Further Refinements of Jensen’s Type Inequalities for the Function Defined on the Rectangle

Abstract

We give refinement of Jensen’s type inequalities given by Bakula and Pečarić (2006) for the co-ordinate convex function. Also we establish improvement of Jensen’s inequality for the convex function of two variables.

1. Introduction

Jensen’s inequality for convex functions plays a crucial role in the theory of inequalities due to the fact that other inequalities such as the arithmetic mean-geometric mean inequality, the Hölder and Minkowski inequalities, and the Ky Fan inequality, can be obtained as particular cases of it. Therefore, it is worth studying it thoroughly and refining it from different point of view. There are many refinements of Jensen’s inequality; see, for example, [1–14] and the references in them.

A function𝑓∶[𝑎,𝑏]×[𝑐,𝑑]→ℝ,[𝑎,𝑏]×[𝑐,𝑑]⊂ℝ² with𝑎<𝑏and𝑐<𝑑 is called convex on the co-ordinates if the partial mappings𝑓_𝑦∶[𝑎,𝑏]→ℝ defined as𝑓_𝑦(𝑡)=𝑓(𝑡,𝑦) and𝑓_𝑥∶[𝑐,𝑑]→ℝdefined as𝑓_𝑥(𝑠)=𝑓(𝑥,𝑠) are convex for all𝑥∈[𝑎,𝑏],𝑦∈[𝑐,𝑑]. Note that every convex function 𝑓∶[𝑎,𝑏]×[𝑐,𝑑]→ℝ is co-ordinate convex, but the converse is not generally true [8].

The following theorem has been given in [4].

Theorem 1. Let𝑓∶[𝑎,𝑏]×[𝑐,𝑑]→ℝ be a convex function on the co-ordinates on[𝑎,𝑏]×[𝑐,𝑑]. If 𝐱 is an n-tuple in[𝑎,𝑏], 𝐲 is m-tuple in[𝑐,𝑑], 𝐩 is a nonnegative n-tuple, and 𝐰 is a nonnegative m-tuple such that𝑃_𝑛=∑^𝑛_𝑖=1𝑝_𝑖>0and𝑊_𝑚=∑^𝑚_𝑖=1𝑤_𝑗>0, then 𝑓1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖,1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑦_𝑗≤121𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗≤1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗, (1) where𝑥=(1/𝑃_𝑛)∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖, and𝑦=(1/𝑊_𝑚)∑^𝑚_𝑗=1𝑤_𝑗𝑦_𝑗.

Recently Dragomir has given new refinement for Jensen inequality in [9]. The purpose of this paper is to give related refinements of Jensen’s type inequalities (1) for the co-ordinate convex function. We will also discuss some particular interesting cases. We establish improvement of Jensen’s inequality for the convex function defined on the rectangles. For related improvements of Jensen’s inequality, see, for example, [1, 2, 9, 13, 14]. For further several related integral inequalities, see [15].

