Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 214123, 8 pages
http://dx.doi.org/10.1155/2013/214123
Research Article
Further Refinements of Jensen’s Type Inequalities for
the Function Defined on the Rectangle
M. Adil Khan,1 G. Ali Khan,1 T. Ali,1 T. Batbold,2 and A. Kiliçman3
1
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
Institute of Mathematics, National University of Mongolia, P.O. Box 46A/104, 14201 Ulaanbaatar, Mongolia
3
Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM),
43400 Serdang, Selangor, Malaysia
2
Correspondence should be addressed to A. Kiliçman; kilicman@yahoo.com
Received 26 September 2013; Accepted 10 November 2013
Academic Editor: Abdullah Alotaibi
Copyright © 2013 M. Adil Khan et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → R, [𝑎, 𝑏] × [𝑐, 𝑑] ⊂ R2
with 𝑎 < 𝑏 and 𝑐 < 𝑑 is called convex on the co-ordinates
if the partial mappings 𝑓𝑦 : [𝑎, 𝑏] → R defined as 𝑓𝑦 (𝑡) =
𝑓(𝑡, 𝑦) and 𝑓𝑥 : [𝑐, 𝑑] → Rdefined as 𝑓𝑥 (𝑠) = 𝑓(𝑥, 𝑠) are
convex for all 𝑥 ∈ [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. Let 𝑓 : [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 that 𝑃𝑛 = ∑𝑛𝑖=1 𝑝𝑖 > 0 and 𝑊𝑚 =
∑𝑚
𝑖=1 𝑤𝑗 > 0, then
𝑓(
1 𝑛
1 𝑚
∑𝑝𝑖 𝑥𝑖 ,
∑𝑤 𝑦 )
𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗 𝑗
≤
}
1{1 𝑛
1 𝑚
𝑝
𝑓
(𝑥
,
𝑦)
+
∑
∑𝑤 𝑓 (𝑥, 𝑦𝑗 )}
𝑖
𝑖
2 { 𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
}
{
≤
1 𝑛 𝑚
∑ ∑𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) ,
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
(1)
where 𝑥 = (1/𝑃𝑛 ) ∑𝑛𝑖=1 𝑝𝑖 𝑥𝑖 , and 𝑦 = (1/𝑊𝑚 ) ∑𝑚
𝑗=1 𝑤𝑗 𝑦𝑗 .
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,
2
Abstract and Applied Analysis
see, for example, [1, 2, 9, 13, 14]. For further several related
integral inequalities, see [15].
𝐷𝑘 (𝑓, p, x)
:= 𝐷 (𝑓, p, x, {𝑘})
2. Main Results
Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → R be convex on the co-ordinate
on [𝑎, 𝑏] × [𝑐, 𝑑]. If 𝑥𝑖 ∈ [𝑎, 𝑏], 𝑦𝑗 ∈ [𝑐, 𝑑], 𝑝𝑖 , 𝑤𝑗 > 0, 𝑖 ∈
{1, 2, . . . , 𝑛}, 𝑗 ∈ {1, 2, . . . , 𝑚} with 𝑃𝑛 = ∑𝑛𝑖=1 𝑝𝑖 , and 𝑊𝑚 =
∑𝑚
𝑗=1 𝑤𝑗 , then for any subsets 𝐼 ⊂ {1, 2, . . . , 𝑛} and 𝐽 ⊂
{1, 2, . . . , 𝑚}, we assume that 𝐼 := {1, 2, . . . , 𝑛} \ 𝐼 and 𝐽 :=
{1, 2, . . . , 𝑚} \ 𝐽. Define 𝑃𝐼 = ∑𝑖∈𝐼 𝑝𝑖 , 𝑃𝐼 = ∑𝑖∈𝐼 𝑝𝑖 , 𝑊𝐽 =
∑𝑗∈𝐽 𝑤𝑗 , and 𝑊𝐽 = ∑𝑗∈𝐽 𝑤𝑗 . For the function 𝑓 and the 𝑛-,
𝑚-tuples, x = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ), y = (𝑦1 , 𝑦2 , . . . , 𝑦𝑚 ), p = (𝑝1 ,
𝑝2 , . . . , 𝑝𝑛 ), and w = (𝑤1 , 𝑤2 , . . . , 𝑤𝑚 ), we define the following
functionals:
𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )
=
∑𝑛 𝑝 𝑥 − 𝑝𝑘 𝑥𝑘
𝑃 − 𝑝𝑘
𝑝𝑘
𝑓 (𝑥𝑘 , 𝑦) + 𝑛
𝑓 ( 𝑖=1 𝑖 𝑖
, 𝑦) ,
𝑃𝑛
𝑃𝑛
𝑃𝑛 − 𝑝𝑘
𝐷𝑙 (𝑓, w, y)
:= 𝐷 (𝑓, w, y, {𝑙})
=
∑𝑚
𝑤𝑙
𝑊 − 𝑤𝑙
𝑗=1 𝑤𝑗 𝑦𝑗 − 𝑤𝑙 𝑦𝑙
𝑓 (𝑥, 𝑦𝑙 ) + 𝑚
𝑓 (𝑥,
).
𝑊𝑚
𝑊𝑚
𝑊𝑚 − 𝑤𝑙
(3)
The following refinement of (1) holds.
