应用数学进展  >> Vol. 8 No. 11 (November 2019)

协同(r,(h,m))-凸函数的Hermite-Hadamard型积分不等式
Hermite-Hadamard Type Integral Inequality for Coordinated (r,(h,m))-Convex Function

DOI: 10.12677/AAM.2019.811202, PDF, HTML, XML, 下载: 241  浏览: 1,544 

作者: 高 爽, 计东海:哈尔滨理工大学理学院,黑龙江 哈尔滨

关键词: r-凸性协同(r(hm))-凸函数Hermite-Hadamard不等式r-convex co-ordinated (r(hm))-Conve Hermite-Hadamard Type Inequalities

摘要: Hermite-Hadamard不等式是凸函数中重要不等式之一,其积分误差估计在优化问题、计算问题等中有着很重要的应用,多元函数的协同凸性的概念引进以来,使得凸性理论进一步发展,本文将定义一个新的二元函数,且其一个分量满足r-凸性,另一个分量为广义(h,m)-凸性的协同(r,(h,m))-凸函数,并研究其Hermite-Hadamard型积分不等式。
Abstract: Hermite-Hadamard inequality is one of the most important inequalities in convex functions. Its integral error estimates have important applications in optimization and computation. Since the concept of co-convexity of multivariate functions was introduced, the convexity theory has been further developed. In this paper, a new binary function is defined, coordinated (r,(h,m))-convex functions of bivariate functions, one of whose components is r-convex and the other is extended (h,m)-convex; Hermite-Hadamard type integral inequalities are studied.

文章引用: 高爽, 计东海. 协同(r,(h,m))-凸函数的Hermite-Hadamard型积分不等式[J]. 应用数学进展, 2019, 8(11): 1722-1731. https://doi.org/10.12677/AAM.2019.811202

1. 引言

1985年,Toader在文 [1] 引进了m-凸函数的概念,2007年Varosanec引入了h-凸函数的概念,引用文献 [2]。2011年Özdemir等进一步推广了h-凸函数与m-凸函数的概念,提出了 ( h , m ) -凸函数的概念,见文献 [3],若无特殊说明,本文均有 = ( , + )

定义1:设 m ( 0 , 1 ] ,函数 h : [ 0 , 1 ] ( 0 , ) ,区间 I ,若函数 f : I R 满足条件,若对任意的 x , y I 和任意的 λ [ 0 , 1 ] ,有

f ( λ x + m ( 1 λ ) y ) h ( λ ) f ( x ) + m h ( 1 λ ) f ( y ) ,

则称f为I上的 ( h , m ) -凸函数。

M. P. Gill等人在文 [4] 中引进了“r-凸函数”的等价形式

定义2:设 I 为区间,实数 r ,函数 f : I + = ( 0 , + ) ,若对任意的点 x , y I 和任意的 λ [ 0 , 1 ] ,有

f ( λ x + ( 1 λ ) y ) { ( λ [ f ( x ) ] r + ( 1 λ ) [ f ( y ) ] r ) 1 / r , r 0 , [ f ( x ) ] λ [ f ( y ) ] 1 λ , r = 0 ,

则称函数 f ( x ) 为区间I上的r-凸函数。

吴善和在文 [5] 中定义了r-平均凸函数的概念:

定义3:设 I 为区间,实数 r ,函数 f : I + + ,若对任意的点 x , y I 及任意的 λ [ 0 , 1 ] ,有

f ( [ λ x r + ( 1 λ ) y r ] 1 / r ) ( λ [ f ( x ) ] r + ( 1 λ ) [ f ( y ) ] r ) 1 / r , r 0

f ( x λ y 1 λ ) [ f ( x ) ] λ [ f ( y ) ] 1 λ , r = 0

则称函数 f ( x ) 为区间I上的r-平均凸函数。

2001年,Dragomir.S.S在文 [6] 中引入多元函数的协同凸性的概念。

定义4:设函数 f : Δ = [ a , b ] × [ c , d ] 2 ,其中 a < b , c < d ,若对任意的点 ( x , y ) ( z , w ) Δ 和任意的 t , λ [ 0 , 1 ] ,有

f ( t x + ( 1 t ) z , λ y + ( 1 λ ) w ) t λ f ( x , y ) + t ( 1 λ ) f ( x , w ) + ( 1 t ) λ f ( z , y ) + ( 1 t ) ( 1 λ ) f ( z , w )

