# Some New Inequalities of Hermite-Hadamard Type for (α1,m1)-(α2,m2)-Convex Functions on Coordinates

Authors: Ling Chun
Journal: Journal of Function Spaces
 (2014)

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## Abstract

The author introduces a new concept “(α1,m1)-(α2,m2)-convex functions on coordinates” and establishes some new inequalities of Hermite-Hadamard type for (α1,m1)-(α2,m2)-convex functions of two variables on co-ordinates.


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## Body

## 1. Introduction

The following definitions are well known in the literature.Definition 1.
Letf:I⊆ℝ=(-∞,∞)→ℝ be a function on an interval I. If
(1)f(λx+(1-λ)y)≤λf(x)+(1-λ)f(y)
is valid for all x,y∈I and λ∈[0,1], then we say that f is a convex function on I.Definition 2 (see [1]).
Forf:[0,b]→ℝ and m∈(0,1], if
(2)f(tx+m(1-t)y)≤tf(x)+m(1-t)f(y)
is valid for all x,y∈[0,b] and t∈[0,1], then we say that f is an m-convex function on [0,b].Definition 3 (see [2]).
Forf:[0,b]→ℝ and (α,m)∈(0,1]2, if
(3)f(tx+m(1-t)y)≤tαf(x)+m(1-tα)f(y)
is valid for all x,y∈[0,b] and t∈[0,1], then we say that f is an (α,m)-convex function on [0,b].Definition 4 (see [3, 4]).
Fora<b and c<d, a function f:Δ=[a,b]×[c,d]⊆ℝ2→ℝ is said to be convex on coordinates, if the partial mappings
(4)fy:[a,b]∋u↦f(u,y)∈ℝ,fx:[c,d]∋v↦f(x,v)∈ℝ
are convex for all x∈(a,b) and y∈(c,d).A formal definition for coordinated convex functions may be stated as follows.Definition 5.
A functionf:Δ→ℝ is said to be convex on coordinates, if
(5)f(tx+(1-t)z,λy+(1-λ)w)≤tλf(x,y)+t(1-λ)f(x,w)+(1-t)λf(z,y)+(1-t)(1-λ)f(z,w)
holds for all t,λ∈[0,1], (x,y),(z,w)∈Δ.Some inequalities of Hermite-Hadamard type for convex functions on coordinates may be recited as follows.Theorem 6 (see Theorem 1 in [3, 4]).
Letf:Δ→ℝ be a convex function on coordinates. Then
(6)f(a+b2,c+d2)≤12[1b-a∫abf(x,c+d2)dx+1d-c∫cdf(a+b2,y)dy]≤1(b-a)(d-c)∫ab∫cdf(x,y)dydx≤14[1b-a(∫abf(x,c)dx+∫abf(x,d)dx)+1d-c(∫cdf(a,y)dy+∫cdf(b,y)dy)]≤14[f(a,c)+f(b,c)+f(a,d)+f(b,d)].Theorem 7 (see [5, Theorem 2.3]).
Letf:Δ→ℝ be a partial differentiable function. If |∂2f/∂t∂λ| is a convex function on coordinates, then
(7)|×∫ab∫cdf(x,y)dydx-A19[f(a,c+d2)+f(b,c+d2)+4f(a+b2,c+d2)+f(a+b2,c)+f(a+b2,d)]+f(a,c)+f(a,d)+f(b,c)+f(b,d)36+1(b-a)(d-c)∫ab∫cdf(x,y)dydx-A|≤25(b-a)(d-c)5184×{|∂2∂t∂λf(a,c)|+|∂2∂t∂λf(a,d)|+|∂2∂t∂λf(b,c)|+|∂2∂t∂λf(b,d)|},
where
(8)A=1b-a×∫ab{16[f(x,c)+4f(x,c+d2)+f(x,d)]}dx+1d-c×∫cd{16[f(a,y)+4f(a+b2,y)+f(b,y)]}dy.Theorem 8 (see [6, Theorem 2]).
Letf:Δ→ℝ be a partial differentiable function. If |∂2f/∂t∂λ| is a convex function on coordinates, then
(9)|1(b-a)(d-c)∫ab‍∫cd‍f(x,y)dydx+f(a+b2,c+d2)-A|≤(b-a)(d-c)16·((|∂2∂t∂λf(a,c)|+|∂2∂t∂λf(a,d)|+|∂2∂t∂λf(b,c)|+|∂2∂t∂λf(b,d)|)×(4)-1),
where
(10)A=1b-a∫abf(x,c+d2)dx+1d-c∫cdf(a+b2,y)dy.For more information on inequalities of Hermite-Hadamard type for various(s1,m1)-(s2,m2)-convex functions on coordinates and for extended s-convex functions on coordinates, please refer to the recently published articles [7, 8] and related references therein.The aim of this paper is to introduce a new concept “(α1,m1)-(α2,m2)-convex functions on coordinates” and to establish some new inequalities of Hermite-Hadamard type for (α1,m1)-(α2,m2)-convex functions of two variables on co-ordinates.


