, w) | p + q < 0, 2(p − q)2 + (2 + w)(p + q) ≥ 0, w ≥ 0} ∩ (A5 ∪ A6 ).

Then when (p, q, w) ∈ S3 , Hp,q,w (x, y) is Schur-harmonically convex with respect to (x, y) on ℝ2++ , and when (p, q, w) ∈ S4 , Hp,q,w (x, y) is Schur-harmonically concave with respect to (x, y) on ℝ2++ . 3.5.2 Seiffert type mean In 1993, Seiffert [147] proposed the following mean of two positive real numbers x, y: x−y , if x ≠ y, { x P = P(x, y) = { 4 arctan √ y −π if x = y. {x,

(3.5.3)

Theorem 3.5.8 ([86]). The Seiffert mean P(x, y) is both Schur-concave and Schurgeometrically concave on ℝ2++ . Proof. First, we determine the symmetry of P(x, y) on ℝ2++ . For (x, y) ∈ ℝ2++ , x ≠ y, we have x−y

P(x, y) = Let t = u1 . Then u = Therefore,

1 . t

4 arctan √ xy

−π

=

x−y x

√y

4 ∫1

1 1+t 2

dt

.

When t = 1, then u = 1, and when t = √ xy , then u = √ ba .

P(x, y) =

y−x

y x

1 4 ∫1 1+u 2

du

=

y−x

4 arctan √ xy − π

= P(y, x).

So P(x, y) is symmetric with respect to x and y on ℝ2 . Next we prove that P(x, y) satisfies the Schur-condition. For (x, y) ∈ ℝ2++ , x ≠ y, let v = v(x, y) = √ xy , φ = φ(x, y) = 4 arctan v − π. Then P(x, y) =

x−y

4 arctan √ xy

−π

=

x−y x−y = 4 arctan v − π φ

3.5 Schur-convexity of other two-variable means | 131

and then Δ := (x − y)( = = = = = =

𝜕P 𝜕P − ) 𝜕x 𝜕y

𝜕φ 𝜕φ x−y [2φ + (x − y)( − )] 𝜕y 𝜕x φ2

4 𝜕v 4 𝜕v x−y − )] [2φ + (x − y)( φ2 1 + v2 𝜕y 1 + v2 𝜕x x−y 4(x − y) x 1 [2φ + (− − )] 2 2 2 2vy φ 1+v 2vy 2(x − y) x 1 x−y ( 2 + )] [2φ − 2 2 y φ v(1 + v ) y

2(x − y) 1 (φ + − v) 2 v φ

1 2(x − y) (4 arctan v − π + − v). v φ2

Let f (v) = 4 arctan v − π +

1 v

− v. Then 1 4 − 2 −1 2 1+v v −v4 + 2v2 − 1 (v2 − 1)2 = =− 2 ≤ 0. 2 2 v (1 + v ) v (1 + v2 )

f 󸀠 (v) =

Therefore, f (v) is descending with respect to v(v > 0). So f (v) ≥ f (1) = 0 for v < 1 and f (v) ≤ f (1) = 0 for v > 1. It can be seen that f (v) ≥ 0 for x < y, v < 1 and f (v) ≤ 0 for x > y, v > 1. It follows that Δ ≤ 0 holds for all (x, y) ∈ ℝ2++ . From Theorem 2.1.3 (Vol. 1), we know that P(x, y) is Schur-concave with respect to (x, y) ∈ ℝ2++ . We have Λ := (log x − log y)(x = (log x − log y){ = = = =

𝜕P 𝜕P −y ) 𝜕x 𝜕y

𝜕φ 𝜕φ x y [φ − (x − y) ] − 2 [−φ − (x − y) ]} 𝜕x 𝜕y φ2 φ

𝜕φ 𝜕φ log x − log y [(x + y)φ + (x − y)(y − x )] 𝜕y 𝜕x φ2

4x 𝜕v log x − log y 4y 𝜕v − )] ⋅ [(x + y)φ + (x − y)( φ2 1 + v2 𝜕y 1 + v2 𝜕x

𝜕φ 𝜕φ log x − log y 4 ⋅ [(x + y)φ − (x − y) (y − x )] 𝜕y 𝜕x φ2 1 + v2

4u log x − log y ⋅ [(x + y)φ − (x − y) ] φ2 1 + u2

132 | 3 Schur-convex functions and mean value inequalities for two variables

= = = =

y(log x − log y) 4v [(v2 + 1)φ − (v2 − 1) ] φ2 1 + v2 y(log x − log y) 2 4v(v2 − 1) (v + 1)[φ − ] 2 φ (1 + u2 )2

4v3 4v y(log x − log y)(v2 + 1) ⋅ [φ − + ] 2 φ (1 + v2 )2 (1 + v2 )2

4v 4v3 y(x − y)(v2 + 1) log x − log y + ]. ⋅ [4 arctan v − π − 2 x−y φ (1 + v2 )2 (1 + v2 )2

Let g(v) = 4 arctan v − π − g 󸀠 (v) =

4v3 4v + , 2 2 (1 + v ) (1 + v2 )2

3v2 (1 + v2 )2 − 2u3 (1 + v2 )2u (1 + v2 )2 − 4v2 (1 + v2 ) 4 − 4 + 4 1 + v2 (1 + v2 )4 (1 + v2 )4

(1 + v2 )2 3v2 + 3v4 − 4v4 1 + v2 − 4v2 − + ] (1 + v2 )3 (1 + v2 )3 (1 + v2 )3 4 (v4 + 2v2 + 1 − 3v2 + v4 + 1 + v2 − 4v2 ) = (1 + v2 )3 8 = (v4 − 2v2 + 1) (1 + v2 )3

= 4[

=

8(v2 − 1)2 ≥ 0. (1 + v2 )3

Therefore, g(v) is increasing with respect to v(v > 0), and then g(v) ≤ g(1) = 0 for v < 1 and g(v) ≥ g(1) = 0 for v > 1. Thus, when a < b, v < 1, g(v) ≤ 0, Λ ≥ 0, when a > b, that is, v > 1, we have g(v) ≥ 0; this implies Λ ≥ 0, for all (a, b) ∈ ℝ2++ with a ≠ b. By Theorem 2.4.3 (Vol. 1), it follows that P(a, b) is Schur-geometrically convex with respect to (a, b) ∈ ℝ2++ . In 2012, Deng He [59] proved the following. Theorem 3.5.9 ([59]). The Seiffert mean P(x, y) is m-order Schur-power convex (or Schur-power concave, respectively) on ℝ2++ if and only if m ≤ 21 (or m ≥ 32 , respectively). In 1995, Seiffert [148] also defined the following arc tangent Seiffert type mean: x−y { 2 arctan x−y , if x ≠ y, x+y T = T(x, y) = { x, if x = y. {

(3.5.4)

Theorem 3.5.10 ([89]). The Seiffert mean T(x, y) is both Schur-convex and Schurgeometrically convex on ℝ2++ .

3.5 Schur-convexity of other two-variable means | 133

He and Shen [65] pointed out that the following inverse hyperbolic tangent Seiffert type mean can also be defined: x−y

{ −1 N2 = N2 (x, y) = { 2 tanh {x,

x−y x+y

if x ≠ y,

,

if x = y.

(3.5.5)

They studied its Schur-convexity and Schur-geometric convexity. In 2003, Neuman and Sàndor [120] defined the so-called Neuman–Sàndor mean of two positive real numbers x, y, i. e., x−y

{ −1 N1 = N3 (x, y) = { 2 sinh {x,

x−y x+y

if x ≠ y,

,

if x = y.

(3.5.6)

In 2012, Qian [143] proved the following. Theorem 3.5.11. The Seiffert mean N1 (x, y) is both Schur-convex and Schur-geometrically convex on ℝ2++ . In 2015, Witkowski [218] defined the following general form of the Seiffert type mean: |x−y| { 2f ( |x−y| ) , if x ≠ y, S(f )(x, y) = { x+y if x = y. {x,

(3.5.7)

The following conclusions were obtained. Theorem 3.5.12. Let f : (0, 1) → ℝ satisfy z z ≤ f (z) ≤ . 1+z 1−z

(3.5.8)

If f : (0, 1) → ℝ is concave (or convex, respectively), then S(f )(x, y) is Schur-convex (or Schur-concave, respectively) on ℝ2++ . The reference [218] is interesting and rich in content, and readers are advised to read the original text in detail.

3.5.3 Exponent type mean For (x, y) ∈ ℝ2++ , the following exponent type mean is introduced in [195]: E = E(x, y) =

yey −xex y x { e −e

x,

− 1,

if x ≠ y,

if x = y.

(3.5.9)

134 | 3 Schur-convex functions and mean value inequalities for two variables The following basic result is given: E(x, y) >

x+y = A(x, y). 2

(3.5.10)

Another type of exponential mean was introduced in [197], i. e., y

x

xe −ye ̃ y) = { ey −ex + 1, if x ≠ y, Ẽ = E(x, x, if x = y.

(3.5.11)

It is easy to verify that Ẽ = 2A − E,

(3.5.12)

and combining with (3.5.12), the following inequality can be obtained: Ẽ < A.

(3.5.13)

Theorem 3.5.13 ([70]). The function E(x, y) is both Schur-convex and Schur-geometrically convex on ℝ2++ . Proof. Obviously, E(x, y) is symmetric with respect to x, y. We have 𝜕E yex+y − xex+y − ex+y + e2x , = 𝜕x (ey − ex )2

(3.5.14)

𝜕E xex+y − yex+y − ex+y + e2y , = 𝜕y (ey − ex )2

Δ := (x − y)(

(3.5.15)

𝜕E 𝜕E (x − y)2 e2x − e2y ( − )= y − 2ex+y ). 𝜕x 𝜕y x−y (e − ex )2 2x

2y

−e ≥ 2ex+y It is easy to prove that f (x) = e2x is convex on (0, +∞), and the inequality e x−y can be obtained from the left inequality in (2.1.1), thus Δ ≥ 0, so E(x, y) is Schur-convex on ℝ2++ with respect to (x, y). From (3.5.14) and (3.5.15), we obtain

Λ := (log x − log y)(x =

𝜕E 𝜕E −y ) 𝜕x 𝜕y

log x − log y (x − y)2 xe2x − ye2y − (x + y − 1)2ex+y ). ( x−y x−y (ey − ex )2

It is easy to prove that f (x) = (2x + 1)e2x is convex on (0, +∞), and the inequality xe2x − ye2y ≥ (x + y + 1)ex+y > (x + y − 1)ex+y x−y can be obtained from inequality (2.1.1), thus Λ ≥ 0, so E(x, y) is Schur-geometrically convex on ℝ2++ with respect to (x, y).

