Open Access

On some Hermite–Hadamard type inequalities for (s, QC)-convex functions

SpringerPlus20165:49

https://doi.org/10.1186/s40064-016-1676-9

Received: 8 October 2015

Accepted: 6 January 2016

Published: 20 January 2016

Abstract

In the paper, the authors introduce a new notion “\((s,\text {QC})\)-convex function on the co-ordinates” and establish some Hermite–Hadamard type integral inequalities for \((s,\text {QC})\)-convex functions on the co-ordinates.

Keywords

Convex function (s, QC)-Convex function on the co-ordinates Hermite–Hadamard’s integral inequality

Mathematics Subject Classification

26A51 26D15 26D20 26E60 41A55

Background

Let \(f:I\subseteq \mathbb {R}\rightarrow \mathbb {R}\) be a convex function and \(a, b\in I\) with \(a < b\). The double inequality
$$\begin{aligned} f\biggl (\frac{a+b}{2}\biggr ) \le \frac{1}{b-a} \int _{a}^{b}f(x)\mathrm{d}x \le \frac{f(a)+f(b)}{2} \end{aligned}$$
(1)
is known in the literature as Hermite–Hadamard’s inequality for convex functions.

Definition 1

(Dragomir and Pearce 1998; Pečarić et al. 1992) A function \(f:I\subseteq \mathbb {R}\rightarrow \mathbb {R}\) is said to be quasi-convex (QC), if
$$\begin{aligned} f(\lambda x+(1-\lambda )y)\le \max \{f(x),f(y)\} \end{aligned}$$
(2)
holds for all \(x,y\in I\) and \(\lambda \in [0,1]\).

Definition 2

(Dragomir and Pearce 1998) The function \(f:I\subseteq \mathbb {R}\rightarrow \mathbb {R}\) is Jensen- or J-quasi-convex (JQC) if
$$\begin{aligned} f\biggl (\frac{x+y}{2}\biggl )\le \max \{f(x),f(y)\} \end{aligned}$$
(3)
holds for all \(x,y\in I\).

Definition 3

(Hudzik and Maligranda 1994) Let \(s\in (0, 1]\). A function \(f:I\subseteq \mathbb {R}\rightarrow \mathbb {R}\) is said to be s-convex (in the second sense) if
$$\begin{aligned} f(\lambda x+(1-\lambda )y)\le \lambda ^sf(x)+(1-\lambda )^sf(y) \end{aligned}$$
(4)
holds for all \(x,y\in I\) and \(\lambda \in [0,1].\)

Definition 4

(Xi and Qi 2015a) For some \(s\in [-1,1]\), a function \(f:I\subseteq \mathbb {R}\rightarrow \mathbb {R}\) is said to be extended s-convex if
$$\begin{aligned} f(\lambda x+(1-\lambda )y)\le \lambda ^sf(x)+(1-\lambda )^sf(y) \end{aligned}$$
(5)
is valid for all \(x,y\in I\) and \(\lambda \in (0,1)\).

Definition 5

(Dragomir 2001; Dragomir and Pearce 2000) A function \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}\) is said to be convex on co-ordinates on \(\Delta\) if the partial functions
$$\begin{aligned} f_y:[a,b]\rightarrow \mathbb {R},\quad f_y(u)=f_y(u,y)\quad \text {and}\quad f_x:[c,d]\rightarrow \mathbb {R},\quad f_x(v)= f_x(x,v) \end{aligned}$$
(6)
are convex for all \(x\in (a,b)\) and \(y \in (c,d)\).

Definition 6

A function \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}\) is said to be convex on co-ordinates on \(\Delta\) if the inequality
$$\begin{aligned}&f(tx+(1-t)z,\lambda y+(1-\lambda )w)\nonumber \\&\quad \le t\lambda f(x,y)+t(1-\lambda )f(x,w)+(1-t)\lambda f(z,y)+(1-t)(1-\lambda )f(z,w) \end{aligned}$$
(7)
holds for all \(t,\lambda \in [0,1]\) and \((x,y),(z,w)\in \Delta\).

Definition 7

(Alomari and Darus 2008) A function \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}_0=[0,\infty )\) is s-convex on \(\Delta\) for some fixed \(s\in (0, 1]\) if
$$\begin{aligned} f(\lambda x + (1-\lambda )z, \lambda y + (1-\lambda )w)\le \lambda ^sf(x, y)+(1-\lambda )^sf(z,w) \end{aligned}$$
(8)
holds for all \((x, y), (z,w)\in \Delta\) and \(\lambda \in [0, 1]\).

