# Hermite-Hadamard Type Integral Inequalities for Functions Whose Second-Order Mixed Derivatives Are Coordinated(s,m)-P-Convex

Authors: Yu-Mei Bai; Shan-He Wu; Ying Wu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We establish some new Hermite-Hadamard type integral inequalities for functions whose second-order mixed derivatives are coordinated(s,m)-P-convex. An expression form of Hermite-Hadamard type integral inequalities via the beta function and the hypergeometric function is also presented. Our results provide a significant complement to the work of Wu et al. involving the Hermite-Hadamard type inequalities for coordinated (s,m)-P-convex functions in an earlier article.


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## Body

## 1. Introduction

Letf:I→R be a convex mapping. Then for any a,b∈I with a<b, we have the following double inequality:(1)fa+b2≤1b-a∫abfxdx≤fa+fb2.This celebrated inequality is known in the literature as the Hermite-Hadamard inequality. As we all know, some of the classical inequalities for means can be derived from (1) for appropriate particular selections of the mapping f. Indeed, Hermite-Hadamard’s inequality (1) has already found many applications in mathematical analysis and optimization (see, for example, [1–9]).In recent years, the applications of various properties of extended convex functions in establishing and improving Hermite-Hadamard type inequalities have attracted the attention of many researchers (see [10–15] and references cited therein).In [16], Wu et al. established some Hermite-Hadamard type inequalities under the assumption that the function f is a coordinated (s,m)-P-convex function. Motivated by the ideas of work [16], in this paper we study Hermite-Hadamard type inequalities related to the convexity of second-order mixed derivatives of f. More precisely, we focus on establishing some new Hermite-Hadamard type inequalities for functions whose second-order mixed derivatives are coordinated (s,m)-P-convex. For convenience of our discussions in subsequent sections, we begin with recalling some relevant definitions.Definition 1.
A functionf:I⊆R→R is said to be convex function if (2)ftx+1-ty≤tfx+1-tfyholds for all x,y∈I and t∈[0,1].Definition 2 (see [5]).
We say that a mapf:I⊆R→R belongs to the class P(I) if it is nonnegative and for all x,y∈I and t∈[0,1] satisfies the following inequality: (3)ftx+1-ty≤fx+fy.
In [17], the concept of m-convex functions was introduced as follows.Definition 3 (see [17]).
Forf:0,b⊆R0=0,+∞→R and m∈0,1, if (4)ftx+m1-ty≤tfx+m1-tfyis valid for all x,y∈0,b and t∈0,1, then we say that f is a m-convex function on 0,b.
In [18], the concept of s-convex functions was presented as follows.Definition 4 (see [18]).
Lets∈0,1. A function f:I⊆R→R is said to be s-convex (in the second sense) if (5)ftx+1-ty≤tsfx+1-tsfyholds for all x,y∈I and t∈[0,1].Definition 5 (see [19]).
Fors,m∈0,1×0,1, a function f:0,b→R is said to be s,m-convex if (6)ftx+m1-ty≤tsfx+m1-tsfyholds for all x,y∈I and t∈0,1.Definition 6 (see [20]).
For somes∈-1,1, a function f:I⊆R→R is said to be extended s-convex if (7)ftx+1-ty≤tsfx+1-tsfyis valid for all x,y∈I and t∈0,1.
Dragomir [21] and Dragomir and Pearce [22] considered the convexity of a function on the coordinates and put forward the following definition.Definition 7 (see [21, 22]).
A functionf:Δ=a,b×c,d⊆R2→R is said to be convex on the coordinates on Δ with a<b and c<d if the partial functions (8)fy:a,b⟶R,fyu=fu,y,fx:c,d⟶R,fxv=fx,v,are convex for all x∈a,b and y∈c,d.
It should be noted that a formal definition for coordinated convex functions is stated as follows.Definition 8 (see [21, 22]).
A functionf:Δ=a,b×c,d⊆R2→R is said to be convex on the coordinates on Δ with a<b and c<d if the partial function (9)ftx+1-tz,λy+1-λw≤tλfx,y+t1-λfx,w+1-tλfz,y+1-t1-λfz,wholds for all t,λ∈0,1,x,y,z,w∈Δ.Definition 9 (see [16]).
For somem∈0,1 and s∈-1,1, a function f:0,b×c,d→R is said to be coordinated (s,m)-P-convex on 0,b×c,d with 0<b and c<d, if (10)ftx+m1-tz,λy+1-λw≤tsfx,y+fx,w+m1-tsfz,y+fz,wholds for all t∈0,1, λ∈0,1 and x,y,z,w∈0,b×c,d.
Dragomir [21] and Dragomir and Pearce [22] established the following result.Theorem 10 (see [21, 22]).
Letf:Δ=[a,b]×c,d be convex on the coordinates on Δ=a,b×c,d with a<b and c<d. Then, one has the inequalities: (11)fa+b2,c+d2≤121b-a∫abfx,c+d2dx+1d-c∫cdfa+b2,ydy≤1b-ad-c∫ab∫cdfx,ydydx≤141b-a∫abfx,cdx+∫abfx,ddx+1d-c∫cdfa,ydy+∫cdfb,ydy≤fa,c+fb,c+fa,d+fb,d4.In this paper, we shall establish some new integral inequalities of Hermite-Hadamard type for coordinated(s,m)-P-convex functions.