3.5 Schur-convexity of other two-variable means | 135

̃ y) is Schur-concave on ℝ2 . Theorem 3.5.14. The function E(x, ++ ̃ y) is Schur-geometrically convex on (0, 3 )2 and Theorem 3.5.15. The function E(x, 2 Schur-geometrically concave on ( 32 , +∞)2 . Proof. After a simple calculation, we obtain Λ := (log x − log y)(x =

𝜕Ẽ 𝜕Ẽ −y ) 𝜕x 𝜕y

log x − log y (x − y)2 2x+2y e s(x, y), x−y (ey − ex )2

where s(x, y) =

x e2x

−

y e2y

x−y

−

1 − (x + y) . ex+y

. By computation, h󸀠󸀠 (x) = 4(3−2x) . Then when 0 < x < 32 , h(x) is a Let h(x) = 1−2x e2x e2x convex function. From (2.1.1), it follows that h(x, y) ≥ 0, when x > 32 , h(x) is concave. From (2.1.1), it follows that h(x, y) ≤ 0. Thus, when (x, y) ∈ (0, 32 )2 , Λ ≥ 0, and when (x, y) ∈ ( 32 , +∞)2 , Λ ≤ 0. Theorem 3.5.15 is proved. 3.5.4 Trigonometric mean For (x, y) ∈ [0, π2 ]2 , two triangular means are introduced in [197], i. e., y sin y−x sin x

Msin (x, y) = { sin y−sin x x,

− tan( x+y ), 2

if x ≠ y,

if x = y

(3.5.16)

and cos y y tan y−x tan x+log( cos ) x , tan y−tan x

{ Mtan (x, y) = { {x,

if x ≠ y,

if x = y.

(3.5.17)

The following double inequality is given: Msin (x, y) < A(x, y) < Mtan (x, y).

(3.5.18)

Bi and Jiang [5] proved the following theorem. Theorem 3.5.16. The function Msin (x, y) is Schur-concave with respect to (x, y) on [0, π2 ]2 and Mtan (x, y) is Schur-convex with respect to (x, y) on (0, π2 ]2 .

136 | 3 Schur-convex functions and mean value inequalities for two variables Corollary 3.5.1. Let (x, y) ∈ (0, π2 ]2 , x ≤ y. Then 3x + y x + 3y , ) ≤ A(x, y) a 4 3x + y x + 3y , ) ≤ Mtan (x, y). ≤ Mtan ( a 4

Msin (x, y) ≤ Msin (

(3.5.19)

Inequality (3.5.19) is a refinement of inequality (3.5.18). He et al. [64] proved the following. Theorem 3.5.17. The function Msin (x, y) is m-order Schur-power convex on [0, π2 ]2 if and only if m ≥ 1 and Mtan (x, y) is Schur-power convex on (0, π2 ]2 if and only if m ≤ 1. After [5], Li and He [90] for (x, y) ∈ ℝ2+ defined the following two means, which are related to hyperbolic functions: Msinh (x, y) = {

y sinh y−x sinh x sinh y−sinh x

x,

− tanh( x+y ), if x ≠ y, 2 if x = y

(3.5.20)

and cosh y

{ y tanh y−x tanh x+log( cosh x ) , if x ≠ y, tanh y−tanh x Mtanh (x, y) = { x, if x = y. {

(3.5.21)

They proved the following two theorems. Theorem 3.5.18. The function Msinh (x, y) is Schur-convex with respect to (x, y) on ℝ2++ , Mtanh (x, y) is Schur-concave with respect to (x, y) on ℝ2++ . Theorem 3.5.19. The function Msinh (x, y) is m-order Schur-power convex on ℝ2+ if and only if m ≤ 1 and Mtanh (x, y) is Schur-power convex on ℝ2+ if and only if m ≥ 1. Two new triangular means are defined in [266], i. e., y cos y−x cos x

Mcos (x, y) = { cos y−cos x x,

− coth( x+y ), 2

if x ≠ y, if x = y

(3.5.22)

and sin y

{ y cot y−x cot x+log( sin x ) , if x ≠ y, cot y−cot x Mcot (x, y) = { x, if x = y. {

(3.5.23)

Theorem 3.5.20. The function Mcos (x, y) is Schur-convex on [0, π2 ]2 and Mcot (x, y) is Schur-concave on [0, π2 ]2 . In 2016, Xu [243] further proved the following theorem.

3.5 Schur-convexity of other two-variable means | 137

Theorem 3.5.21. The function Mcos (x, y) is Schur-geometrically convex on [0, π2 ]2 and Mcot (x, y) is not Schur-geometrically convex (or Schur-geometrically concave) on [0, π2 ]2 . Problem 3.5.1. What are the Schur-harmonic convexity and Schur-power convexity of the means Mcos (x, y) and Mcot (x, y)? Two inverse trigonometric functions were also introduced in [265], i. e., Marcsin (x, y) =

√1 − y2 − √1 − x2

arcsin y − arcsin x

,

x, y ∈ [0, 1],

(3.5.24)

and Marctan (x, y) =

log √1 + y2 − log √1 + x2 arctan y − arctan x

,

x, y ≥ 0.

(3.5.25)

Zhang and Qian [265] proved the following theorem. Theorem 3.5.22. The function Marcsin (x, y) is Schur-convex on [0, 1]2 and Marctan (x, y) is Schur-concave on [0, +∞)2 . 3.5.5 Lehmer mean For (x, y) ∈ ℝ2++ the Lehmer mean [126] of (x, y) be defined as Lp (x, y) =

xp + yp , −∞ < p < +∞. xp−1 + yp−1

(3.5.26)

Many mean values are special cases of the Lehmer mean values; for example, x+y = L1 (x, y) is the arithmetic mean, 2 G(x, y) = √xy = L 1 (x, y) is the geometric mean,

A(x, y) =

2

2xy = L0 (x, y) is the harmonic mean, H(x, y) = x+y

2 2 ̃ y) = x + y = L2 (x, y) is the antiharmonic mean. H(x, x+y

For Lp (x, y), Witkowski [214] gave the following conclusions. (a) For a fixed (x, y) ∈ ℝ2++ , Lp (x, y) is strictly increasing with respect to p, L+∞ (x, y) = lim Lp (x, y) = max(x, y) p→+∞

and L−∞ (x, y) = lim Lp (x, y) = min(x, y). p→−∞

138 | 3 Schur-convex functions and mean value inequalities for two variables (b) For x ≠ y, when p < (>) − 21 , Lp (x, y) is a logarithmically convex function of p. If p > − 21 , then for any real number t, we have Lp0 −t (x, y)Lp0 +t (x, y) ≤ (≥)L2p0 (x, y). Wang et al. [206] gave the best estimate of the two types of Seiffert mean P(x, y) and T(x, y) by Lp (x, y). For x, y ∈ ℝ++ , x ≠ y, we have L− 1 (x, y) < P(x, y) < L0 (x, y),

(3.5.27)

6

and L− 1 (x, y) and L0 (x, y) are the lower and upper bounds of P(x, y), respectively. We 6 also have L0 (x, y) < T(x, y) < L 1 (x, y),

(3.5.28)

3

and L0 (x, y) and L 1 (x, y) are the lower and upper bounds of T(x, y), respectively. 3

Gu and Shi [44] completely solved Schur-convexity and Schur-geometric convexity of the binary Lehmer mean Lp (x, y) with respect to (x, y) on ℝ2++ , and they discussed preliminarily the Schur-convexity of the Lehmer mean with n variables on ℝn++ . The main conclusions are the following two theorems. Theorem 3.5.23. When p ≥ 1, Lp (x, y) is Schur-convex with respect to (x, y) on ℝ2++ , and when p ≤ 1, Lp (x, y) is Schur-concave with respect to (x, y) on ℝ2++ . Theorem 3.5.24. When p ≥ 21 , Lp (x, y) is Schur-geometrically convex with respect to (x, y) on ℝ2++ , and when p ≤ 21 , Lp (x, y) is Schur-geometrically concave with respect to (x, y) on ℝ2++ . Let x = (x1 , . . . , xn ) ∈ ℝ2+ . For the Lehmer mean with n variables Lp (x) = Lp (x1 , . . . , xn ) =

∑ni=1 xip

∑ni=1 xip−1

,

Gu and Shi [44] obtained the following result. Theorem 3.5.25. Let x = (x1 , . . . , xn ) ∈ ℝn+ and p ∈ ℝ. If 1 ≤ p ≤ 2, then Lp (x) is Schurconvex with x ∈ ℝn+ ; if 0 ≤ p ≤ 1, then Lp (x) is Schur-concave with x ∈ ℝn+ . They proposed the following conjecture. Conjecture 3.5.1. When p ≥ 2, Lp (x) is Schur-convex on ℝn++ , and when p ≤ 0, Lp (x) is Schur-concave on ℝn++ . In [44], Theorem 3.5.23 and Theorem 3.5.24 are proved using Theorem 2.1.3 (Vol. 1) and Theorem 2.4.3 (Vol. 1), respectively. Now we prove Theorem 3.5.23 and Theorem 3.5.24 using Theorem 2.1.6 (Vol. 1) and Theorem 2.4.5 (Vol. 1), respectively.

3.5 Schur-convexity of other two-variable means | 139

Proof of Theorem 3.5.23. For Lp (x, a − x) =

xp + (a − x)p , + (a − x)p−1

xp−1

by computation 𝜕Lp 𝜕x

=

[xp−1

h(x) , + (a − x)p−1 ]2

where h(x) = p[xp−1 − (a − x)p−1 ][x p−1 + (a − x)p−1 ]

− (p − 1)[x p−2 − (a − x)p−2 ][x p + (a − x)p ]

= px2p−2 − p(a − x)2p−2 − (p − 1)x 2p−2 − (p − 1)xp−2 (a − x)p + (p − 1)x p (a − x)p−2 + (p − 1)(a − x)2p−2

= x2p−2 − (a − x)2p−2 − (p − 1)x p−2 (a − x)p + (p − 1)x p (a − x)p−2

= x2 (p − 1) − (a − x)2(p−1) + (p − 1)xp−2 (a − x)p−2 (x2 − (a − x)2 ). For x ≤ a2 , we have x ≤ a − x, so x2 − (a − x)2 ≤ 0. When p ≥ 1 (or ≤ 1, respectively), we have x2(p−1) − (a − x)2(p−1) ≤ 0 (or ≥ 0, respectively), so h(x) ≤ 0 (or ≥ 0, respectively); Lp (x, a − x) is decreasing (or increasing, respectively) on (−∞, a2 ), and from Theorem 2.1.6 (Vol. 1), it follows that Lp (a, b) is Schur-convex (or Schur-concave, respectively) with respect to (a, b) on ℝ2++ . Proof of Theorem 3.5.24. For xp + ( ax )p a Lp (x, ) = p−1 , x x + ( ax )p−1 by computation 𝜕Lp 𝜕x

=

k(x) , [xp−1 + ( ax )p−1 ]2

where k(x) = p(x p−1 −

p−1

ap a )[xp−1 + ( ) x xp+1

]

p

ap−1 a − (p − 1)[xp − ( ) ](xp−2 − p ) x x

ap a2p−1 − 2p ] x2 x ap a2p−1 − (p − 1)(x2(p−1) − ap−1 + 2 − 2p ) x x

= p[x 2(p−1) + ap−1 −

140 | 3 Schur-convex functions and mean value inequalities for two variables = x2(p−1) − For x ≤ √a, we have ap−1 −

ap x2

a2p−1 ap + (2p − 1)(ap−1 − 2 ). 2p x x

≤ 0, and when p ≥

1 2

(or ≤ 21 , respectively), we have 2p−1

x2(2p−1) ≤ a2p−1 (or ≥ a2p−1 , respectively) so x2(p−1) − ax2p ≤ 0 (or ≥ 0, respectively) and then k(x) ≤ 0 (or ≥ 0, respectively). This means that Lp (x, ax ) is decreasing (or increasing, respectively) on (−∞, √a), and from Theorem 2.4.5 (Vol. 1), it follows that Lp (x, ax ) is Schur-geometrically convex (or Schur-geometrically concave, respectively) with respect to (a, b) on ℝ2++ . In 2016, Fu et al. [36] pointed out that Conjecture 3.5.1 was not valid through counterexamples. In fact, for n = 3, p = 3, by computation, we have Δ := (x1 − x2 )(

𝜕L3 (x) 𝜕L3 (x) (x − x2 )2 λ(x) − ) = 21 , 𝜕x1 𝜕x2 (x1 + x22 + x32 )2

(3.5.29)

where λ(x) = λ(x1 , x2 , x3 ) = 3(x1 + x2 )(x12 + x22 + x32 ) − 2(x13 + x23 + x33 ). If taking x = (1, 3, 7), then λ(x) = −34, so that Δ < 0, but taking y = (1, 2, 3), then λ(y) = 54, so that Δ > 0. By Theorem 2.1.3 (Vol. 1), we assert that the Schur-convexity of L3 (x1 , x2 , x3 ) is not determined on the whole ℝ3+ . 1 It can easily be shown that L−2 (x1 , x2 , x3 ) = 1 1 1 , and since the Schurconvexity of L3 (x1 , x2 , x3 ) is not determined on the Fu et al. [36] obtained the following results.