Definition 8

(Özdemir et al. 2012a, Definition 7) A function \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}\) is called a Jensen- or J-quasi-convex function on the co-ordinates on \(\Delta\) if
$$\begin{aligned} f\left( \frac{x +z}{2},\frac{y +w}{2}\right) \le \max \{f(x, y), f(z,w)\} \end{aligned}$$
(9)
holds for all \((x, y), (z,w)\in \Delta\).

Definition 9

(Özdemir et al. 2012a, Definition 5) A function \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}\) is called a quasi-convex function on the co-ordinates on \(\Delta\) if
$$\begin{aligned} f(\lambda x + (1-\lambda )z, \lambda y + (1-\lambda )w)\le \max \{f(x, y), f(z,w)\} \end{aligned}$$
(10)
holds for all \((x, y), (z,w)\in \Delta\) and \(\lambda \in [0, 1]\).

Theorem 1

(Dragomir 2001; Dragomir and Pearce 2000 Theorem 2.2) Let \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}\) be convex on the co-ordinates on \(\Delta\) with \(a<b\) and \(c<d\). Then
$$\begin{aligned}&f\biggl (\frac{a+b}{2},\frac{c + d}{2}\biggr )\\&\quad \le \frac{1}{2}\biggl [ \frac{1}{b - a}\int _a^b f\biggl (x,\frac{c + d}{2}\biggr )\mathrm{d}x + \frac{1}{d -c}\int _c^df\biggl (\frac{a + b}{2},y\biggr ) \mathrm{d}y\biggr ]\\&\quad \le \frac{1}{(b - a)(d - c)}\int _a^b \int _c^d f(x,y)\mathrm{d}y\mathrm{d}x\\&\quad \le \frac{1}{4}\biggl [\frac{1}{b - a}\biggl (\int _a^b f(x,c)\mathrm{d}x + \int _a^bf(x,d)\mathrm{d}x\biggr ) + \frac{1}{d - c}\biggl (\int _c^d f(a,y) \mathrm{d}y + \int _c^df(b,y)\mathrm{d}y\biggr )\biggr ]\\&\quad \le \frac{f(a,c) + f(b,c) + f(a,d) + f(b,d)}{4}. \end{aligned}$$

Theorem 2

(Özdemir et al. 2012a, Lemma 8) Every J-quasi-convex mapping \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}\) is J-quasi-convex on the co-ordinates.

Theorem 3

(Özdemir et al. 2012a, Lemma 6) Every quasi-convex mapping \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}\) is quasi-convex on the coordinates.

For more information on this topic, please refer to Bai et al. (2016), Hwang et al. (2007), Özdemir et al. (2011, 2012a, b, c, 2014), Qi and Xi (2013), Roberts and Varberg (1973), Sarikaya et al. (2012), Wu et al. (2016), Xi et al. (2012, 2015), Xi and Qi (2012, 2013, 2015a, b, c) and related references therein.

In this paper, we introduce a new concept “\((s, \text {QC})\)-convex functions on the co-ordinates on the rectangle of \(\mathbb {R}^2\)” and establish some new integral inequalities of Hermite–Hadamard type for \((s, \text {QC})\)-convex functions on the co-ordinates.

Definitions and Lemmas

We now introduce three new definitions

Definition 10

For \(s \in [-1,1]\), a function \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}_0\) is said to be \((\text {J}s, \text {JQC})\)-convex on the co-ordinates on \(\Delta\) with \(a<b\) and \(c<d\), if
$$\begin{aligned} f\biggl (\frac{x+z}{2},\frac{y+w}{2}\biggr ) \le \frac{1}{2^{s}}\bigl [\max \{f(x,y),f(x,w)\} +\max \{f(z,y),f(z,w)\}\bigr ] \end{aligned}$$
(11)
holds for all \(t, \lambda \in [0,1]\) and \((x,y),(z,w)\in \Delta\).

Remark 1

By Definitions 8 and 10 and Lemma 1, we see that, for \(s \in [-1,1]\) and \(f:\Delta \subseteq \mathbb {R}^2 \rightarrow \mathbb {R}_0\),
  1. 1.

    If \(f:\Delta \rightarrow \mathbb {R}_0\) is a J-quasi-convex function on the co-ordinates on \(\Delta\), then f is a \((\text {J}s, \text {JQC})\)-convex function on the co-ordinates on \(\Delta\);

     
  2. 2.

