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Article

Schur-Convexity of the Mean of Convex Functions for Two Variables

1
Department of Electronic Information, Teacher’s College, Beijing Union University, Beijing 100011, China
2
Basic Courses Department, Beijing Polytechnic, Beijing 100176, China
3
Applied College of Science and Technology, Beijing Union University, Beijing 102200, China
*
Author to whom correspondence should be addressed.
Axioms 2022, 11(12), 681; https://doi.org/10.3390/axioms11120681
Submission received: 17 September 2022 / Revised: 20 November 2022 / Accepted: 21 November 2022 / Published: 29 November 2022

Abstract

:
The results of Schur convexity established by Elezovic and Pecaric for the average of convex functions are generalized relative to the case of the means for two-variable convex functions. As an application, some binary mean inequalities are given.

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
g d + c 2 1 d c c d g ( t ) d t g ( d ) + g ( c ) 2 .
This is the famous Hadamard’s inequality for convex functions.
In 2000, utilizing Hadamard’s inequality, Elezovic and Pecaric [1] researched Schur-convexity on the lower and upper limit of the integral for the mean of the convex functions and obtained the following important and profound theorem.
Theorem 1
([1]). Let I be an interval with nonempty interior on R and g be a continuous function on I. Then,
Φ ( c , d ) = 1 d c c d g ( s ) d s , c , d I , d c g ( c ) , d = c
is S c h u r c o n v e x ( S c h u r c o n c a v e , r e s p . ) on I × I iff g is convex (concave, resp.) on I.
In recent years, this result attracted the attention of many scholars (see references [2,3,4,5,6,7,8,9,10,11,12] and Chapter II of the monograph [13] and its references).
In this paper, the result of theorem 1 is generalized to the case of bivariate convex functions, and some bivariate mean inequalities are established.
Theorem 2.
Let I be an interval with non-empty interior on R and g ( s , t ) be a continuous function on I × I . If g is convex (or concave resp.) on I × I , then
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, resp.) on I × I .

2. Definitions and Lemmas

To prove Theorem 2, we provide the following lemmas and definitions.
Definition 1.
Let ( x 1 , x 2 ) and ( y 1 , y 2 ) R × R .
(1)
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 .
(2)
Let Ω R × R be 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
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, resp.) iff ψ is convex (or strictly convex, resp.)
Definition 2.
([14,15]). Let Ω R × R , ( x 1 , x 2 ) and ( y 1 , y 2 ) Ω , and let φ : Ω R :
(1)
( 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 .
(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 ) , and ψ is said to be a Schur-concave function on Ω iff ψ is a Schur-convex function.
Lemma 1
([14] (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 2
([14] (p. 5)). Let Ω R × R be symmetric set with a nonempty interior Ω . ψ : Ω R is continuous on Ω and differentiable in Ω . Then, function ψ is Schur convex (or Schur concave, resp.) iff ψ is symmetric on Ω and
x 1 x 2 ψ x 1 ψ x 2 0 ( o r 0 , r e s p . )
holds for any x 1 , x 2 Ω .
Lemma 3
([16]). Let φ x , w and φ x , w w be continuous on
D = ( x , w ) : a x b , c w d ; l e t
a ( w ) , b ( w ) and their derivatives be continuous on [ c , d ] ; v [ c , d ] implies a ( w ) , b ( w ) [ a , b ] . Then,
d d w a ( w ) b ( w ) φ ( x , w ) d x = a ( w ) b ( w ) φ ( x , w ) w d x + φ ( b ( w ) , u ) b ( w ) φ ( a ( w ) , w ) a ( w ) .
Lemma 4.
Let g ( s , t ) be continuous on rectangle [ a , p ; a , q ] , G c , d = c d c d g ( s , t ) d s d t . If c = c ( b ) and d = d ( b ) are differentiable with b, a c ( b ) p and a d ( b ) q , then
G b = c d g ( s , d ) d ( b ) d s c d g ( s , c ) c ( b ) d s + d ( b ) c d g ( d , t ) d t c ( b ) c d g ( c , t ) d t .
Proof. 
Let φ ( s , b ) = c d g ( s , t ) d t . Then,
φ ( s , b ) b = g ( s , d ) d ( b ) g ( s , c ) c ( b ) .
By Lemma 3, we have
G b = d d b c d φ ( s , b ) d s = c d φ ( s , b ) b d s + φ ( d , b ) d ( b ) φ ( c , b ) c ( b ) = c d g ( s , d ) d ( b ) d s c d g ( s , c ) c ( b ) d s + d ( b ) c d g ( d , s ) d s c ( b ) c d g ( c , s ) d s .
Remark 1.
In passing, it is pointed out that (9) in Lemma 5 of reference [2] is incorrect and should be replaced by (4) of this paper.
Lemma 5.
Let I be an interval with nonempty 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,
G v = u v g ( s , v ) d s + u v g ( v , t ) d t ,
G u = u v g ( s , u ) d s + u v g ( u , t ) d t .
Proof. 
By taking c ( b ) = a and d ( b ) = b , we have c ( b ) = 0 and d ( b ) = 1 . By (5) in Lemma 4, we obtain (6).
Notice that G u , v = v u v u g ( s , t ) d s d t ; from (5), we have
G u = v u g ( s , u ) d s + v u g ( u , t ) d t = u v g ( s , u ) d s + u v g ( u , t ) d t .
Lemma 6
([14] (p. 38, Proposition 4.3) and [15] (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 iff the Hessian matrix
H ( x , y ) = 2 ψ x x 2 ψ x y 2 ψ y x 2 ψ y y
is non-negative definite on Ω. If H ( x ) is positive definite on Ω, then ψ is strictly convex on Ω.