## 2. Lemma

Lemma 11 (see [23]).
Iff:Δ=a,b×c,d⊆R2→R has partial derivatives and ∂2f/∂x∂y∈L1(Δ) with a<b and c<d, then (12)Pa,b,c,d≜1b-ad-c∫ab∫cdfx,ydydx+fa+b2,c+d2-1b-a∫abfx,c+d2dx-1d-c∫cdfa+b2,ydy=b-ad-c∫01∫01Kt,λ∂2∂x∂yfta+1-tb,λc+1-λddtdλ,where (13)Kt,λ=tλ,t,λ∈0,12×0,12,tλ-1,t,λ∈0,12×12,1,t-1λ,t,λ∈12,1×0,12,t-1λ-1,t,λ∈12,1×12,1.


## 3. Main Results

In this section, we establish some Hermite-Hadamard type integral inequalities for functions whose second-order mixed derivatives are coordinated(s,m)-P-convex on the plane R0×R.Theorem 12.
Suppose that the functionf:R0×R→R has continuous partial derivatives of the second-order and ∂2f/∂x∂y∈L10,b∗/m×c,d with 0≤a<b≤b∗, c<d, for some m∈(0,1] and s∈[-1,1]. If ∂2f/∂x∂yq is coordinated (s,m)-P-convex functions on 0,b∗/m×c,d for q≥1, then(1)
ifs∈-1,1, we have (14)Pa,b,c,d≤b-ad-c23×2s+1/q1s+2Δ1q+m2s+2-s-3s+1s+2Δ2m,q1/q+2s+2-s-3s+1s+2Δ1q+ms+2Δ2m,q1/q,(2)
ifs=-1, we have(15)Pa,b,c,d≤b-ad-c8×Δ1q+m2ln⁡2-1Δ2m,q1/q+2ln⁡2-1Δ1q+mΔ2m,q1/q,where (16)Δ1q=∂2fa,c∂x∂yq+∂2fa,d∂x∂yq,Δ2m,q=∂2fb/m,c∂x∂yq+∂2fb/m,d∂x∂yq.Proof.
By Lemma11 and Hölder’s integral inequality, we have (17)Pa,b,c,d≤b-ad-c∫01∫01Kt,λdtdλ1-1/q×∫01/2∫01/2tλ∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ1/q+∫1/21∫01/2t1-λ∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ1/q+∫01/2∫1/211-tλ∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ1/q+∫1/21∫1/211-t1-λ∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ1/q.A straightforward computation gives (18)∫01/2λdλ=∫1/211-λdλ=18,∫01/2ts+1dt=∫1/211-ts+1dt=12s+5s+2,fors∈-1,1,∫01/2t1-tsdt=∫1/211-ttsdt=2s+2-s-32s+5s+1s+2,fors∈-1,1,∫01/2t-1+1dt=∫1/211-t-1+1dt=12,∫01/2t1-t-1dt=∫1/211-tt-1dt=ln⁡2-12,∫01∫01Kt,λdtdλ=116.Now, by using the coordinated (s,m)-P-convexity of ∂2f/∂x∂yq, it follows that if -1<s≤1, we have (19)∫01/2∫01/2tλ∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ≤∫01/2∫01/2tλtsΔ1q+m1-tsΔ2m,qdtdλ=12s+5s+2Δ1q+m2s+2-s-32s+5s+1s+2Δ2m,q,and if s=-1, we have (20)∫01/2∫01/2tλ∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ≤∫01/2∫01/2tλt-1Δ1q+m1-t-1Δ2m,qdtdλ=116Δ1q+m2ln⁡2-1Δ2m,q.By a similar argument, we obtain (21)∫1/21∫01/2t1-λ∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ≤12s+5×1s+2Δ1q+m2s+2-s-3s+1s+2Δ2m,q,-1<s≤1,Δ1q+m2ln⁡2-1Δ2m,q,s=-1,∫01/2∫1/211-tλ∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ≤12s+5×2s+2-s-3s+1s+2Δ1q+ms+2Δ2m,q,-1<s≤1,2ln⁡2-1Δ1q+mΔ2m,q,s=-1,∫1/21∫1/211-t1-λ∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ≤12s+5×2s+2-s-3s+1s+2Δ1q+ms+2Δ2m,q,-1<s≤1,2ln⁡2-1Δ1q+mΔ2m,q,s=-1.Applying (18) and inequalities (19)–(21) into inequality (17), we get (14) and (15). This completes the proof of Theorem 12.Corollary 13.
Under the assumptions of Theorem12, if q=1, then (1)
ifs∈-1,1, then (22)Pa,b,c,d≤b-ad-c2s+1-12s+3s+1s+2Δ11+mΔ2m,1,(2)
ifs=-1, then (23)Pa,b,c,d≤b-ad-cln⁡24Δ11+mΔ2m,1.Corollary 14.
