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

Authors: M. Adil Khan; G. Ali Khan; T. Ali; T. Batbold; A. Kiliçman
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 mappingsfy:[a,b]→ℝ defined asfy(t)=f(t,y) andfx:[c,d]→ℝdefined asfx(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 thatPn=∑i=1npi>0andWm=∑i=1mwj>0, then
(1)f(1Pn∑i=1n‍pixi,1Wm∑j=1m‍wjyj)≤12{1Pn∑i=1n‍pif(xi,y¯)+1Wm∑j=1m‍wjf(x¯,yj)}≤1PnWm∑i=1n‍∑j=1m‍piwjf(xi,yj),
wherex¯=(1/Pn)∑i=1n‍pixi, andy¯=(1/Wm)∑j=1m‍wjyj.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]. Ifxi∈[a,b],yj∈[c,d], pi,wj>0,i∈{1,2,…,n},j∈{1,2,…,m}withPn=∑i=1npi, andWm=∑j=1mwj, then for any subsetsI⊂{1,2,…,n}andJ⊂{1,2,…,m}, we assume thatI¯:={1,2,…,n}∖IandJ¯:={1,2,…,m}∖J. DefinePI=∑i∈Ipi, PI¯=∑i∈I¯pi, WJ=∑j∈Jwj, andWJ¯=∑j∈J¯wj. For the functionfand then-, m-tuples,x=(x1,x2,…,xn),y=(y1,y2,…,ym), p=(p1,p2,…,pn), and w=(w1,w2,…,wm), we define the following functionals:
(2)D(f,p,x,I,yj)=PIPnf(1PI∑i∈I‍pixi,yj)+PI¯Pnf(1PI¯∑i∈I¯‍pixi,yj),D(f,w,y,J,xi)=WJWmf(xi,1WJ∑j∈J‍wjyj)+WJ¯Wmf(xi,1WJ¯∑j∈J¯‍wjyj),D(f,p,x,I)=PIPnf(1PI∑i∈I‍pixi,y¯)+PI¯Pnf(1PI¯∑i∈I¯‍pixi,y¯),D(f,w,y,J)=WJWmf(x¯,1WJ∑j∈J‍wjyj)+WJ¯Wmf(x¯,1WJ¯∑j∈J¯‍wjyj),
wherex¯=(1/Pn)∑i=1n‍pixi, and y¯=(1/Wm)∑j=1m‍wjyj.It is worth to observe that forI={k}, k∈{1,…,n}, andJ={l}, l∈{1,…,m}, we have the functionals
(3)Dk(f,p,x,yj):=D(f,p,x,{k},yj)=pkPnf(xk,yj)+Pn-pkPnf(∑i=1npixi-pkxkPn-pk,yj),Dl(f,w,y,xi):=D(f,w,y,{l},xi)=wlWmf(xi,yl)+Wm-wlWmf(xi,∑j=1mwjyj-wlylWm-wl),Dk(f,p,x):=D(f,p,x,{k})=pkPnf(xk,y¯)+Pn-pkPnf(∑i=1npixi-pkxkPn-pk,y¯),Dl(f,w,y):=D(f,w,y,{l})=wlWmf(x¯,yl)+Wm-wlWmf(x¯,∑j=1mwjyj-wlylWm-wl).
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]. Ifxi∈[a,b],yj∈[c,d], pi, wj>0,i∈{1,2,…,n}, j∈{1,2,…,m}with Pn=∑i=1npi, and Wm=∑j=1mwj, then for any subsets I⊂{1,2,…,n}and J⊂{1,2,…,m}, one has
(4)f(x¯,y¯)≤12[D(f,w,y,J)+D(f,p,x,I)]≤12[1Pn∑i=1n‍pif(xi,y¯)+1Wm∑j=1m‍wjf(x¯,yj)]≤12[1Pn∑i=1n‍piD(f,w,y,J,xi)+1Wm∑j=1m‍wjD(f,p,x,I,yj)]≤1PnWm∑i=1n‍∑j=1m‍piwjf(xi,yj),
wherex¯=(1/Pn)∑i=1n‍pixi, and y¯=(1/Wm)∑j=1m‍wjyj.Proof.
One-dimensional Jensen’s inequality gives us(5)f(xi,y¯)≤1Wm∑j=1m‍wjf(xi,yj),f(x¯,yj)≤1Pn∑i=1n‍pif(xi,yj).