Theorem 2. Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → R be a co-ordinate
convex function on [𝑎, 𝑏] × [𝑐, 𝑑]. If 𝑥𝑖 ∈ [𝑎, 𝑏], 𝑦𝑗 ∈ [𝑐, 𝑑], 𝑝𝑖 ,
𝑤𝑗 > 0, 𝑖 ∈ {1, 2, . . . , 𝑛}, 𝑗 ∈ {1, 2, . . . , 𝑚} with 𝑃𝑛 = ∑𝑛𝑖=1 𝑝𝑖 ,
and 𝑊𝑚 = ∑𝑚
𝑗=1 𝑤𝑗 , then for any subsets 𝐼 ⊂ {1, 2, . . . , 𝑛} and
𝐽 ⊂ {1, 2, . . . , 𝑚}, one has
𝑃
𝑃𝐼
1
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 ) + 𝐼 𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 ) ,
𝑃𝑛
𝑃𝐼 𝑖∈𝐼
𝑃𝑛
𝑃𝐼
𝑖∈𝐼
𝐷 (𝑓, w, y, 𝐽, 𝑥𝑖 )
𝑊𝐽
𝑊𝐽
1
1
=
𝑓 (𝑥𝑖 ,
𝑓 (𝑥𝑖 ,
∑𝑤𝑗 𝑦𝑗 ) +
∑𝑤𝑗 𝑦𝑗 ) ,
𝑊𝑚
𝑊𝐽 𝑗∈𝐽
𝑊𝑚
𝑊𝐽
𝑗∈𝐽
𝐷 (𝑓, p, x, 𝐼)
=
=
𝑃
𝑃𝐼
1
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦) + 𝐼 𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦) ,
𝑃𝑛
𝑃𝐼 𝑖∈𝐼
𝑃𝑛
𝑃𝐼
𝑖∈𝐼
𝑓 (𝑥, 𝑦)
≤
1
[𝐷 (𝑓, w, y, 𝐽) + 𝐷 (𝑓, p, x, 𝐼)]
2
≤
1 𝑚
1[1 𝑛
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) +
∑ 𝑤 𝑓 (𝑥, 𝑦𝑗 )]
2 𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
]
[
≤
1[1 𝑛
1 𝑚
∑𝑝𝑖 𝐷 (𝑓, w, y, 𝐽, 𝑥𝑖 ) +
∑ 𝑤 𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )]
2 𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
]
[
≤
1 𝑛 𝑚
∑ ∑𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) ,
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
𝐷 (𝑓, w, y, 𝐽)
𝑊𝐽
𝑊𝐽
1
1
=
𝑓 (𝑥,
𝑓 (𝑥,
∑𝑤𝑗 𝑦𝑗 ) +
∑𝑤𝑗 𝑦𝑗 ) ,
𝑊𝑚
𝑊𝐽 𝑗∈𝐽
𝑊𝑚
𝑊𝐽
𝑗∈𝐽
(4)
(2)
where 𝑥 = (1/𝑃𝑛 ) ∑𝑛𝑖=1 𝑝𝑖 𝑥𝑖 , and 𝑦 = (1/𝑊𝑚 ) ∑𝑚
𝑗=1 𝑤𝑗 𝑦𝑗 .
It is worth to observe that for 𝐼 = {𝑘}, 𝑘 ∈ {1, . . . , 𝑛},
and 𝐽 = {𝑙}, 𝑙 ∈ {1, . . . , 𝑚}, we have the functionals
where 𝑥 = (1/𝑃𝑛 ) ∑𝑛𝑖=1 𝑝𝑖 𝑥𝑖 , and 𝑦 = (1/𝑊𝑚 ) ∑𝑚
𝑗=1 𝑤𝑗 𝑦𝑗 .
Proof. One-dimensional Jensen’s inequality gives us
𝐷𝑘 (𝑓, p, x, 𝑦𝑗 )
𝑓 (𝑥𝑖 , 𝑦) ≤
:= 𝐷 (𝑓, p, x, {𝑘} , 𝑦𝑗 )
=
1 𝑛
𝑓 (𝑥, 𝑦𝑗 ) ≤ ∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦𝑗 ) .
𝑃𝑛 𝑖=1
∑𝑛 𝑝 𝑥 − 𝑝𝑘 𝑥𝑘
𝑝𝑘
𝑃 − 𝑝𝑘
𝑓 (𝑥𝑘 , 𝑦𝑗 ) + 𝑛
𝑓 ( 𝑖=1 𝑖 𝑖
, 𝑦𝑗 ) ,
𝑃𝑛
𝑃𝑛
𝑃𝑛 − 𝑝𝑘
𝐷𝑙 (𝑓, w, y, 𝑥𝑖 )
:= 𝐷 (𝑓, w, y, {𝑙} , 𝑥𝑖 )
∑𝑚
𝑤𝑙
𝑊𝑚 − 𝑤𝑙
𝑗=1 𝑤𝑗 𝑦𝑗 − 𝑤𝑙 𝑦𝑙
=
𝑓 (𝑥𝑖 , 𝑦𝑙 ) +
𝑓 (𝑥𝑖 ,
),
𝑊𝑚
𝑊𝑚
𝑊𝑚 − 𝑤𝑙
1 𝑚
∑𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) ,
𝑊𝑚 𝑗=1 𝑗
(5)
As we have
𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )
=
(6)
𝑃
𝑃𝐼
1
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 ) + 𝐼 𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 ) ,
𝑃𝑛
𝑃𝐼 𝑖∈𝐼
𝑃𝑛
𝑃𝐼
𝑖∈𝐼
Abstract and Applied Analysis
3
Multiplying (9) and (10), respectively, by 𝑤𝑗 and 𝑝𝑖 and summing over 𝑖 and 𝑗, we obtain
so by Jensen’s inequality, we have
𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )
=
1 𝑚
1 𝑚
∑ 𝑤𝑗 𝑓 (𝑥, 𝑦𝑗 ) ≤
∑𝑤 𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )
𝑊𝑚 𝑗=1
𝑊𝑚 𝑗=1 𝑗
𝑃
𝑃𝐼
1
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 ) + 𝐼 𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛
𝑃𝐼 𝑖∈𝐼
𝑃𝑛
𝑃𝐼
1 𝑛 𝑚
≤
∑∑ 𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) ,
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
𝑖∈𝐼
≤
𝑃 1
𝑃𝐼 1
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦𝑗 ) + 𝐼 ∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛 𝑃𝐼 𝑖∈𝐼
𝑃𝑛 𝑃𝐼
𝑖∈𝐼
=
1
1
∑𝑝 𝑓 (𝑥𝑖 , 𝑦𝑗 ) + ∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛 𝑖∈𝐼 𝑖
𝑃𝑛
(7)
1 𝑛
1 𝑛
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) ≤ ∑𝑝𝑖 𝐷 (𝑓, w, y, 𝐽, 𝑥𝑖 )
𝑃𝑛 𝑖=1
𝑃𝑛 𝑖=1
1 𝑛 𝑚
≤
∑∑𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) .
