# Generalizations of Inequalities for Differentiable Co-Ordinated Convex Functions

Authors: Feixiang Chen
Journal: Chinese Journal of Mathematics
 (2014)

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## Abstract

A generalized lemmas is proved and several new inequalities for differentiable co-ordinated convex and concave functions in two variables are obtained.


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## Body

## 1. Introduction

Letf:I⊆ℝ→ℝ be a convex function and a,b∈I with a<b; we have the following double inequality:
(1)f(a+b2)≤1b-a∫ab‍f(t)dt≤f(a)+f(b)2.
This remarkable result is well known in the literature as the Hermite-Hadamard inequality for convex mapping.Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1–4]).A modification for convex functions which is also known as coordinated convex functions was introduced as follows by Dragomir in [5].Let us consider the bidimensional intervalΔ:=[a,b]×[c,d] in ℝ2 with a<b and c<d; a mapping f:Δ→ℝ is said to be convex on Δ if the inequality
(2)f(λx+(1-λ)z,λy+(1-λ)w)≤λf(x,y)+(1-λ)f(z,w)
holds for all (x,y), (z,w)∈Δ, and λ∈[0,1].A functionf:Δ→ℝ is said to be coordinated convex on Δ if the partial mappings fy:[a,b]→ℝ, fy(u)=f(u,y) and fx:[c,d]→ℝ, fx(v)=f(x,v) are convex for all y∈[c,d] and x∈[a,b].A formal definition for coordinated convex functions may be stated as follows.Definition 1.
A functionf:Δ→ℝ is said to be convex on coordinates on Δ if the inequality
(3)f(λx+(1-λ)z,ty+(1-t)w)≤λtf(x,y)+λ(1-t)f(x,w)+(1-λ)tf(z,y)+(1-t)(1-λ)f(z,w)
holds for all (x,y), (z,y), (x,w), (z,w)∈Δ, and t,λ∈[0,1].Dragomir in [5] established the following Hadamard-type inequalities for coordinated convex functions in a rectangle from the plane ℝ2.Theorem 2.
Suppose thatf:Δ=[a,b]×[c,d]→ℝ is convex on the coordinates on Δ. Then one has the inequalities as follows:
(4)f(a+b2,c+d2)≤1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx≤f(a,c)+f(a,d)+f(b,c)+f(b,d)4.Some new integral inequalities that are related to the Hermite-Hadamard type for coordinated convex functions are also established by many authors.In ([6], 2008), Alomari and Darus defined coordinated s-convex functions and proved some inequalities based on this definition. In ([7], 2009), analogous results for h-convex functions on the coordinates were proved by Latif and Alomari. In ([8], 2009), Alomari and Darus established some Hadamard-type inequalities for coordinated log-convex functions.In ([9], 2012), Latif and Dragomir obtained some new Hadamard type inequalities for differentiable coordinated convex and concave functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for coordinated convex functions in two variables based on the following lemma.Lemma 3.
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If ∂2f/∂u∂v∈L(Δ), then the following equality holds:
(5)1(b-a)(d-c)∫ab∫cd‍‍f(x,y)dydx+f(a+b2,c+d2)-1b-a∫ab‍f(x,c+d2)dx-1d-c∫cd‍f(a+b2,y)dy=(b-a)(d-c)×∬01K(u,v)∂2f∂u∂v(ua+(1-u)b,vc+(1-v)d)dudv,
where
(6)K(u,v)={uv,(u,v)∈[0,12]×[0,12]u(v-1),(u,v)∈[0,12]×(12,1](u-1)v,(u,v)∈(12,1]×[0,12](u-1)(v-1),(u,v)∈(12,1]×(12,1].Theorem 4 (see [9]).
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v| is convex on the coordinates on Δ, then the following equality holds:
(7)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+f(a+b2,c+d2)-A|≤(b-a)(d-c)16×((|∂2f∂u∂v(a,c)|+|∂2f∂u∂v(a,d)|+|∂2f∂u∂v(b,c)|aaaaaaaa+|∂2f∂u∂v(b,d)|)×(4)-1),
where
(8)A=1b-a∫ab‍f(x,c+d2)dx+1d-c∫cd‍f(a+b2,y)dy.Theorem 5 (see [9]).
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v|q is convex on the coordinates on Δ and p, q>1, 1/p+1/q=1, then the following equality holds:
(9)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+f(a+b2,c+d2)-A|≤(b-a)(d-c)4(p+1)2/p×((|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|q+|∂2f∂u∂v(b,c)|qaaaaaaaa+|∂2f∂u∂v(b,d)|q)×(4)-1)1/q,