As we have
(6)D(f,p,x,I,yj)=PIPnf(1PI∑i∈I‍pixi,yj)+PI¯Pnf(1PI¯∑i∈I¯‍pixi,yj),
so by Jensen’s inequality, we have
(7)D(f,p,x,I,yj)=PIPnf(1PI∑i∈I‍pixi,yj)+PI¯Pnf(1PI¯∑i∈I¯‍pixi,yj)≤PIPn1PI∑i∈I‍pif(xi,yj)+PI¯Pn1PI¯∑i∈I¯‍pif(xi,yj)=1Pn∑i∈I‍pif(xi,yj)+1Pn∑i∈I¯‍pif(xi,yj)=1Pn∑i∈I∪I¯‍pif(xi,yj)⟹D(f,p,x,I,yj)≤1Pn∑i=1n‍pif(xi,yj).
As the functionfis convex on the first co-ordinate, so we have
(8)D(f,p,x,I,yj)=PIPnf(1PI∑i∈I‍pixi,yj)+PI¯Pnf(1PI¯∑i∈I¯‍pixi,yj)≥f(PIPn1PI∑i∈I‍pixi+PI¯Pn1PI¯∑i∈I¯‍pixi,yj)=f(1Pn∑i∈I‍pixi+1Pn∑i∈I¯‍pixi,yj)=f(1Pn∑i∈I∪I¯‍pixi,yj)=f(1Pn∑i=1n‍pixi,yj)⟹D(f,p,x,I,yj)≥f(1Pn∑i=1n‍pixi,yj).
Now, from (7) and (8), we have
(9)f(x¯,yj)≤D(f,p,x,I,yj)≤1Pn∑i=1n‍pif(xi,yj).
Similarly, we can write
(10)f(xi,y¯)≤D(f,w,y,J,xi)≤1Wm∑j=1m‍wjf(xi,yj).
Multiplying (9) and (10), respectively, bywjandpiand summing overiandj, we obtain
(11)1Wm∑j=1m‍wjf(x¯,yj)≤1Wm∑j=1m‍wjD(f,p,x,I,yj)1Wm∑j=1m‍wjf(x¯,yj)≤1PnWm∑i=1n‍∑j=1m‍piwjf(xi,yj),(12)1Pn∑i=1n‍pif(xi,y¯)≤1Pn∑i=1n‍piD(f,w,y,J,xi)1Pn∑i=1n‍pif(xi,y¯)≤1PnWm∑i=1n‍∑j=1m‍piwjf(xi,yj).
Adding (11) and (12), we have
(13)12[1Pn∑i=1n‍pif(xi,y¯)+1Wm∑j=1m‍wjf(x¯,yj)]≤12[+1Wm∑j=1m‍wjD(f,p,x,I,yj)1Pn∑i=1n‍piD(f,w,y,J,xi)m.mm.m+1Wm∑j=1m‍wjD(f,p,x,I,yj)]≤1PnWm∑i=1n‍∑j=1mpiwj‍f(xi,yj).
Again by one-dimensional Jensen’s inequality, we have
(14)f(x¯,y¯)≤1Pn∑i=1n‍pif(xi,y¯),f(x¯,y¯)≤1Wm∑j=1m‍wjf(x¯,yj).
As we have the functional
(15)D(f,p,x,I)=PIPnf(1PI∑i∈I‍pixi,y¯)+PI¯Pnf(1PI¯∑i∈I¯‍pixi,y¯),
so by Jensen’s inequality, we get
(16)D(f,p,x,I)=PIPnf(1PI∑i∈I‍pixi,y¯)D(f,p,x,I)+PI¯Pnf(1PI¯∑i∈I¯‍pixi,y¯)D(f,p,x,I)≤PIPn1PI∑i∈I‍pif(xi,y¯)+PI¯Pn1PI¯∑i∈I¯‍pif(xi,y¯)D(f,p,x,I)=1Pn∑i∈I‍pif(xi,y¯)+1Pn∑i∈I¯‍pif(xi,y¯)D(f,p,x,I)=1Pn∑i∈I∪I¯‍pif(xi,y¯)D(f,p,x,I)⟹D(f,p,x,I)≤1Pn∑i=1n‍pif(xi,y¯),
and as the functionfis convex on the first co-ordinate, so we have
(17)D(f,p,x,I)=PIPnf(1PI∑i∈I‍pixi,y¯)D(f,p,x,I)+PI¯Pnf(1PI¯∑i∈I¯‍pixi,y¯)D(f,p,x,I)≥f(PIPn1PI∑i∈I‍pixi+PI¯Pn1PI¯∑i∈I¯‍pixi,y¯)D(f,p,x,I)=f(1Pn∑i∈I‍pixi+1Pn∑i∈I¯‍pixi,y¯)D(f,p,x,I)=f(1Pn∑i∈I∪I¯‍pixi,y¯)=f(1Pn∑i=1n‍pixi,y¯)D(f,p,x,I)⟹D(f,p,x,I)≥f(x¯,y¯).
