Schur-power convexity of integral mean for convex functions on the coordinates

Huannan Shi; Jing Zhang

doi:10.1515/math-2023-0157

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Schur-power convexity of integral mean for convex functions on the coordinates

Huannan Shi und Jing Zhang

Veröffentlicht/Copyright: 1. Dezember 2023

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Abstract

In this article, we investigate the concepts of monotonicity, Schur-geometric convexity, Schur-harmonic convexity, and Schur-power convexity for the lower and upper limits of the integral mean, focusing on convex functions on coordinate axes. Furthermore, we introduce novel and fascinating inequalities for binary means as a practical application.

Keywords: Schur-geometric convexity; Schur-harmonic convexity; monotonicity; Schur-power convexity; convex functions on the coordinates; inequality; binary means

MSC 2010: 26B25; 26D15; 26A51

1 Introduction

Let R be a set of real numbers, g be a convex function defined on the interval I ⊆ R → R , and c , d ∈ I , c < d . Then the following double inequality

(1) g c + d 2 ≤ 1 d − c ∫ c d g ( t ) d t ≤ g ( c ) + g ( d ) 2

is known in the literature as Hermite-Hadamard inequality for convex functions.

In 2000, Elezović and Pecccarić conducted research on the Schur-convexity of the integral arithmetic mean of a convex function, based on the Hermite-Hadamard inequality. They derived the following significant and profound theorem.

Theorem A

[1] Let I be an interval with non-empty interior on R and g ( u ) be a continuous function on I. Then

Φ ( u , v ) = 1 v − u ∫ u v g ( s ) d s , u , v ∈ I , u ≠ v g ( u ) , u , v ∈ I , u = v

is Schur-convex (or Schur-concave, respectively) on I × I if and only if g ( u ) is convex (or concave, respectively) on I .

In the recent past, this result has attracted and improved by several researchers and by now there exists a considerable literature on this theorem, see the articles [2–12] and Chapter II of the book [13] and references therein.

In [14], the authors generalize the result of Theorem 1 to the case of bivariate convex functions, leading to the derivation of the following Theorem 1.

Theorem B

[14] Let I be an interval with non-empty interior on R , and g ( u , v ) be a continuous function on I × I . If g ( u , v ) is convex (or concave, respectively) on I × I , then

(2) G ( u , v ) = 1 ( v − u ) 2 ∫ u v ∫ u v g ( s , t ) d s d t , ( u , v ) ∈ I × I , u ≠ v g ( u , u ) , ( u , v ) ∈ I × I , u = v

is Schur-convex (or Schur-concave, respectively) on I × I .

Long et al. [3] and Sun et al. [4] proved the following theorem in different ways.

Theorem C

[3,4] Suppose g ( u ) is a continuous convex (or concave, respectively) function that is increasing (or decreasing, respectively) on I ⊆ ( 0 , + ∞ ) . The function Φ ( u , v ) mentioned in Theorem A, then 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 × I .

In [15], an inequality of Hadamard’s type is provided for convex functions on the coordinates defined in a plane rectangle. Subsequently, [16] extends the corresponding results. In this article, we explore the monotonicity, Schur-geometric convexity, Schur-harmonic convexity, and Schur-power convexity of G ( u , v ) in Theorem B, considering g ( s , t ) to be an increasing convex function on the coordinates. Our main result is stated as follows.

Theorem 1

Let I be an interval with non-empty interior on R , and g ( s , t ) be a continuous function on I × I . If g ( s , t ) is increasing convex on the coordinates on I × I , then

G ( u , v ) is decreasing with respect to u on I ; G ( u , v ) is increasing with respect to v on I.
G ( u , v ) is Schur-geometrically convex, Schur-harmonically convex, and Schur-power convex on I × I ⊆ ( 0 , + ∞ ) × ( 0 , + ∞ ) .

In this article, in Section 2, the necessary definitions and lemmas are presented. The proof of the main result is provided in Sections 3 and 4, two intriguing applications are derived based on our main result. In Section 5, we briefly discuss possible future works.