L3 ( x , x , x ) 1 2 3 whole ℝ3+ , L−2 (x1 , x2 , x3 )

so is.

Theorem 3.5.26. Let x = (x1 , . . . , xn ) ∈ ℝn+ , n ≥ 2 and p ∈ ℝ. , a]n ; (a) If p ≥ 2, then for any a > 0, Lp (x) is Schur-convex with x ∈ [ (p−2)a p

(b) if p < 0, then for any a > 0, Lp (x) is Schur-concave with x ∈ [a, (p−2)a ]n . p

Theorem 3.5.27. Let x = (x1 , . . . , xn ) ∈ ℝn+ , n ≥ 2, and p ∈ ℝ. (a) If p < 21 and p ≠ 0, then for any a > 0, Lp (x) is Schur-geometrically concave with )2 a]n ; x ∈ [a, ( p−1 p

(b) if p > 21 , then for any a > 0, Lp (x) is Schur-geometrically convex with x ∈ [( p−1 )2 a, a]n . p (c) If p = 0, then Lp (x) is Schur-geometrically convex with x ∈ ℝn+ .

Theorem 3.5.28. Let x = (x1 , . . . , xn ) ∈ ℝn+ , n ≥ 2, and p ∈ ℝ. (a) If 0 ≤ p ≤ 1, then Lp (x) is Schur-harmonically convex with x ∈ ℝn+ ; if −1 ≤ p ≤ 0, then Lp (x) is Schur-harmonically concave with x ∈ ℝn+ ; , a]n ; (b) if p > 1, then for any a > 0, Lp (x) is Schur-harmonically convex with x ∈ [ (p−1)a p+1

3.5 Schur-convexity of other two-variable means | 141

(c) if p < −1, then for any a > 0, Lp (x) is Schur-harmonically concave with x ∈ [a, (p−1)a ]n . p+1 It is easy to see that Lemmas 3.5.1, 3.5.2, and 3.5.3 hold. Lemma 3.5.1. The Schur-condition (2.1.1) (Vol. 1) in Theorem 2.1.3 (Vol. 1) is equivalent to 𝜕(φx) 𝜕φ(x) ≤ (or ≥, respectively) , 𝜕xi 𝜕xi+1

i = 1, . . . , n − 1, for all x ∈ 𝔻 ∩ Ω,

(3.5.30)

where 𝔻 = {x : x1 ≤ x2 ≤ ⋅ ⋅ ⋅ ≤ xn }. The condition (2.1.1) (Vol. 1) is also equivalent to 𝜕φ(x) 𝜕φ(x) ≥ (or ≤, respectively) , 𝜕xi 𝜕xi+1

i = 1, . . . , n − 1, for all x ∈ 𝔼 ∩ Ω,

(3.5.31)

where 𝔼 = {x : x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn }. Lemma 3.5.2. The condition (2.4.1) (Vol. 1) in Theorem 2.4.3 (Vol. 1) is equivalent to xi

𝜕φ(x) 𝜕φ(x) ≤ (or ≥, respectively) xi+1 , 𝜕xi 𝜕xi+1

i = 1, . . . , n − 1, for all x ∈ 𝔻 ∩ Ω,

(3.5.32)

where 𝔻 = {x : x1 ≤ x2 ≤ ⋅ ⋅ ⋅ ≤ xn }. The condition (2.4.28) (Vol. 1) is also equivalent to xi

𝜕φ(x) 𝜕φ(x) ≥ (or ≤, respectively) xi+1 , 𝜕xi 𝜕xi+1

i = 1, . . . , n − 1, for all x ∈ 𝔼 ∩ Ω,

(3.5.33)

where 𝔼 = {x : x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn }. Lemma 3.5.3. The condition (2.4.28) (Vol. 1) in Theorem 2.4.19 (Vol. 1) is equivalent to xi2

𝜕φ(x) 2 𝜕φ(x) ≤ (or ≥, respectively) xi+1 , 𝜕xi 𝜕xi+1

i = 1, . . . , n−1, for all x ∈ 𝔻∩Ω,

(3.5.34)

where 𝔻 = {x : x1 ≤ x2 ≤ ⋅ ⋅ ⋅ ≤ xn }. The condition (2.4.28) (Vol. 1) is also equivalent to xi2

𝜕φ(x) 2 𝜕φ(x) ≥ (or ≤, respectively) xi+1 , 𝜕xi 𝜕xi+1

i = 1, . . . , n − 1, for all x ∈ 𝔼 ∩ Ω,

(3.5.35)

where 𝔼 = {x : x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn }. Lemma 3.5.4. Let x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn > 0, m ∈ ℝ. Then x1 ≥

x1m + x2m + ⋅ ⋅ ⋅ + xnm ≥ xn . x1m−1 + x2m−1 + ⋅ ⋅ ⋅ + xnm−1

(3.5.36)

142 | 3 Schur-convex functions and mean value inequalities for two variables Proof. We have x1 (x1m−1 + x2m−1 + ⋅ ⋅ ⋅ + xnm−1 ) − (x1m + x2m + ⋅ ⋅ ⋅ + xnm )

= x1m−1 (x1 − x1 ) + x2m−1 (x1 − x2 ) + ⋅ ⋅ ⋅ + xnm−1 (x1 − xn ) ≥ 0, xn (x1m−1 + x2m−1 + ⋅ ⋅ ⋅ + xnm−1 ) − (x1m + x2m + ⋅ ⋅ ⋅ + xnm )

= x1m−1 (xn − x1 ) + x2m−1 (xn − x2 ) + ⋅ ⋅ ⋅ + xnm−1 (xn − xn ) ≤ 0. We have thus proved Lemma 3.5.4. Proof of Theorem 3.5.26. Straightforward computation gives 𝜕Lp (x) 𝜕xi

=

pxip−1 ∑nj=1 xjp−1 − (p − 1)xip−2 ∑nj=1 xjp (∑nj=1 xjp−1 )2

,

i = 1, 2, . . . , n,

(3.5.37)

and then 𝜕Lp (x) 𝜕xi

−

𝜕Lp (x) 𝜕xi+1

fi (x) , n (∑i=1 xip−1 )2

=

i = 1, 2, . . . , n − 1,

(3.5.38)

where n

n

j=1

j=1

p−2 p−1 ) ∑ xjp . ) ∑ xjp−1 − (p − 1)(xip−2 − xi+1 fi (x) = p(xip−1 − xi+1

(3.5.39)

It is clear that Lp (x) is symmetric with x ∈ ℝn+ . Without loss of generality, we may assume that x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn > 0. For any a > 0, according to the integral mean value theorem, there is a ξ lying between xi and xi+1 , such that p−2 p−1 ) ) − a(p − 1)(xip−2 − xi+1 p(xip−1 − xi+1 xi

= (p − 1)p ∫ x xi+1

p−2

xi

dx − a(p − 2)(p − 1) ∫ xp−3 dx xi+1

xi

= (p − 1) ∫ [pxp−2 − a(p − 2)xp−3 ]dx xi+1

= (p − 1)[pξ p−2 − a(p − 2)ξ p−3 ](xi − xi+1 ) = (p − 1)pξ p−3 (ξ −

(p − 2)a )(xi − xi+1 ). p

(I) When p ≥ 2 and a ≥ x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn ≥

(p−2)a p

(3.5.40)

> 0, from (3.5.40), we have

p−1 p−2 p(xip−1 − xi+1 ) − a(p − 1)(xip−2 − xi+1 ) ≥ 0,

3.5 Schur-convexity of other two-variable means | 143

that is, p−1 ) p(xip−1 − xi+1

p−2 ) (p − 1)(xip−2 − xi+1

≥ a,

and then from (1.3.39) (Vol. 1), it follows that p−1 ) p(xip−1 − xi+1

p−2 ) (p − 1)(xip−2 − xi+1

namely, fi (x) ≥ 0, and then

𝜕Lp (x) 𝜕xi n

a, a] Schur-convex with x ∈ [ p−2 p (p−2)a (II) When p < 0 and p ≥

.

≥

≥ x1 ≥

𝜕Lp (x) . 𝜕xi+1

∑nj=1 xjp

∑nj=1 xjp−1

,

By Lemma 3.5.1, it follows that Lp (x) is

x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn ≥ a > 0, from (3.5.40), we have

p−2 p−1 ) ≤ 0, ) − a(p − 1)(xip−2 − xi+1 p(xip−1 − xi+1

that is, p−1 ) p(xip−1 − xi+1

p−2 ) (p − 1)(xip−2 − xi+1

≤ a,

and then from Lemma 3.5.4, it follows that p−1 ) p(xip−1 − xi+1

p−2 ) (p − 1)(xip−2 − xi+1

namely, fi (x) ≤ 0, and then

𝜕Lp (x) 𝜕xi p−2 n a] . p

≤

≤ xn ≤

𝜕Lp (x) . 𝜕xi+1

Schur-concave with x ∈ [a, The proof of Theorem 3.5.26 is complete.

∑nj=1 xjp

∑nj=1 xjp−1

,

By Lemma 3.5.1, it follows that Lp (x) is

Proof of Theorem 3.5.27. From (3.5.37), we have xi

𝜕Lp (x) 𝜕xi

− xi+1

𝜕Lp (x) 𝜕xi+1

=

gi (x) , n (∑i=1 xip−1 )2

i = 1, 2, . . . , n − 1,

(3.5.41)

where n

n

j=1

j=1

p p−1 ) ∑ xjp−1 − (p − 1)(xip−1 − xi+1 ) ∑ xjp . gi (x) = p(xip − xi+1

(3.5.42)

It is clear that Lp (x) is symmetric with x ∈ ℝn+ . Without loss of generality, we may assume that x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn > 0.