## 2. A Definition and a Lemma

We now introduce a new notion “(α1,m1)-(α2,m2)-convex functions on coordinates.”Definition 9.
For some(α1,m1),(α2,m2)∈(0,1]2, a function f:[0,b]×[0,d]→ℝ is said to be (α1,m1)-(α2,m2)-convex on coordinates, if
(11)f(tx+m1(1-t)z,λy+m2(1-λ)w)≤tα1λα2f(x,y)+m2tα1(1-λα2)f(x,w)+m1(1-tα1)λα2f(z,y)+m1m2(1-tα1)(1-λα2)f(z,w)
holds for all t,λ∈[0,1], (x,y),(z,w)∈[0,b]×[0,d].In order to establish some new inequalities of Hermite-Hadamard type for(α1,m1)-(α2,m2)-convex functions of two variables on coordinates, we need the following lemma.Lemma 10.
Letf:Δ→ℝ be a partial differentiable function and L(Δ) denote the set of all Lebesgue integrable functions on Δ. If ∂2f/∂x∂y∈L(Δ), then
(12)I(f)∶=f(a,c)-1d-c∫cdf(a,y)dy-1b-a×∫abf(x,c)dx+1(d-c)(b-a)×∫cd∫abf(x,y)dydx=(b-a)(d-c)×∫01∫01tλ∂2f∂x∂y×(ta+(1-t)b,λc+(1-λ)d)dtdλ.Proof.
Integrating by part yields(13)∫01∫01tλ∂2f∂x∂y(ta+(1-t)b,λc+(1-λ)d)dtdλ=∫01λ{t1a-b∂f∂y(ta+(1-t)b,λc+(1-λ)d)|01-1a-b∫01∂f∂y(ta+(1-t)b,λc+(1-λ)d)dtt1a-b∂f∂y(ta+(1-t)b,λc+(1-λ)d)}dλ=∫01λ{1a-b∂f∂y(a,λc+(1-λ)d)-1a-b×∫01∂f∂y(ta+(1-t)b,λc+(1-λ)d)dt1a-b∂f∂y}dλ=1a-b{∫01λ∂f∂y(a,λc+(1-λ)d)dλ-∫01∫01λ∂f∂y(ta+(1-t)b,λc+(1-λ)d)dtdλ∫01λ∂f∂y(a,λc+(1-λ)d)dλ},∫01λ∂f∂y(a,λc+(1-λ)d)dλ-∫01∫01λ∂f∂y(ta+(1-t)b,λc+(1-λ)d)dtdλ=λf(a,λc+(1-λ)d)c-d|01-1c-d×∫01f(a,λc+(1-λ)d)dλ-∫01{λf(ta+(1-t)b,λc+(1-λ)d)c-d|01-1c-d∫01f(ta+(1-t)b,λc+(1-λ)d)dλλf(ta+(1-t)b,λc+(1-λ)d)c-d|01}dt=f(a,c)c-d-1c-d∫01f(a,λc+(1-λ)d)dλ-∫01{f(ta+(1-t)b,c)c-d-1c-d×∫01f(ta+(1-t)b,λc+(1-λ)d)dλf(ta+(1-t)b,c)c-d}dt=f(a,c)c-d-1c-d∫01f(a,λc+(1-λ)d)dλ-1c-d∫01f(ta+(1-t)b,c)dt+1c-d∫01∫01f(ta+(1-t)b,λc+(1-λ)d)dλdt.
Choosing x=ta+(1-t)b and y=λc+(1-λ)d for (t,λ)∈[0,1]2 and multiplying by (b-a)(d-c) on both sides of the above equations lead to the identity (12). The proof of Lemma 10 is complete.