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
𝑖∈𝐼
=
1
∑ 𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛
𝑖∈𝐼∪𝐼
󳨐⇒ 𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 ) ≤
(12)
Adding (11) and (12), we have
𝑛
1
∑𝑝 𝑓 (𝑥𝑖 , 𝑦𝑗 ) .
𝑃𝑛 𝑖=1 𝑖
As the function 𝑓 is convex on the first co-ordinate, so we
have
1[1 𝑛
1 𝑚
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) +
∑𝑤 𝑓 (𝑥, 𝑦𝑗 )]
2 𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
]
[
1[1 𝑛
∑𝑝 𝐷 (𝑓, w, y, 𝐽, 𝑥𝑖 )
2 𝑃𝑛 𝑖=1 𝑖
[
≤
𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )
=
(11)
𝑃
𝑃𝐼
1
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 ) + 𝐼 𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛
𝑃𝐼 𝑖∈𝐼
𝑃𝑛
𝑃𝐼
+
𝑖∈𝐼
1
∑ 𝑤 𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )]
𝑊𝑚 𝑗=1 𝑗
]
1 𝑛 𝑚
∑ ∑𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) .
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
≤
𝑃 1
𝑃 1
≥ 𝑓 ( 𝐼 ∑𝑝𝑖 𝑥𝑖 + 𝐼 ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛 𝑃𝐼 𝑖∈𝐼
𝑃𝑛 𝑃𝐼
(13)
𝑚
𝑖∈𝐼
1
1
= 𝑓 ( ∑𝑝𝑖 𝑥𝑖 + ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛 𝑖∈𝐼
𝑃𝑛
𝑖∈𝐼
= 𝑓(
Again by one-dimensional Jensen’s inequality, we have
(8)
𝑓 (𝑥, 𝑦) ≤
1
∑ 𝑝𝑖 𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛
(14)
1 𝑚
𝑓 (𝑥, 𝑦) ≤
∑𝑤 𝑓 (𝑥, 𝑦𝑗 ) .
𝑊𝑚 𝑗=1 𝑗
𝑖∈𝐼∪𝐼
= 𝑓(
1 𝑛
∑𝑝 𝑓 (𝑥𝑖 , 𝑦) ,
𝑃𝑛 𝑖=1 𝑖
1 𝑛
∑𝑝 𝑥 , 𝑦 )
𝑃𝑛 𝑖=1 𝑖 𝑖 𝑗
As we have the functional
𝐷 (𝑓, p, x, 𝐼) =
1 𝑛
󳨐⇒ 𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 ) ≥ 𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦𝑗 ) .
𝑃𝑛 𝑖=1
𝑃
𝑃𝐼
1
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦) + 𝐼 𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦) ,
𝑃𝑛
𝑃𝐼 𝑖∈𝐼
𝑃𝑛
𝑃𝐼
𝑖∈𝐼
(15)
Now, from (7) and (8), we have
so by Jensen’s inequality, we get
𝑓 (𝑥, 𝑦𝑗 ) ≤ 𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 ) ≤
1 𝑛
∑𝑝 𝑓 (𝑥𝑖 , 𝑦𝑗 ) .
𝑃𝑛 𝑖=1 𝑖
(9)
𝐷 (𝑓, p, x, 𝐼) =
𝑃𝐼
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦)
𝑃𝑛
𝑃𝐼 𝑖∈𝐼
+
Similarly, we can write
𝑃𝐼
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦)
𝑃𝑛
𝑃𝐼
𝑖∈𝐼
𝑓 (𝑥𝑖 , 𝑦) ≤ 𝐷 (𝑓, w, y, 𝐽, 𝑥𝑖 ) ≤
𝑚
1
∑ 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) .
𝑊𝑚 𝑗=1 𝑗
(10)
≤
𝑃 1
𝑃𝐼 1
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) + 𝐼 ∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦)
𝑃𝑛 𝑃𝐼 𝑖∈𝐼
𝑃𝑛 𝑃𝐼
𝑖∈𝐼
4
Abstract and Applied Analysis
=
1
1
∑𝑝 𝑓 (𝑥𝑖 , 𝑦) + ∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦)
𝑃𝑛 𝑖∈𝐼 𝑖
𝑃𝑛
Combining (13) and (20), we have
1
∑ 𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦)
𝑃𝑛
𝑓 (𝑥, 𝑦)
𝑖∈𝐼
=
𝑖∈𝐼∪𝐼
≤
1
[𝐷 (𝑓, w, y, 𝐽) + 𝐷 (𝑓, p, x, 𝐼)]
2
(16)
≤
1 𝑚
1[1 𝑛
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) +
∑ 𝑤 𝑓 (𝑥, 𝑦𝑗 )]
2 𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
]
[
and as the function 𝑓 is convex on the first co-ordinate, so
we have
≤
1[1 𝑛
∑𝑝 𝐷 (𝑓, w, y, 𝐽, 𝑥𝑖 )
2 𝑃𝑛 𝑖=1 𝑖
[
󳨐⇒ 𝐷 (𝑓, p, x, 𝐼) ≤
𝐷 (𝑓, p, x, 𝐼) =
𝑛
1
∑𝑝 𝑓 (𝑥𝑖 , 𝑦) ,
𝑃𝑛 𝑖=1 𝑖
𝑃𝐼
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦)
𝑃𝑛
𝑃𝐼 𝑖∈𝐼
+
+
𝑃𝐼
1
𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦)
𝑃𝑛
𝑃𝐼
≤
𝑖∈𝐼
≥ 𝑓(
𝑃 1
𝑃𝐼 1
∑𝑝𝑖 𝑥𝑖 + 𝐼 ∑𝑝𝑖 𝑥𝑖 , 𝑦)
𝑃𝑛 𝑃𝐼 𝑖∈𝐼
𝑃𝑛 𝑃𝐼
1
1
∑𝑝 𝑥 + ∑𝑝𝑖 𝑥𝑖 , 𝑦)
𝑃𝑛 𝑖∈𝐼 𝑖 𝑖 𝑃𝑛
1
1 𝑛
∑ 𝑝𝑖 𝑥𝑖 , 𝑦) = 𝑓 ( ∑𝑝𝑖 𝑥𝑖 , 𝑦)
𝑃𝑛
𝑃𝑛 𝑖=1
𝑖∈𝐼∪𝐼
󳨐⇒ 𝐷 (𝑓, p, x, 𝐼) ≥ 𝑓 (𝑥, 𝑦) .