3. Proofs of Main Results

Proof of Theorem 2.
Let g ( s , t ) be convex on I × I . G ( u , v ) is evidently symmetric. By Lemma 5, we have
G ( u , v ) v = 2 ( v u ) 3 u v u v g ( s , t ) d s d t + 1 ( v u ) 2 u v g ( s , v ) d s + u v g ( v , t ) d t .
G ( u , v ) u = 2 ( v u ) 3 u v u v g ( s , t ) d s d t 1 ( v u ) 2 u v g ( s , u ) d s + u v g ( u , t ) d t .
Δ : = ( v u ) G ( u , v ) v G ( u , v ) u = 4 ( v u ) 2 u v u v g ( s , t ) d s d t + 1 v u u v ( g ( s , v ) + g ( s , u ) ) d s + 1 v u u v ( g ( u , t ) + g ( v , t ) ) d t
By Hadamards inequality, we have
2 ( v u ) 2 u v u v g ( s , t ) d s d t = 2 v u u v 1 v u u v g ( s , t ) d s d t 2 v u a u v g ( u , t ) + g ( v , t ) 2 d t = 1 v u u v a ( g ( u , t ) + g ( v , t ) ) d t
and
2 ( v u ) 2 u v u v g ( s , t ) d s d t = 2 v u u v 1 v u u v g ( s , t ) d t d s 2 v u u v g ( s , u ) + g ( s , v ) 2 d s = 1 v u u v ( g ( s , u ) + g ( s , v ) ) d s .
Moreover, we have
4 ( v u ) 2 u v u v g ( s , t ) d s d t 1 v u u v ( g ( s , v ) + g ( s , u ) ) d s + 1 v u u v ( g ( u , t ) + g ( v , t ) ) d t .
Therefore, Δ 0 , so G ( u , v ) is Schur-convex on I × I .
When g ( s , t ) is a concave function on I × I , it can be proved with similar methods. □