则称二元函数 f ( x , y ) 为矩形区域 Δ 上的协同凸函数。

文 [7] 中定义了协同r-凸函数的概念。

下面介绍引进Stolarsky平均数:

( u , v ; r , s ) R + 2 × R 2 ,Stolarsky平均数 E ( u , v ; r , s ) 定义为:

E ( u , v ; r , s ) = ( r ( v s u s ) s ( v r u r ) ) 1 / ( s r ) , r s ( r s ) ( u v ) 0 , E ( u , v ; 0 , s ) = ( v s u s s ( ln v ln u ) ) 1 / s , s ( u v ) 0 , E ( u , v ; r , r ) = 1 e 1 / r ( u u r v v r ) 1 / ( u r v r ) , r ( u v ) 0 , E ( u , v ; 0 , 0 ) = u v , u v , E ( u , v ; 0 , 0 ) = u , u = v .

其中 L ( u , v ) E ( u , v ; 0 , 1 ) L r ( u , v ) E ( u , v ; r , r + 1 ) 分别称为对数平均数和广义对数平均数。

文 [8] 中建立了协同对数凸函数的Hermite-Hadamard型积分不等式。

定理1 [4] 设函数 f : [ a , b ] + 为对数凸函数,且 a < b ,则

1 b a a b f ( x ) d x L ( f ( a ) , f ( b ) ) ,

其中 L ( u , v ) 为对数平均数。

定理2 [4] 设一元函数 f : [ a , b ] + 为r-凸函数,且 a < b r ,若 f L 1 ( [ a , b ] ) ,则

1 b a a b f ( x ) d x L r ( f ( a ) , f ( b ) ) ,

其中 L r ( x , y ) 为广义对数平均数。

定理3 设函数 f : Δ = [ a , b ] × [ c , d ] 2 + 为矩形区域 Δ 上的协同对数凸函数,其中 a < b , c < d ,则

1 ( b a ) ( d c ) c d a b f ( x , y ) d x d y 1 2 [ 1 b a a b L ( f ( x , c ) , f ( x , d ) ) d x + 1 d c c d L ( f ( a , y ) , f ( b , y ) ) d y ] 1 4 [ 1 b a a b [ f ( x , c ) + f ( x , d ) ] d x + 1 d c c d [ f ( a , y ) + f ( b , y ) ] d y ] Ψ f ( Δ ) 1 4 [ f ( a , c ) + f ( b , c ) + f ( a , d ) + f ( b , d ) ] ,

其中 L ( u , v ) 为对数平均,且

Ψ f ( Δ ) = 1 4 [ L ( f ( a , c ) , f ( b , c ) ) + L ( f ( a , d ) , f ( b , d ) ) + L ( f ( a , c ) , f ( a , d ) ) + L ( f ( b , c ) , f ( b , d ) ) ] .

2. 主要结果

2.1. 协同 ( r , ( h , m ) ) -凸函数概念及引理

本节将定义二元函数的一个分量满足r-凸性,另一个分量为具有广义 ( h , m ) -凸性的协同 ( r , ( h , m ) ) -凸函数概念和协同 ( ( h , m ) , r ) -凸函数概念。

定义1.1设常数 0 < m 1 ,函数 h : [ 0 , 1 ] [ 0 , ) ,实数 r ,函数 f : Δ = [ a , b ] × [ 0 , d ] + × 0 + ,其中 a < b , 0 < d 。称二元函数 f ( x , y ) 为区域 Δ 上的协同 ( r , ( h , m ) ) -凸函数,若对任意点 ( x , y ) , ( z , w ) Δ 和任意的 ( t , λ ) [ 0 , 1 ] 2 ,有

r 0 ,有

f ( [ t x r + ( 1 t ) z r ] 1 / r , λ y + m ( 1 λ ) w ) h ( λ ) { t [ f ( x , y ) ] r + ( 1 t ) [ f ( z , y ) ] r } 1 / r + m h ( 1 λ ) { t [ f ( x , w ) ] r + ( 1 t ) [ f ( z , w ) ] r } 1 / r , (3.1.1)

r = 0 ,有

f ( x t z 1 t , λ y + m ( 1 λ ) w ) h ( λ ) [ f ( x , y ) ] t [ f ( z , y ) ] 1 t + h ( 1 λ ) [ f ( x , w ) ] t [ f ( z , w ) ] 1 t .