(17)
1 𝑛
∑𝑝 𝑓 (𝑥𝑖 , 𝑦) .
𝑃𝑛 𝑖=1 𝑖
1[1 𝑛
∑𝑝 𝐷 (𝑓, w, y, 𝐽, 𝑥𝑖 )
𝐼⊂{1,...,𝑛} 2
𝑃𝑛 𝑖=1 𝑖
𝐽⊂{1,...,𝑚} [
+
(18)
1 𝑚
∑ 𝑤 𝑓 (𝑥, 𝑦𝑗 ) .
𝑊𝑚 𝑗=1 𝑗
(19)
1 𝑛
≤ min [ ∑𝑝𝑖 𝐷 (𝑓, w, y, 𝐽, 𝑥𝑖 )
𝐼⊂{1,...,𝑛}
𝑃𝑛 𝑖=1
𝐽⊂{1,...,𝑚} [
Adding (18) and (19), we get
𝑓 (𝑥, 𝑦) ≤
≤
1 𝑚
∑ 𝑤 𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )] ,
𝑊𝑚 𝑗=1 𝑗
]
1 𝑛
1 𝑚
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) +
∑ 𝑤 𝑓 (𝑥, 𝑦𝑗 )
𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
Similarly, we can prove that
𝑓 (𝑥, 𝑦) ≤ 𝐷 (𝑓, w, y, 𝐽) ≤
1 𝑛 𝑚
∑∑ 𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
≥ max
Now from (16) and (17), we have
𝑓 (𝑥, 𝑦) ≤ 𝐷 (𝑓, p, x, 𝐼) ≤
1 𝑛 𝑚
∑∑ 𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) .
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
Remark 3. We observe that the inequalities in (4) can be
written equivalently as
𝑖∈𝐼
= 𝑓(
1 𝑚
∑𝑤 𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )]
𝑊𝑚 𝑗=1 𝑗
]
The following cases from the above inequalities are of
interest [6, 7].
𝑖∈𝐼
= 𝑓(
(21)
1
[𝐷 (𝑓, w, y, 𝐽) + 𝐷 (𝑓, p, x, 𝐼)]
2
1 𝑚
1[1 𝑛
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) +
∑ 𝑤 𝑓 (𝑥, 𝑦𝑗 )] .
2 𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
]
[
(20)
+
1 𝑚
∑𝑤 𝐷 (𝑓, p, x, 𝐼, 𝑦𝑗 )] ,
𝑊𝑚 𝑗=1 𝑗
]
1 𝑛
1 𝑚
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) +
∑ 𝑤 𝑓 (𝑥, 𝑦𝑗 )
𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
≥ max [𝐷 (𝑓, w, y, 𝐽) + 𝐷 (𝑓, p, x, 𝐼)] ,
𝐼⊂{1,...,𝑛}
𝐽⊂{1,...,𝑚}
Abstract and Applied Analysis
5
𝑓 (𝑥, 𝑦)
min 𝐷𝑘 (𝑓, p, x, 𝑦𝑗 ) ≥ min 𝐷 (𝑓, p, x; 𝐼, 𝑦𝑗 ) ,
𝐼⊂{1,...,𝑛}
𝑘∈{1,...,𝑛}
1
≤ min
[𝐷 (𝑓, w, y, 𝐽) + 𝐷 (𝑓, p, x, 𝐼)] .
𝐼⊂{1,...,𝑛} 2
min 𝐷𝑙 (𝑓, w, y, 𝑥𝑖 ) ≥ min 𝐷 (𝑓, w, y; 𝐽, 𝑥𝑖 ) .
𝐽⊂{1,...,𝑚}
𝑙∈{1,...,𝑚}
𝐽⊂{1,...,𝑚}
(24)
(22)
These inequalities imply the following results:
We discuss the following particular cases of the above
inequalities which is of interest [6].
In the case when 𝑝𝑖 = 1 and 𝑤𝑗 = 1 for 𝑖 ∈ {1, . . . , 𝑛}
and 𝑗 ∈ {1, . . . , 𝑚}, consider the natural numbers 𝑘, 𝑙 with
1 ≤ 𝑘 ≤ 𝑛 − 1 and 1 ≤ 𝑙 ≤ 𝑚 − 1 and define
𝑛 𝑚
1
∑ ∑𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 )
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
1[1 𝑛
∑𝑝 𝐷 (𝑓, w, y, 𝑥𝑖 )
𝑘∈{1,...,𝑛} 2
𝑃𝑛 𝑖=1 𝑖 𝑙
𝑙∈{1,...,𝑚} [
≥ max
+
𝐷𝑘 (𝑓, x, 𝑦𝑗 )
𝑚
1
∑𝑤 𝐷 (𝑓, p, x, 𝑦𝑗 )] ,
𝑊𝑚 𝑗=1 𝑗 𝑘
]
1 𝑛
1 𝑚
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) +
∑𝑤 𝑓 (𝑥, 𝑦𝑗 )
𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
𝐷𝑙 (𝑓, y, 𝑥𝑖 )
=
1 𝑛
≤ min [ ∑𝑝𝑖 𝐷𝑙 (𝑓, w, y, 𝑥𝑖 )
𝑘∈{1,...,𝑛} 𝑃𝑛
𝑖=1
𝑙∈{1,...,𝑚} [
+
𝐷𝑙 (𝑓, y) =
1 𝑛
1 𝑚
∑𝑝𝑖 𝑓 (𝑥𝑖 , 𝑦) +
∑𝑤 𝑓 (𝑥, 𝑦𝑗 )
𝑃𝑛 𝑖=1
𝑊𝑚 𝑗=1 𝑗
𝑘∈{1,...,𝑛}
𝑙∈{1,...,𝑚}
1
𝑛
𝑘
𝑛−𝑘
1 𝑘
1
𝑓 ( ∑𝑥𝑖 , 𝑦) +
𝑓(
∑ 𝑥𝑖 , 𝑦) ,
𝑛
𝑘 𝑖=1
𝑛
𝑛 − 𝑘 𝑖=𝑘+1
𝑙
1 𝑙
𝑚−𝑙
1 𝑚
𝑓 (𝑥, ∑𝑦𝑗 ) +
𝑓 (𝑥,
∑ 𝑦 ).