    Every J-quasi-convex function \(f:\Delta \rightarrow \mathbb {R}_0\) is a \((\text {J}s, \text {JQC})\)-convex function on the co-ordinates on \(\Delta\).

     

Definition 11

A function \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2\rightarrow \mathbb {R}_0\) is called \((s, \text {JQC})\)-convex on the co-ordinates on \(\Delta\) with \(a<b\) and \(c<d\), if
$$\begin{aligned} f\biggl (tx+(1-t)z,\frac{y+w}{2}\biggr ) \le t^s\max \{f(x,y),f(x,w)\} +(1-t)^s\max \{f(z,y),f(z,w)\} \end{aligned}$$
(12)
holds for all \(t\in (0,1)\), \((x,y),(z,w) \in \Delta\), and some \(s \in [-1,1]\).

Definition 12

For some \(s \in [-1,1]\), a function \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2\rightarrow \mathbb {R}_0\) is called \((s, \text {QC})\)-convex on the co-ordinates on \(\Delta\) with \(a<b\) and \(c<d\), if
$$\begin{aligned}&f(tx+(1-t)z,\lambda y+(1-\lambda )w)\nonumber \\&\quad \le t^s\max \{f(x,y),f(x,w)\} +(1-t)^s\max \{f(z,y),f(z,w)\} \end{aligned}$$
(13)
is valid for all \(t\in (0,1)\), \(\lambda \in [0,1]\), and \((x,y),(z,w) \in \Delta\).

Remark 2

For \(s \in (0,1]\) and \(f:\Delta \subseteq \mathbb {R}^2 \rightarrow \mathbb {R}_0\),
  1. 1.

    If taking \(\lambda =\frac{1}{2}\) and \(t=\lambda =\frac{1}{2}\) in (13), then \((\text {J}s,\text {JQC})\subseteq (s, \text {JQC})\subseteq (s, \text {QC})\);

     
  2. 2.

    If \(f:\Delta \rightarrow \mathbb {R}_0\) is a s-convex function on \(\Delta\), then f is an \((s, \text {QC})\)-convex function on the co-ordinates on \(\Delta\).

     

Remark 3

Considering Definitions 9 and 12 and Lemma 1, for \(s \in [-1,1]\) and \(f:\Delta \subseteq \mathbb {R}^2 \rightarrow \mathbb {R}_0\),
  1. 1.

    If \(f:\Delta \rightarrow \mathbb {R}_0\) is a quasi-convex function on the co-ordinates on \(\Delta\), then it is an \((s, \text {QC})\)-convex function on the co-ordinates on \(\Delta\);

     
  2. 2.

    Every quasi-convex function \(f:\Delta \rightarrow \mathbb {R}_0\) is an \((s, \text {QC})\)-convex function on the co-ordinates on \(\Delta\).

     

Lemma 1

(Latif and Dragomir 2012) If \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2 \rightarrow \mathbb {R}\) has partial derivatives and \(\frac{\partial ^2f}{\partial x\partial y}\in L_1(\Delta )\) with \(a<b\) and \(c<d\), then
$$\begin{aligned} \Phi (f;a,b,c,d)\triangleq&\frac{1}{(b-a)(d-c)}\int _a^b\int _c^df(x,y)\mathrm{d}y\mathrm{d}x+f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \\&-\frac{1}{b-a}\int _a^bf\biggl (x,\frac{c+d}{2}\biggr )\mathrm{d}x -\frac{1}{d-c}\int _c^df\biggl (\frac{a+b}{2},y\biggr ) \mathrm{d}y\\ &=(b-a)(d-c)\int _0^1 \int _0^1 K(t,\lambda ) \frac{\partial ^2}{\partial x\partial y}f(ta+(1 - t)b,\lambda c+(1-\lambda )d)\mathrm{d}t\mathrm{d}\lambda , \end{aligned}$$
where
$$\begin{aligned} K(t,\lambda )= {\left\{ \begin{array}{ll} t\lambda , &{}(t,\lambda ) \in \bigr [0,\frac{1}{2}\bigr ]\times \bigr [0,\frac{1}{2}\bigr ], \\ t\bigr (\lambda - 1), &{}(t,\lambda ) \in \bigr [0,\frac{1}{2}\bigr ]\times \bigr (\frac{1}{2},1\bigr ], \\ \bigr (t - 1)\lambda , &{}(t,\lambda ) \in \bigr (\frac{1}{2},1\bigr ]\times \bigr [0,\frac{1}{2}\bigr ], \\ \bigr (t - 1)\bigr (\lambda - 1),&{}(t,\lambda ) \in \bigr (\frac{1}{2},1\bigr ]\times \bigr (\frac{1}{2},1\bigr ]. \end{array}\right. } \end{aligned}$$
(14)