Now from (16) and (17), we have
(18)f(x¯,y¯)≤D(f,p,x,I)≤1Pn∑i=1n‍pif(xi,y¯).
Similarly, we can prove that
(19)f(x¯,y¯)≤D(f,w,y,J)≤1Wm∑j=1m‍wjf(x¯,yj).
Adding (18) and (19), we get
(20)f(x¯,y¯)≤12[D(f,w,y,J)+D(f,p,x,I)]f(x¯,y¯)≤12[1Pn∑i=1n‍pif(xi,y¯)+1Wm∑j=1m‍wjf(x¯,yj)].
Combining (13) and (20), we have
(21)f(x¯,y¯)≤12[D(f,w,y,J)+D(f,p,x,I)]≤12[1Pn∑i=1n‍pif(xi,y¯)+1Wm∑j=1m‍wjf(x¯,yj)]≤12[+1Wm∑j=1m‍wjD(f,p,x,I,yj)1Pn∑i=1n‍piD(f,w,y,J,xi)mmmm+1Wm∑j=1m‍wjD(f,p,x,I,yj)]≤1PnWm∑i=1n‍∑j=1m‍piwjf(xi,yj).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
(22)1PnWm∑i=1n‍∑j=1m‍piwjf(xi,yj)≥maxI⊂{1,…,n}J⊂{1,…,m}12[∑j=1m‍1Pn∑i=1n‍piD(f,w,y,J,xi)mmmmmmmm+1Wm∑j=1m‍wjD(f,p,x,I,yj)],1Pn∑i=1n‍pif(xi,y¯)+1Wm∑j=1m‍wjf(x¯,yj)≤minI⊂{1,…,n}J⊂{1,…,m}[∑j=1m‍1Pn∑i=1n‍piD(f,w,y,J,xi)mmmmmmm+1Wm∑j=1m‍wjD(f,p,x,I,yj)],1Pn∑i=1n‍pif(xi,y¯)+1Wm∑j=1m‍wjf(x¯,yj)≥maxI⊂{1,…,n}J⊂{1,…,m}[D(f,w,y,J)+D(f,p,x,I)],f(x¯,y¯)≤minI⊂{1,…,n}J⊂{1,…,m}12[D(f,w,y,J)+D(f,p,x,I)].
These inequalities imply the following results:
(23)1PnWm∑i=1n‍∑j=1m‍piwjf(xi,yj)≥maxk∈{1,…,n}l∈{1,…,m}12[1Wm∑j=1m‍wjDk(f,p,x,yj)1Pn∑i=1n‍piDl(f,w,y,xi)mmmmmm+1Wm∑j=1m‍wjDk(f,p,x,yj)],1Pn∑i=1n‍pif(xi,y¯)+1Wm∑j=1m‍wjf(x¯,yj)≤mink∈{1,…,n}l∈{1,…,m}[∑j=1m‍1Pn∑i=1n‍piDl(f,w,y,xi)mmmmm+1Wm∑j=1m‍wjDk(f,p,x,yj)],1Pn∑i=1n‍pif(xi,y¯)+1Wm∑j=1m‍wjf(x¯,yj)≥maxk∈{1,…,n}l∈{1,…,m}[Dl(f,w,y)+Dk(f,p,x)],f(x¯,y¯)≤mink∈{1,…,n}l∈{1,…,m}12[Dl(f,w,y)+Dk(f,p,x)].Moreover, from the above, we also have(24)maxI⊂{1,…,n}D(f,p,x;I)≥maxk∈{1,…,n}Dk(f,p,x),maxJ⊂{1,…,m}D(f,w,y;J)≥maxl∈{1,…,m}Dl(f,w,y),maxI⊂{1,…,n}D(f,p,x;I,yj)≥maxk∈{1,…,n}Dk(f,p,x,yj),maxJ⊂{1,…,m}D(f,w,y;J,xi)≥maxl∈{1,…,m}Dl(f,w,y,xi),mink∈{1,…,n}Dk(f,p,x)≥minI⊂{1,…,n}D(f,p,x;I),minl∈{1,…,m}Dl(f,w,y)≥minJ⊂{1,…,m}D(f,w,y;J),mink∈{1,…,n}Dk(f,p,x,yj)≥minI⊂{1,…,n}D(f,p,x;I,yj),minl∈{1,…,m}Dl(f,w,y,xi)≥minJ⊂{1,…,m}D(f,w,y;J,xi).We discuss the following particular cases of the above inequalities which is of interest [6].In the case whenpi=1 andwj=1fori∈{1,…,n}andj∈{1,…,m}, consider the natural numbersk,lwith1≤k≤n-1 and1≤l≤m-1and define
(25)Dk(f,x,yj)=knf(1k∑i=1k‍xi,yj)+n-knf(1n-k∑i=k+1n‍xi,yj),Dl(f,y,xi)=lmf(xi,1l∑j=1l‍yj)+m-lmf(xi,1m-l∑j=l+1m‍yj),Dk(f,x)=knf(1k∑i=1k‍xi,y¯)+n-knf(1n-k∑i=k+1n‍xi,y¯),Dl(f,y)=lmf(x¯,1l∑j=1l‍yj)+m-lmf(x¯,1m-l∑j=l+1m‍yj).