2 Definitions and lemmas

To prove Theorem 1, we give the following lemmas and definitions.

Definition 1

Let ( x 1 , x 2 ) and ( y 1 , y 2 ) ∈ R × R .

( x 1 , x 2 ) ≤ ( y 1 , y 2 ) means x 1 ≤ y 1 , x 2 ≤ y 2 .
Let Ω ⊆ R × R , ψ : Ω → R said to be increasing if ( x 1 , x 2 ) ≤ ( y 1 , y 2 ) implies ψ ( x 1 , x 2 ) ≤ ψ ( y 1 , y 2 ) . ψ is said to be decreasing if and only if − ψ is increasing.
A set Ω ⊆ R × R is said to be convex if ( x 1 , x 2 ) , ( y 1 , y 2 ) ∈ Ω , and 0 ≤ β ≤ 1 implies
( β x 1 + ( 1 − β ) y 1 , β x 2 + ( 1 − β ) y 2 ) ∈ Ω .
Let Ω ⊆ R × R be a convex set. A function ψ : Ω → R is said to be a convex function on Ω if for all β ∈ [ 0 , 1 ] and all ( x 1 , x 2 ) , ( y 1 , y 2 ) ∈ Ω , inequality
(3) ψ ( β x 1 + ( 1 − β ) y 1 , β x 2 + ( 1 − β ) y 2 ) ≤ β ψ ( x 1 , x 2 ) + ( 1 − β ) ψ ( y 1 , y 2 )
holds. If for all β ∈ [ 0 , 1 ] and all ( x 1 , x 2 ) , ( y 1 , y 2 ) ∈ Ω , the strict inequality in (3) holds, then ψ is said to be strictly convex. ψ is called concave (or strictly concave, respectively) if and only if − ψ is convex (or strictly convex, respectively).

Lemma 1

[17, p. 39, Proposition 4.5], [18, p. 644, B.3.d] Let Ω ⊆ R × R be an open convex set and let ψ ( x , y ) : Ω → R be twice differentiable. Then ψ is convex on Ω if and only if the Hessian matrix

H ( x , y ) = ∂ 2 ψ ∂ x 2 ∂ 2 ψ ∂ x ∂ y ∂ 2 ψ ∂ y ∂ x ∂ 2 ψ ∂ y 2

is nonnegative definite on Ω . If H ( x ) is positive definite on Ω , then ψ is strictly convex on Ω .

Definition 2

[15] Let the bidimensional interval Δ ≔ [ a , b ] × [ c , d ] in R × R with a < b and c < d . A function f : Δ → R will be called convex on the coordinates, if the partial mappings f y : [ a , b ] → R , f y ( u ) ≔ f ( u , y ) , and f x : [ c , d ] → R , f x ( v ) ≔ f ( u , v ) , are convex where defined for all y ∈ [ c , d ] and x ∈ [ a , b ] .

The following lemma holds:

Lemma 2

[15] Every convex mapping f : Δ → R is convex on the coordinates, but the converse is not generally true.

For example obviously, for x > 0 , y > 0 , m ≥ 2 , the function f ( x , y ) = x m y m is convex on the coordinates, but is not convex. In fact, the Hessian matrix of f ( x , y )

H ( x , y ) = m ( m − 1 ) x m − 2 y m m 2 x m − 1 y m − 1 m 2 x m − 1 y m − 1 m ( m − 1 ) x m y m − 2

det H ( x , y ) = m 2 ( m − 1 ) 2 x 2 m − 2 y 2 m − 2 − m 4 x 2 m − 2 y 2 m − 2 < 0 , but m ( m − 1 ) x m − 2 y m > 0 , so f ( x , y ) is not convex.

Lemma 3

[15] Suppose that g : Δ = [ a , b ] × [ c , d ] → R is convex on the coordinates on Δ . Then one has the inequalities

(4) 1 ( b − a ) ( d − c ) ∫ c d ∫ a b g ( s , t ) d s d t ≤ 1 4 1 b − a ∫ a b ( g ( s , c ) + g ( s , d ) ) d s + 1 d − c ∫ c d ( g ( a , t ) + g ( b , t ) ) d t .