定义1.2设常数 0 < m 1 ,函数 h : [ 0 , 1 ] ( 0 , ) r , 函数 f : Δ = [ 0 , b ] × [ c , d ] + ,其中 0 < b , 0 < c < d 。称二元函数 f ( x , y ) 为区域 Δ 上的协同 ( ( h , m ) , r ) -凸函数,若对任意点 ( x , y ) , ( z , w ) Δ 和任意的 ( t , λ ) ( 0 , 1 ) × [ 0 , 1 ] ,有

r 0 ,有

f ( t x + m ( 1 t ) z , [ λ y r + ( 1 λ ) w r ] 1 / r ) h ( t ) { λ [ f ( x , y ) ] r + ( 1 λ ) [ f ( x , w ) ] r } 1 / r + m h ( 1 t ) { λ [ f ( z , y ) ] r + ( 1 λ ) [ f ( z , w ) ] r } 1 / r

r = 0 ,有

f ( t x + m ( 1 t ) z , y λ w 1 λ ) h ( t ) [ f ( x , y ) ] λ [ f ( x , w ) ] 1 λ + m h ( 1 t ) [ f ( z , y ) ] λ [ f ( z , w ) ] 1 λ .

2.2. 协同 ( r , ( h , m ) ) -凸函数的Hermite-Hadamard型积分不等式

本节将研究协同 ( r , ( h , m ) ) 凸函数的积分估计问题,建立协同 ( r , ( h , m ) ) -凸函数的几个Hermite-Hadamard型积分不等式。

定理2.1设 r , r 0 0 < m 1 ,函数 h : [ 0 , 1 ] ( 0 , ) ,且函数 f : Δ = [ a , b ] × [ 0 , d m ] +

( r , ( h , m ) ) -凸函数, 0 < a < b , 0 < c < d ,若 f L 1 ( Δ ) ,则

r ( b r a r ) ( d c ) c d a b f ( x , y ) x 1 r d x d y [ L r ( f ( a , c ) , f ( b , c ) ) + m L r ( f ( a , d m ) , f ( b , d m ) ) ] 0 1 h ( λ ) d λ ,

其中 L r ( u , v ) 为广义对数平均数。

证 作变换 x = t a r + ( 1 t ) b r r y = λ c + ( 1 λ ) d ( t , λ ) [ 0 , 1 ] 2 ,由 f ( x , y ) 的协同 ( r , ( h , m ) ) -凸性,有

r ( b r a r ) ( d c ) c d a b f ( x , y ) x 1 r d x d y = 0 1 0 1 f ( [ t a r + ( 1 t ) b r ] 1 / r , λ c + ( 1 λ ) d ) d t d λ = 0 1 0 1 f ( [ t a r + ( 1 t ) b r ] 1 / r , λ c + m ( 1 λ ) d m ) d t d λ 0 1 0 1 [ h ( λ ) { t [ f ( a , c ) ] r + ( 1 t ) [ f ( b , c ) ] r } 1 / r + m h ( 1 λ ) { t [ f ( a , d m ) ] r + ( 1 t ) [ f ( b , d m ) ] r } 1 / r ] d t d λ = 0 1 h ( λ ) d λ 0 1 { t [ f ( a , c ) ] r + ( 1 t ) [ f ( b , c ) ] r } 1 / r d t + m 0 1 h ( 1 λ ) d λ 0 1 { t [ f ( a , d m ) ] r + ( 1 t ) [ f ( b , d m ) ] r } 1 / r d t = { 0 1 ( t [ f ( a , c ) ] r + ( 1 t ) [ f ( b , c ) ] r ) 1 / r d t + m 0 1 ( t [ f ( a , d m ) ] r + ( 1 t ) [ f ( b , d m ) ] r ) 1 / r d t } 0 1 h ( λ ) d λ .