## 3. Some Integral Inequalities of Hermite-Hadamard Type

We are now in a position to establish new inequalities of Hermite-Hadamard type for(α1,m1)-(α2,m2)-convex functions on coordinates on a rectangle in the plane ℝ02=[0,∞)×[0,∞).Theorem 11.
Letf:Δ=[0,b*]×[0,d*]→ℝ be a partial differentiable function and ∂2f/∂x∂y∈L(Δ). If |∂2f/∂x∂y|q is (α1,m1)-(α2,m2)-convex on coordinates on [a,b/m1]×[c,d/m2]⊆Δ with a<b and c<d for some (α1,m1),(α2,m2)∈(0,1]2 and q≥1, then
(14)|I(f)|≤(b-a)(d-c)4[1(α1+2)(α2+2)]1/q×[4|∂2f∂x∂y(a,c)|q+2α2m2|∂2f∂x∂y(a,dm2)|q+2α1m1|∂2f∂x∂y(bm1,c)|q+α1α2m1m2|∂2f∂x∂y(bm1,dm2)|q]1/q.Proof.
Using Lemma10, the coordinated (α1,m1)-(α2,m2)-convexity of |∂2f/∂x∂y|q, and Hölder integral inequality yields
(15)|I(f)|(b-a)(d-c)≤∫01∫01tλ|∂2f∂x∂y(ta+(1-t)b,λc+(1-λ)d)|dtdλ≤(∫01∫01tλdtdλ)1-1/q×{∫01∫01tλ[tα1λα2|∂2f∂x∂y(a,c)|q+m2tα1(1-λα2)×|∂2f∂x∂y(a,dm2)|q+m1(1-tα1)λα2|∂2f∂x∂y(bm1,c)|q+m1m2(1-tα1)(1-λα2)×|∂2f∂x∂y(bm1,dm2)|q]dtdλ}1/q=14(1(α1+2)(α2+2))1/q×[4|∂2f∂x∂y(a,c)|q+2α2m2|∂2f∂x∂y(a,dm2)|q+2α1m1|∂2f∂x∂y(bm1,c)|q+α1α2m1m2|∂2f∂x∂y(bm1,dm2)|q]1/q.
Theorem 11 is thus proved.Corollary 12.
Under conditions of Theorem11, when q=1, one has
(16)|I(f)|≤(b-a)(d-c)4(α1+2)(α2+2)×[4|∂2f∂x∂y(a,c)|+2α2m2|∂2f∂x∂y(a,dm2)|+2α1m1|∂2f∂x∂y(bm1,c)|+α1α2m1m2|∂2f∂x∂y(bm1,dm2)|].Corollary 13.
Under conditions of Theorem11,  (1)
ifα1=α2=α and m1=m2=m, then
(17)|I(f)|≤(b-a)(d-c)4(1(α+2)2)1/q×[4|∂2f∂x∂y(a,c)|q+2αm|∂2f∂x∂y(a,dm)|q+2αm|∂2f∂x∂y(bm,c)|q+α2m2|∂2f∂x∂y(bm,dm)|q]1/q;(2)
ifα1=α2=m1=m2=1, then
(18)|I(f)|≤(b-a)(d-c)4(19)1/q×[4|∂2f∂x∂y(a,c)|q+2|∂2f∂x∂y(a,d)|q+2|∂2f∂x∂y(b,c)|q+|∂2f∂x∂y(b,d)|q]1/q.Theorem 14.