Definition 3

[17,18] Let Ω ⊆ R × R , ( x 1 , x 2 ) and ( y 1 , y 2 ) ∈ Ω , and let ψ : Ω → R .

( x 1 , x 2 ) is said to be majorized by ( y 1 , y 2 ) (in symbols ( x 1 , x 2 ) ≺ ( y 1 , y 2 ) ) if max { x 1 , x 2 } ≤ max { y 1 , y 2 } and x 1 + x 2 = y 1 + y 2 .
ψ is said to be a Schur-convex function on Ω if ( x 1 , x 2 ) ≺ ( y 1 , y 2 ) on Ω implies ψ ( x 1 , x 2 ) ≤ ψ ( y 1 , y 2 ) , ψ is said to be a Schur-concave function on Ω if and only if − ψ is Schur-convex function.

Definition 4

[19]

Ω ⊆ ( 0 , + ∞ ) × ( 0 , + ∞ ) is called a geometrically convex set if ( x 1 α y 1 β , x 2 α y 2 β ) ∈ Ω for any ( x 1 , x 2 ) and ( y 1 , y 2 ) ∈ Ω , where α and β ∈ [ 0 , 1 ] with α + β = 1 .
Let Ω ⊆ ( 0 , + ∞ ) × ( 0 , + ∞ ) be a geometrically convex set, ψ : Ω → [ 0 , + ∞ ) is said to be a Schur-geometrically convex function on Ω if ( log x 1 , log x 2 ) ≺ ( log y 1 , log y 2 ) implies ψ ( x 1 , x 2 ) ≤ ψ ( y 1 , y 2 ) for any ( x 1 , x 2 ) and ( y 1 , y 2 ) ∈ Ω . ψ is said to be a Schur-geometrically concave function on Ω if and only if − ψ is Schur-geometrically convex function.

Definition 5

[20,21] Let Ω ⊆ ( 0 , + ∞ ) × ( 0 , + ∞ ) .

A set Ω is said to be a harmonically convex set if
x 1 y 1 λ x 1 + ( 1 − λ ) y 1 , x 2 y 2 λ x 2 + ( 1 − λ ) y 2 ∈ Ω
for any ( x 1 , x 2 ) , ( y 1 , y 2 ) ∈ Ω and λ ∈ [ 0 , 1 ] .
Let Ω be a harmonically convex set, a function ψ : Ω → [ 0 , + ∞ ) is said to be a Schur-harmonically convex function on Ω if ( 1 x 1 , 1 x 2 ) ≺ ( 1 y 1 , 1 y 2 ) implies ψ ( x 1 , x 2 ) ≤ ψ ( y 1 , y 2 ) for any ( x 1 , x 2 ) , ( y 1 , y 2 ) ∈ Ω . A function ψ is said to be a Schur-harmonically concave function on Ω if and only if − ψ is a Schur-harmonically convex function.

Definition 6

[22] Let f : ( 0 , + ∞ ) → R be defined by

(5) f ( x ) = x m − 1 m , m ≠ 0 ; ln x , m = 0 .

Then a function ψ : Ω ⊆ ( 0 , + ∞ ) × ( 0 , + ∞ ) → R is said to be Schur- m -power convex on Ω if

( f ( x 1 ) , f ( x 2 ) ) ≺ ( f ( y 1 ) , f ( y 2 ) )

implies ψ ( x 1 , x 2 ) ≤ ψ ( y 1 , y 2 ) for all ( x 1 , x 2 ) ∈ Ω and ( y 1 , y 2 ) ∈ Ω . If − ψ is Schur- m -power convex, then we say that ψ is Schur- m -power concave.

If making f ( x ) = x , log x , 1 x in Definition 6, then definitions of the Schur-convex, Schur-geometrically convex, and Schur-harmonically convex functions can be deduced, respectively.