2. Main Results

Let𝑓∶[𝑎,𝑏]×[𝑐,𝑑]→ℝ be convex on the co-ordinate on[𝑎,𝑏]×[𝑐,𝑑]. If𝑥_𝑖∈[𝑎,𝑏],𝑦_𝑗∈[𝑐,𝑑], 𝑝_𝑖,𝑤_𝑗>0,𝑖∈{1,2,…,𝑛},𝑗∈{1,2,…,𝑚}with𝑃_𝑛=∑^𝑛_𝑖=1𝑝_𝑖, and𝑊_𝑚=∑^𝑚_𝑗=1𝑤_𝑗, then for any subsets𝐼⊂{1,2,…,𝑛}and𝐽⊂{1,2,…,𝑚}, we assume that𝐼∶={1,2,…,𝑛}⧵𝐼and𝐽∶={1,2,…,𝑚}⧵𝐽. Define𝑃_𝐼=∑_𝑖∈𝐼𝑝_𝑖, 𝑃_𝐼=∑_𝑖∈𝐼𝑝_𝑖, 𝑊_𝐽=∑_𝑗∈𝐽𝑤_𝑗, and𝑊_𝐽=∑_𝑗∈𝐽𝑤_𝑗. For the function𝑓and the𝑛-, 𝑚-tuples,𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛),𝐲=(𝑦₁,𝑦₂,…,𝑦_𝑚), 𝐩=(𝑝₁,𝑝₂,…,𝑝_𝑛), and 𝐰=(𝑤₁,𝑤₂,…,𝑤_𝑚), we define the following functionals: 𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗=𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗+𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗,𝐷𝑓,𝐰,𝐲,𝐽,𝑥_𝑖=𝑊_𝐽𝑊_𝑚𝑓𝑥_𝑖,1𝑊_{𝐽𝑗∈𝐽}𝑤_𝑗𝑦_𝑗+𝑊_𝐽𝑊_𝑚𝑓𝑥_𝑖,1𝑊_{𝐽𝑗∈𝐽}𝑤_𝑗𝑦_𝑗,𝐷=𝑃𝑓,𝐩,𝐱,𝐼_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦+𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦,𝐷=𝑊𝑓,𝐰,𝐲,𝐽_𝐽𝑊_𝑚𝑓1𝑥,𝑊_{𝐽𝑗∈𝐽}𝑤_𝑗𝑦_𝑗+𝑊_𝐽𝑊_𝑚𝑓1𝑥,𝑊_{𝐽𝑗∈𝐽}𝑤_𝑗𝑦_𝑗, (2) where𝑥=(1/𝑃_𝑛)∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖, and 𝑦=(1/𝑊_𝑚)∑^𝑚_𝑗=1𝑤_𝑗𝑦_𝑗.

It is worth to observe that for𝐼={𝑘}, 𝑘∈{1,…,𝑛}, and𝐽={𝑙}, 𝑙∈{1,…,𝑚}, we have the functionals 𝐷_𝑘𝑓,𝐩,𝐱,𝑦_𝑗∶=𝐷𝑓,𝐩,𝐱,{𝑘},𝑦_𝑗=𝑝_𝑘𝑃_𝑛𝑓𝑥_𝑘,𝑦_𝑗+𝑃_𝑛−𝑝_𝑘𝑃_𝑛𝑓∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖−𝑝_𝑘𝑥_𝑘𝑃_𝑛−𝑝_𝑘,𝑦_𝑗,𝐷_𝑙𝑓,𝐰,𝐲,𝑥_𝑖∶=𝐷𝑓,𝐰,𝐲,{𝑙},𝑥_𝑖=𝑤_𝑙𝑊_𝑚𝑓𝑥_𝑖,𝑦_𝑙+𝑊_𝑚−𝑤_𝑙𝑊_𝑚𝑓𝑥_𝑖,∑^𝑚_𝑗=1𝑤_𝑗𝑦_𝑗−𝑤_𝑙𝑦_𝑙𝑊_𝑚−𝑤_𝑙,𝐷_𝑘{𝑘}=𝑝𝑓,𝐩,𝐱∶=𝐷𝑓,𝐩,𝐱,_𝑘𝑃_𝑛𝑓𝑥_𝑘,𝑦+𝑃_𝑛−𝑝_𝑘𝑃_𝑛𝑓∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖−𝑝_𝑘𝑥_𝑘𝑃_𝑛−𝑝_𝑘,𝑦,𝐷_𝑙{𝑙}=𝑤𝑓,𝐰,𝐲∶=𝐷𝑓,𝐰,𝐲,_𝑙𝑊_𝑚𝑓𝑥,𝑦_𝑙+𝑊_𝑚−𝑤_𝑙𝑊_𝑚𝑓∑𝑥,^𝑚𝑗=1𝑤_𝑗𝑦_𝑗−𝑤_𝑙𝑦_𝑙𝑊_𝑚−𝑤_𝑙. (3) The following refinement of (1) holds.