where A is as given in Theorem 4.Theorem 6 (see [9]).
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v|q is convex on the coordinates on Δ and q≥1, then the following equality holds:
(10)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+f(a+b2,c+d2)-A|≤(b-a)(d-c)16×((|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|q+|∂2f∂u∂v(b,c)|qaaaaaaaa+|∂2f∂u∂v(b,d)|q)×(4)-1)1/q,
where A is as given in Theorem 4.In ([10], 2012), analogous results which are related to the right-hand side of Hermite-Hadamard type inequality for coordinated convex functions in two variables were proved by Sarıkaya et al. based on the following lemma.Lemma 7.
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If ∂2f/∂u∂v∈L(Δ), then the following equality holds:
(11)f(a,c)+f(a,d)+f(b,c)+f(b,d)4+1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx-12[1b-a∫ab‍[f(x,c)+f(x,d)]dx+1d-c∫cd‍[f(a,y)+f(b,y)]dy]=(b-a)(d-c)4×∬01(1-2u)(1-2v)∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv.Theorem 8 (see [10]).
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v| is convex on the coordinates on Δ, then the following equality holds:
(12)|f(a,c)+f(a,d)+f(b,c)+f(b,d)4+1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx-A|≤(b-a)(d-c)16×((|∂2f∂u∂v(a,c)|+|∂2f∂u∂v(a,d)|+|∂2f∂u∂v(b,c)|aaaaaaaa+|∂2f∂u∂v(b,d)|)×(4)-1),
where
(13)A=12[1b-a∫ab‍[f(x,c)+f(x,d)]dxaaaaaa+1d-c∫cd‍[f(a,y)+f(b,y)]dy].Theorem 9 (see [10]).
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v|q is convex on the coordinates on Δ and p, q>1, 1/p+1/q=1, then the following equality holds:
(14)|f(a,c)+f(a,d)+f(b,c)+f(b,d)4+1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx-A|≤(b-a)(d-c)4(p+1)2/p×((|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|qaaaaaaaa+|∂2f∂u∂v(b,c)|q+|∂2f∂u∂v(b,d)|q)×(4)-1)1/q,
where A is as given in Theorem 8.Theorem 10 (see [10]).
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v|q is convex on the coordinates on Δ and q≥1, then the following equality holds:
(15)|f(a,c)+f(a,d)+f(b,c)+f(b,d)4+1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx-A|≤(b-a)(d-c)16×((|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|qaaaaaaaa+|∂2f∂u∂v(b,c)|q+|∂2f∂u∂v(b,d)|q)×(4)-1)1/q,
where A is as given in Theorem 8.In [11], Ozdemir et al. established some Simpson’s inequalities for coordinated convex functions based on the following lemma.Lemma 11.
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If ∂2f/∂u∂v∈L(Δ), then the following equality holds:
(16)(f(a,c+d2)+f(b,c+d2)+4f(a+b2,c+d2)+f(a+b2,c)+f(a+b2,d)×(9)-1)+f(a,c)+f(a,d)+f(b,c)+f(b,d)36-16(b-a)∫ab‍(f(x,c)+4f(x,c+d2)+f(x,d))dx-16(d-c)∫cd‍(f(a,y)+4f(a+b2,y)+f(b,y))dy+1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx=(b-a)(d-c)×∬01p(x,u)p(y,v)∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv,
where
(17)p(x,u)={(u-16),t∈[0,12](u-56),t∈(12,1],p(y,v)={(v-16),s∈[0,12](v-56),s∈(12,1].Theorem 12 (see [11]).
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d]. If |∂2f/∂u∂v| is convex on the coordinates on Δ, then the following equality holds:
(18)|(f(a,c+d2)+f(b,c+d2)+4f(a+b2,c+d2)+f(a+b2,c)+f(a+b2,d)×(9)-1)+f(a,c)+f(a,d)+f(b,c)+f(b,d)36+1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx-A|≤(572)2(b-a)(d-c)×(|∂2f∂u∂v(a,c)|+|∂2f∂u∂v(a,d)|+|∂2f∂u∂v(b,c)|aaaaaaa+|∂2f∂u∂v(b,d)|),
where
(19)A=16(b-a)∫ab‍(f(x,c)+4f(x,c+d2)+f(x,d))dx+16(d-c)∫cd‍(f(a,y)+4f(a+b2,y)+f(b,y))dy.For recent results and generalizations concerning Hermite-Hadamard type inequality for differentiable coordinated convex functions see ([12], 2012) and the references given therein.In this paper, a generalized lemma is proved and several new inequalities for differentiable coordinated convex and concave functions in two variables are obtained.