Lemma 4

[17, p. 5] Let ( x 1 , x 2 ) ∈ R × R . Then

x 1 + x 2 2 , x 1 + x 2 2 ≺ ( x 1 , x 2 ) .

Lemma 5

[17, p. 57] Let Ω ⊆ R × R be a symmetric convex set with a non-empty interior Ω ∘ . ψ : Ω → R is continuous on Ω and differentiable in Ω ∘ . Then the function ψ is the Schur-convex (or Schur-concave, respectively), if and only if ψ is symmetric on Ω and

( x 1 − x 2 ) ∂ ψ ∂ x 1 − ∂ ψ ∂ x 2 ≥ 0 ( o r ≤ 0 , r e s p e c t i v e l y )

holds for any ( x 1 , x 2 ) ∈ Ω ∘ .

Lemma 6

[19] Let Ω ⊆ ( 0 , + ∞ ) × ( 0 , + ∞ ) be geometrically symmetric convex set, and has a non-empty interior set Ω ∘ . Let ψ : Ω → [ 0 , + ∞ ) be continuous on Ω and differentiable in Ω ∘ . Then ψ is the Schur-geometrically convex (or Schur-geometrically concave, respectively) function, if and only if it is symmetric on Ω and

(6) ( log x 1 − log x 2 ) x 1 ∂ ψ ∂ x 1 − x 2 ∂ ψ ∂ x 2 ≥ 0 ( o r ≤ 0 , r e s p e c t i v e l y )

holds for any ( x 1 , x 2 ) ∈ Ω ∘ .

Lemma 7

[20,21] Let Ω ⊆ ( 0 , + ∞ ) × ( 0 , + ∞ ) be a harmonically symmetric convex set with a non-empty interior Ω ∘ . Let ψ : Ω → [ 0 , + ∞ ) be continuous on Ω and differentiable on Ω . Then ψ is a Schur-harmonically convex (or Schur-harmonically concave, respectively) function if and only if ψ is symmetric on Ω and

(7) ( x 1 − x 2 ) x 1 2 ∂ ψ ∂ x 1 − x 2 2 ∂ ψ ∂ x 2 ≥ 0 ( o r ≤ 0 , r e s p e c t i v e l y )

holds for any ( x 1 , x 2 ) ∈ Ω ∘ .

Lemma 8

[22] Let Ω ⊆ ( 0 , + ∞ ) × ( 0 , + ∞ ) be a symmetric convex set with a non-empty interior Ω ∘ and ψ : Ω → [ 0 , + ∞ ) be continuous on Ω and differentiable in Ω ∘ . Then ψ is a Schur-m-power convex (or Schur-m-power concave, respectively) function if and only if ψ is symmetric on Ω and

(8) ( x 1 − x 2 ) x 1 1 − m ∂ ψ ∂ x 1 − x 2 1 − m ∂ ψ ∂ x 2 ≥ 0 ( o r ≤ 0 , r e s p e c t i v e l y )

holds for any ( x 1 , x 2 ) ∈ Ω ∘ .

The concept of Schur- m -power convex was initially introduced by Z.-H. Yang, who established the aforementioned decision theorem (Lemma 8). The inequality (8) within this theorem adopts a simplified form recommended by Ming Li. Li conveyed this form to Yang, who, in turn, shared it with the author (J. Zhang).

Obviously, Lemma 8 contains Lemmas 5–7.

Lemma 9

[14] Let I be an interval with non-empty interior on R and g ( s , t ) be a continuous function on I × I . For ( u , v ) ∈ I × I , u ≠ v , let G ( u , v ) = ∫ u v ∫ u v g ( s , t ) d s d t . Then

(9) ∂ G ∂ v = ∫ u v g ( s , v ) d s + ∫ u v g ( v , t ) d t ,

(10) ∂ G ∂ u = − ∫ u v g ( s , u ) d s + ∫ u v g ( u , t ) d t .