𝑚
𝑙 𝑗=1
𝑚
𝑚 − 𝑙 𝑗=𝑙+1 𝑗
(25)
≥ max [𝐷𝑙 (𝑓, w, y) + 𝐷𝑘 (𝑓, p, x)] ,
𝑘∈{1,...,𝑛} 2
𝑙∈{1,...,𝑚}
𝑙
1 𝑙
𝑚−𝑙
1 𝑚
𝑓 (𝑥𝑖 , ∑𝑦𝑗 ) +
𝑓 (𝑥𝑖 ,
∑ 𝑦 ),
𝑚
𝑙 𝑗=1
𝑚
𝑚 − 𝑙 𝑗=𝑙+1 𝑗
𝐷𝑘 (𝑓, x) =
1 𝑚
∑ 𝑤 𝐷 (𝑓, p, x, 𝑦𝑗 )] ,
𝑊𝑚 𝑗=1 𝑗 𝑘
]
𝑓 (𝑥, 𝑦) ≤ min
𝑛
𝑘
𝑛−𝑘
1 𝑘
1
𝑓 ( ∑𝑥𝑖 , 𝑦𝑗 ) +
𝑓(
∑ 𝑥𝑖 , 𝑦𝑗 ) ,
𝑛
𝑘 𝑖=1
𝑛
𝑛 − 𝑘 𝑖=𝑘+1
=
We can give the following result.
Corollary 4. Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → R be a co-ordinate
convex function on [𝑎, 𝑏]×[𝑐, 𝑑]. If 𝑥𝑖 ∈ [𝑎, 𝑏] and 𝑦𝑗 ∈ [𝑐, 𝑑],
then for any 𝑘 ∈ {1, . . . , 𝑛 − 1} and 𝑙 ∈ {1, . . . , 𝑚 − 1}, one has
[𝐷𝑙 (𝑓, w, y) + 𝐷𝑘 (𝑓, p, x)] .
(23)
Moreover, from the above, we also have
max 𝐷 (𝑓, p, x; 𝐼) ≥ max 𝐷𝑘 (𝑓, p, x) ,
𝐼⊂{1,...,𝑛}
𝑘∈{1,...,𝑛}
1 𝑚
1 𝑛
𝑓 ( ∑𝑥𝑖 , ∑𝑦𝑗 )
𝑛 𝑖=1 𝑚 𝑗=1
max 𝐷 (𝑓, w, y; 𝐽) ≥ max 𝐷𝑙 (𝑓, w, y) ,
≤
1
[𝐷 (𝑓, y) + 𝐷𝑘 (𝑓, x)]
2 𝑙
max 𝐷 (𝑓, p, x; 𝐼, 𝑦𝑗 ) ≥ max 𝐷𝑘 (𝑓, p, x, y𝑗 ) ,
≤
1 𝑚
1 [1 𝑛
∑𝑓 (𝑥𝑖 , 𝑦) + ∑𝑓 (𝑥, 𝑦𝑗 )]
2 𝑛 𝑖=1
𝑚 𝑗=1
]
[
≤
1 [1 𝑛
1 𝑚
∑𝐷𝑙 (𝑓, y, 𝑥𝑖 ) + ∑𝐷𝑘 (𝑓, x, 𝑦𝑗 )]
2 𝑛 𝑖=1
𝑚 𝑗=1
]
[
≤
1 𝑛 𝑚
∑∑ 𝑓 (𝑥𝑖 , 𝑦𝑗 ) .
𝑚𝑛 𝑖=1𝑗=1
𝐽⊂{1,...,𝑚}
𝐼⊂{1,...,𝑛}
𝑙∈{1,...,𝑚}
𝑘∈{1,...,𝑛}
max 𝐷 (𝑓, w, y; 𝐽, 𝑥𝑖 ) ≥ max 𝐷𝑙 (𝑓, w, y, 𝑥𝑖 ) ,
𝐽⊂{1,...,𝑚}
𝑙∈{1,...,𝑚}
min 𝐷𝑘 (𝑓, p, x) ≥ min 𝐷 (𝑓, p, x; 𝐼) ,
𝑘∈{1,...,𝑛}
𝐼⊂{1,...,𝑛}
min 𝐷𝑙 (𝑓, w, y) ≥ min 𝐷 (𝑓, w, y; 𝐽) ,
𝑙∈{1,...,𝑚}
𝐽⊂{1,...,𝑚}
(26)
6
Abstract and Applied Analysis
−𝑞/𝑝
Proof. Using the functions 𝑓(𝑥) = 𝑥𝑝 , 𝑝 > 1, 𝑥𝑖 → 𝑥𝑖 𝑦𝑖
𝑞
and 𝑝𝑖 → 𝑦𝑖 in (28), we get (29).