## 2. Lemmas

To establish our results, we need the following lemma.Lemma 13.
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If ∂2f/∂u∂v∈L(Δ) and λ∈[0,1], then the following equality holds:
(20)1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ(1-λ)2×[f(a,c+d2)+f(a,c+d2)+f(a+b2,c)+f(a+b2,d)]+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-12(b-a)×∫ab‍(λf(x,c)+2(1-λ)f(x,c+d2)+λf(x,d))dx-12(d-c)×∫cd‍(λf(a,y)+2(1-λ)f(a+b2,y)+λf(b,y))dy=(b-a)(d-c)×∬01M(u,v)∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv,
where
(21)M(u,v)={(u-λ2)(v-λ2),(u,v)∈[0,12]×[0,12](u-λ2)(v-(1-λ2)),(u,v)∈[0,12]×(12,1](u-(1-λ2))(v-λ2),(u,v)∈(12,1]×[0,12](u-(1-λ2))(v-(1-λ2)),(u,v)∈(12,1]×(12,1].Proof.
Since(22)∬01M(u,v)∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv=∬01/2(u-λ2)(v-λ2)∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv+∫01/2‍∫1/21‍(u-λ2)(v-(1-λ2))∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv+∫1/21‍∫01/2‍(u-(1-λ2))(v-λ2)∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv+∬1/21(u-(1-λ2))(v-(1-λ2))∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv,
thus, by integration by parts, it follows that
(23)∬01/2(u-λ2)(v-λ2)∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv=∫01/2‍(u-λ2)×{(v-λ2)1c-d∂f∂u×(ua+(1-u)b,vc+(1-v)d)(v-λ2)|01/2-1c-d×∫01/2‍∂f∂u×(ua+(1-u)b,vc+(1-v)d)dv(v-λ2)}du=1-λ21c-d∫01/2‍(u-λ2)∂f∂u(ua+(1-u)b,c+d2)du+λ21c-d∫01/2‍(u-λ2)∂f∂u(ua+(1-u)b,d)du-1c-d∬01/2(u-λ2)∂f∂u×(ua+(1-u)b,vc+(1-v)d)dvdu=1-λ21c-d1a-b(u-λ2)×f(ua+(1-u)b,c+d2)(u-λ2)|01/2-1-λ21c-d1a-b∫01/2‍f(ua+(1-u)b,c+d2)du+λ21c-d1a-b(u-λ2)f(ua+(1-u)b,d)|01/2-λ21c-d1a-b∫01/2‍f(ua+(1-u)b,d)du-1c-d1a-b×∫01/2‍{(u-λ2)f(ua+(1-u)b,vc+(1-v)d)(u-λ2)|01/2-∫01/2‍f(ua+(1-u)b,vc+(1-v)d)du}dv=(1-λ)24(c-d)(a-b)f(a+b2,c+d2)+λ(1-λ)4(c-d)(a-b)f(b,c+d2)-1-λ2(c-d)(a-b)∫01/2‍f(ua+(1-u)b,c+d2)du+λ(1-λ)4(c-d)(a-b)f(a+b2,d)+λ24(c-d)(a-b)f(b,d)-λ2(c-d)(a-b)∫01/2‍f(ua+(1-u)b,d)du-1-λ2(c-d)(a-b)∫01/2‍f(a+b2,vc+(1-v)d)dv-λ2(c-d)(a-b)∫01/2‍f(b,vc+(1-v)d)dv+1(c-d)(a-b)×∬01/2f(ua+(1-u)b,vc+(1-v)d)dudv.