Theorem 2. Let𝑓∶[𝑎,𝑏]×[𝑐,𝑑]→ℝ be a co-ordinate convex function on[𝑎,𝑏]×[𝑐,𝑑]. If𝑥_𝑖∈[𝑎,𝑏],𝑦_𝑗∈[𝑐,𝑑], 𝑝_𝑖, 𝑤_𝑗>0,𝑖∈{1,2,…,𝑛}, 𝑗∈{1,2,…,𝑚}with 𝑃_𝑛=∑^𝑛_𝑖=1𝑝_𝑖, and 𝑊_𝑚=∑^𝑚_𝑗=1𝑤_𝑗, then for any subsets 𝐼⊂{1,2,…,𝑛}and 𝐽⊂{1,2,…,𝑚}, one has 𝑓𝑥,𝑦≤12𝐷≤1𝑓,𝐰,𝐲,𝐽+𝐷𝑓,𝐩,𝐱,𝐼21𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗≤121𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝐷𝑓,𝐰,𝐲,𝐽,𝑥_𝑖+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗≤1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗, (4) where𝑥=(1/𝑃_𝑛)∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖, and 𝑦=(1/𝑊_𝑚)∑^𝑚_𝑗=1𝑤_𝑗𝑦_𝑗.

Proof. One-dimensional Jensen’s inequality gives us 𝑓𝑥_𝑖,𝑦≤1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗,𝑓𝑥,𝑦_𝑗≤1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦_𝑗. (5) As we have 𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗=𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗+𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗, (6) so by Jensen’s inequality, we have 𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗=𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗+𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗≤𝑃_𝐼𝑃_𝑛1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦_𝑗+𝑃_𝐼𝑃_𝑛1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦_𝑗=1𝑃_{𝑛𝑖∈𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦_𝑗+1𝑃_{𝑛𝑖∈𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦_𝑗=1𝑃_{𝑛𝑖∈𝐼∪𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦_𝑗⟹𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗≤1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦_𝑗. (7) As the function𝑓is convex on the first co-ordinate, so we have 𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗=𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗+𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗𝑃≥𝑓^𝐼𝑃_𝑛1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖+𝑃_𝐼𝑃_𝑛1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗1=𝑓𝑃_{𝑛𝑖∈𝐼}𝑝_𝑖𝑥_𝑖+1𝑃_{𝑛𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗1=𝑓𝑃_{𝑛𝑖∈𝐼∪𝐼}𝑝_𝑖𝑥_𝑖,𝑦_𝑗1=𝑓𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖,𝑦_𝑗⟹𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗1≥𝑓𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖,𝑦_𝑗. (8) Now, from (7) and (8), we have 𝑓𝑥,𝑦_𝑗≤𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗≤1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦_𝑗. (9) Similarly, we can write 𝑓𝑥_𝑖,𝑦≤𝐷𝑓,𝐰,𝐲,𝐽,𝑥_𝑖≤1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗. (10) Multiplying (9) and (10), respectively, by𝑤_𝑗and𝑝_𝑖and summing over𝑖and𝑗, we obtain 1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗≤1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗≤1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗,1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦≤1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝐷𝑓,𝐰,𝐲,𝐽,𝑥_𝑖≤1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗. (11)(12) Adding (11) and (12), we have 121𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗≤121𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝐷𝑓,𝐰,𝐲,𝐽,𝑥_𝑖+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗≤1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗. (13) Again by one-dimensional Jensen’s inequality, we have 𝑓𝑥,𝑦≤1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦,𝑓𝑥,𝑦≤1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗. (14) As we have the functional 𝐷=𝑃𝑓,𝐩,𝐱,𝐼^𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦+𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦, (15) so by Jensen’s inequality, we get 𝐷=𝑃𝑓,𝐩,𝐱,𝐼^𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦+𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦≤𝑃_𝐼𝑃_𝑛1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦+𝑃_𝐼𝑃_𝑛1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦=1𝑃_{𝑛𝑖∈𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑃_{𝑛𝑖∈𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦=1𝑃_{𝑛𝑖∈𝐼∪𝐼}𝑝_𝑖𝑓𝑥_𝑖,𝑦≤1⟹𝐷𝑓,𝐩,𝐱,𝐼𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦, (16) and as the function𝑓is convex on the first co-ordinate, so we have 𝐷=𝑃𝑓,𝐩,𝐱,𝐼^𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦+𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦𝑃≥𝑓^𝐼𝑃_𝑛1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖+𝑃_𝐼𝑃_𝑛1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦1=𝑓𝑃_{𝑛𝑖∈𝐼}𝑝_𝑖𝑥_𝑖+1𝑃_{𝑛𝑖∈𝐼}𝑝_𝑖𝑥_𝑖,𝑦1=𝑓𝑃_{𝑛𝑖∈𝐼∪𝐼}𝑝_𝑖𝑥_𝑖,𝑦1=𝑓𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖,𝑦⟹𝐷𝑓,𝐩,𝐱,𝐼≥𝑓𝑥,𝑦. (17) Now from (16) and (17), we have 𝑓𝑥,𝑦≤1≤𝐷𝑓,𝐩,𝐱,𝐼𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦. (18) Similarly, we can prove that 𝑓𝑥,𝑦≤1≤𝐷𝑓,𝐰,𝐲,𝐽𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗. (19) Adding (18) and (19), we get 𝑓𝑥,𝑦≤12𝐷≤1𝑓,𝐰,𝐲,𝐽+𝐷𝑓,𝐩,𝐱,𝐼21𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗. (20)
Combining (13) and (20), we have 𝑓𝑥,𝑦≤12𝐷≤1𝑓,𝐰,𝐲,𝐽+𝐷𝑓,𝐩,𝐱,𝐼21𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗≤121𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝐷𝑓,𝐰,𝐲,𝐽,𝑥_𝑖+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗≤1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗. (21)

The following cases from the above inequalities are of interest [6, 7].

Remark 3. We observe that the inequalities in (4) can be written equivalently as 1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗≥max_{{}𝐼⊂1,…,𝑛𝐽⊂{1,…,𝑚}}121𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝐷𝑓,𝐰,𝐲,𝐽,𝑥_𝑖+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗,1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗≤min_{{}𝐼⊂1,…,𝑛𝐽⊂{1,…,𝑚}}1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝐷𝑓,𝐰,𝐲,𝐽,𝑥_𝑖+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝐷𝑓,𝐩,𝐱,𝐼,𝑦_𝑗,1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗≥max_{{}𝐼⊂1,…,𝑛𝐽⊂{1,…,𝑚}}𝐷,𝑓𝑓,𝐰,𝐲,𝐽+𝐷𝑓,𝐩,𝐱,𝐼𝑥,𝑦≤min_{{}𝐼⊂1,…,𝑛𝐽⊂{1,…,𝑚}}12𝐷𝑓,𝐰,𝐲,𝐽+𝐷𝑓,𝐩,𝐱,𝐼. (22) These inequalities imply the following results: 1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗≥max_{{}𝑘∈1,…,𝑛𝑙∈{1,…,𝑚}}121𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝐷_𝑙𝑓,𝐰,𝐲,𝑥_𝑖+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝐷_𝑘𝑓,𝐩,𝐱,𝑦_𝑗,1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗≤min_{𝑘∈{1,…,𝑛}𝑙∈{1,…,𝑚}}1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝐷_𝑙𝑓,𝐰,𝐲,𝑥_𝑖+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝐷_𝑘𝑓,𝐩,𝐱,𝑦_𝑗,1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖,𝑦+1𝑊_𝑚^𝑚_𝑗=1𝑤_𝑗𝑓𝑥,𝑦_𝑗≥max_{{}{}𝑘∈1,…,𝑛𝑙∈1,…,𝑚}𝐷_𝑙𝑓,𝐰,𝐲+𝐷_𝑘,𝑓𝑓,𝐩,𝐱𝑥,𝑦≤min_{{}𝑘∈1,…,𝑛𝑙∈{1,…,𝑚}}12𝐷_𝑙𝑓,𝐰,𝐲+𝐷_𝑘𝑓,𝐩,𝐱. (23)