In particular, we have the bounds
𝑛 𝑚
1
∑∑ 𝑓 (𝑥𝑖 , 𝑦𝑗 )
𝑚𝑛 𝑖=1𝑗=1
≥ max
Remark 7. As mentioned above from the inequalities in (29),
we can write
𝑛
1 [1
∑𝐷𝑙 (𝑓, y, 𝑥𝑖 )
𝑛
[ 𝑖=1
𝑘∈{1,...,𝑛} 2
𝑙∈{1,...,𝑚}
+
𝑛
𝑝
(∑𝑥𝑖 )
𝑖=1
1 𝑚
∑ 𝐷 (𝑓, x, 𝑦𝑗 )] ,
𝑚 𝑗=1 𝑘
]
𝑛
1/𝑝
1/𝑞
𝑞
(∑𝑦𝑖 )
𝑖=1
𝑛
𝑛
𝑞
≥ max [(∑𝑦𝑖 )
𝐼⊂{1,...,𝑛}
[ 𝑖=1
𝑚
1
1
∑𝑓 (𝑥𝑖 , 𝑦) + ∑𝑓 (𝑥, 𝑦𝑗 )
𝑛 𝑖=1
𝑚 𝑗=1
𝑝/𝑞
−𝑝/𝑞
{
𝑞
{(∑𝑦𝑖 )
{ 𝑖∈𝐼
+
(27)
𝑛
1
≤ min [ ∑𝐷𝑙 (𝑓, y, 𝑥𝑖 )
𝑘∈{1,...,𝑛} 𝑛
𝑙∈{1,...,𝑚} [ 𝑖=1
(∑𝑥𝑖 𝑦𝑖 )
𝑖∈𝐼
1/𝑝
−𝑝/𝑞
𝑝
}
𝑞
(∑𝑦𝑖 )
(∑𝑥𝑖 𝑦𝑖) }] ,
𝑖∈𝐼
𝑖∈𝐼
}]
𝑛
𝑖=1
𝑛
𝑞
≤ min [(∑𝑦𝑖 )
𝐼⊂{1,...,𝑛}
[ 𝑖=1
𝑝/𝑞
{
𝑞
{(∑𝑦𝑖 )
{ 𝑖∈𝐼
−𝑝/𝑞
≥ max [𝐷𝑙 (𝑓, y) + 𝐷𝑘 (𝑓, x)] ,
(∑𝑥𝑖 𝑦𝑖 )
𝑖∈𝐼
𝑞
+(∑𝑦𝑖 )
𝑖∈𝐼
1
[𝐷𝑙 (𝑓, y) + 𝐷𝑘 (𝑓, x)] .
𝑘∈{1,...,𝑛} 2
𝑓 (𝑥, 𝑦) ≤ min
𝑙∈{1,...,𝑚}
Remark 5. Note that if we substitute 𝑚 = 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 ∑𝑛𝑖=1 𝑝𝑖 = 𝑃𝑛 > 0,
𝑃
𝑃
1
1
1 𝑛
∑𝑝 𝑥 ) ≤ 𝐼 𝑓 ( ∑𝑝𝑖 𝑥𝑖 ) + 𝐼 𝑓 ( ∑𝑝𝑖 𝑥𝑖 )
𝑃𝑛 𝑖=1 𝑖 𝑖
𝑃𝑛
𝑃𝐼 𝑖∈𝐼
𝑃𝑛
𝑃𝐼
𝑖∈𝐼
𝑛
1
≤ ∑𝑝𝑖 𝑓 (𝑥𝑖 ) .
𝑃𝑛 𝑖=1
(28)
The following refinement of Hölder inequality holds.
Corollary 6. Let x = (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
𝑛
𝑝/𝑞
𝑛
𝑞
≤ [ ( ∑𝑦 )
[
≤
𝑖=1
𝑖
−𝑝/𝑞
{
𝑞
{(∑𝑦𝑖 )
{ 𝑖∈𝐼
1/𝑝
}
(∑𝑥𝑖 𝑦𝑖) }] .
𝑖∈𝐼
}]
(30)
Theorem 8. Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → R be convex on the coordinates of [𝑎, 𝑏] × [𝑐, 𝑑] ⊆ R2 . If x is an n-tuple in [𝑎, 𝑏], y is
an m-tuple in [𝑐, 𝑑], p is a nonnegative n-tuple such that 𝑃𝑛 =
∑𝑛𝑖=1 𝑝𝑖 > 0, and w is a nonnegative m-tuple such that 𝑊𝑚 =
∑𝑚
𝑗=1 𝑤𝑗 > 0, then
1 𝑛 𝑚
∑ ∑𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) − 𝑓 (𝑥, 𝑦)
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
󵄨󵄨
󵄨󵄨 1 𝑛 𝑚
󵄨
󵄨
≥ 󵄨󵄨󵄨󵄨
∑ ∑𝑝𝑖 𝑤𝑗 󵄨󵄨󵄨󵄨𝑓 (𝑥𝑖 , 𝑦𝑗 ) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨󵄨
󵄨󵄨 𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1
󵄨
󵄨󵄨
󵄨󵄨 𝜕𝑓 (𝑥, 𝑦)
1 𝑛 𝑚
−
(𝑥𝑖 − 𝑥)
∑ ∑𝑝𝑖 𝑤𝑗 󵄨󵄨󵄨󵄨 +
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1
󵄨󵄨 𝜕𝑤
󵄨
(31)
󵄨󵄨 󵄨󵄨
𝜕𝑓+ (𝑥, 𝑦)
󵄨󵄨 󵄨󵄨
+
(𝑦𝑗 − 𝑦) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 ,
𝜕𝑧
󵄨󵄨 󵄨󵄨
󵄨󵄨
𝑝
(∑𝑥𝑖 𝑦𝑖 )
𝑖∈𝐼
1/𝑝
−𝑝/𝑞
𝑝
}
𝑞
]
+ (∑𝑦𝑖 )
(∑𝑥𝑖 𝑦𝑖 ) }
𝑖∈𝐼
𝑖∈𝐼
}]
1/𝑝
1/𝑞
𝑛
𝑛
𝑝
𝑞
(∑𝑥𝑖 ) (∑𝑦𝑖 ) .
𝑖=1
𝑖=1
𝑝
The following improvement of Jensen’s inequality is valid.
∑𝑥𝑖 𝑦𝑖
𝑖=1
𝑝
−𝑝/𝑞
𝑘∈{1,...,𝑛}
𝑙∈{1,...,𝑚}
𝑓(
𝑝
∑𝑥𝑖 𝑦𝑖
𝑚
1
∑𝐷 (𝑓, x, 𝑦𝑗 )] ,
𝑚 𝑗=1 𝑘
]
1 𝑛
1 𝑚
∑𝑓 (𝑥𝑖 , 𝑦) + ∑𝑓 (𝑥, 𝑦𝑗 )
𝑛 𝑖=1
𝑚 𝑗=1
+
,
(29)
where 𝑥 = (1/𝑃𝑛 ) ∑𝑛𝑖=1 𝑝𝑖 𝑥𝑖 , and 𝑦 = (1/𝑊𝑚 ) ∑𝑚
𝑗=1 𝑤𝑗 𝑦𝑗 .