4. Application on Binary Mean

Theorem 3.
Let c > 0 and d > 0 . If c d , 0 < s < 1 , then
A ( d , c ) S s + 1 s ( d , c ) S s s 1 ( d , c ) ( c + d ) 2 s 1 s ( s + 1 ) ,
where A ( d , c ) = c + d 2 and S s ( d , c ) = d s c s s ( d c ) 1 s 1 are the arithmetic mean and the s-order Stolarsky mean of positive numbers c and d, respectively.
Proof. 
Let x > 0 , y > 0 and 0 < s < 1 . From Theorem 4 in the reference [17], we know that g ( x , y ) = x s y 1 s is concave on ( 0 , + ) × ( 0 , + ) . For c d , by Theorem 2, from ( d + c 2 , d + c 2 ) ( c , d ) ( d + c , 0 ) , it follows that
G ( d + c , 0 ) = 1 ( d + c 0 ) 2 c d 0 d + c x s y 1 s d x d y = 1 ( d + c ) 2 0 d + c x s d x 0 d + c y 1 s d y = 1 ( d + c ) 2 ( c + d ) s + 1 s + 1 ( c + d ) s s = ( c + d ) 2 s 1 s ( s + 1 ) G ( c , d ) = 1 ( d c ) 2 c d c d x s y 1 s d x d y = 1 ( d c ) 2 c d x s d x c d y 1 s d y = 1 ( d c ) 2 d s + 1 c s + 1 s + 1 d s c s s G d + c 2 , d + c 2 = d + c 2 ,
That is, we obtain the following.
( c + d ) 2 s 1 s ( s + 1 ) S s + 1 s ( d , c ) S s s 1 ( d , c ) = d s + 1 c s + 1 ( s + 1 ) ( d c ) · d s c s s ( d c ) d + c 2 = A ( d , c ) .
Theorem 4.
Let c > 0 , d > 0 . Then,
log A ( d , c ) B ( d , c ) 2 c d d + c 2 ,
where B ( d , c ) = d c is the geometric mean of of positive numbers c and d.
Proof. 
From reference [17], we know that the function g ( x , y ) = 1 ( x + y ) 2 is convex on ( 0 , + ) × ( 0 , + ) . For c > 0 , d > 0 and d c , by Theorem 2, from ( d + c 2 , d + c 2 ) ( d , c ) , it follows that
G ( c , d ) = 1 ( d c ) 2 c d c d 1 ( x + y ) 2 d x d y = 1 ( d c ) 2 c d 1 c + y 1 d + y d y = 1 ( d c ) 2 [ ( log ( d + c ) log ( 2 c ) ) ( log ( 2 d ) log ( d + c ) ) ] G d + c 2 , d + c 2 = 1 ( d + c ) 2 ,
That is, we obtain the following.
log A ( d , c ) B ( d , c ) 2 = log ( d + c ) 2 4 d c c d d + c 2 .
Theorem 5.
Let c > 0 , d > 0 . Then,
H e ( c 2 , d 2 ) A 2 ( c , d ) ,
where H e ( c , d ) = c + c d + d 3 is the Heronian mean of positive numbers c and d.
Proof. 
From reference [18], 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 and r > 0 . For d > 0 , c > 0 , and c d , by Theorem 2, from ( d + c 2 , d + c 2 ) ( d , c ) , 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 d + c 2 , d + c 2 = ( c + d ) 2 8 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 ) ( d + c ) 2 4 = A 2 ( d , c ) .
Theorem 6.
Let c > 0 , d > 0 . We have
H e ( c 2 , d 2 ) L ( d , c ) A ( d , c ) ,
where L ( d , c ) = d c log d log c is the logarithmic mean of positive numbers c and d.
Proof. 
Let g ( x , y ) = y 2 x 1 , x > 0 , y > 0 . Then,
g x x = 2 x 3 y 2 , g x y = 2 x 2 y = g y x , g y y = 2 x 1 .
The Hesse matrix of g ( x , y ) is
H = 2 x 3 y 2 2 x 2 y 2 x 2 y 2 x 1 .
det ( H λ I ) = det 2 x 3 y 2 λ 2 x 2 y 2 x 2 y 2 x 1 λ = 0
λ ( λ 2 x 3 y 2 2 x 1 ) = 0 λ 1 = 0 , λ 2 = 2 x 3 y 2 + 2 x 1 > 0 .
Therefore, matrix H is positive semidefinite, so it is known that g ( x , y ) is a convex function on ( 0 , + ) × ( 0 , + ) . For d > 0 , c > 0 and d c , by Theorem 2, from ( d + c 2 , d + c 2 ) ( d , c ) , it follows that
G ( c , d ) = 1 ( d c ) 2 c d c d y 2 x 1 d x d y = log d log c d c · d 2 + c d + c 2 3 d + c 2 2 c + c 2 = d + c 2 ,
which is
H e ( c 2 , d 2 ) L ( d , c ) A ( d , c ) .
Theorem 7.
Let d > 0 , c > 0 , d c . Then
E ˜ ( d , c ) A ( d , c ) e ( d + c ) d c e d e c 2 A ( d , c ) ,
where
E ˜ ( d , c ) = c e d d e c e d e c + 1 , d , c I , d c c , c = d
is exponent type mean of positive numbers c and d (see [13] (p. 134)).
Proof. 
Let g ( x , y ) = x e ( x + y ) , y > 0 , x > 0 . From reference [19], we know that function g ( x , y ) is convex on R × R . For d > 0 , c > 0 , and d c by Theorem 2 from ( d + c 2 , d + c 2 ) ( d , c ) , it follows that
G ( c , d ) = 1 ( c d ) 2 c d c d x e x y d x d y = 1 ( c d ) 2 c d x e x d x c d e y d y = 1 ( c d ) 2 c + 1 e c d + 1 e d · 1 e c 1 e d = 1 ( d c ) 2 ( c e d d e c ) + ( e d e c ) e ( c + d ) · e d e c e ( c + d ) G d + c 2 , d + c 2 = c + d 2 1 e ( d + c ) ,
which is
c e d d e c e d e c + 1 d + c 2 e ( d + c ) d c e d e c 2 .
For the rest, we only need to prove that
e ( c + d ) d c e d e c 2 1 .
We write e d = u and e c = v ; then, the above inequality is equivalent to the well-known log-geometric mean inequality.
L ( v , u ) = v u log v log u v u = B ( v , u ) .