Similarly, we can get(24)∫01/2‍∫1/21‍(u-λ2)(v-(1-λ2))∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv=(1-λ)24(c-d)(a-b)f(a+b2,c+d2)+λ(1-λ)4(c-d)(a-b)f(b,c+d2)-1-λ2(c-d)(a-b)×∫01/2‍f(ua+(1-u)b,c+d2)du+λ(1-λ)4(c-d)(a-b)f(a+b2,c)+λ24(c-d)(a-b)f(b,c)-λ2(c-d)(a-b)×∫01/2‍f(ua+(1-u)b,c)du-1-λ2(c-d)(a-b)×∫1/21‍f(a+b2,vc+(1-v)d)dv-λ2(c-d)(b-a)×∫1/21‍f(a,vc+(1-v)d)dv+1(c-d)(a-b)×∫01/2‍∫1/21‍f(ua+(1-u)b,vc+(1-v)d)dudv,∫1/21‍∫01/2‍(u-(1-λ2))(v-λ2)∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv=(1-λ)24(c-d)(a-b)f(a+b2,c+d2)+λ(1-λ)4(c-d)(a-b)f(a,c+d2)-1-λ2(c-d)(a-b)∫1/21‍f(ua+(1-u)b,c+d2)du+λ(1-λ)4(c-d)(a-b)f(a+b2,d)+λ24(d-c)(b-a)f(a,d)-λ2(c-d)(a-b)×∫1/21‍f(ua+(1-u)b,d)du-1-λ2(c-d)(a-b)×∫01/2‍f(a+b2,vc+(1-v)d)dv-λ2(d-c)(b-a)×∫01/2‍f(a,vc+(1-v)d)dv+1(c-d)(a-b)×∫1/21‍∫01/2‍f(ua+(1-u)b,vc+(1-v)d)dudv,∬1/21(u-(1-λ2))(v-(1-λ2))∂2f∂u∂v×(ua+(1-u)b,vc+(1-v)d)dudv,=(1-λ)24(c-d)(a-b)f(a+b2,c+d2)+λ(1-λ)4(c-d)(a-b)f(a,c+d2)-1-λ2(c-d)(a-b)∫1/21‍f(ua+(1-u)b,c+d2)du+λ(1-λ)4(c-d)(a-b)f(a+b2,c)+λ24(c-d)(a-b)f(a,c)-λ2(c-d)(a-b)×∫1/21‍f(ua+(1-u)b,c)du-1-λ2(c-d)(a-b)×∫1/21‍f(a+b2,vc+(1-v)d)dv-λ2(c-d)(a-b)∫1/21‍f(a,vc+(1-v)d)dv+1(c-d)(a-b)×∬1/21f(ua+(1-u)b,vc+(1-v)d)dudv.
Now(25)∬01M(u,v)∂2f∂u∂v(ua+(1-u)b,vc+(1-v)d)dudv=(1-λ)2(d-c)(b-a)f(a+b2,c+d2)+λ(1-λ)2(d-c)(b-a)×[f(a,c+d2)+f(b,c+d2)+f(a+b2,c)+f(a+b2,d)]+λ24(d-c)(b-a)×[f(a,c)+f(b,c)+f(a,d)+f(b,d)]+12(d-c)(b-a)×∫01‍[(ua+(1-u)b,c+d2)λf(ua+(1-u)b,c)+2(1-λ)f(ua+(1-u)b,c+d2)+λf(ua+(1-u)b,d)(ua+(1-u)b,c+d2)]du+12(d-c)(b-a)×∫01‍[(a+b2,vc+(1-v)d)λf(a,vc+(1-v)d)+2(1-λ)f(a+b2,vc+(1-v)d)+λf(b,vc+(1-v)d)(a+b2,vc+(1-v)d)]dv+1(b-a)(d-c)×∬01∂2f∂u∂v(ua+(1-u)b,vc+(1-v)d)dudv.
Multiplying both sides by (b-a)(d-c) and using the change of the variable x=ua+(1-u)b and y=vc+(1-v)d, which completes the proof.Remark 14.
Applying Lemma13 for λ=0, 1, 1/3, we get the results of Lemmas 3, 7, and 11, respectively.