Letf:Δ=[0,b*]×[0,d*]→ℝ be a partial differentiable function and ∂2f/∂x∂y∈L(Δ). If |∂2f/∂x∂y|q is (α1,m1)-(α2,m2)-convex on coordinates on [a,b/m1]×[c,d/m2]⊆Δ with a<b and c<d for (α1,m1),(α2,m2)∈(0,1]2 and q>1, then
(19)|I(f)|≤(b-a)(d-c)(q-12q-1)2(1-1/q)×(1(α1+1)(α2+1))1/q×[|∂2f∂x∂y(a,c)|q+α2m2|∂2f∂x∂y(a,dm2)|q+α1m1|∂2f∂x∂y(bm1,c)|q+α1α2m1m2|∂2f∂x∂y(bm1,dm2)|q]1/q.Proof.
By Lemma10, the coordinated (α1,m1)-(α2,m2)-convexity of |∂2f/∂x∂y|q, and Hölder integral inequality, we have
(20)|I(f)|(b-a)(d-c)≤∫01∫01tλ|∂2f∂x∂y(ta+(1-t)b,λc+(1-λ)d)|dtdλ≤(∫01∫01(tλ)q/(q-1)dtdλ)1-1/q×{∫01∫01[tα1λα2|∂2f∂x∂y(a,c)|q+m2tα1(1-λα2)|∂2f∂x∂y(a,dm2)|q+m1(1-tα1)λα2|∂2f∂x∂y(bm1,c)|q+m1m2(1-tα1)(1-λα2)×|∂2f∂x∂y(bm1,dm2)|q]dtdλ}1/q=(q-12q-1)2(1-1/q)(1(α1+1)(α2+1))1/q×[|∂2f∂x∂y(a,c)|q+α2m2|∂2f∂x∂y(a,dm2)|q+α1m1|∂2f∂x∂y(bm1,c)|q+α1α2m1m2|∂2f∂x∂y(bm1,dm2)|q]1/q.
Theorem 14 is thus proved.Corollary 15.
Under conditions of Theorem14,   if α1=α2=α and m1=m2=m, then
(21)|I(f)|≤(b-a)(d-c)(q-12q-1)2(1-1/q)(1(α+1)2)1/q×[|∂2f∂x∂y(a,c)|q+αm|∂2f∂x∂y(a,dm)|q+αm|∂2f∂x∂y(bm,c)|q+α2m2|∂2f∂x∂y(bm,dm)|q]1/q;
if α1=α2=m1=m2=1, then
(22)|I(f)|≤(b-a)(d-c)(q-12q-1)2(1-1/q)(14)1/q×[|∂2f∂x∂y(a,c)|q+|∂2f∂x∂y(a,d)|q+|∂2f∂x∂y(b,c)|q+|∂2f∂x∂y(b,d)|q]1/q.Theorem 16.
Letf:Δ=[0,b*]×[0,d*]→ℝ be a partial differentiable function and ∂2f/∂x∂y∈L(Δ). If |∂2f/∂x∂y|q is (α1,m1)-(α2,m2)-convex on co-ordinates on [a,b/m1]×[c,d/m2]⊆Δ with a<b and c<d for (α1,m1),(α2,m2)∈(0,1]2, q>1, and q≥r, s>0, then
(23)|I(f)|≤(b-a)(d-c)[(q-1)2(2q-r-1)(2q-s-1)]1-1/q×[1(r+1)(α1+r+1)(s+1)(α2+s+1)]1/q×[(r+1)(s+1)|∂2f∂x∂y(a,c)|q+α2m2(r+1)|∂2f∂x∂y(a,dm2)|q+α1m1(s+1)|∂2f∂x∂y(bm1,c)|q+α1α2m1m2|∂2f∂x∂y(bm1,dm2)|q]1/q.Proof.