144 | 3 Schur-convex functions and mean value inequalities for two variables For any a > 0, according to the integral mean value theorem, there is a ξ lying between xi and xi+1 , such that p−1 p ) ) − a(p − 1)(xip−1 − xi+1 p(xip − xi+1 xi

2

=p ∫ x xi

p−1

xi

dx − a(p − 1) ∫ xp−2 dx 2

xi+1

xi+1

= ∫ [p2 xp−1 − a(p − 12 )x p−2 ]dx xi+1

= [p2 ξ p−1 − a(p − 1)2 ξ p−2 ](xi − xi+1 ) = p2 ξ p−2 [ξ − ( (I) When p ≥

1 2

2

p−1 ) a](xi − xi+1 ). p

(3.5.43)

and a ≥ x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn ≥ ( p−1 )2 a > 0, from (3.5.43), we have p p−1 p ) ≥ 0, ) − a(p − 1)(xip−1 − xi+1 p(xip − xi+1

that is, p p(xip − xi+1 )

p−1 ) (p − 1)(xip−1 − xi+1

≥ a,

and then from Lemma 3.5.4, it follows that p ) p(xip − xi+1

p−1 ) (p − 1)(xip−1 − xi+1

namely, gi (x) ≥ 0, and then xi

𝜕Lp (x) 𝜕xi

≥ x1 ≥

∑nj=1 xjp

∑nj=1 xjp−1

≥ xi+1

is Schur-geometrically convex with x ∈

𝜕Lp (x) . By Lemma 𝜕xi+1 p−1 2 [( p ) a, a]n .

, 3.5.2, it follows that Lp (x)

)2 a ≥ x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn ≥ a > 0, from (3.5.43), we (II) When p < 21 , p ≠ 0, and ( p−1 p have p−1 p ) ≤ 0, ) − a(p − 1)(xip−1 − xi+1 p(xip − xi+1

that is, p ) p(xip − xi+1

p−1 ) (p − 1)(xip−1 − xi+1

≤ a,

and then from (1.3.39) (Vol. 1), it follows that p ) p(xip − xi+1

p−1 ) (p − 1)(xip−1 − xi+1

≤ xn ≤

∑nj=1 xjp

∑nj=1 xjp−1

,

3.5 Schur-convexity of other two-variable means | 145

namely, gi (x) ≤ 0, and then xi

𝜕Lp (x) 𝜕xi

≤ xi+1

𝜕Lp (x) . By Lemma 𝜕xi+1 p−1 2 n [a, ( p ) a] .

3.5.2, it follows that Lp (x)

is Schur-geometrically concave with x ∈ (III) When p = 0, gi (x) ≤ 0, it follows that Lp (x) is Schur-geometrically concave with x ∈ ℝn+ . The proof of Theorem 3.5.27 is complete.

Proof of Theorem 3.5.28. From (3.5.37), we have xi2

𝜕Lp (x) 𝜕xi

2 − xi+1

𝜕Lp (x) 𝜕xi+1

=

hi (x)

(∑ni=1 xip−1 )2

i = 1, 2, . . . , n − 1,

,

(3.5.44)

where n

n

j=1

j=1

p p+1 ) ∑ xjp . ) ∑ xjp−1 − (p − 1)(xip − xi+1 hi (x) = p(xip+1 − xi+1

(3.5.45)

It is clear that Lp (x) is symmetric with x ∈ ℝn+ . Without loss of generality, we may assume that x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn > 0. (I) According to the integral mean value theorem, there is a ξ lying between xi and xi+1 , such that hi (x) =

n

∑ xjp−1 [p(xip+1 j=1 n

=

∑ xjp−1 [(p j=1 n

xi

j=1

xi+1

−

p+1 ) xi+1 xi

− (p −

p

+ 1)p ∫ x dx − p(p − 1) xi+1

=

∑ xjp−1 p[(p j=1

p

+ 1)ξ − (p − 1)

n

= ∑ xjp−1 (p + 1)pξ p−1 [ξ − j=1

Note that for −1 < p ≤ 1, ξ −

n p p−1 ∑j=1 xj n p−1 p+1 ∑j=1 x j

−

p ) xi+1

∑nj=1 xjp

∑nj=1 xjp−1

∑nj=1 xjp

]

xi

∫ xp−1 dx] ∑nj=1 xjp−1 x i+1

= ∑ xjp−1 p ∫ [(p + 1)xp − (p − n

1)(xip

∑nj=1 xjp p−1 1) n p−1 x ]dx ∑j=1 xj ∑nj=1 xjp

∑nj=1 xjp−1

ξ p−1 ](xi − xi+1 )

p n p − 1 ∑j=1 xj ](xi − xi+1 ). p + 1 ∑nj=1 xp−1 j

(3.5.46)

≥ 0.

2 p p ≥ xi+1 . By When 0 < p ≤ 1, from (3.5.46), we have hi (x) ≥ 0, and then xi2 𝜕x 𝜕xi+1 i n Lemma 3.5.3, it follows that Lp (x) is Schur-harmonically convex with x ∈ ℝ+ . 𝜕L (x)

𝜕L (x)

2 p p When −1 < p ≤ 0, hi (x) ≤ 0, and then xi2 𝜕x ≤ xi+1 . By Lemma 3.5.3, it 𝜕xi+1 i n follows that Lp (x) is Schur-harmonically concave with x ∈ ℝ+ . −1 When p = −1, hi (x) = 2 ∑nj=1 xj−1 (xi−1 − xi+1 ) ≤ 0, it follows that Lp (x) is Schurn harmonically concave with x ∈ ℝ+ . 𝜕L (x)

𝜕L (x)

146 | 3 Schur-convex functions and mean value inequalities for two variables (II) For any a > 0, according to the integral mean value theorem, there is a ξ lying between xi and xi+1 , such that p+1 p p(xip+1 − xi+1 ) − a(p − 1)(xip − xi+1 ) xi

xi

= p(p + 1) ∫ x dx − a(p − 1)p ∫ xp−1 dx p

xi+1

xi+1

xi

= p ∫ [(p + 1)xp − a(p − 1)x p−1 ]dx xi+1

= p[(p + 1)ξ p − a(p − 1)ξ p−1 ](xi − xi+1 ) = p(p + 1)ξ p−1 [ξ −

(p − 1)a ](xi − xi+1 ). p+1

When p ≥ 1 and a ≥ x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn ≥

p−1 a p+1

(3.5.47)

> 0, from (3.5.47), we have

p p+1 ) ≥ 0, ) − a(p − 1)(xip − xi+1 p(xip+1 − xi+1

that is, p+1 ) p(xip+1 − xi+1

p ) (p − 1)(xip − xi+1

≥ a,

and then from Lemma 3.5.4, it follows that p+1 ) p(xip+1 − xi+1

p ) (p − 1)(xip − xi+1

namely, hi (x) ≥ 0, and then xi2

𝜕Lp (x) 𝜕xi

2 ≥ xi+1

is Schur-harmonically convex with x ∈ (III) When p < −1 and

p−1 a p+1

≥ x1 ≥

∑nj=1 xjp

∑nj=1 xjp−1

𝜕Lp (x) . By 𝜕xi+1 p−1 [ p+1 a, a]n .

,

Lemma 3.5.3, it follows that Lp (x)

≥ x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ xn ≥ a > 0, from (3.5.47), we have

p p+1 ) ≤ 0, ) − a(p − 1)(xip − xi+1 p(xip+1 − xi+1

that is, p+1 ) p(xip+1 − xi+1

p ) (p − 1)(xip − xi+1

≤ a,

and then from (1.3.39) (Vol. 1), it follows that p+1 ) p(xip+1 − xi+1

p (p − 1)(xip − xi+1 )

≤ xn ≤

∑nj=1 xjp

∑nj=1 xjp−1

,

3.5 Schur-convexity of other two-variable means | 147

namely, hi (x) ≤ 0, and then xi2

𝜕Lp (x) 𝜕xi

2 ≤ xi+1

𝜕Lp (x) . By 𝜕xi+1 p−1 n [a, p+1 a] .

is Schur-harmonically concave with x ∈ The proof of Theorem 3.5.28 is complete.

Lemma 3.5.3, it follows that Lp (x)

In 2018 Ku and Zhao [75] studied the m-order Schur-power convexity of the Lehmer mean with n variables. First, we define six regions of the double-parameter space (m, p) in ℝ2 . We have D1 = {(m.p) | p(p − m) ≥ 0 ≥ (p − 1)(p − m − 1)},

D2 = {(m.p) | p(p − m) > (p − 1)(p − m − 1) > 0},

D3 = {(m.p) | 0 > p(p − m) > (p − 1)(p − m − 1)}, E1 = {(m.p) | p(p − m) ≤ 0 ≤ (p − 1)(p − m − 1)},

E2 = {(m.p) | 0 < p(p − m) < (p − 1)(p − m − 1)},

E3 = {(m.p) | p(p − m) < (p − 1)(p − m − 1) < 0}. Theorem 3.5.29. Let x = (x1 , . . . , xn ) ∈ ℝn+ , n ≥ 2, and p ∈ ℝ. (a) If (m, p) ∈ D1 , then Lp (x) is m-order Schur-power convex with x ∈ ℝn++ ; (b) if (m, p) ∈ E1 , then Lp (x) is m-order Schur-power concave with x ∈ ℝn++ ; (c) if (m, p) ∈ D2 , then for any a > 0, Lp (x) is m-order Schur-power convex with x [ (p−1)(p−m−1) a, a]n ; p(p−m) (d) if (m, p) ∈ E2 , then for any a > 0, Lp (x) is m-order Schur-power concave with x [a, (p−1)(p−m−1) a]n ; p(p−m) (e) if (m, p) ∈ D3 , then for any a > 0, Lp (x) is m-order Schur-power convex with x [a, (p−1)(p−m−1) a]n ; p(p−m) (f) if (m, p) ∈ E3 , then for any a > 0, Lp (x) is m-order Schur-power concave with x a, a]n . [ (p−1)(p−m−1) p(p−m)

∈ ∈ ∈ ∈

3.5.6 A strange mean Question 11031 of the American Mathematical Monthly Let x, y > 0, mean M(x, y) = log N(x, y). Define the monster mean M(x, y) of two positive real numbers to be log N(x, y), where N(x, y) is the function 1 + log(√1 + f (x, y) + √f (x, y))

1 − log(√1 + f (x, y) + √f (x, y))

,

and f (x, y) =

(e2(e

x

−1)⋅(ex +1)−1

4e((e

− 1)(e2(e

y

−1)⋅(ey +1)−1

x −1)⋅(ex +1)−1 +(ey −1)⋅(ey +1)−1 )

− 1)

.

148 | 3 Schur-convex functions and mean value inequalities for two variables Prove or disprove M(x, y) ≤ G(x, y) = √xy.

(3.5.48)

In [272, pp. 118–121] Zhang proved the above inequality by investigating the Schurgeometric concavity of the function g(x, y) = √1 + f (x, y) + √f (x, y). Li and Shi [83] pointed out that M(x, y) can be expressed in the composite form of the hyperbolic function and the arc-hyperbolic function as follows: y x M(x, y) = 2 tanh−1 sinh−1 √sinh(tanh( )) sinh(tanh( )). 2 2

(3.5.49)

By using geometric concavity of the hyperbolic function sinh(tanh x) they also gave another proof. Using the expression (3.5.49), the following results can be proved. Theorem 3.5.30. The function x y M(x, y) = 2 tanh−1 sinh−1 √sinh(sin(tanh )) sinh(tanh ) 2 2 is increasing, Schur-concave and Schur-geometrically concave on ℝ2+ . To prove this, we first prove two lemmas. Lemma 3.5.5. The function g(x) = [cosh2 x tanh(tanh x)]

−1

is decreasing on ℝ++ . Proof. Because g 󸀠 (x) = −

2 sinh x cosh x tanh(tanh x) + cosh2 x[cosh(tanh x) cosh x]−2 cosh4 x tanh2 (tanh x)

2

=−

sinh x ⋅ tanh(tanh x) ⋅ cosh2 (tanh x) + 1 cosh4 x ⋅ sinh2 (tanh x)

≤ 0,

g(x) is decreasing on ℝ++ . Lemma 3.5.6. The function h(x) = xg(x) = x[cosh2 x tanh(tanh x)]

−1

is decreasing on ℝ++ .