Moreover, from the above, we also have max_{𝐼⊂{1,…,𝑛}}𝐷𝑓,𝐩,𝐱;𝐼≥max_{𝑘∈{1,…,𝑛}}𝐷_𝑘𝑓,𝐩,𝐱,max_{{}𝐽⊂1,…,𝑚}𝐷𝑓,𝐰,𝐲;𝐽≥max_{{}𝑙∈1,…,𝑚}𝐷_𝑙𝑓,𝐰,𝐲,max_{{}𝐼⊂1,…,𝑛}𝐷𝑓,𝐩,𝐱;𝐼,𝑦_𝑗≥max_{{}𝑘∈1,…,𝑛}𝐷_𝑘𝑓,𝐩,𝐱,𝐲_𝑗,max_{{}𝐽⊂1,…,𝑚}𝐷𝑓,𝐰,𝐲;𝐽,𝑥_𝑖≥max_{{}𝑙∈1,…,𝑚}𝐷_𝑙𝑓,𝐰,𝐲,𝑥_𝑖,min_{{}𝑘∈1,…,𝑛}𝐷_𝑘𝑓,𝐩,𝐱≥min_{{}𝐼⊂1,…,𝑛}𝐷𝑓,𝐩,𝐱;𝐼,min_{{}𝑙∈1,…,𝑚}𝐷_𝑙𝑓,𝐰,𝐲≥min_{{}𝐽⊂1,…,𝑚}𝐷𝑓,𝐰,𝐲;𝐽,min_{{}𝑘∈1,…,𝑛}𝐷_𝑘𝑓,𝐩,𝐱,𝑦_𝑗≥min_{{}𝐼⊂1,…,𝑛}𝐷𝑓,𝐩,𝐱;𝐼,𝑦_𝑗,min_{𝑙∈{1,…,𝑚}}𝐷_𝑙𝑓,𝐰,𝐲,𝑥_𝑖≥min_{{}𝐽⊂1,…,𝑚}𝐷𝑓,𝐰,𝐲;𝐽,𝑥_𝑖. (24)

We discuss the following particular cases of the above inequalities which is of interest [6].

In the case when𝑝_𝑖=1 and𝑤_𝑗=1for𝑖∈{1,…,𝑛}and𝑗∈{1,…,𝑚}, consider the natural numbers𝑘,𝑙with1≤𝑘≤𝑛−1 and1≤𝑙≤𝑚−1and define 𝐷_𝑘𝑓,𝐱,𝑦_𝑗=𝑘𝑛𝑓1𝑘^𝑘_𝑖=1𝑥_𝑖,𝑦_𝑗+𝑛−𝑘𝑛𝑓1𝑛−𝑘^𝑛_𝑖=𝑘+1𝑥_𝑖,𝑦_𝑗,𝐷_𝑙𝑓,𝐲,𝑥_𝑖=𝑙𝑚𝑓𝑥_𝑖,1𝑙^𝑙_𝑗=1𝑦_𝑗+𝑚−𝑙𝑚𝑓𝑥_𝑖,1𝑚−𝑙^𝑚_𝑗=𝑙+1𝑦_𝑗,𝐷_𝑘=𝑘𝑓,𝐱𝑛𝑓1𝑘^𝑘_𝑖=1𝑥_𝑖,𝑦+𝑛−𝑘𝑛𝑓1𝑛−𝑘^𝑛_𝑖=𝑘+1𝑥_𝑖,𝑦,𝐷_𝑙=𝑙𝑓,𝐲𝑚𝑓1𝑥,𝑙^𝑙_𝑗=1𝑦_𝑗+𝑚−𝑙𝑚𝑓1𝑥,𝑚−𝑙^𝑚_𝑗=𝑙+1𝑦_𝑗. (25) We can give the following result.