Lemma 2

Let \(r\ge 0\) and \(q>1\). Then
$$\begin{aligned} \int _0^{1/2}u^{r}\mathrm{d}u&=\int _{1/2}^1(1-u )^{r}\mathrm{d}u =\frac{1}{2^{r+1}(r+1)} \end{aligned}$$
(15)
and
$$\begin{aligned} \int _0^1\int _0^1|K(t,\lambda )|^{q/(q-1)}\mathrm{d}t\mathrm{d}\lambda =\biggl (\frac{q-1}{2q-1}\biggl )^2\biggl (\frac{1}{4}\biggr )^{q/(q-1)}. \end{aligned}$$
(16)
where \(K(t,\lambda )\) is defined by (14).

Proof

This follows from a straightforward computation.\(\square\)

Some integral inequalities of Hermite–Hadamard type

In this section, we will establish Hermite–Hadamard type integral inequalities for \((s, \text {QC})\)-convex functions on the co-ordinates on rectangle from the plane \(\mathbb {R}^2\).

Theorem 4

Let \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2\rightarrow \mathbb {R}\) have partial derivatives and \(\frac{\partial ^2f}{\partial x\partial y}\in L_1(\Delta )\). If \(\bigl | \frac{\partial ^2f}{\partial x\partial y}\bigr |^q\) is an (s, QC)-convex function on the co-ordinates on \(\Delta\) with \(a<b\) and \(c<d\) for some \(s \in [-1,1]\) and \(q\ge 1\), then
  1. 1.
    When \(s \in (-1,1]\),
    $$\begin{aligned} \begin{array}{ll} |\Phi (f;a,b,c,d)| &{}\le \frac{(b-a)(d-c)}{8}\biggl (\frac{1}{2^{s+1}(s+1)(s+2)}\biggr )^{1/q}\\ &{}\quad \times \Bigl \{\bigl [(s+1)M_q(a,c,d)+(2^{s+2}-s-3)M_q(b,c,d)\bigl ]^{1/q}\\ &{}\quad +\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigl ]^{1/q}\Bigl \}; \end{array} \end{aligned}$$
    (17)
     
  2. 2.
    When \(s=-1\),
    $$\begin{aligned} |\Phi (f;a,b,c,d)|\le & {} \frac{(b-a)(d-c)}{8}\Bigl \{\bigl [M_q(a,c,d)+(2\ln 2-1)M_q(b,c,d)\bigl ]^{1/q}\nonumber \\&+\bigl [(2\ln 2-1)M_q(a,c,d)+M_q(b,c,d)\bigl ]^{1/q}\Bigl \}; \end{aligned}$$
    (18)
     
where
$$\begin{aligned} M_q(u,c,d)=\max \biggl \{\biggl |\frac{\partial ^2}{\partial x\partial y}f(u,c)\biggl |^q, \biggl | \frac{\partial ^2}{\partial x\partial y}f(u,d)\biggl |^q\biggl \}. \end{aligned}$$
(19)