3 Proofs of main results

Proof of Theorem 1

Since the function g ( s , t ) is coordinate convex, so for c = a = u , d = b = v , from (4), we have

(11) G ( u , v ) = ∫ u v ∫ u v g ( s , t ) d s d t ( v − u ) 2 ≤ 1 4 1 v − u ∫ u v g ( s , u ) d s + 1 v − u ∫ u v g ( s , v ) d s + 1 4 1 v − u ∫ u v g ( u , t ) d t + 1 v − u ∫ u v g ( v , t ) d t .

G ( u , v ) is evidently symmetric, without loss of generality, we may assume that u ≤ v . Since g ( s , t ) is increasing, g ( s , u ) ≤ g ( s , v ) and g ( u , t ) ≤ g ( v , t ) , from inequality (11), it follows that

(12) G ( u , v ) = ∫ u v ∫ u v g ( s , t ) d s d t ( v − u ) 2 ≤ 1 2 1 v − u ∫ u v g ( s , v ) d s + ∫ u v g ( v , t ) d t .

By Lemma 9, we have

∂ G ( u , v ) ∂ v = − 2 ∫ u v ∫ u v g ( s , t ) d s d t ( v − u ) 3 + 1 ( v − u ) 2 ∫ u v g ( s , v ) d s + ∫ u v g ( v , t ) d t = 1 v − u 1 v − u ∫ u v g ( s , v ) d s + ∫ u v g ( v , t ) d t − 2 ∫ u v ∫ u v g ( s , t ) d s d t ( v − u ) 2 .

From u ≤ v and (12), it follows that ∂ G ( u , v ) ∂ v ≥ 0 , this means that G ( u , v ) is increasing with respect to v on I .

∂ G ( u , v ) ∂ u = 2 ∫ u v ∫ u v g ( s , t ) d s d t ( v − u ) 3 − 1 ( v − u ) 2 ∫ u v g ( s , u ) d s + ∫ u v g ( u , t ) d t = 1 v − u 2 ∫ u v ∫ u v g ( s , t ) d s d t ( v − u ) 2 − 1 v − u ∫ u v g ( s , u ) d s + ∫ u v g ( u , t ) d t .

From u ≤ v and (12), it follows that ∂ G ( u , v ) ∂ u ≤ 0 , this means that G ( u , v ) is decreasing with respect to u on I . Thus, ( a ) is proved.

From ∂ G ∂ u ≤ 0 , ∂ G ∂ v ≥ 0 and 0 < u ≤ v , it follows that

( log u − log v ) u ∂ G ∂ u − v ∂ G ∂ v ≥ 0 ,

( u − v ) u 2 ∂ G ∂ u − v 2 ∂ G ∂ v ≥ 0 ,

and

( u − v ) u 1 − m ∂ G ∂ u − v 1 − m ∂ G ∂ v ≥ 0 .

Thus, ( b ) is proved.□

The proof of Theorem 1 is complete.

4 Application on binary mean

Theorem 2

Let c > 0 , d > 0 , and m ≠ 0 . Then

(13) H e ( c 2 , d 2 ) ≥ ( M m ( c , d ) ) 2 ,

where H e ( c , d ) = c + c d + d 3 and M m ( c , d ) = c m + d m 2 m are the Heronian mean and power mean of positive numbers c and d, respectively.

Proof

From [23], we know that the function of two variables

ψ ( x , y ) = x 2 2 r 2 + y 2 2 s 2

is a convex function on ( 0 , + ∞ ) × ( 0 , + ∞ ) , where s > 0 , r > 0 . By Lemma 2, convex function must be convex on the coordinates. The function ψ ( x , y ) is obviously increasing, by Theorem 1, G ( c , d ) is Schur-power convex function. For c > 0 , d > 0 , and c ≠ d , from