Corollary 4. Let𝑓∶[𝑎,𝑏]×[𝑐,𝑑]→ℝ be a co-ordinate convex function on[𝑎,𝑏]×[𝑐,𝑑]. If𝑥_𝑖∈[𝑎,𝑏] and 𝑦_𝑗∈[𝑐,𝑑], then for any𝑘∈{1,…,𝑛−1}and𝑙∈{1,…,𝑚−1}, one has 𝑓1𝑛^𝑛_𝑖=1𝑥_𝑖,1𝑚^𝑚_𝑗=1𝑦_𝑗≤12𝐷_𝑙𝑓,𝐲+𝐷_𝑘≤1𝑓,𝐱21𝑛^𝑛_𝑖=1𝑓𝑥_𝑖,𝑦+1𝑚^𝑚_𝑗=1𝑓𝑥,𝑦_𝑗≤121𝑛^𝑛_𝑖=1𝐷_𝑙𝑓,𝐲,𝑥_𝑖+1𝑚^𝑚_𝑗=1𝐷_𝑘𝑓,𝐱,𝑦_𝑗≤1𝑚𝑛^𝑛𝑚_𝑖=1𝑗=1𝑓𝑥_𝑖,𝑦_𝑗. (26)

In particular, we have the bounds 1𝑚𝑛^𝑛𝑚_𝑖=1𝑗=1𝑓𝑥_𝑖,𝑦_𝑗≥max_{𝑘∈{1,…,𝑛}𝑙∈{1,…,𝑚}}121𝑛^𝑛_𝑖=1𝐷_𝑙𝑓,𝐲,𝑥_𝑖+1𝑚^𝑚_𝑗=1𝐷_𝑘𝑓,𝐱,𝑦_𝑗,1𝑛^𝑛_𝑖=1𝑓𝑥_𝑖,𝑦+1𝑚^𝑚_𝑗=1𝑓𝑥,𝑦_𝑗≤min_{𝑘∈{1,…,𝑛}𝑙∈{1,…,𝑚}}1𝑛^𝑛_𝑖=1𝐷_𝑙𝑓,𝐲,𝑥_𝑖+1𝑚^𝑚_𝑗=1𝐷_𝑘𝑓,𝐱,𝑦_𝑗,1𝑛^𝑛_𝑖=1𝑓𝑥_𝑖,𝑦+1𝑚^𝑚_𝑗=1𝑓𝑥,𝑦_𝑗≥max_{𝑘∈{1,…,𝑛}𝑙∈{1,…,𝑚}}𝐷_𝑙𝑓,𝐲+𝐷_𝑘,𝑓𝑓,𝐱𝑥,𝑦≤min_{𝑘∈{1,…,𝑛}𝑙∈{1,…,𝑚}}12𝐷_𝑙𝑓,𝐲+𝐷_𝑘𝑓,𝐱. (27)

Remark 5. Note that if we substitute 𝑚=1, 𝑓(𝑥,𝑦₁)→𝑓(𝑥), 𝐷(𝑓,𝐰,𝐲,𝐽,𝑥_𝑖)=𝑓(𝑥), 𝐷(𝑓,𝐩,𝐱,𝐼,𝑦_𝑗)=𝐷(𝑓,𝐩,𝐱,𝐼), and 𝐷(𝑓,𝐰,𝐲,𝐽)=𝑓(𝑥) in Theorem 2, we get the following result of Dragomir [9] for convex function defined on the interval and ∑^𝑛_𝑖=1𝑝_𝑖=𝑃_𝑛>0, 𝑓1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖≤𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖+𝑃_𝐼𝑃_𝑛𝑓1𝑃_{𝐼𝑖∈𝐼}𝑝_𝑖𝑥_𝑖≤1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖. (28)

The following refinement of Hölder inequality holds.