Proof

By Lemma 1 and Hölder’s integral inequality, we have
$$\begin{aligned} \begin{array}{ll} |\Phi &{}(f;a,b,c,d)| \\ &{}\le (b-a)(d-c)\int _0^1\int _0^1|K(t,\lambda )| \biggl |\frac{\partial ^2}{\partial x\partial y}f(ta+(1-t)b, \lambda c+(1-\lambda )d)\biggr |\mathrm{d}t\mathrm{d}\lambda \\ &{}\le (b-a)(d-c)\biggl (\int _0^{1}\int _0^{1}|K(t,\lambda )|\mathrm{d}t\mathrm{d}\lambda \biggr )^{1-1/q}\\ &{}\quad \times \biggl \{ \biggl [\int _0^{1/2}\int _0^{1/2}t\lambda \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\ &{}\quad +\biggl [\int _{1/2}^1\int _0^{1/2}t(1-\lambda )\biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\ &{}\quad +\biggl [\int _0^{1/2}\int _{1/2}^1(1-t)\lambda \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\ &{}\quad +\biggl [ \int _{1/2}^1\int _{1/2}^1(1-t)(1-\lambda )\biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\biggl \}. \end{array} \end{aligned}$$
(20)
When \(s \in (-1,1]\), using the co-ordinated \((s, \text {QC})\)-convexity of \(\bigl | \frac{\partial ^2f}{\partial x\partial y}\bigr |^q\) and by Lemma 2, we obtain
$$\begin{aligned} \begin{array}{ll} \int _0^{1/2}&{}\int _0^{1/2}t\lambda \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \\ &{}\le \biggl (\int _0^{1/2}\lambda \mathrm{d}\lambda \biggr )\int _0^{1/2} \biggl [t^{s+1}\max \biggl \{\biggl |\frac{\partial ^2}{\partial x\partial y}f(a,c)\biggl |^q, \biggl | \frac{\partial ^2}{\partial x\partial y}f(a,d)\biggl |^q\biggl \}\\ &{}\quad +t(1-t)^{s}\max \biggl \{\biggl |\frac{\partial ^2}{\partial x\partial y}f(b,c)\biggl |^q, \biggl | \frac{\partial ^2}{\partial x\partial y}f(b,d)\biggl |^q\biggl \} \biggr ]\mathrm{d}t\\ &{}=\frac{1}{2^{s+5}(s+1)(s+2)}\bigl [(s+1)M_q(a,c,d) +(2^{s+2}-s-3)M_q(b,c,d)\bigr ]. \end{array} \end{aligned}$$
(21)
Similarly, we also have
$$\begin{aligned} \begin{array}{ll} &{}\int _{1/2}^1\int _0^{1/2}t(1-\lambda )\biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \\ &{}\quad \le \frac{1}{2^{s+5}(s+1)(s+2)}\bigl [(s+1)M_q(a,c,d)+ (2^{s+2}-s-3)M_q(b,c,d)\bigr ], \end{array} \end{aligned}$$
(22)
$$\begin{aligned} \begin{array}{ll} &{}\int _0^{1/2}\int _{1/2}^1(1-t)\lambda \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \\ &{}\quad \le \frac{1}{2^{s+5}(s+1)(s+2)}\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigr ], \end{array} \end{aligned}$$
(23)
$$\begin{aligned} \begin{array}{ll} &{}\int _{1/2}^1\int _{1/2}^1(1-t)(1-\lambda )\biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \\ &{}\quad \le \frac{1}{2^{s+5}(s+1)(s+2)}\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigr ]. \end{array} \end{aligned}$$
(24)
Applying inequalities (21) to (24) into the inequality (20) yields
$$\begin{aligned}&|\Phi (f;a,b,c,d)| \le (b-a)(d-c)\biggl (\frac{1}{16}\biggr )^{1-1/q}\\&\qquad \times \biggl \{ 2\biggl [\frac{1}{2^{s+5}(s+1)(s+2)}\bigl [(s+1)M_q(a,c,d)+(2^{s+2}-s-3)M_q(b,c,d)\biggl ]^{1/q}\\&\qquad +2\biggl [\frac{1}{2^{s+5}(s+1)(s+2)}\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigr ]\biggl ]^{1/q}\biggl \}\\&\quad =\frac{(b-a)(d-c)}{8}\biggl (\frac{1}{2^{s+1}(s+1)(s+2)}\biggr )^{1/q}\\&\qquad \times \Bigl \{\bigl [(s+1)M_q(a,c,d)+(2^{s+2}-s-3)M_q(b,c,d)\bigl ]^{1/q}\\&\qquad +\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigl ]^{1/q}\Bigl \}. \end{aligned}$$
When \(s=-1\), similar to the proof of inequalities (21) to (24), we can write
$$\begin{aligned} \begin{array}{ll} &{}\int _0^{1/2}\int _0^{1/2}t\lambda \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \\ &{}\quad \le \frac{1}{2^{4}}\bigl [M_q(a,c,d)+(2\ln 2-1)M_q(b,c,d)\bigl ], \end{array} \end{aligned}$$
(25)
$$\begin{aligned} \begin{array}{ll} &{}\int _{1/2}^1\int _0^{1/2}t(1-\lambda )\biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \\ &{}\quad \le \frac{1}{2^{4}}\bigl [M_q(a,c,d)+(2\ln 2-1)M_q(b,c,d)\bigl ], \end{array} \end{aligned}$$
(26)
$$\begin{aligned} \begin{array}{ll} &{}\int _0^{1/2}\int _{1/2}^1(1-t)\lambda \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \\ &{}\quad \le \frac{1}{2^{4}}\bigl [(2\ln 2-1)M_q(a,c,d)+M_q(b,c,d)\bigl ], \end{array} \end{aligned}$$
(27)
$$\begin{aligned} \begin{array}{ll} &{}\int _{1/2}^1\int _{1/2}^1(1-t)(1-\lambda )\biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \\ &{}\quad \le \frac{1}{2^{4}}\bigl [(2\ln 2-1)M_q(a,c,d)+M_q(b,c,d)\bigl ]. \end{array} \end{aligned}$$
(28)
Substituting inequalities (25) to (28) into (20) leads to the inequality (18). Theorem 4 is thus proved.\(\square\)