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]. Ifxi∈[a,b] and yj∈[c,d], then for anyk∈{1,…,n-1}andl∈{1,…,m-1}, one has
(26)f(1n∑i=1n‍xi,1m∑j=1m‍yj)≤12[Dl(f,y)+Dk(f,x)]≤12[1n∑i=1n‍f(xi,y¯)+1m∑j=1m‍f(x¯,yj)]≤12[1n∑i=1n‍Dl(f,y,xi)+1m∑j=1m‍Dk(f,x,yj)]≤1mn∑i=1n‍∑j=1mf(xi,yj).‍In particular, we have the bounds(27)1mn∑i=1n‍∑j=1m‍f(xi,yj)≥maxk∈{1,…,n}l∈{1,…,m}12[1m∑j=1m‍Dk(f,x,yj)1n∑i=1n‍Dl(f,y,xi)mmmmmmmm+1m∑j=1m‍Dk(f,x,yj)],1n∑i=1n‍f(xi,y¯)+1m∑j=1m‍f(x¯,yj)≤mink∈{1,…,n}l∈{1,…,m}[1m∑j=1m‍Dk(f,x,yj)1n∑i=1n‍Dl(f,y,xi)mmmmmmm+1m∑j=1m‍Dk(f,x,yj)],1n∑i=1n‍f(xi,y¯)+1m∑j=1m‍f(x¯,yj)≥maxk∈{1,…,n}l∈{1,…,m}[Dl(f,y)+Dk(f,x)],f(x¯,y¯)≤mink∈{1,…,n}l∈{1,…,m}12[Dl(f,y)+Dk(f,x)].Remark 5.
Note that if we substitutem=1, f(x,y1)→f(x), D(f,w,y,J,xi)=f(x¯), D(f,p,x,I,yj)=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 ∑i=1npi=Pn>0,
(28)f(1Pn∑i=1n‍pixi)≤PIPnf(1PI∑i∈I‍pixi)+PI¯Pnf(1PI¯∑i∈I¯‍pixi)f(1Pn∑i=1n‍pixi)≤1Pn∑i=1n‍pif(xi).The following refinement of Hölder inequality holds.Corollary 6.
Letx=(x1,x2,…,xn) and y=(y1,y2,…,yn) be two positive n-tuples. Then for (1/p)+(1/q)=1, p, q>1, one has
(29)∑i=1n‍xiyi≤[(∑i∈I¯‍yiq)-p/q(∑i=1n‍yiq)p/q{(∑i∈I¯‍yiq)-p/q(∑i∈I‍yiq)-p/q(∑i∈I‍xiyi)pmmmmmm,m,mm+(∑i∈I¯‍yiq)-p/q(∑i∈I¯‍xiyi)p}]1/p≤(∑i=1n‍xip)1/p(∑i=1n‍yiq)1/q.Proof.
Using the functionsf(x)=xp,p>1, xi→xiyi-q/p, andpi→yiqin (28), we get (29).Remark 7.
As mentioned above from the inequalities in (29), we can write
(30)(∑i=1n‍xip)1/p(∑i=1n‍yiq)1/q≥maxI⊂{1,…,n}[(∑i=1n‍yiq)p/q{(∑i∈I‍yiq)-p/q(∑i∈I‍xiyi)pmmmmmmmmbbbbmm,m+(∑i∈I¯‍yiq)-p/q(∑i∈I¯‍xiyi)p}]1/p,∑i=1n‍xiyi≤minI⊂{1,…,n}[(∑i=1n‍yiq)p/q{(∑i∈I‍yiq)-p/q(∑i∈I‍xiyi)pmmmmmmmmmbbbmmm+(∑i∈I¯‍yiq)-p/q(∑i∈I¯‍xiyi)p}]1/p.The following improvement of Jensen’s inequality is valid.Theorem 8.