3.5 Schur-convexity of other two-variable means | 149

Proof. We have h(x)󸀠 =

4

q(x)

cosh x tanh2 (tanh x)

,

where q(x) = cosh2 x tanh(tanh x) − x sinh 2x tanh(tanh x) − x cosh2 x[cosh(tanh x) ⋅ cosh x] . −2

We only prove that q(x) ≤ 0. We have q(x) = (cosh2 x − x sinh2 x) tanh(tanh x) − x[cosh(tanh x)]

−2

sinh(tanh x) −2 − x[cosh(tanh x)] cosh(tanh x) sinh(tanh x) cosh(tanh x) −2 − x[coth(tanh x)] = (cosh2 x − x sinh 2x) 2 cosh (tanh x)

= (cosh2 x − x sinh 2x)

= (coth2 x − x sinh 2x) sinh(tanh x) cosh(tanh x) − x = sinh(2 tanh x)(cosh2 x − x sinh 2x) − 2x

= sinh(2 tanh x) cosh2 x(1 − 2x tanh x) − 2x

≤ sinh(2 tanh x) cosh2 x(1 − 2 tanh2 x) − 2 tanh x (note tanh x < x)

= sinh(2 tanh x)(cosh2 x − 2 sinh2 x) − 2 tanh x cosh2 x − 2 sinh2 x

− 2 tanh x cosh2 x − sinh2 x tanh2 x = sinh(2 tanh x)(1 − ) − 2 tanh x 1 − tanh2 x +∞ (2 tanh x)2k+1 (1 − tanh2 x − tanh4 x − ⋅ ⋅ ⋅ − tanh2n x − ⋅ ⋅ ⋅) − 2 tanh x = ∑ (2k + 1)! k=0

= sinh(2 tanh x)

(2 tanh x)2k+1 (1 − tanh2 x) − 2 tanh x] (2k + 1)! k=0 +∞

= [∑

+∞

(2 tanh x)2k+1 (2k + 1)! k=0 +∞

− ( ∑ tanh2k x) ∑ k=2

(2 tanh x)2k+1 +∞ 22k+1 (tanh x)2k+3 −∑ − 2 tanh x] (2k + 1)! (2k + 1)! k=0 k=0 +∞

= [∑

+∞

(2 tanh x)2k+1 (2k + 1)! k=0 +∞

− ( ∑ tanh2k x) ∑ k=2

+∞

= [∑

k=1

22k+1 (tanh x)2k+1 +∞ 22k+1 (tanh x)2k+3 −∑ ] (2k + 1)! (2k + 1)! k=0

+∞

(2 tanh x)2k+1 (2k + 1)! k=0 +∞

− ( ∑ tanh2k x) ∑ k=2

150 | 3 Schur-convex functions and mean value inequalities for two variables 22k+3 (tanh x)2k+3 +∞ 22k+1 (tanh x)2k+3 −∑ ] (2k + 3)! (2k + 1)! k=0 k=0 +∞

= [∑

+∞

(2 tanh x)2k+1 (2k + 1)! k=0 +∞

− ( ∑ tanh2k x) ∑ k=2 2k+3

[

− 22k+1 (2k + 2)(2k + 3) ](tanh x)2k+3 (2k + 3)!

2

+∞

(2 tanh x)2k+1 (2k + 1)! k=0 +∞

− ( ∑ tanh2k x) ∑ k=2 +∞ 2k+1

= − ∑[

2

k=0

+∞

(4k 2 + 10k + 2) ](tanh x)2k+3 (2k + 3)! (2 tanh x)2k+1 ≤ 0, (2k + 1)! k=0 +∞

− ( ∑ tanh2k x) ∑ k=2

so h(x) is decreasing on ℝ2++ . Proof of Theorem 3.5.30. We have y x 𝜕M cosh(tanh 2 ) sinh(tanh 2 ) = 𝜕x F(x, y) cosh2 x2

and y x 𝜕M cosh(tanh 2 ) sinh(tanh 2 ) = , 𝜕y F(x, y) cosh2 y2

where 2

x y F(x, y) = 2{1 − [sinh−1 √sinh(tanh ) sinh(tanh )] } 2 2 x x y y ⋅ √1 + sinh(tanh ) sinh(tanh ) ⋅ √sinh(tanh ) sinh(tanh ). 2 2 2 2 Since cosh t > 0, sinh t > 0, tanh t > 0 for t > 0, and 0 < tanh t < 1 implies 0 < sinh(tanh t) < sinh 1, for x > 0, y > 0 we have y x 0 < sinh(tanh ) sinh(tanh ) < sinh2 1 2 2

2

y x ⇒ 1 − [sinh−1 √sinh(tanh ) sinh(tanh )] > 0 2 2 ⇒ F(x, y) > 0.

3.5 Schur-convexity of other two-variable means | 151

So

𝜕M 𝜕x

≥ 0,

𝜕M 𝜕y

≥ 0. Therefore M(x, y) is decreasing on ℝ2++ . We have

Δ := (x − y)( = =

x−y [ F(x, y)

=

cosh2

x 2

(x − y) sinh(tanh x2 ) sinh(tanh y2 ) [

=

𝜕M 𝜕M − ) 𝜕x 𝜕y cosh(tanh x2 ) sinh(tanh y2 )

F(x, y) cosh tanh x2

cosh2

x 2

sinh(tanh x2 )

−

cosh(tanh y2 ) sinh(tanh x2 ) cosh2

cosh2

y 2

]

⋅

cosh(tanh y2 )

(x − y) sinh(tanh x2 ) sinh(tanh F(x, y)

−

y sinh(tanh y2 ) 2 y ) 2

]

[g(x) − g(y)]

(x − y)2 sinh(tanh x2 ) sinh(tanh y2 ) g(x) − g(y) ⋅ . F(x, y) x−y

By Lemma 3.5.5, when x > 0, the function g(x) = [cosh2 g(x)−g(y) x−y

x 2

tanh(tanh x2 )]−1 is decreas-

≤ 0. Furthermore, Δ ≤ 0, so M(x, y) is a Schur-concave function on ℝ2+ . ing, so Theorem 3.5.30 is proved. In 2016, He [60] defined the following function, which is similar to M(x, y): x π y H(x, y) = 2 tan−1 sin−1 √sin(tan ) sin(tan ), (x, y) ∈ (0, 2 tan−1 ). 2 2 2

(3.5.50)

The Schur-power convexity of M(x, y) and H(x, y) was also studied by means of maple mathematics and polynomial discriminants [250], [249]. The following results were obtained. Theorem 3.5.31. The function M(x, y) is an m-order Schur-power concave function on ℝ2++ if and only if m ≥ 0. Theorem 3.5.32. The function H(x, y) is an m-order Schur-power concave function on (0, 2 tan−1 π2 )2 if and only if m ≥ 1 − p(t0 ) ≈ 0.0862, where p(t) =

t[2 − sin 2t sin(2 tan t)] , cos 2t sin(2 tan t)

and t0 is the only stable point of p(t) in the interval (0, 2 tan−1 π2 ). In 2017, He and Li [63] defined the following two complex means of hyperbolic functions and inverse hyperbolic functions. By using the analytical method, their Schur-power convexity was studied, and the necessary and sufficient conditions for

152 | 3 Schur-convex functions and mean value inequalities for two variables their determination were given. We have 1 x 1 y M1 (x, y) = 2 tanh−1 sinh−1 [ sinh(tanh ) + sinh(tanh )] x, y ∈ (0, +∞), 2 2 2 2 (3.5.51) x 1 y 1 M2 (x, y) = 2 sinh−1 tanh−1 [ tanh(sinh ) + tanh(sinh )] x, y ∈ (0, +∞). 2 2 2 2 (3.5.52) Theorem 3.5.33. Functions M1 (x, y) and M2 (x, y) are Schur-m-order power concave with (x, y) ∈ ℝ2++ if and only if m ≥ 1. In 2018, combining the power mean Mp (x, y) with the strange mean M(x, y), He [61] gave a parametric generalization of M(x, y), and for (x, y) ∈ ℝ2++ , defined the following new hyperbolic mean: x y Mp∗ (x, y) = 2 tanh−1 sinh−1 [Mp (sinh(tanh ), sinh(tanh ))] 2 2

(3.5.53) 1

{2 tanh−1 sinh−1 {[ 21 sinhp (tanh x2 ) + 21 sinhp (tanh y2 )] p }, if p > 0, ={ 2 tanh−1 sinh−1 √sinh(tanh x2 ) sinh(tanh y2 ), if p = 0. { He studied the Schur-power convexity of Mp∗ (x, y) on ℝ2++ and gave sufficient and necessary conditions for its determination. Theorem 3.5.34. The function Mp∗ (x, y) is Schur-m-order power concave with (x, y) ∈ ℝ2++ if and only if m ≥ p. He also proposed the following three questions. Problem 3.5.1. For the mean Mp∗ (x, y), where the range of p can be extended to ℝ, give the necessary and sufficient conditions for Schur-m-power convexity of Mp∗ (x, y) (p ∈ ℝ) with (x, y) ∈ ℝ2++ . Problem 3.5.2. Similar to the form of Mp∗ (x, y), for (x, y) ∈ ℝ2++ , we can define the following new hyperbolic mean: y x Mp∗∗ (x, y) = 2 sinh−1 tanh−1 [Mp (tanh(sinh ), tanh(sinh ))] 2 2

(3.5.54) 1

{2 sinh−1 tanh−1 {[ 21 tanhp (sinh x2 ) + 21 tanhp (sinh y2 )] p }, ={ 2 sinh−1 tanh−1 √tanh(sinh x2 ) tanh(sinh y2 ), {

if p ≠ 0,

if p = 0.

Give the necessary and sufficient conditions for Schur-m-power convexity of Mp∗∗ (x, y) (p ∈ ℝ) with (x, y) ∈ ℝ2++ .

3.5 Schur-convexity of other two-variable means | 153

Problem 3.5.3. Combining the power mean Mp (x, y) with the strange mean H(x, y), for (x, y) ∈ (0, 2 tan−1 π2 )2 , a new triangle mean is defined as follows: y x Hp∗ (x, y) = 2 tan−1 sin−1 [Mp (sin(tan ), sin(tan ))] 2 2

(3.5.55) 1

{2 tan−1 sin−1 {[ 21 sinp (tan x2 ) + 21 sinp (tan y2 )] p }, if p > 0, ={ 2 tan−1 sin−1 √sin(tan x2 ) sin(tan y2 ), if p = 0, { give the necessary and sufficient conditions for Schur-m-power convexity of Hp∗ (x, y) (p ∈ ℝ) with (x, y) ∈ (0, 2 tan−1 π2 )2 . Problem 3.5.4. Similar to the form of Hp∗ (x, y), for (x, y) ∈ (0, π)2 , we can define the following new triangle mean: y x Hp∗∗ (x, y) = 2 sin−1 tan−1 [Mp (tan(sin ), tan(sin ))] 2 2

(3.5.56) 1

{2 sin−1 tan−1 {[ 21 tanp (sin x2 ) + 21 tanp (sin y2 )] p }, ={ 2 sin−1 tan−1 √tan(sin x2 ) tan(sin y2 ), {

if p ≠ 0,

if p = 0.