Using Lemma10, the coordinated (α1,m1)-(α2,m2)-convexity of |∂2f/∂x∂y|q, and Hölder integral inequality reveals
(24)|I(f)|(b-a)(d-c)≤∫01∫01tλ|∂2f∂x∂y(ta+(1-t)b,λc+(1-λ)d)|dtdλ≤(∫01∫01t(q-r)/(q-1)λ(q-s)/(q-1)dtdλ)1-1/q×{∫01∫01trλs[tα1λα2|∂2f∂x∂y(a,c)|q+m2tα1(1-λα2)|∂2f∂x∂y(a,dm2)|q+m1(1-tα1)λα2|∂2f∂x∂y(bm1,c)|q+m1m2(1-tα1)(1-λα2)×|∂2f∂x∂y(bm1,dm2)|q]dtdλ}1/q=[(q-1)2(2q-r-1)(2q-s-1)]1-1/q×[1(α1+r+1)(α2+s+1)|∂2f∂x∂y(a,c)|q+α2m2(α1+r+1)(s+1)(α2+s+1)|∂2f∂x∂y(a,dm2)|q+α1m1(r+1)(α1+r+1)(α2+s+1)|∂2f∂x∂y(bm1,c)|q+α1α2m1m2(r+1)(α1+r+1)(s+1)(α2+s+1)×|∂2f∂x∂y(bm1,dm2)|q]1/q=[(q-1)2(2q-r-1)(2q-s-1)]1-1/q×[1(r+1)(α1+r+1)(s+1)(α2+s+1)]1/q×[(r+1)(s+1)|∂2f∂x∂y(a,c)|q+α2m2(r+1)|∂2f∂x∂y(a,dm2)|q+α1m1(s+1)|∂2f∂x∂y(bm1,c)|q+α1α2m1m2|∂2f∂x∂y(bm1,dm2)|q]1/q.
Hence, the proof of Theorem 16 is completed.Corollary 17.
Under conditions of Theorem16,  (1)
ifr=s=q, then
(25)|I(f)|≤(b-a)(d-c)[1(q+1)2(α1+q+1)(α2+q+1)]1/q×[(q+1)2|∂2f∂x∂y(a,c)|q+α2m2(q+1)|∂2f∂x∂y(a,dm2)|q+α1m1(q+1)|∂2f∂x∂y(bm1,c)|q+α1α2m1m2|∂2f∂x∂y(bm1,dm2)|q]1/q;(2)
ifr=s=q, α1=α2=α, and m1=m2=m, one has
(26)|I(f)|≤(b-a)(d-c)[1(q+1)2(α+q+1)2]1/q×[(q+1)2|∂2f∂x∂y(a,c)|q+αm(q+1)|∂2f∂x∂y(a,dm)|q+αm(q+1)|∂2f∂x∂y(bm,c)|q+α2m2|∂2f∂x∂y(bm,dm)|q]1/q;(3)
ifr=s=q and α1=α2=m1=m2=1, one has
(27)|I(f)|≤(b-a)(d-c)[1(q+1)2(q+2)2]1/q×[(q+1)2|∂2f∂x∂y(a,c)|q+(q+1)|∂2f∂x∂y(a,d)|q+(q+1)|∂2f∂x∂y(b,c)|q+|∂2f∂x∂y(b,d)|q]1/q.Corollary 18.
Under conditions of Theorem16, if α1=α2=m1=m2=1, then
(28)|I(f)|≤(b-a)(d-c)[(q-1)2(2q-r-1)(2q-s-1)]1-1/q×[1(r+1)(r+2)(s+1)(s+2)]1/q×[(r+1)(s+1)|∂2f∂x∂y(a,c)|q+(r+1)|∂2f∂x∂y(a,d)|q+(s+1)|∂2f∂x∂y(b,c)|q+|∂2f∂x∂y(b,d)|q]1/q.



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