Author Contributions

Conceptualization, H.-N.S., D.-S.W. and C.-R.F.; Methodology, H.-N.S.; Validation, C.-R.F.; Formal analysis, H.-N.S. and D.-S.W.; Investigation, D.-S.W.; Resources, C.-R.F.; Writing—original draft, D.-S.W.; Funding acquisition, C.-R.F. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors sincerely thanks Chen Dirong and Chen Jihang for their valuable opinions and suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Elezovic, N.; Pečarić, J. A note on schur-convex fuctions. Rocky Mt. J. Math. 2000, 30, 853–856. [Google Scholar] [CrossRef]
  2. Shi, H.N. Schur-convex functions relate to Hadamard-type inequalities. J. Math. Inequal. 2007, 1, 127–136. [Google Scholar] [CrossRef] [Green Version]
  3. Čuljak, V.; Franjić, I.; Ghulam, R.; Pečarić, J. Schur-convexity of averages of convex functions. J. Inequal. Appl. 2011, 2011, 581918. [Google Scholar] [CrossRef] [Green Version]
  4. Long, B.-Y.; Jiang, Y.-P.; Chu, Y.-M. Schur convexity properties of the weighted arithmetic integral mean and Chebyshev functional. Rev. Anal. Numr. Thor. Approx. 2013, 42, 72–81. [Google Scholar]
  5. Sun, J.; Sun, Z.-L.; Xi, B.-Y.; Qi, F. Schur-geometric and Schur harmonic convexity of an integral mean for convex functions. Turk. J. Anal. Number Theory 2015, 3, 87–89. [Google Scholar] [CrossRef] [Green Version]
  6. Zhang, X.-M.; Chu, Y.-M. Convexity of the integral arithmetic mean of a convex function. Rocky Mt. J. Math. 2010, 40, 1061–1068. [Google Scholar] [CrossRef]
  7. Chu, Y.-M.; Wang, G.-D.; Zhang, X.-H. Schur convexity and Hadmards inequality. Math. Inequal. Appl. 2010, 13, 725–731. [Google Scholar]
  8. Sun, Y.-J.; Wang, D.; Shi, H.-N. Two Schur-convex functions related to the generalized integral quasiarithmetic means. Adv. Inequal. Appl. 2017, 2017, 7. [Google Scholar]
  9. Nozar, S.; Ali, B. Schur-convexity of integral arithmetic means of co-ordinated convex functions in R3. Math. Anal. Convex Optim. 2020, 1, 15–24. [Google Scholar] [CrossRef]
  10. Sever, D.S. Inequalities for double integrals of Schur convex functions on symmetric and convex domains. Mat. Vesnik 2021, 73, 63–74. [Google Scholar]
  11. Kovač, S. Schur-geometric and Schur-harmonic convexity of weighted integral mean. Trans. Razmadze Math. Inst. 2021, 175, 225–233. [Google Scholar]
  12. Dragomir, S.S. Operator Schur convexity and some integral inequalities. Linear Multilinear Algebra 2019, 69, 2733–2748. [Google Scholar] [CrossRef]
  13. Shi, H.-N. Schur-Convex Functions and Inequalities: Volume 2: Applications in Inequalities; Harbin Institute of Technology Press Ltd.: Harbin, China, 2019. [Google Scholar]
  14. Wang, B.Y. Foundations of Majorization Inequalities; Beijing Normal University Press: Beijing, China, 1990. (In Chinese) [Google Scholar]
  15. Marshall, A.W.; Olkin, I. Inequalities: Theory of Majorization and Its Application; Academies Press: New York, NY, USA, 1979. [Google Scholar]
  16. Ye, Q.; Shen, Y. Handbook of Practical Mathematics, 2nd ed.; Science Press: Beijing, China, 2019; pp. 246–247. [Google Scholar]
  17. Shi, H.-N.; Wang, P.; Zhang, J.; Du, W.-S. Notes on judgment criteria of convex functions of several variables. Results Nonlinear Anal. 2021, 4, 235–243. [Google Scholar] [CrossRef]
  18. Shi, H.-N. Schur-Convex Functions and Inequalities: Volume 1: Concepts, Properties, and Applications in Symmetric Function Inequalities; Harbin Institute of Technology Press Ltd.: Harbin, China, 2019. [Google Scholar]
  19. You, X. The properties and applications of convex function of many variables. J. Beijing Inst. Petrochem. Technol. 2008, 16, 61–64. (In Chinese) [Google Scholar]
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Shi, H.-N.; Wang, D.-S.; Fu, C.-R. Schur-Convexity of the Mean of Convex Functions for Two Variables. Axioms 2022, 11, 681. https://doi.org/10.3390/axioms11120681

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Shi H-N, Wang D-S, Fu C-R. Schur-Convexity of the Mean of Convex Functions for Two Variables. Axioms. 2022; 11(12):681. https://doi.org/10.3390/axioms11120681

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Shi, Huan-Nan, Dong-Sheng Wang, and Chun-Ru Fu. 2022. "Schur-Convexity of the Mean of Convex Functions for Two Variables" Axioms 11, no. 12: 681. https://doi.org/10.3390/axioms11120681

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