Give the necessary and sufficient conditions for Schur-m-power convexity of Mp∗∗ (x, y) (p ∈ ℝ) with (x, y) ∈ (0, π)2 . 3.5.7 Toader type integral mean In 1998, Gh. Toader defined an integral mean Mg,n (a, b) as follows [196]: Mg,n (a, b) = g −1 [f (a, b; g, n)],

(3.5.57)

where g : ℝ++ → ℝ is a strictly monotonic function, and f (a, b; g, n) is defined as follows: 1

f (a, b; g, r) = { 2π 1 2π

2π

1

∫0 g[(an cos2 θ + br sin2 θ) r ]dθ, 2π

2

2

∫0 g(acos θ bsin θ )dθ,

r ≠ 0,

r = 0.

(3.5.58)

In 2014, Li and Zhang [92], taking a kernel function as linear function g(x) = kx + c(k ≠ 0), obtained the following linear kernel Toader mean: Mr (a, b) = where r ∈ (−∞, +∞).

1 { 2π 1 2π

2π

1

∫0 (ar cos2 θ + br sin2 θ) r dθ, 2π ∫0

cos2 θ sin2 θ

a

b

dθ,

r ≠ 0,

r = 0,

(3.5.59)

154 | 3 Schur-convex functions and mean value inequalities for two variables By using the property of definite integral, Mr (a, b) can equivalently be transformed into π

1

{ 2 ∫ 2 (ar cos2 θ + br sin2 θ) r dθ, r ≠ 0 Mr (a, b) = { π 0π 2 2 cos2 θ sin2 θ b dθ, r = 0, { π ∫0 a

(3.5.60)

where r ∈ (−∞, +∞). Li and Zhang proved the following. Theorem 3.5.35 ([92]). When r ≥ 1, Mr (a, b) is Schur-convex on ℝ2++ ; when r ≤ 1, Mr (a, b) is Schur-concave on ℝ2++ . Theorem 3.5.36 ([92]). When r ≥ 21 , Mr (a, b) is Schur-geometrically convex on ℝ2++ . In the second issue of the 2001 Communication Research on Inequalities (China), Li and Zhang proposed a Toader type integral mean Tr,m (a, b) as follows: π

m

m

1

2 cos θ sin θ r r { π ∫02 (a cosm θ+sinm θ + b cosm θ+sinm θ ) r dθ, r ≠ 0, f (a, b; g, n) = { m m π sin θ cos θ 2 2 m m m m r = 0, { π ∫0 g(a cos θ+sin θ b cos θ+sin θ )dθ,

(3.5.61)

where r ∈ (−∞, +∞), m ∈ [0, +∞). Li and Zhang obtained the following results. Theorem 3.5.37. When r ≥ 1, Tr,m (a, b) is Schur-convex on ℝ2++ ; when r ≤ 1, Tr,m (a, b) is Schur-concave on ℝ2++ . Corollary 3.5.2. Let the circumference of the ellipse

x2 a2

+

y2 b2

= 1 (a ≥ b > 0) be L. Then

π(a + b) ≤ L ≤ 4a + πb.

(3.5.62)

Proof. From (a, b) ≺ (a +

b b , ), 2 2

it follows that T2,2 (a, b) ≤ T2,2 (a +

b b , ), 2 2

and then π 2

L = 4 ∫ √a2 cos2 θ + b2 sin2 θdθ = 2πT2,2 (a, b) ≤ 2πT2,2 (a + 0

π 2

= 4 ∫ √(a + 0

2

2

b b ) cos2 θ + ( ) sin2 θdθ 2 2

b b , ) 2 2

3.5 Schur-convexity of other two-variable means | 155 π 2

≤ 4 ∫(a cos θ + 0

b )dθ = 4a + πb. 2

In 2018, Li and Shan [91], taking a kernel function as logarithmic function g(x) = logp (x) (0 < p ≠ 1), obtained the following logarithmic kernel Toader mean: 2π

1 r ≠ 0, exp[ 2πr ∫0 log(ar cos2 θ + br sin2 θ)dθ], Lr (a, b) = { 1 2π 2 2 exp[ 2π ∫0 (log a ⋅ cos θ + log b ⋅ sin θ)dθ], r = 0,

(3.5.63)

where r ∈ (−∞, +∞). By using the property of definite integral, Lr (a, b) can equivalently be transformed into π

{exp[ 2 ∫ 2 log(ar cos2 θ + br sin2 θ)dθ], r ≠ 0 πr 0 Lr (a, b) = { √ab, r = 0, {

(3.5.64)

where r ∈ (−∞, +∞). Li and Shan proved the following. Theorem 3.5.38 ([91]). When r ≤ Lr (a, b) is Schur-concave on ℝ2++ .

1 , 2

Lr (a, b) is Schur-concave on ℝ2++ ; when r ≤ 1,

Theorem 3.5.39 ([91]). When r ≥ 0, Lr (a, b) is Schur-geometrically convex on ℝ2++ ; when r ≤ 0, Lr (a, b) is Schur-geometrically concave on ℝ2++ . 3.5.8 Elliptic Neuman mean For (x, y) ∈ ℝ2++ and k ∈ [0, 1], the elliptic Neuman mean is defined as √y2 −x 2 { { x < y, { −1 ( y ,k) , { cn { x { Nk (x, y) = {x, x = y, { { { 2 2 { { √x −y , y < x, −1 y { nc ( x ,k)

(3.5.65)

where 1

cn (x, k) = ∫

du

−1

x

√(1 −

u2 )(k 󸀠 2

+ k 2 u2 )

(3.5.66)

and x

nc (x, k) = ∫ −1

1

√(u2

du

− 1)(k 2 + k 󸀠 2 u2 )

(3.5.67)

156 | 3 Schur-convex functions and mean value inequalities for two variables are the inverse functions of Jacobian elliptic functions cn and nc (see [8]), respectively, and k 󸀠 = √1 − k 2 . In particular, π 2

cn (0, k) = K(k) = ∫ −1

0

du . √1 − k 2 sin2 t

(3.5.68)

Song et al. [186] studied the Schur-convexity of the elliptic Neuman mean. Theorem 3.5.40. The elliptic Neuman mean Nk (x, y) is strictly Schur-convex on ℝ2++

if and only if k ≤ 22 and strictly Schur-concave on ℝ2++ if and only if k > k0 ∈ (0.897, 0.898). Here, k0 is the unique solution of the equation K(k) − √ 1 2 = 0. √

1−k

Theorem 3.5.41. The elliptic Neuman mean Nk (x, y) is strictly Schur-geometrically con√ vex on ℝ2++ if and only if k ≤ 22 , and Nk (x, y) is not Schur-geometrically concave on ℝ2++ for any

√2 2

< k ≤ 1.

Theorem 3.5.42. The elliptic Neuman mean Nk (x, y) is strictly Schur-harmonically con√ vex on ℝ2++ if and only if k ≤ 22 , and Nk (x, y) is not Schur harmonically concave on ℝ2++ for any

√2 2

< k ≤ 1.

3.5.9 Oscillatory mean For a > b > 0 and α ∈ (0, 1), the oscillatory mean and its dual are defined as follows: O = O(a, b; α) = α√ab + (1 − α)(

a+b ) 2

and α

O(d) = O(d) (a, b; α) = (ab) 2 (

1−α

a+b ) 2

.

For a > b > 0, r being a real number, and α ∈ (0, 1), the rth oscillatory mean and its dual are defined as follows: 1

ar + br r a+b O = O(a, b; α, r) = α( ) + (1 − α)( ) 2 2 and α

O(d) = O(d) (a, b; α, r) = (

1−α

ar + br r a + b ) ( ) 2 2

.

Nagaraja and Sampathkumar [118] studied the Schur-convexity, Schur-geometrical convexity, and Schur-harmonic convexity of the oscillatory mean and its dual and obtained the following conclusions.

3.5 Schur-convexity of other two-variable means | 157

Theorem 3.5.43. For a > b > 0 and α ∈ (0, 1), the oscillatory mean O(a, b; α) and its dual are Schur-concave, Schur-geometrically convex, and Schur-harmonically convex. Theorem 3.5.44. For a > b > 0, r ≥ 1, and α ∈ (0, 1), the rth oscillatory mean O(a, b; α, r) and its dual are Schur-convex. Theorem 3.5.45. For a > b > 0, r ≥ 0, and α ∈ (0, 1), the rth oscillatory mean O(a, b; α, r) and its dual are Schur-geometrically convex. Theorem 3.5.46. For a > b > 0, r ≥ −1, and α ∈ (0, 1), the rth oscillatory mean O(a, b; α, r) and its dual are Schur-harmonically convex.

3.5.10 Information entropy Let pi ≥ 0 (i = 1, 2, . . . , n) and ∑ni=1 pi = 1. The function (see [109, p. 101]) n

H(p1 , . . . , pn ) = − ∑ pi log pi i=1

(3.5.69)

is called the entropy of p, or the Shannon information entropy of p (here x log x = 0 for x = 0). Then H(p) is strictly Schur-concave, and consequently, H(p) ≥ H(q) whenever p ≺ q, and in particular, 1 1 H(1, 0, . . . , 0) ≤ H(p) ≤ ( . . . , ). n n

(3.5.70)

Let t ∈ ℝ and t ≠ 0. The function is defined as follows [229]: Ht (p1 , p2 ) = −

pt1 log pt1 + pt2 log pt2 pt1 + pt2

(3.5.71)

for (p1 , p2 ) ∈ ℝ2++ . Particularly, if pi > 0 (i = 1, 2) and p1 +p2 = 1, then Ht (p1 , p2 ) is information entropy. In 2011, Xi et al. [229] studied Schur-convexity, Schur-geometric convexity, and Schur-harmonic convexity of Ht (p1 , p2 ) on ℝ2++ . Theorem 3.5.47. Let (p1 , p2 ) ∈ ℝ2++ and t < 0 or t ≥ 21 . Then Ht (p1 , p2 ) is Schur-concave on ℝ2++ . Theorem 3.5.48. Let (p1 , p2 ) ∈ ℝ2++ and t ∈ ℝ, t ≠ 0. Then Ht (p1 , p2 ) is Schurgeometrically concave on ℝ2++ . Theorem 3.5.49. Let (p1 , p2 ) ∈ ℝ2++ and t > 0 or t < − 21 . Then Ht (p1 , p2 ) is Schurharmonically concave on ℝ2++ .

158 | 3 Schur-convex functions and mean value inequalities for two variables

3.6 Schur-convexity for difference of some means 3.6.1 Convexity and Schur-convexity for difference of some means Taneja [192] gave the following inequality chain about the binary means: H(x, y) ≤ G(x, y) ≤ N1 (x, y) ≤ N3 (x, y) ≤ N2 (x, y) ≤ A(x, y) ≤ S(x, y),

(3.6.1)

where a+b , 2 G(a, b) = √ab,

A(a, b) =

H(a, b) = 2

2ab , a+b

√a + √b A(a, b) + G(a, b) ) = , 2 2 a + √ab + b 2A(a, b) + G(a, b) = , N3 (a, b) = 3 3 N1 (a, b) = (

N2 (a, b) = (

√a + √b a+b )(√ ), 2 2

S(a, b) = √

a2 + b2 . 2

The means A(a, b), G(a, b), H(a, b), S(a, b), N1 (a, b), and N3 (a, b) are arithmetic, geometric, harmonic, root-square, square root, and Heron’s means, respectively. Furthermore, the following differences of means are considered in [27]: MSA (a, b) = S(a, b) − A(a, b),

MSN2 (a, b) = S(a, b) − N2 (a, b),

(3.6.2) (3.6.3)

MSN3 (a, b) = S(a, b) − N3 (a, b),

(3.6.4)

MSN1 (a, b) = S(a, b) − N1 (a, b),

(3.6.5)

MSG (a, b) = S(a, b) − G(a, b),

(3.6.6)

MAN2 (a, b) = A(a, b) − N2 (a, b),

(3.6.8)

MAH (a, b) = A(a, b) − H(a, b),

(3.6.10)

MN2 G (a, b) = N2 (a, b) − G(a, b),

(3.6.12)

MSH (a, b) = S(a, b) − H(a, b),

MAG (a, b) = A(a, b) − G(a, b),

MN2 N1 (a, b) = N2 (a, b) − N1 (a, b), and the following theorems are established.