Under the assumptions of Theorem12, if q=m=1, then (1)
ifs∈(-1,1], then (24)Pa,b,c,d≤b-ad-c2s+1-12s+3s+1s+2×∂2fa,c∂x∂y+∂2fa,d∂x∂y+∂2fb,c∂x∂y+∂2fb,d∂x∂y,(2)
ifs=-1, then (25)Pa,b,c,d≤b-ad-cln⁡24×∂2fa,c∂x∂y+∂2fa,d∂x∂y+∂2fb,c∂x∂y+∂2fb,d∂x∂y.Furthermore, if q=m=1,s=0, then (26)Pa,b,c,d≤b-ad-c16×∂2fa,c∂x∂y+∂2fa,d∂x∂y+∂2fb,c∂x∂y+∂2fb,d∂x∂y.Theorem 15.
Suppose that the functionf:R0×R→R has continuous partial derivatives of the second-order and ∂2f/∂x∂y∈L1[0,b∗/m]×[c,d] with 0≤a<b≤b∗, c<d, and 0≤r≤q, -1<l≤q. If ∂2f/∂x∂yq is coordinated (s,m)-P-convex functions on [0,b∗/m]×[c,d] for some m∈0,1, s∈-1,1, and q>1, then (27)Pa,b,c,d≤b-ad-c2ll+11/qq-122q-r-12q-l-1124q-r-l-2/q-11-1/q×Δ1q2r+s+1r+s+1+2-smΔ2m,qr+1F21-s,r+1,r+2,2-11/q+Br+1,s+1-2-sF21-r,s+1,s+2,2-1Δ1qs+1+mΔ2m,q2r+s+1r+s+11/q,where Δ1(q) and Δ2m,q are defined as in (16), and B(α,β) is the beta function defined by (28)Bα,β=∫01tα-11-tβ-1dt,α,β>0,and F21c,d,e,z is the hypergeometric function defined by (29)F21c,d,e,z=ΓeΓdΓe-d∫01td-11-te-d-11-zt-cdtfor e>d>0, |z|<1, c∈R, u>0.Proof.
Using Lemma11 and Hölder’s integral inequality, we obtain (30)Pa,b,c,d≤b-ad-c∫01/2∫01/2tq-r/q-1λq-l/q-1dtdλ1-1/q×∫01/2∫01/2trλl∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ1/q+∫1/21∫01/2tq-r/q-11-λq-l/q-1dtdλ1-1/q×∫1/21∫01/2tr1-λl∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ1/q+∫01/2∫1/211-tq-r/q-1λq-l/q-1dtdλ1-1/q×∫01/2∫1/211-trλl∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ1/q+∫1/21∫1/211-tq-r/q-11-λq-l/q-1dtdλ1-1/q×∫1/21∫1/211-tr1-λl∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ1/q.After some calculations, it follows that (31)∫01/2∫01/2tq-r/q-1λq-l/q-1dtdλ=∫1/21∫01/2tq-r/q-11-λq-l/q-1dtdλ=∫01/2∫1/211-tq-r/q-1λq-l/q-1dtdλ=∫1/21∫1/211-tq-r/q-11-λq-l/q-1dtdλ=q-122q-r-12q-l-1×2-4q-r-l-2/q-1.From the coordinated (s,m)-P-convexity of ∂2f/∂x∂yq, we deduce that (32)∫01/2∫01/2trλl∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ≤2-l+1l+1∫01/2trtsΔ1q+m1-tsΔ2m,qdt=2-l+1l+1Δ1q2r+s+1r+s+1+2-smΔ2m,qr+1F21-s,r+1,r+2,2-1,∫1/21∫01/2tr1-λl∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ≤2-l+1l+1Δ1q2r+s+1r+s+1+2-smΔ2m,qr+1F21-s,r+1,r+2,2-1,∫01/2∫1/211-trλl∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ≤2-l+1l+1Br+1,s+1-2-sF21-r,s+1,s+2,2-1×Δ1qs+1+mΔ2m,q2r+s+1r+s+1,∫1/21∫1/211-tr1-λl∂2∂x∂yfta+1-tb,λc+1-λdqdtdλ≤2-l+1l+1Br+1,s+1-2-sF21-r,s+1,s+2,2-1×Δ1qs+1+mΔ2m,q2r+s+1r+s+1.
Applying (31) and inequalities (32) into inequality (30), we get inequality (27). The proof of Theorem 15 is complete.Corollary 16.
Under the assumptions of Theorem15, if r=0, then (33)Pa,b,c,d≤b-ad-c2s+l+1s+1l+11/qq-122q-12q-l-1124q-l-2/q-11-1/q×Δ1q+m2s+1-1Δ2m,q1/q+2s+1-1Δ1q+mΔ2m,q1/q.In particular, if r=l=0, then (34)Pa,b,c,d≤b-ad-c2s+1s+11/qq-12q-1122q-1/q-121-1/q×Δ1q+m2s+1-1Δ2m,q1/q+2s+1-1Δ1q+mΔ2m,q1/q.



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