## 3. Main Results

Theorem 15.
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v| is convex on the coordinates on Δ and λ∈[0,1], then the following equality holds:
(26)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)(2λ2-2λ+18)2×(|∂2f∂u∂v(a,c)|+|∂2f∂u∂v(a,d)|+|∂2f∂u∂v(b,c)|+|∂2f∂u∂v(b,d)|),
where
(27)A=12(b-a)×∫ab‍(λf(x,c)+2(1-λ)f(x,c+d2)+λf(x,d))dx+12(d-c)×∫cd‍(λf(a,y)+2(1-λ)f(a+b2,y)+λf(b,y))dy-λ(1-λ)2×[f(a,c+d2)+f(a,c+d2)+f(a+b2,c)+f(a+b2,d)].Proof.
From Lemma13, we obtain
(28)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A∫cd‍|≤(b-a)(d-c)×∬01|M(u,v)|×|∂2f∂u∂v(ua+(1-u)b,vc+(1-v)d)|dudv.
Because |∂2f/∂u∂v| is a convex function on the coordinates on Δ, then one has
(29)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)×∬01|M(u,v)|×{uv|∂2f∂u∂v(a,c)|+u(1-v)|∂2f∂u∂v(a,d)|+(1-u)v|∂2f∂u∂v(b,c)|+(1-u)(1-v)|∂2f∂u∂v(b,d)|}dudv=(b-a)(d-c)×{|∂2f∂u∂v(a,c)|∬01|M(u,v)|uvdudv+|∂2f∂u∂v(a,d)|∬01|M(u,v)|u(1-v)dudv+|∂2f∂u∂v(b,c)|∬01|M(u,v)|(1-u)vdudv+|∂2f∂u∂v(b,d)|∬01|M(u,v)|(1-u)(1-v)dudv|∂2f∂u∂v(b,d)|}=(b-a)(d-c)(2λ2-2λ+18)2×(|∂2f∂u∂v(a,c)|+|∂2f∂u∂v(a,d)|+|∂2f∂u∂v(b,c)|+|∂2f∂u∂v(b,d)|).
On the other hand, we have
(30)∬01|M(u,v)|uvdudv=∬01|M(u,v)|u(1-v)dudv=∬01|M(u,v)|(1-u)vdudv=∬01|M(u,v)|(1-u)(1-v)dudv,∬01|M(u,v)|uvdudv=∬01/2|(u-λ2)(v-λ2)|uvdudv+∫01/2‍∫1/21‍|(u-λ2)(v-(1-λ2))|uvdudv+∫1/21‍∫01/2‍|(u-(1-λ2))(v-λ2)|uvdudv+∬1/21|(u-(1-λ2))(v-(1-λ2))|uvdudv=(2λ2-2λ+18)2,
which completes the proof.Remark 16.
Applying Theorem15 for λ=0, 1, 1/3, we get the results of Theorems 4, 8, and 12, respectively.Theorem 17.
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v|q is convex on the coordinates on Δ and q>1, one gets the following inequality:
(31)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)4(λp+1+(1-λ)p+1p+1)2/p×((|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|q+|∂2f∂u∂v(b,c)|q+|∂2f∂u∂v(b,d)|q)×(4)-1)1/q,
where λ∈[0,1] and A is as given in Theorem 15 and 1/p+1/q=1.Proof.
From Lemma13, we obtain
(32)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)×∬01|M(u,v)|×|∂2f∂u∂v(ua+(1-u)b,vc+(1-v)d)|dudv.
By using the well-known Hölder inequality for double integrals, then one has
(33)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)(∬01|M(u,v)|pdudv)1/p×(∬01|∂2f∂u∂v(+(1-v)d)∬01|∂2f∂u∂v|qduua+(1-u)b,vcaaaaaaaaaaaaaaaaaa+(1-v)d)∬01|∂2f∂u∂v|qdudv)1/q.
Because |∂2f/∂u∂v|q is a convex function on the coordinates on Δ, by (4), then one has
(34)∬01|∂2f∂u∂v(ua+(1-u)b,vc+(1-v)d)|qdudv≤((|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|q+|∂2f∂u∂v(b,c)|q+|∂2f∂u∂v(b,d)|q)×(4)-1).
We note that
(35)∬01|M(u,v)|pdudv=∬01/2|(u-λ2)(v-λ2)|pdudv+∫01/2‍∫1/21‍|(u-λ2)(v-(1-λ2))|pdudv+∫1/21‍∫01/2‍|(u-(1-λ2))(v-λ2)|pdudv+∬1/21|(u-(1-λ2))(v-(1-λ2))|pdudv=(λp+1+(1-λ)p+1(p+1)2p)2.
Hence, it follows that
(36)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)4(λp+1+(1-λ)p+1p+1)2/p×((|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|q+|∂2f∂u∂v(b,c)|q+|∂2f∂u∂v(b,d)|q)×(4)-1)1/q.Remark 18.