经计算可得

0 1 ( t [ f ( a , c ) ] r + ( 1 t ) [ f ( b , c ) ] r ) 1 / r d t = L r ( f ( a , c ) , f ( b , c ) ) , 0 1 ( t [ f ( a , d m ) ] r + ( 1 t ) [ f ( b , d m ) ] r ) 1 / r d t = L r ( f ( a , d m ) , f ( b , d m ) ) .

由上述三个公式,我们有

r ( b r a r ) ( d c ) c d a b f ( x , y ) x 1 r d x d y { 0 1 ( t [ f ( a , c ) ] r + ( 1 t ) [ f ( b , c ) ] r ) 1 / r d t + m 0 1 ( t [ f ( a , d m ) ] r + ( 1 t ) [ f ( b , d m ) ] r ) 1 / r d t } 0 1 h ( λ ) d λ = [ L r ( f ( a , c ) , f ( b , c ) ) + m L r ( f ( a , d m ) , f ( b , d m ) ) ] 0 1 h ( λ ) d λ

推论2.1.1在定理2.1的条件下,若 m = 1 ,有

r ( b r a r ) ( d c ) c d a b f ( x , y ) x 1 r d x d y [ L r ( f ( a , c ) , f ( b , c ) ) + L r ( f ( a , d ) , f ( b , d ) ) ] 0 1 h ( λ ) d λ ,

其中 L r ( u , v ) 为广义对数平均数。

定理2.2 设常数 0 < m 1 ,函数 h : [ 0 , 1 ] [ 0 , ) ,函数 f : Δ = [ a , b ] × [ 0 , d m ] + 为协同 ( 0 , ( h , m ) ) -凸函数,若 f L 1 ( Δ ) ,则

1 ( ln b ln a ) ( d c ) c d a b f ( x , y ) x d x d y 0 1 h ( λ ) d λ [ L ( f ( a , c ) , f ( b , c ) ) + m L ( f ( a , d m ) , f ( b , d m ) ) ] ,

其中 L ( u , v ) 为对数平均数。

证 作变换 x = a t b 1 t y = λ c + ( 1 λ ) d ( t , λ ) [ 0 , 1 ] 2 ,再由 f ( x , y ) 的协同 ( 0 , ( h , m ) ) -凸性,有

1 ( ln b ln a ) ( d c ) c d a b f ( x , y ) x d x d y = 0 1 0 1 f ( a t b 1 t , λ c + ( 1 λ ) d ) d t d λ 0 1 0 1 { h ( λ ) [ f ( a , c ) ] t [ f ( b , c ) ] 1 t + m h ( 1 λ ) [ f ( a , d m ) ] t [ f ( b , d m ) ] 1 t } d t d λ = 0 1 h ( λ ) d λ 0 1 [ f ( a , c ) ] t [ f ( b , c ) ] 1 t d t + 0 1 h ( 1 λ ) d λ 0 1 m [ f ( a , d m ) ] t [ f ( b , d m ) ] 1 t d t = 0 1 h ( λ ) d λ 0 1 { [ f ( a , c ) ] t [ f ( b , c ) ] 1 t + m [ f ( a , d m ) ] t [ f ( b , d m ) ] 1 t } d t .

0 1 [ f ( a , c ) ] t [ f ( b , c ) ] 1 t d t = L ( f ( a , c ) , f ( b , c ) ) , 0 1 [ f ( a , d m ) ] t [ f ( b , d m ) ] 1 t d t = L ( f ( a , d m ) , f ( b , d m ) ) .

于是,有

1 ( ln b ln a ) ( d c ) c d a b f ( x , y ) x d x d y 0 1 h ( λ ) d λ 0 1 { [ f ( a , c ) ] t [ f ( b , c ) ] 1 t + m [ f ( a , d m ) ] t [ f ( b , d m ) ] 1 t } d t = [ L ( f ( a , c ) , f ( b , c ) ) + m L ( f ( a , d m ) , f ( b , d m ) ) ] 0 1 h ( λ ) d λ .