c m − 1 m + d m − 1 m 2 , c m − 1 m + d m − 1 m 2 ≺ c m − 1 m , d m − 1 m ,

i.e.,

( M m ( c , d ) ) m − 1 m , ( M m ( c , d ) ) m − 1 m ≺ c m − 1 m , d m − 1 m ,

it follows that

G ( c , d ) = 1 ( d − c ) 2 ∫ c d ∫ c d x 2 2 r 2 + y 2 2 s 2 d x d y = 1 ( d − c ) 2 ∫ c d d 3 − c 3 6 r 2 + y 2 ( d − c ) 2 s 2 d y = 1 ( d − c ) 2 ( d 3 − c 3 ) ( d − c ) 6 r 2 + ( d 3 − c 3 ) ( d − c ) 6 s 2 = 1 ( d − c ) 2 ⋅ ( d 3 − c 3 ) ( d − c ) 6 1 r 2 + 1 s 2 ≥ G ( M m ( c , d ) , M m ( c , d ) ) = ( M m ( c , d ) ) 2 2 1 r 2 + 1 s 2 ,

namely

□ H e ( c 2 , d 2 ) = c 2 + c d + d 2 3 = d 3 − c 3 3 ( d − c ) ≥ ( M m ( c , d ) ) 2 .

Theorem 3

Let c > 0 , d > 0 , and m ≥ 2 . Then

(14) S m + 1 ( c , d ) ≥ M m ( c , d ) ,

where S m + 1 ( c , d ) = d m + 1 − c m + 1 ( m + 1 ) ( d − c ) 1 m and M m ( c , d ) = c m + d m 2 m are m + 1 -order Stolarsky mean and m-order power mean of positive numbers c and d, respectively.

Proof

For m ≥ 2 , it is obvious that the function ψ ( x , y ) = x m y m is increasing and convex on the coordinates on ( 0 , + ∞ ) × ( 0 , + ∞ ) , by Theorem 1, G ( c , d ) is Schur- m -power convex function. For c > 0 , d > 0 , and c ≠ d , from

( M m ( c , d ) ) m − 1 m , ( M m ( c , d ) ) m − 1 m ≺ c m − 1 m , d m − 1 m ,

it follows that

G ( c , d ) = 1 ( d − c ) 2 ∫ c d ∫ c d x m y m d x d y = 1 ( d − c ) 2 ∫ c d x m d x ∫ c d y m d y = d m + 1 − c m + 1 ( m + 1 ) ( d − c ) 2 ≥ G ( M m ( c , d ) , M m ( c , d ) ) = ( M m ( c , d ) ) 2 m ,

extracting the 2 m -th root on both sides, we obtain

□ S m + 1 ( c , d ) = d m + 1 − c m + 1 ( m + 1 ) ( d − c ) 1 m ≥ M m ( c , d ) .

5 Conclusion

We explore the ideas of monotonicity, Schur-geometric convexity, Schur-harmonic convexity, and Schur-power convexity in relation to the lower and upper bounds of integral means. Our focus is on convex functions on coordinate axes.

Through the utilization of two coordinate convex functions and the application of Theorem 1, we have established two binary mean inequalities. We firmly believe that by seeking out other interesting coordinate convex functions and utilizing Theorem 1, we can generate a plethora of innovative binary mean inequalities. We encourage interested readers to explore and experiment with these possibilities.

sfthuannan@buu.edu.cn, ldtzhangjing1@buu.edu.cn

Acknowledgments

Both authors wish to express their gratitude to Professor Yang Zhenhang for his invaluable and practical constructive suggestions during the Tenth National Mathematical Inequality Academic Conference in Guangzhou in 2023. We would like to extend our heartfelt appreciation to the anonymous referee for their valuable comments, which will undoubtedly enhance the overall quality of our manuscript.

Funding information: The work was supported by Beijing Union University 2022 Liberal Arts Education Core Curriculum Construction Project.
Author contributions: The inception of the main idea in this article can be attributed to Huannan SHI. The execution of this work was a collaborative effort among both the authors. We have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-07-05

Revised: 2023-11-10

Accepted: 2023-11-13

Published Online: 2023-12-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2023-0157

Schlagwörter für diesen Artikel

Schur-geometric convexity; Schur-harmonic convexity; monotonicity; Schur-power convexity; convex functions on the coordinates; inequality; binary means

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