Proof. Since 𝑓 is convex on [𝑎, 𝑏] × [𝑐, 𝑑], therefore we have
𝑓 (𝑥, 𝑦) − 𝑓 (𝑤, 𝑧)
≥
𝜕𝑓+ (𝑤, 𝑧)
𝜕𝑓 (𝑤, 𝑧)
(𝑦 − 𝑧) .
(𝑥 − 𝑤) + +
𝜕𝑤
𝜕𝑧
(32)
Abstract and Applied Analysis
7
One has
From the above inequality, we have
𝑛 𝑚
𝑓 (𝑥, 𝑦) − 𝑓 (𝑤, 𝑧) −
𝜕𝑓+ (𝑤, 𝑧)
𝜕𝑓 (𝑤, 𝑧)
(𝑦 − 𝑧)
(𝑥 − 𝑤) − +
𝜕𝑤
𝜕𝑧
󵄨󵄨
𝜕𝑓 (𝑤, 𝑧)
󵄨
= 󵄨󵄨󵄨𝑓 (𝑥, 𝑦) − 𝑓 (𝑤, 𝑧) − +
(𝑥 − 𝑤)
󵄨󵄨
𝜕𝑤
󵄨󵄨
𝜕𝑓 (𝑤, 𝑧)
󵄨
− +
(𝑦 − 𝑧)󵄨󵄨󵄨
󵄨󵄨
𝜕𝑧
󵄨󵄨
󵄨
󵄨󵄨
≥ 󵄨󵄨󵄨 󵄨󵄨󵄨𝑓 (𝑥, 𝑦) − 𝑓 (𝑤, 𝑧)󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝜕𝑓 (𝑤, 𝑧)
󵄨󵄨 󵄨󵄨
𝜕𝑓 (𝑤, 𝑧)
󵄨
󵄨󵄨
− 󵄨󵄨󵄨 +
(𝑦 − 𝑧)󵄨󵄨󵄨 󵄨󵄨󵄨 .
(𝑥 − 𝑤) + +
󵄨󵄨 𝜕𝑤
󵄨󵄨 󵄨󵄨
𝜕𝑧
𝑖=1 𝑗=1
𝜕𝑓+ (𝑥, 𝑦)
(𝑥𝑖 − 𝑥)
𝜕𝑤
𝑚
= ∑ 𝑤𝑗
𝑗=1
𝜕𝑓+ (𝑥, 𝑦) 𝑛
1 𝑛
(∑𝑝𝑖 𝑥𝑖 − 𝑃𝑛 ⋅ ∑𝑝𝑖 𝑥𝑖 ) = 0, (36)
𝜕𝑤
𝑃𝑛 𝑖=1
𝑖=1
𝑛 𝑚
∑∑ 𝑝𝑖 𝑤𝑗
𝑖=1 𝑗=1
𝜕𝑓+ (𝑥, 𝑦)
(𝑦𝑗 − 𝑦) = 0.
𝜕𝑧
Therefore (35) becomes
𝑛 𝑚
𝑛 𝑚
𝑖=1 𝑗=1
𝑖=1 𝑗=1
∑ ∑𝑝𝑖 𝑤𝑗 𝑓 (𝑥𝑖 , 𝑦𝑗 ) − ∑∑ 𝑝𝑖 𝑤𝑗 𝑓 (𝑥, 𝑦)
(33)
Let 𝑥 → 𝑥𝑖 , 𝑦 → 𝑦𝑗 , 𝑤 → ∑𝑛𝑖=1 𝑝𝑖 𝑥𝑖 /𝑃𝑛 := 𝑥, and 𝑧 →
∑𝑛𝑗=1 𝑤𝑗 𝑦𝑗 /𝑊𝑚 := 𝑦, then (33) becomes
󵄨󵄨 𝑛 𝑚
󵄨󵄨
󵄨
󵄨
≥ 󵄨󵄨󵄨󵄨∑ ∑𝑝𝑖 𝑤𝑗 󵄨󵄨󵄨󵄨𝑓 (𝑥𝑖 , 𝑦𝑗 ) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨󵄨
󵄨󵄨𝑖=1𝑗=1
󵄨
󵄨󵄨
𝑛 𝑚
󵄨󵄨 𝜕𝑓 (𝑥, 𝑦)
− ∑∑ 𝑝𝑖 𝑤𝑗 󵄨󵄨󵄨󵄨 +
(𝑥𝑖 − 𝑥)
󵄨󵄨 𝜕𝑤
𝑖=1 𝑗=1
󵄨
+
𝜕𝑓 (𝑥, 𝑦)
𝜕𝑓+ (𝑥, 𝑦)
(𝑥𝑖 − 𝑥) − +
(𝑦𝑗 − 𝑦)
𝜕𝑤
𝜕𝑧
󵄨󵄨
(34)
󵄨󵄨
󵄨
≥ 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑓 (𝑥𝑖 , 𝑦𝑗 ) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝜕𝑓 (𝑥, 𝑦)
󵄨󵄨 󵄨󵄨
𝜕𝑓 (𝑥, 𝑦)
󵄨
󵄨󵄨
− 󵄨󵄨󵄨󵄨 +
(𝑥𝑖 − 𝑥) + +
(𝑦𝑗 − 𝑦)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 .
𝜕𝑧
󵄨󵄨 𝜕𝑤
󵄨󵄨 󵄨󵄨
Multiplying (34) by 𝑝𝑖 and 𝑤𝑗 and summing over 𝑖 and 𝑗, we
have
1 𝑛 𝑚
∑∑ 𝑝 𝑤 𝑓 (𝑥𝑖 , 𝑦𝑗 ) − 𝑓 (𝑥, 𝑦)
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1 𝑖 𝑗
󵄨󵄨
󵄨󵄨 1 𝑛 𝑚
󵄨
󵄨
≥ 󵄨󵄨󵄨󵄨
∑∑ 𝑝𝑖 𝑤𝑗 󵄨󵄨󵄨󵄨𝑓 (𝑥𝑖 , 𝑦𝑗 ) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨󵄨
󵄨󵄨 𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1
󵄨
󵄨󵄨 𝜕𝑓 (𝑥, 𝑦)
1 𝑛 𝑚
󵄨
−
(𝑥𝑖 − 𝑥)
∑ ∑𝑝𝑖 𝑤𝑗 󵄨󵄨󵄨󵄨 +
𝑃𝑛 𝑊𝑚 𝑖=1𝑗=1
󵄨󵄨 𝜕𝑤
+
𝑛 𝑚
𝑛 𝑚
𝑖=1 𝑗=1
𝑖=1 𝑗=1
This completes the proof.