故定理证毕。

推论2.2.1 在定理2.2的条件下,若 m = 1 ,有

1 ( ln b ln a ) ( d c ) c d a b f ( x , y ) x d x d y [ L ( f ( a , c ) , f ( b , c ) ) + L ( f ( a , d ) , f ( b , d ) ) ] 0 1 h ( λ ) d λ ,

其中 L ( u , v ) 为对数平均数。

定理2.3 设 r , r 0 ,常数 m ( 0 , 1 ] ,函数 h : [ 0 , 1 ] [ 0 , ) ,函数 f : Δ = [ a , b ] × [ 0 , d m ] +

( r , ( h , m ) ) -凸函数, 0 < a < b , 0 < c < d ,且 f L 1 ( Δ )

(i) 若 r 1 ,则

f ( ( a r + b r 2 ) 1 / r , c + d 2 ) h ( 1 2 ) r ( b r a r ) ( d c ) c d a b ( f ( x , y ) x 1 r + m f ( x , y m ) x 1 r ) d x d y ,

(ii) 若 r < 1 ,则

f ( ( a r + b r 2 ) 1 / r , c + d 2 ) h ( 1 2 ) [ r ( b r a r ) ( d c ) c d a b ( [ f ( x , y ) ] r x 1 r + m [ f ( x , y m ) ] r x 1 r ) d x d y ] 1 / r ,

证 对任意的 ( t , λ ) [ 0 , 1 ] 2 ,由函数 f ( x , y ) 的区域 Δ 上协同 ( r , ( h , m ) ) -凸性,有

f ( ( a r + b r 2 ) 1 / r , c + d 2 ) = f ( ( t a r + ( 1 t ) b r + ( 1 t ) a r + t b r 2 ) 1 / r , λ c + ( 1 λ ) d + ( ( 1 λ ) c + λ d ) 2 ) = f ( 1 2 [ ( t a r + ( 1 t ) b r ) 1 / r ] r + 1 2 [ ( ( 1 t ) a r + t b r ) 1 / r ] r , 1 2 ( λ c + ( 1 λ ) d ) + 1 2 m ( ( 1 λ ) c + λ d m ) ) h ( 1 2 ) { ( 1 2 [ f ( [ t a r + ( 1 t ) b r ] 1 / r , λ c + ( 1 λ ) d ) ] r + 1 2 [ f ( [ ( 1 t ) a r + t b r ] 1 / r , λ c + ( 1 λ ) d ) ] r ) 1 / r + m ( 1 2 [ f ( [ t a r + ( 1 t ) b r ] 1 / r , ( 1 λ ) c + λ d m ) ] r + 1 2 [ f ( [ ( 1 t ) a r + t b r ] 1 / r , ( 1 λ ) c + λ d m ) ] r ) 1 / r } .

r 1 ,时,对两边求 ( t , λ ) 的积分,并作积分变换

x = [ t a r + ( 1 t ) b r ] 1 / r , y = λ c + ( 1 λ ) d , ( t , λ ) [ 0 , 1 ] × ( 0 , 1 ) ,

可得

f ( ( a r + b r 2 ) 1 / r , c + d 2 ) h ( 1 2 ) { 0 1 0 1 [ 1 2 f ( [ t a r + ( 1 t ) b r ] 1 / r , λ c + ( 1 λ ) d ) + 1 2 f ( [ ( 1 t ) a r + t b r ] 1 / r , λ c + ( 1 λ ) d ) + m 2 f ( [ t a r + ( 1 t ) b r ] 1 / r , ( 1 λ ) c + λ d m ) + m 2 f ( [ ( 1 t ) a r + t b r ] 1 / r , ( 1 λ ) c + λ d m ) ] d t d λ } = h ( 1 2 ) r ( b r a r ) ( d c ) c d a b ( f ( x , y ) x 1 r + m f ( x , d m ) x 1 r ) d x d y .