(3.6.7) (3.6.9) (3.6.11)

3.6 Schur-convexity for difference of some means | 159

Theorem 3.6.1. The differences of means given by (3.6.2)–(3.6.12) are nonnegative and convex on R2++ . Theorem 3.6.2. The following inequalities among the mean differences hold: 1 1 1 M (a, b) ≤ MAH (a, b) ≤ MSG (a, b) ≤ MAG (a, b), 3 SH 2 2 1 1 1 M (a, b) ≤ MN2 N1 (a, b) ≤ MN2 G (a, b) ≤ MAG (a, b) ≤ MAN2 (a, b), 8 AH 3 4 4 MSA (a, b) ≤ MSN2 (a, b) ≤ 4MAN2 (a, b), 5 3 MSH (a, b) ≤ 2MSN1 (a, b) ≤ MSG (a, b), 2 3 2 MSA (a, b) ≤ MSN3 (a, b) ≤ MSN1 (a, b). 4 3 MSA (a, b) ≤

(3.6.13) (3.6.14) (3.6.15) (3.6.16) (3.6.17)

Remark 3.6.1. The differences between the mean values given by equations (3.6.2)– (3.6.12) are clearly symmetric. It is known from Theorem 3.6.1 and Corollary 2.2.1 (Vol. 1) that the differences between these mean values are Schur-convex on ℝ2++ . 3.6.2 Schur-geometric convexity for difference of some means Shi et al. [172] obtained the following theorem and, based on this, established some inequalities about the mean difference. Theorem 3.6.3. The mean differences in (3.6.2)–(3.6.12) are all Schur-geometrically convex on ℝ2++ . Theorem 3.6.4. Let x > 0, y > 0,

1 2

≤ t ≤ 1 or 0 ≤ t ≤ 21 . Then

0≤√

xt y1−t + x1−t yt √ x2 + y2 x + y xt y(1−t) + x(1−t) yt − ≤ − , 2 2 2 2

(3.6.18)

0≤√

2 2 2 2 √xt y1−t + √x1−t yt xt y(1−t) + x(1−t) yt −( ) 2 2

(3.6.19)

2

⋅ (√ 0≤√ ≤√ 0≤

2

2

2

xt y1−t + x1−t yt x2 + y2 √x + √y x+y )≤√ − ⋅√ , 2 2 2 2

xt y1−t + √xy + x1−t yt xt y(1−t) + x(1−t) yt − 2 3 2

2

2

2

(3.6.20)

x2 + y2 x + √xy + y − , 2 3

√xt y1−t + √x1−t yt xt y1−t + x1−t yt −( ) 2 2

(3.6.21)

160 | 3 Schur-convex functions and mean value inequalities for two variables

⋅ (√

x + y √x + √y xt y1−t + x1−t yt x+y )≤ − ⋅√ , 2 2 2 2

and 0≤(

√xt y1−t + √x1−t yt

−(

2

) ⋅ (√

√xt y1−t + √x1−t yt 2

2

) ≤

xt y1−t + x1−t yt ) 2

(3.6.22) 2

√x + √y √x + √y x+y ⋅√ −( ). 2 2 2

Proof. By (1.4.32) (Vol. 1) we have (log √xy, log √xy) ≺ (log(yt x1−t ), log(x t y1−t )) ≺ (log x, log y),

(3.6.23)

and by Theorem 3.6.3, the difference of the means in (3.6.2) MSA (x, y) = S(x, y) − A(x, y) = √

x2 + y2 x + y − 2 2

is Schur-geometrically convex on R2++ , so we obtain MSA (xy, xy) ≤ MSA (yt x1−t , xt y1−t ) ≤ MSA (x, y). That is, (3.6.18) holds. Similarly, according to the Schur-geometric convexity of the mean differences in (3.6.3), (3.6.4), (3.6.8), and (3.6.11) on ℝ2++ , from (3.6.23), the inequalities (3.6.20), (3.6.21), and (3.6.22) can be obtained. Remark 3.6.2. The inequalities (3.6.18) and (3.6.19) strengthen the inequalities A(x, y) ≤ S(x, y) and N2 (x, y) ≤ A(x, y) in (3.6.1), respectively. Wu et al. [226] defined a binary mean, i. e., M1 (x, y) =

xy A(x, y) + H(x, y) x + y = + , 2 4 x+y

(3.6.24)

and considered the following mean difference with M1 (x, y): MSM1 (x, y) = S(x, y) − M1 (x, y),

(3.6.25)

MAM1 (x, y) = A(x, y) − M1 (x, y),

(3.6.26)

MN3 M1 (x, y) = N3 (x, y) − M1 (x, y),

(3.6.28)

MN1 M1 (x, y) = N1 (x, y) − M1 (x, y),

(3.6.29)

MN2 M1 (x, y) = N2 (x, y) − M1 (x, y),

MM1 G (x, y) = M1 (x, y) − G(x, y),

MM1 H (x, y) = M1 (x, y) − H(x, y).

(3.6.27)

(3.6.30) (3.6.31)

Theorem 3.6.5. The differences of means in (3.6.25)–(3.6.31) are all Schur-geometrically convex on ℝ2++ .

3.6 Schur-convexity for difference of some means | 161

3.6.3 Schur-geometric and harmonic convexity for difference of some means For the differences of means in Theorem 3.6.2, Shi et al. [172] further studied the Schurgeometric convexity of the difference between these differences in order to further improve the inequalities in Theorem 3.6.2. The main result of this paper reads as follows. Theorem 3.6.6. The following differences of means are Schur-geometrically convex on ℝ2++ : 1 DSH−SA (a, b) = MSH (a, b) − MSA (a, b), (3.6.32) 3 1 1 DAH−SH (a, b) = MAH (a, b) − MSH (a, b), (3.6.33) 2 3 DSG−AH (a, b) = MSG (a, b) − MAH (a, b), (3.6.34) 1 (3.6.35) DAG−SG (a, b) = MAG (a, b) − MSG (a, b), 2 1 (3.6.36) DN2 N1 −AH (a, b) = MN2 N1 (a, b) − MAH (a, b), 8 1 DN2 G−N2 N1 (a, b) = MN2 G (a, b) − MN2 N1 (a, b), (3.6.37) 3 1 1 DAG−N2 G (a, b) = MAG (a, b) − MN2 G (a, b), (3.6.38) 4 3 1 (3.6.39) DAN2 −AG (a, b) = MAN2 (a, b) − MAG (a, b), 4 4 DSN2 −SA (a, b) = MSN2 (a, b) − MSA (a, b), (3.6.40) 5 4 DAN2 −SN2 (a, b) = 4MAN2 (a, b) − MSN2 (a, b), (3.6.41) 5 (3.6.42) DSN1 −SH (a, b) = 2MSN1 (a, b) − MSH (a, b), 3 (3.6.43) DSG−SN1 (a, b) = MSG (a, b) − 2MSN1 (a, b), 2 3 DSN3 −SA (a, b) = MSN3 (a, b) − MSA (a, b), (3.6.44) 4 3 2 (3.6.45) DSN1 −SN3 (a, b) = MSN1 (a, b) − MSN3 (a, b). 3 4

Theorem 3.6.7. Let 0 < a ≤ b, 1/2 ≤ t ≤ 1 or 0 ≤ t ≤ 1/2, u = at b1−t , and v = bt a1−t . Then 1 1 1 MSA (a, b) ≤ MSH (a, b) − ( MSH (u, v) − MSA (u, v)) ≤ MSH (a, b) (3.6.46) 3 3 3 1 1 1 1 ≤ MAH (a, b) − ( MAH (u, v) − MSH (u, v)) ≤ MAH (a, b) 2 2 3 2 1 1 1 1 ≤ MSG (a, b) − ( MSG (u, v) − MAH (u, v)) ≤ MSG (a, b) 2 2 2 2 1 ≤ MAG (a, b) − (MAG (u, v) − MSG (u, v)) ≤ MAG (a, b), 2

162 | 3 Schur-convex functions and mean value inequalities for two variables 1 1 M (a, b) ≤ MN2 N1 (a, b) − (MN2 N1 (u, v) − MAH (u, v)) ≤ MN2 N1 (a, b) 8 AH 8 1 1 1 ≤ MN2 G (a, b) − ( MN2 G (u, v) − MN2 N1 (u, v)) ≤ MN2 G (a, b) 3 3 3 1 1 1 1 ≤ MAG (a, b) − ( MAG (u, v) − MN2 G (u, v)) ≤ MAG (a, b) 4 4 3 4 1 ≤ MAN2 (a, b) − (MAN2 (u, v) − MAG (u, v)) ≤ MAN2 (a, b), 4 4 4 4 4 MSA (a, b) ≤ MSN2 (a, b) − ( MSN2 (u, v) − MSN2 (u, v)) ≤ MSN2 (a, b) 5 5 5 5 4 ≤ 4MAN2 (a, b) − (4MAN2 (u, v) − MSN2 (u, v)) ≤ 4MAN2 (a, b), 5 MSH (a, b) ≤ 2MSN1 (a, b) − (2MSN1 (u, v) − MSH (u, v)) ≤ 2MSN1 (a, b)

3 3 3 3 M (a, b) − ( MSG (u, v) − MSG (u, v)) ≤ MSG (a, b), 2 SG 2 2 2 3 3 3 MSA (a, b) ≤ MSN3 (a, b) − ( MSN3 (u, v) − MSA (u, v)) ≤ MSN3 (a, b) 4 4 4 2 3 2 2 ≤ MSN1 (a, b) − ( MSN1 (u, v) − MSN3 (u, v)) ≤ MSN1 (a, b). 3 3 4 3

(3.6.47)

(3.6.48)

(3.6.49)

≤

(3.6.50)

Remark 3.6.3. The inequalities (3.6.46)–(3.6.50) are a refinement of (3.6.13)–(3.6.17), respectively. Wu et al. [225] also proved the following. Theorem 3.6.8. The differences of means in (3.6.2)–(3.6.12) and (3.6.32)–(3.6.45) are all Schur-harmonically convex on ℝ2++ . In addition to the differences of means shown in (3.6.2)–(3.6.12), Wu and Qi [224] also introduced the following differences of means: MAN3 (x, y) = A(x, y) − N3 (x, y),

(3.6.51)

MAN1 (x, y) = A(x, y) − N1 (x, y),

(3.6.52)

MN2 N3 (x, y) = N2 (x, y) − N3 (x, y),

(3.6.53)

MN2 H (x, y) = N2 (x, y) − H(x, y),

(3.6.54)

MN3 N1 (x, y) = N3 (x, y) − N1 (x, y),

(3.6.55)

MN3 G (x, y) = N3 (x, y) − G(x, y),

(3.6.56)

MN3 H (x, y) = N3 (x, y) − H(x, y),

(3.6.57)

MN1 G (x, y) = N1 (x, y) − G(x, y),

(3.6.58)

MGH (x, y) = G(x, y) − H(x, y).