Corollary 6. Let 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛) and 𝐲=(𝑦₁,𝑦₂,…,𝑦_𝑛) be two positive n-tuples. Then for (1/𝑝)+(1/𝑞)=1, 𝑝, 𝑞>1, one has ^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖≤^𝑛_𝑖=1𝑦^𝑞_𝑖^𝑝/𝑞_𝑖∈𝐼𝑦^𝑞_𝑖^−𝑝/𝑞_𝑖∈𝐼𝑥_𝑖𝑦_𝑖^𝑝+_𝑖∈𝐼𝑦^𝑞_𝑖^−𝑝/𝑞_𝑖∈𝐼𝑥_𝑖𝑦_𝑖^𝑝1/𝑝≤^𝑛_𝑖=1𝑥^𝑝_𝑖^1/𝑝𝑛_𝑖=1𝑦^𝑞_𝑖^1/𝑞. (29)

Proof. Using the functions𝑓(𝑥)=𝑥^𝑝,𝑝>1, 𝑥_𝑖→𝑥_𝑖𝑦_𝑖^−𝑞/𝑝, and𝑝_𝑖→𝑦^𝑞_𝑖in (28), we get (29).

Remark 7. As mentioned above from the inequalities in (29), we can write ^𝑛_𝑖=1𝑥^𝑝_𝑖^1/𝑝𝑛_𝑖=1𝑦^𝑞_𝑖^1/𝑞≥max_{𝐼⊂{1,…,𝑛}}^𝑛_𝑖=1𝑦^𝑞_𝑖^𝑝/𝑞_𝑖∈𝐼𝑦^𝑞_𝑖^−𝑝/𝑞_𝑖∈𝐼𝑥_𝑖𝑦_𝑖^𝑝+_𝑖∈𝐼𝑦^𝑞_𝑖^−𝑝/𝑞_𝑖∈𝐼𝑥_𝑖𝑦_𝑖^𝑝1/𝑝,_𝑛𝑖=1𝑥_𝑖𝑦_𝑖≤min_{𝐼⊂{1,…,𝑛}}^𝑛_𝑖=1𝑦^𝑞_𝑖^𝑝/𝑞_𝑖∈𝐼𝑦^𝑞_𝑖^−𝑝/𝑞_𝑖∈𝐼𝑥_𝑖𝑦_𝑖^𝑝+_𝑖∈𝐼𝑦^𝑞_𝑖^−𝑝/𝑞_𝑖∈𝐼𝑥_𝑖𝑦_𝑖^𝑝1/𝑝. (30)

The following improvement of Jensen’s inequality is valid.

Theorem 8. Let 𝑓∶[𝑎,𝑏]×[𝑐,𝑑]→ℝ be convex on the co-ordinates of[𝑎,𝑏]×[𝑐,𝑑]⊆ℝ². If𝐱is an n-tuple in[𝑎,𝑏], 𝐲 is an m-tuple in[𝑐,𝑑], 𝐩 is a nonnegative n-tuple such that𝑃_𝑛=∑^𝑛_𝑖=1𝑝_𝑖>0, and 𝐰 is a nonnegative m-tuple such that𝑊_𝑚=∑^𝑚_𝑗=1𝑤_𝑗>0, then 1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗−𝑓𝑥,𝑦≥1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗−𝑓𝑥,𝑦−1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝜕𝑓₊𝑥,𝑦𝑥𝜕𝑤^𝑖−𝑥+𝜕𝑓₊𝑥,𝑦𝑦𝜕𝑧^𝑗−𝑦, (31) where𝑥=(1/𝑃_𝑛)∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖, and𝑦=(1/𝑊_𝑚)∑^𝑚_𝑗=1𝑤_𝑗𝑦_𝑗.