r < 1 时,有

h ( 1 2 ) ( 1 2 [ f ( [ t a r + ( 1 t ) b r ] 1 / r , λ c + ( 1 λ ) d ) ] r + 1 2 [ f ( [ ( 1 t ) a r + t b r ] 1 / r , λ c + ( 1 λ ) d ) ] r ) 1 / r + h ( 1 2 ) m ( 1 2 [ f ( [ t a r + ( 1 t ) b r ] 1 / r , ( 1 λ ) c + λ d m ) ] r + 1 2 [ f ( [ ( 1 t ) a r + t b r ] 1 / r , ( 1 λ ) c + λ d m ) ] r ) 1 / r h ( 1 2 ) { 1 2 [ f ( [ t a r + ( 1 t ) b r ] 1 / r , λ c + ( 1 λ ) d ) ] r + 1 2 [ f ( [ ( 1 t ) a r + t b r ] 1 / r , λ c + ( 1 λ ) d ) ] r + m 2 [ f ( [ t a r + ( 1 t ) b r ] 1 / r , ( 1 λ ) c + λ d m ) ] r + m 2 [ f ( [ ( 1 t ) a r + t b r ] 1 / r , ( 1 λ ) c + λ d m ) ] r } 1 / r .

对上述不等式两边求 ( t , λ ) 的积分,并作变换

x = [ t a r + ( 1 t ) b r ] 1 / r , y = λ c + ( 1 λ ) d , ( t , λ ) [ 0 , 1 ] × ( 0 , 1 )

可得

[ f ( ( a r + b r 2 ) 1 / r , c + d 2 ) ] r h ( 1 2 ) r 0 1 0 1 { 1 2 [ f ( [ t a r + ( 1 t ) b r ] 1 / r , λ c + ( 1 λ ) d ) ] r + 1 2 [ f ( [ ( 1 t ) a r + t b r ] 1 / r , λ c + ( 1 λ ) d ) ] r + m 2 [ f ( [ t a r + ( 1 t ) b r ] 1 / r , ( 1 λ ) c + λ d m ) ] r + m 2 [ f ( [ ( 1 t ) a r + t b r ] 1 / r , ( 1 λ ) c + λ d m ) ] r } d t d λ = h ( 1 2 ) r r ( b r a r ) ( d c ) c d a b ( [ f ( x , y ) ] r x 1 r + m [ f ( x , y m ) ] r x 1 r ) d x d y

故定理2.3证毕。

同理,可证得

定理2.4 设函数 f : Δ = [ a , b ] × [ c , d ] + × + 为协同 ( r , ( h , 1 ) ) -凸函数, r , r 0 ,函数 h : [ 0 , 1 ] ( 0 , ) ,且 f L 1 (Δ)

(i) 若 r 1 ,则

f ( ( a r + b r 2 ) 1 / r , c + d 2 ) h ( 1 2 ) 2 r ( b r a r ) ( d c ) c d a b f ( x , y ) x 1 r d x d y ,

(ii) 若 r < 1 ,则

f ( ( a r + b r 2 ) 1 / r , c + d 2 ) h ( 1 2 ) [ 2 r ( b r a r ) ( d c ) c d a b [ f ( x , y ) ] r x 1 r d x d y ] 1 / r

定理2.5 设常数 m ( 0 , 1 ] ,函数 h : [ 0 , 1 ] ( 0 , ) ,正值函数 f : Δ = [ a , b ] × [ 0 , d m ] + ( 0 , ( h , m ) ) -

凸函数, 0 < a < b , 0 < c < d ,若 f L 1 ( Δ ) ,则

f ( ( a b ) 1 / 2 , c + d 2 ) h ( 1 2 ) 1 ( ln b ln a ) ( d c ) c d a b ( f ( x , y ) x + m f ( x , y m ) x ) d x d y .

证 对任意的 ( t , λ ) [ 0 , 1 ] × ( 0 , 1 ) ,利用基本不等式以及 f ( x , y ) ( 0 , ( h , m ) ) -凸性,有

f ( ( a b ) 1 / 2 , c + d 2 ) = f ( ( a t b 1 t a 1 t b t ) 1 / 2 , λ c + ( 1 λ ) d + ( 1 λ ) c + λ d 2 ) = f ( ( a t b 1 t ) 1 / 2 ( a 1 t b t ) 1 1 / 2 , 1 2 [ λ c + ( 1 λ ) d ] + m 2 ( 1 λ ) c + λ d m ) h ( 1 2 ) [ f ( a t b 1 t , λ c + ( 1 λ ) d ) f ( a 1 t b t , λ c + ( 1 λ ) d ) ] 1 / 2 + m h ( 1 2 ) [ f ( a t b 1 t , ( 1 λ ) c + λ d m ) f ( a 1 t b t , ( 1 λ ) c + λ d m ) ] 1 / 2