Applying Theorem17 for λ=0, 1, we get the results of Theorems 5 and 9, respectively.Theorem 19.
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v|q is convex on the coordinates on Δ and q≥1, then
(37)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)(2λ2-2λ+14)2×((|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|q+|∂2f∂u∂v(b,c)|q+|∂2f∂u∂v(b,d)|q)×(4)-1)1/q,
where λ∈[0,1] and A is as given in Theorem 15.Proof.
From Lemma13, we obtain
(38)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)×∬01|M(u,v)|×|∂2f∂u∂v(ua+(1-u)b,vc+(1-v)d)|dudv.
By using the well-known power mean inequality for double integrals, then one has
(39)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)(∬01|M(u,v)|dudv)1-(1/q)×(∬01|M(u,v)|×|∂2f∂u∂v(+(1-v)d)∂2f∂u∂v|qdudv)ua+(1-u)b,vcaaaaaaaaaaaaaaaaaa+(1-v)d)∂2f∂u∂v|qdudv)1/q.
Because |∂2f/∂u∂v|q is a convex function on the coordinates on Δ, then one has
(40)|∂2f∂u∂v(ua+(1-u)b,vc+(1-v)d)|q≤uv|∂2f∂u∂v(a,c)|q+u(1-v)|∂2f∂u∂v(a,d)|q+(1-u)v|∂2f∂u∂v(b,c)|q+(1-u)(1-v)|∂2f∂u∂v(b,d)|q.
Thus, it follows that
(41)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)(2λ2-2λ+14)2-(2/q)×(∬01|M(u,v)|×{uv|∂2f∂u∂v(a,c)|q+u(1-v)|∂2f∂u∂v(a,d)|q+(1-u)v|∂2f∂u∂v(b,c)|q+(1-u)(1-v)|∂2f∂u∂v(b,d)|q}dudv)1/q.
On the other hand, we obtain
(42)∬01|M(u,v)|×{uv|∂2f∂u∂v(a,c)|q+u(1-v)|∂2f∂u∂v(a,d)|q+(1-u)v|∂2f∂u∂v(b,c)|q+(1-u)(1-v)|∂2f∂u∂v(b,d)|q}dudv={|∂2f∂u∂v(a,c)|q∬01|M(u,v)|uvdudv+|∂2f∂u∂v(a,d)|q∬01|M(u,v)|u(1-v)dudv+|∂2f∂u∂v(b,c)|q∬01|M(u,v)|(1-u)vdudv+|∂2f∂u∂v(b,d)|q∬01|M(u,v)|(1-u)(1-v)dudv}=(2λ2-2λ+18)2×(|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|q+|∂2f∂u∂v(b,c)|q+|∂2f∂u∂v(b,d)|q).
Thus, we get the following inequality:
(43)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A|≤(b-a)(d-c)(2λ2-2λ+14)2×((|∂2f∂u∂v(a,c)|q+|∂2f∂u∂v(a,d)|q+|∂2f∂u∂v(b,c)|q+|∂2f∂u∂v(b,d)|q)×(4)-1)1/q,
which completes the proof.Remark 20.
Applying Theorem19 for λ=0, 1, we get the result of Theorems 6 and 10, respectively.Theorem 21.
Letf:Δ⊆ℝ2→ℝ be a partial differentiable mapping on Δ:=[a,b]×[c,d] in ℝ2 with a<b and c<d. If |∂2f/∂u∂v|q is concave on the coordinates on Δ and q>1, then
(44)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24×[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A1(b-a)(d-c)|≤(b-a)(d-c)4(λp+1+(1-λ)p+1p+1)2/p×|∂2f∂u∂v(a+b2,c+d2)|,
where λ∈[0,1] and A is as given in Theorem 15 and 1/p+1/q=1.Proof.
Similarly as in Theorem17, because |∂2f/∂u∂v|q is a concave function on the coordinates on Δ, by the reversed direction of (4), we get
(45)∬01|∂2f∂u∂v(ua+(1-u)b,vc+(1-v)d)|qdudv≤|∂2f∂u∂v(a+b2,c+d2)|q.
Hence, it follows that
(46)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+(1-λ)2f(a+b2,c+d2)+λ24×[f(a,c)+f(a,d)+f(b,c)+f(b,d)]-A1(b-a)(d-c)∫ab‍∫cd‍|≤(b-a)(d-c)4(λp+1+(1-λ)p+1p+1)2/p×|∂2f∂u∂v(a+b2,c+d2)|,
which yields the desired result.



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*Source: 741291-2014-03-19.xml*