Proof. Since𝑓is convex on[𝑎,𝑏]×[𝑐,𝑑], therefore we have 𝑓()≥𝑥,𝑦−𝑓𝑤,𝑧𝜕𝑓₊(𝑤,𝑧)()+𝜕𝑤𝑥−𝑤𝜕𝑓₊(𝑤,𝑧).𝜕𝑧𝑦−𝑧 (32) From the above inequality, we have 𝑓𝑥,𝑦−𝑓(𝑤,𝑧)−𝜕𝑓₊(𝑤,𝑧)()−𝜕𝑤𝑥−𝑤𝜕𝑓₊(𝑤,𝑧)=𝑓()−𝜕𝑧𝑦−𝑧𝑥,𝑦−𝑓𝑤,𝑧𝜕𝑓₊(𝑤,𝑧)()−𝜕𝑤𝑥−𝑤𝜕𝑓₊(𝑤,𝑧)≥𝑓(−𝜕𝑧𝑦−𝑧𝑥,𝑦−𝑓𝑤,𝑧)𝜕𝑓₊(𝑤,𝑧)()+𝜕𝑤𝑥−𝑤𝜕𝑓₊(𝑤,𝑧).𝜕𝑧𝑦−𝑧 (33) Let𝑥→𝑥_𝑖,𝑦→𝑦_𝑗,𝑤→∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖/𝑃_𝑛∶=𝑥, and𝑧→∑^𝑛_𝑗=1𝑤_𝑗𝑦_𝑗/𝑊_𝑚∶=𝑦, then (33) becomes 𝑓𝑥_𝑖,𝑦_𝑗−𝑓𝑥,𝑦−𝜕𝑓₊𝑥,𝑦𝑥𝜕𝑤^𝑖−𝑥−𝜕𝑓₊𝑥,𝑦𝑦𝜕𝑧^𝑗−𝑦≥𝑓𝑥_𝑖,𝑦_𝑗−𝑓𝑥,𝑦−𝜕𝑓₊𝑥,𝑦𝑥𝜕𝑤^𝑖−𝑥+𝜕𝑓₊𝑥,𝑦𝑦𝜕𝑧^𝑗−𝑦. (34) Multiplying (34) by 𝑝_𝑖 and 𝑤_𝑗 and summing over𝑖and𝑗, we have ^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗−^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥,𝑦−^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝜕𝑓₊𝑥,𝑦𝑥𝜕𝑤^𝑖−𝑥−^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝜕𝑓₊𝑥,𝑦𝑦𝜕𝑧^𝑗−𝑦≥^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗−𝑓𝑥,𝑦−^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝜕𝑓₊𝑥,𝑦𝑥𝜕𝑤^𝑖−𝑥+𝜕𝑓₊𝑥,𝑦𝑦𝜕𝑧^𝑗−𝑦. (35) One has ^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝜕𝑓₊𝑥,𝑦𝑥𝜕𝑤^𝑖−𝑥=^𝑚_𝑗=1𝑤_𝑗𝜕𝑓₊𝑥,𝑦𝜕𝑤^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖−𝑃_𝑛⋅1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖=0,_{𝑛𝑚𝑖=1𝑗=1}𝑝_𝑖𝑤_𝑗𝜕𝑓₊𝑥,𝑦𝑦𝜕𝑧^𝑗−𝑦=0. (36) Therefore (35) becomes ^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗−^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥,𝑦≥^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗−𝑓𝑥,𝑦−^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝜕𝑓₊𝑥,𝑦𝑥𝜕𝑤^𝑖−𝑥+𝜕𝑓₊𝑥,𝑦𝑦𝜕𝑧^𝑗−𝑦. (37) Multiplying both hand sides by 1/𝑃_𝑛𝑊_𝑚, we have 1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗−𝑓𝑥,𝑦≥1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝑓𝑥_𝑖,𝑦_𝑗−𝑓𝑥,𝑦−1𝑃_𝑛𝑊_𝑚^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑤_𝑗𝜕𝑓₊𝑥,𝑦𝑥𝜕𝑤^𝑖−𝑥+𝜕𝑓₊𝑥,𝑦𝑦𝜕𝑧^𝑗−𝑦. (38) This completes the proof.

Conflict of Interests

The authors declare that they have no conflict of interests regarding publication of this paper.

Acknowledgments

The authors are grateful to the referees for the useful comments regarding presentation in the early version of the paper. The last author also acknowledges that the present work was partially supported by the University Putra Malaysia (UPM).