h ( 1 2 ) ( 1 2 [ f ( a t b 1 t , λ c + ( 1 λ ) d ) + f ( a 1 t b t , λ c + ( 1 λ ) d ) ] + m 2 [ f ( a t b 1 t , ( 1 λ ) c + λ d m ) + f ( a 1 t b t , ( 1 λ ) c + λ d m ) ] ) = 1 2 h ( 1 2 ) [ f ( a t b 1 t , λ c + ( 1 λ ) d ) + f ( a 1 t b t , λ c + ( 1 λ ) d ) + m f ( a t b 1 t , ( 1 λ ) c + λ d m ) + m f ( a 1 t b t , ( 1 λ ) c + λ d m ) ] ,

对上述不等式两边求 ( t , λ ) 积分,并作积分变换 x = a t b 1 t , y = λ c + ( 1 λ ) d ,有

0 1 0 1 f ( ( a b ) 1 / 2 , c + d 2 ) d t d λ 1 2 h ( 1 2 ) 0 1 0 1 [ f ( a t b 1 t , λ c + ( 1 λ ) d ) + f ( a 1 t b t , λ c + ( 1 λ ) d ) + m f ( a t b 1 t , ( 1 λ ) c + λ d m ) + m f ( a 1 t b t , ( 1 λ ) c + λ d m ) ] d t d λ = h ( 1 2 ) 1 ( ln b ln a ) ( d c ) c d a b ( f ( x , y ) x + m f ( x , y m ) x ) d x d y .

故本定理证毕。

同理可证:

定理2.6 设函数 f : Δ = [ a , b ] × [ c , d ] + × + 是协同 ( 0 , ( h , 1 ) ) -凸函数, 0 < a < b , c < d ,函数 h : [ 0 , 1 ] ( 0 , ) ,若 f L 1 ( Δ ) ,则

f ( ( a b ) 1 / 2 , c + d 2 ) h ( 1 2 ) 2 ( ln b ln a ) ( d c ) c d a b f ( x , y ) x d x d y .

参考文献

[1] Toader, G. (1985) Some Generalization of the Convexity. Proceedings of the Colloquium on Approximation and Optimization, University Cluj-Napoca, 1985, 329-338.
[2] Varosanec, S. (2007) On h-Convexity. Journal of Mathematical Analysis and Applications, 326, 303-311.
https://doi.org/10.1016/j.jmaa.2006.02.086
[3] Özdemir, M.E., Akdemir, A.O. and Set, E. (2011) On -Convexity and Hadamard-Type Inequalities. Mathematics, 2011-12-15. http://arxiv.rg/pdf/1103.6163v1
[4] Gill, P.M., Pearce, C.E.M. and Pěcarić, J. (1997) Hadamard’s Inequality for r-Convex Functions. Journal of Mathematical Analysis and Applications, 215, 461-470.
https://doi.org/10.1006/jmaa.1997.5645
[5] 吴善和. -凸函数与琴生型不等式 [J]. 数学实践与认识, 2005, 35(3): 220-228.
[6] Dragomir, S.S. (2001) On Hadamard’s Inequality for Convex Functions on the Coordinates in a Rectangle from the Plane. Taiwanese Journal of Mathematics, 5, 775-788.
https://doi.org/10.11650/twjm/1500574995
[7] Akdemir A.O. and Ozdemir, E.M. (2010) On Hadamard-Type Inequalities for Coordinated r-Convex Functions. Mathematics, 1309, 7-15.
[8] Bai, Y.M. and Qi, F. (2016) Some Integral Inequalities of the Hermite-Hadamard Type for Log-Convex Functions on Coordinates. Journal of Nonlinear Sciences and Applications, 9, 5900-5908.
https://doi.org/10.22436/jnsa.009.12.01