Letf:[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 thatPn=∑i=1npi>0, and w is a nonnegative m-tuple such thatWm=∑j=1mwj>0, then
(31)1PnWm∑i=1n‍∑j=1m‍piwjf(xi,yj)-f(x¯,y¯)≥|1PnWm∑i=1n‍∑j=1m‍piwj|f(xi,yj)-f(x¯,y¯)|mm,,,-1PnWm∑i=1n‍∑j=1m‍piwj|∑i=1n‍∑j=1m‍∂f+(x¯,y¯)∂w(xi-x¯)mmmmmmmm,,,,,mmm+∂f+(x¯,y¯)∂z(yj-y¯)∑i=1n‍∑j=1m‍||,
wherex¯=(1/Pn)∑i=1n‍pixi, andy¯=(1/Wm)∑j=1m‍wjyj.Proof.
Sincefis convex on[a,b]×[c,d], therefore we have
(32)f(x,y)-f(w,z)≥∂f+(w,z)∂w(x-w)+∂f+(w,z)∂z(y-z).
From the above inequality, we have
(33)f(x,y)-f(w,z)-∂f+(w,z)∂w(x-w)-∂f+(w,z)∂z(y-z)=|f(x,y)-f(w,z)-∂f+(w,z)∂w(x-w)mmlm-∂f+(w,z)∂z(y-z)|≥|∂f+(w,z)∂z|f(x,y)-f(w,z)|mmim-|∂f+(w,z)∂w(x-w)+∂f+(w,z)∂z(y-z)||.
Letx→xi,y→yj,w→∑i=1npixi/Pn:=x¯, andz→∑j=1nwjyj/Wm:=y¯, then (33) becomes
(34)f(xi,yj)-f(x¯,y¯)-∂f+(x¯,y¯)∂w(xi-x¯)-∂f+(x¯,y¯)∂z(yj-y¯)≥|∂f+(x¯,y¯)∂z|f(xi,yj)-f(x¯,y¯)|mmi-|∂f+(x¯,y¯)∂w(xi-x¯)+∂f+(x¯,y¯)∂z(yj-y¯)||.
Multiplying (34) by pi and wj and summing overiandj, we have
(35)∑i=1n‍∑j=1m‍piwjf(xi,yj)-∑i=1n‍∑j=1m‍piwjf(x¯,y¯)-∑i=1n‍∑j=1m‍piwj∂f+(x¯,y¯)∂w(xi-x¯)-∑i=1n‍∑j=1m‍piwj∂f+(x¯,y¯)∂z(yj-y¯)≥|∑i=1n‍∑j=1m‍piwj|f(xi,yj)-f(x¯,y¯)|-∑i=1n‍∑j=1m‍piwj|∂f+(x¯,y¯)∂w(xi-x¯)mmmmmmmmm+∂f+(x¯,y¯)∂z(yj-y¯)|∑j=1m‍|.
One has
(36)∑i=1n‍∑j=1m‍piwj∂f+(x¯,y¯)∂w(xi-x¯)=∑j=1m‍wj∂f+(x¯,y¯)∂w(∑i=1n‍pixi-Pn·1Pn∑i=1n‍pixi)=0,∑i=1n‍∑j=1m‍piwj∂f+(x¯,y¯)∂z(yj-y¯)=0.
Therefore (35) becomes
(37)∑i=1n‍∑j=1m‍piwjf(xi,yj)-∑i=1n‍∑j=1m‍piwjf(x¯,y¯)≥|∑i=1n‍∑j=1m‍piwj|f(xi,yj)-f(x¯,y¯)|-∑i=1n‍∑j=1m‍piwj|∂f+(x¯,y¯j)∂w∂f+(x¯,y¯)∂w(xi-x¯)mmmmmmmmnn+∂f+(x¯,y¯)∂z(yj-y¯)∂f+(x¯,y¯j)∂w|∑j=1m‍|.
Multiplying both hand sides by 1/PnWm, we have
(38)1PnWm∑i=1n‍∑j=1m‍piwjf(xi,yj)-f(x¯,y¯)≥|1PnWm∑i=1n‍∑j=1m‍piwj|f(xi,yj)-f(x¯,y¯)|mmm-1PnWm∑i=1n‍∑j=1m‍piwj|∂f+(x¯,y¯)∂w(xi-x¯)mmmmmmmmmmmmm+∂f+(x¯,y¯)∂z(yj-y¯)|∑j=1m‍|.
This completes the proof.



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*Source: 214123-2013-12-10.xml*