(3.6.60)

MN1 H (x, y) = N1 (x, y) − H(x, y),

(3.6.59)

3.7 Two-parameter homogeneous functions | 163

They proved the following conclusion. Theorem 3.6.9. The mean differences in (3.6.2)–(3.6.12) and (3.6.51)–(3.6.60) are all Schur-harmonically convex on ℝ2++ . 3.6.4 Schur-convexity quotient of some means The differences of means in (3.6.2)–(3.6.12) and (3.6.51)–(3.6.60) are changed to the quotient form; for example, (3.6.2) MSA (x, y) = S(x, y) − A(x, y) is changed to QSA (x, y) =

S(x, y) . A(x, y)

Yin et al. [263] obtained the following conclusions. Theorem 3.6.10. The quotients of means corresponding to (3.6.2)–(3.6.12) and (3.6.51)– (3.6.60) are all Schur-convex, Schur-geometrically convex, and Schur-harmonically convex on ℝ2++ . Yin et al. [264] further proved the following. Theorem 3.6.11. For m ≠ 0, the quotients of means corresponding to (3.6.2)–(3.6.12) and (3.6.51)–(3.6.60) are all m-order Schur-power convex on ℝ2++ .

3.7 Two-parameter homogeneous functions Definition 3.7.1. Let function f (x, y) be defined on Ω. If ∀t ∈ ℝ++ with f (tx, ty) ∈ Ω, we always have f (tx, ty) = t n f (x, y), then the function f (x, y) is called an n-order homogeneous function. In 2005, Yang [254] defined a class of two-parameter homogeneous functions and studied its monotonicity. Definition 3.7.2. Let f : ℝ2++ → ℝ2+ be an n-order homogeneous function and the first-order partial derivative exists, (x, y) ∈ ℝ2++ , (r, s) ∈ ℝ2 . If f (x, y) > 0, ∀(x, y) ∈ ℝ2++ {(x, x) : x ∈ ℝ2++ } and f (x, x) = 0, ∀x ∈ ℝ++ , then we define r

r

1

( f (xs ,ys ) ) r−s , r ≠ s, rs ≠ 0, Hf (r, s; x, y) = { f (x ,y ) Gf ,r (x, y), r = s ≠ 0, where 1

Gf ,r (x, y) = Gfr (x r , yr )

(3.7.1)

164 | 3 Schur-convex functions and mean value inequalities for two variables and Gf (x, y) = exp(

xfx (x, y) log x + yfy (x, y) log y f (x, y)

),

where fx (x, y) and fy (x, y) denote partial derivative with respect to the first and second variable of f (x, y), respectively. If f (x, y) > 0, ∀(x, y) ∈ ℝ2++ , then we further define (a) 1

f (xr , yr ) r ) , Hf (r, 0; x, y) = ( f (1, 1)

r ≠ 0, s = 0,

(3.7.2)

(b) 1

f (xs , ys ) s ) , Hf (0, s; x, y) = ( f (1, 1)

r = 0, s ≠ 0,

(c) fx (1,1)

fy (1,1)

Hf (0, 0; x, y) = x f (1,1) y f (1,1) ,

r = s = 0.

Since f (x, y) is a homogeneous function, Hf (r, s; x, y) is also a homogeneous function, called a homogeneous function with parameters r and s, simply denoted by Hf (r, s), Hf (x, y), or Hf . If taking f (x, y) = L(x, y) =

x−y , log x−log y

then 1

Hf (r, s; x, y) = HL (r, s; x, y) = ( If taking f (x, y) = A(x, y) =

x+y , 2

L(xr , yr ) r−s ) = E(r, s; x, y). L(xs , ys )

then

Hf (r, s; x, y) = HA (r, s; x, y) = G(r, s; x, y), simply denoted by HA (r, s) or HA . The function Hf (r, s) has the following properties. Properties 3.7.1. The function Hf (r, s) is symmetric with respect to r, s, namely, Hf (r, s; x, y) = Hf (s, r; x, y). Properties 3.7.2 ([254]). Let f (x, y) be a positive n-order homogenous function defined on Ω ⊂ ℝ2++ , and let it be second-order differentiable. If I1 = (log f )xy > 0 (or < 0, respectively), then Hf (r, s) is strictly increasing (or decreasing, respectively) with respect to both r and s on (−∞, 0) ∪ (0, +∞).

3.7 Two-parameter homogeneous functions | 165

Properties 3.7.3 ([254]). Let f (x, y) be a positive 1-order homogeneous function defined on Ω ⊂ ℝ2++ , and let it be second-order differentiable. (a) If [(log f )x log(y/x)]y > 0 (or < 0, respectively), then Hf (x, y) is strictly increasing (or decreasing, respectively) with respect to x. (b) If [(log f )y log(x/y)]x > 0 (or < 0, respectively), then Hf (x, y) is strictly increasing (or decreasing, respectively) with respect to y. Properties 3.7.4 ([255]). Let f : Ω ⊂ ℝ2++ → ℝ++ be a homogeneous function and let it be third-order differentiable. If J = (x − y)[x(log f )xy󸀠 ]x < 0 (or > 0, respectively), then Hf (r, s) is strictly log-convex (or log-concave, respectively) with respect to both r and s on (0, +∞) and strictly log-concave (or log-convex, respectively) on (−∞, 0). Properties 3.7.5 ([257]). Let f : ℝ2++ → ℝ++ be a homogeneous differentiable function, and T(t) = T(t; x, y) := log f (x t , yt ), (t; x, y) ∈ ℝ × ℝ2++ .

(3.7.3)

t t t t t t 𝜕T(t; x, y) x fx (x , y ) log x + y fy (x , y ) log y = , 𝜕t f (xt , yt )

(3.7.4)

Then

1

log Hf (r, s; x, y) = ∫ 0

𝜕T(tr + (1 − t)s; x, y) dt. 𝜕t

(3.7.5)

Properties 3.7.6 ([258]). Let f : ℝ2++ → ℝ+ be a homogeneous and m-order differentiable function. Then Hf (r, s; x, y) ∈ C m−1 (ℝ2 × ℝ2++ ). Theorem 3.7.1. Assume that f : ℝ2++ → ℝ+ is a symmetric, n-order homogeneous, continuous, and three-times differentiable function. If for any (x, y) ∈ ℝ2++ with x ≠ y, we have x N(x, y) = (x − y)(x(log f )x − y(log f )y − 2xyC log( )) > 0 (or < 0, respectively), (3.7.6) y where C = (log f )xy󸀠 , then Hf (r, s; x, y) is Schur-geometrically convex on ℝ2++ with respect to (x, y) if and only if r + s > 0 (or < 0, respectively) and Schur-geometrically concave if and only if r + s < 0 (or > 0, respectively). Theorem 3.7.2. Assume that f : ℝ2++ → ℝ+ is a symmetric, homogeneous, continuous, and three-times differentiable function. If J = (x − y)[x(log f )xy ]x < 0 (or > 0, respectively), then for fixed x, y > 0 with x ≠ y, Hf (p, q; a, b) is Schur-convex if and only if r + s > 0 (or < 0, respectively) and Schur-concave if and only if r + s < 0 (or > 0, respectively) with respect to (r, s).

166 | 3 Schur-convex functions and mean value inequalities for two variables In 2010, Yang [256] also defined the following four-parameter homogeneous mean. Definition 3.7.3. Assume (x, y) ∈ ℝ2++ with x ≠ y, (r, s), (p, q) ∈ ℝ2 . Then the fourparameter homogeneous mean denoted by F(r, s; p, q; x, y) is defined as follows: 1

F(r, s; p, q; x, y) = ( or

L(xrp , yrp )L(xsp , ysp ) (r−s)(p−q) ) L(xsq , ysq )L(xrq , yrq )

(3.7.7)

1

xrp − yrp xsq − ysq (r−s)(p−q) ) F(r, s; p, q; x, y) = ( sp . x − ysp xrq − yrq

(3.7.8)

If spq(r − s)(p − q) = 0, then F(r, s; p, q; x, y) are defined as their corresponding limits, for example as follows. If rpq(p − q) ≠ 0, r = s, then 1

I(xrp , yrp ) p(r−s) . F(r, s; p, q; x, y) = lim F(r, s; p, q; x, y) = ( sp sp ) s→r I(x , y )

(3.7.9)

If rpq(p − q) ≠ 0, s = 0, then 1

L(xrp , yrp ) p(r−s) ) F(r, s; p, 0; x, y) = lim = ( . s→0 L(xsp , ysp )

(3.7.10)

If pq(p − q) ≠ 0, r = s = 0, then F(r, s; 0, 0; x, y) = lim = G(x, y),

(3.7.11)

r→0

where L(x, y) and I(x, y) are logarithmic mean and exponential mean, respectively, G(x, y) = √xy. It is easy to verify that F(r, s; p, q; x, y) = R(p, q; r, s; x, y),

(3.7.12)

namely, the R-mean shown in (3.3.11). We have F(r, s; 1, 0; x, y) = E(r, s; x, y), F(r, s; 2, 1; x, y) = G(r, s; x, y), r

r

r

2

x 2 + (√xy) 2 + y 2 r−s 3 1 . F(r, s; , ; x, y) = ( s s s ) 4 4 x 2 + (√xy) 2 + y 2 In particular, r

r

1

xr + (√xy) 2 + y 2 r 3 1 ) = Hr (x, y), F(2r, 0; , ; x, y) = ( 4 4 3 namely, the generalized Heron mean shown in (3.4.9).

3.7 Two-parameter homogeneous functions | 167

Yang [256] obtained the following theorems. Theorem 3.7.3. If (r + s) > 0 (or < 0, respectively), then F(p, q; r, s; x, y) is strictly logarithmically concave (or logarithmically convex, respectively) with respect to p and q on (0, +∞) and strictly logarithmically convex (or logarithmically concave, respectively) on (−∞, 0). Theorem 3.7.4 ([257]). For a fixed (p, q), (r, s) ∈ ℝ2 , the four-parameter homogeneous mean F(p, q; r, s; x, y) is Schur-geometrically convex (or Schur-geometrically concave, respectively) with respect to (x, y) on ℝ2++ if and only if (p+q)(r+s) > 0 (or < 0, respectively). Remark 3.7.1. In fact, Theorem 3.7.4 is Theorem 3.3.7, but the two proofs are different. In 2011, Yang [257] also proved the following theorem. Theorem 3.7.5. For a fixed (x, y) ∈ ℝ2++ , x ≠ y, the four-parameter homogeneous mean F(p, q; r, s; x, y) is Schur-convex (or Schur-concave, respectively) with respect to (r, s) on ℝ2 if and only if (p + q)(r + s) < 0 (or > 0, respectively).

4 Schur-convex functions and mean value inequalities for multivariables 4.1 Schur-convexity of the third k-order symmetric mean 4.1.1 The third k-order symmetric mean For x ∈ ℝ2++ , Peng [129] defined the third k-order symmetric mean, 1 n

k

(k ) 1 k ∏(x) = [ ∏ ( ∑ xij )] , k j−1 n 1≤i