𝜕𝑓+ (𝑥, 𝑦)
(𝑥𝑖 − 𝑥)
𝜕𝑤
Conflict of Interests
∑∑ 𝑝𝑖 𝑤𝑗 𝑓 (𝑥𝑖 , 𝑦𝑗 ) − ∑ ∑𝑝𝑖 𝑤𝑗 𝑓 (𝑥, 𝑦)
𝑛 𝑚
− ∑ ∑𝑝𝑖 𝑤𝑗
𝑖=1 𝑗=1
(37)
󵄨󵄨 󵄨󵄨
󵄨󵄨 󵄨󵄨
𝜕𝑓+ (𝑥, 𝑦)
(𝑦𝑗 − 𝑦) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 .
𝜕𝑧
󵄨󵄨 󵄨󵄨
󵄨󵄨
Multiplying both hand sides by 1/𝑃𝑛 𝑊𝑚 , we have
𝑓 (𝑥𝑖 , 𝑦𝑗 ) − 𝑓 (𝑥, 𝑦)
−
∑∑ 𝑝𝑖 𝑤𝑗
𝜕𝑓 (𝑥, 𝑦)
− ∑ ∑𝑝𝑖 𝑤𝑗 +
(𝑦𝑗 − 𝑦)
𝜕𝑧
𝑖=1 𝑗=1
󵄨󵄨 󵄨󵄨󵄨
𝜕𝑓 (𝑥, 𝑦)
󵄨󵄨
+ +
(𝑦𝑗 − 𝑦)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 .
𝜕𝑧
󵄨󵄨 󵄨󵄨
󵄨
󵄨󵄨 󵄨󵄨󵄨
𝜕𝑓+ (𝑥, 𝑦)
󵄨󵄨
(𝑦𝑗 − 𝑦)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 .
𝜕𝑧
󵄨󵄨 󵄨󵄨
󵄨
The authors declare that they have no conflict of interests
regarding publication of this paper.
𝑛 𝑚
󵄨󵄨 𝑛 𝑚
󵄨󵄨
󵄨
󵄨
≥ 󵄨󵄨󵄨󵄨∑∑ 𝑝𝑖 𝑤𝑗 󵄨󵄨󵄨󵄨𝑓 (𝑥𝑖 , 𝑦𝑗 ) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨󵄨
󵄨󵄨𝑖=1𝑗=1
󵄨
󵄨󵄨 𝜕𝑓 (𝑥, 𝑦)
𝑛 𝑚
󵄨
−∑ ∑𝑝𝑖 𝑤𝑗 󵄨󵄨󵄨󵄨 +
(𝑥𝑖 − 𝑥)
󵄨󵄨 𝜕𝑤
𝑖=1 𝑗=1
(38)
(35)
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).
References
[1] M. Adil Khan, M. Anwar, J. Jakšetić, and J. Pečarić, “On some
improvements of the Jensen inequality with some applications,”
8
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
Abstract and Applied Analysis
Journal of Inequalities and Applications, vol. 2009, Article ID
323615, 15 pages, 2009.
M. Adil Khan, S. Khalid, and J. Pečarić, “Improvement of
Jensen’s inequality in terms of Gâteaux derivatives for convex
functions in linear spaces with applications,” Kyungpook Mathematical Journal, vol. 52, no. 4, pp. 495–511, 2012.
M. A. Khan, S. Khalid, and J. Pečarić, “Refinements of
some majorization type inequalities,” Journal of Mathematical
Inequalities, vol. 7, no. 1, pp. 73–92, 2013.
M. Klaričić Bakula and J. Pečarić, “On the Jensen’s inequality
for convex functions on the co-ordinates in a rectangle from
the plane,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp.
1271–1292, 2006.
L. Horváth, “A parameter-dependent refinement of the discrete
Jensen’s inequality for convex and mid-convex functions,”
Journal of Inequalities and Applications, vol. 2011, article 26, 2011.
S. S. Dragomir, “A refinement of Jensen’s inequality with
applications for f-divergence measures,” Taiwanese Journal of
Mathematics, vol. 14, no. 1, pp. 153–164, 2010.
S. S. Dragomir, “Some refinements of Jensen’s inequality,”
Journal of Mathematical Analysis and Applications, vol. 168, no.
2, pp. 518–522, 1992.
S. S. Dragomir, “On Hadamard’s inequality for the convex
mappings defined on a ball in the space and applications,”
Mathematical Inequalities & Applications, vol. 3, no. 2, pp. 177–
187, 2000.
S. S. Dragomir, “A new refinement of Jensen’s inequality in
linear spaces with applications,” Mathematical and Computer
Modelling, vol. 52, no. 9-10, pp. 1497–1505, 2010.
J. Rooin, “Some refinements of discrete Jensen’s inequality and
some of its applications,” Nonlinear Functional Analysis and
Applications, vol. 12, no. 1, pp. 107–118, 2007.
L.-C. Wang, X.-F. Ma, and L.-H. Liu, “A note on some new
refinements of Jensen’s inequality for convex functions,” Journal
of Inequalities in Pure and Applied Mathematics, vol. 10, no. 2,
article 48, 2009.
X. L. Tang and J. J. Wen, “Some developments of refinned
Jensen’s inequality,” Journal of Southwest University for Nationalities, vol. 29, pp. 20–26, 2003.
S. Hussain and J. Pečarić, “An improvement of Jensen’s inequality with some applications,” Asian-European Journal of Mathematics, vol. 2, no. 1, pp. 85–94, 2009.
G. Zabandan and A. Kılıçman, “A new version of Jensen’s
inequality and related results,” Journal of Inequalities and Applications, vol. 2012, article 238, 2012.
G. Zabandan and A. Kılıçman, “Several integral inequalities
and an upper bound for the bidimensional Hermite-Hadamard
inequality,” Journal of Inequalities and Applications, vol. 2013,
article 27, 2013.
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