Corollary 1

Under the conditions of Theorem 4,
  1. 1.
    If \(q=1\) and \(s \in (-1,1]\), then
    $$\begin{aligned} |\Phi (f;a,b,c,d)|\le & {} \frac{(b-a)(d-c)(2^{s+1}-1)}{2^{s+3}(s+1)(s+2)}\\&\times \biggl [\max \biggl \{\biggl | \frac{\partial ^2f(a,c)}{\partial x\partial y}\biggl |, \biggl | \frac{\partial ^2f(a,d)}{\partial x\partial y}\biggl |\biggl \} +\max \biggl \{\biggl | \frac{\partial ^2f(b,c)}{\partial x\partial y}\biggl |, \biggl | \frac{\partial ^2f(b,d)}{\partial x\partial y}\biggl |\biggl \}\biggl ]; \end{aligned}$$
     
  2. 2.
    If \(q=1\) and \(s=-1\), then
    $$\begin{aligned} |\Phi (f;a,b,c,d)|\le & {} \frac{(b-a)(d-c)\ln 2}{4}\\&\times \biggl [\max \biggl \{\biggl | \frac{\partial ^2f(a,c)}{\partial x\partial y}\biggl |, \biggl | \frac{\partial ^2f(a,d)}{\partial x\partial y}\biggl |\biggl \} +\max \biggl \{\biggl | \frac{\partial ^2f(b,c)}{\partial x\partial y}\biggl |, \biggl | \frac{\partial ^2f(b,d)}{\partial x\partial y}\biggl |\biggl \}\biggl ]. \end{aligned}$$
     

Corollary 2

Under the conditions of Theorem 4,
  1. 1.
    If \(s=0\), then
    $$\begin{aligned} |\Phi (f;a,b,c,d)|\le & {} \frac{(b-a)(d-c)}{4}\biggl (\frac{1}{4}\biggr )^{1/q}\\&\times \biggl [\max \biggl \{\biggl | \frac{\partial ^2f(a,c)}{\partial x\partial y}\biggl |^q, \biggl | \frac{\partial ^2f(a,d)}{\partial x\partial y}\biggl |^q\biggl \} +\max \biggl \{\biggl | \frac{\partial ^2f(b,c)}{\partial x\partial y}\biggl |^q, \biggl | \frac{\partial ^2f(b,d)}{\partial x\partial y}\biggl |^q\biggl \}\biggl ]^{1/q}; \end{aligned}$$
     
  2. 2.
    If \(s=1\), then
    $$\begin{aligned} |\Phi (f;a,b,c,d)|\le & {} \frac{(b-a)(d-c)}{8}\biggl (\frac{1}{12}\biggr )^{1/q}\\&\times \Bigl \{\bigl [M_q(a,c,d)+2M_q(b,c,d)\bigl ]^{1/q} +\bigl [2M_q(a,c,d) +M_q(b,c,d)\bigl ]^{1/q}\Bigl \}. \end{aligned}$$
     

Theorem 5

Let \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2\rightarrow \mathbb {R}\) have partial derivatives and \(\frac{\partial ^2f}{\partial x\partial y}\in L_1(\Delta )\). If \(\bigl | \frac{\partial ^2f}{\partial x\partial y}\bigr |^q\) is an (s, QC)-convex function on the co-ordinates on \(\Delta\) with \(a<b\) and \(c<d\) for some \(s \in [-1,1]\), \(q>1\), and \(0\le \ell \le q\), then
  1. 1.
    When \(s \in (-1,1]\),
    $$\begin{aligned} |\Phi (f;a,b,c,d)|&\le \frac{(b-a)(d-c)}{16}\biggl (\frac{q-1}{2q-\ell -1}\biggr )^{1-1/q} \biggl (\frac{1}{2^{s-1}(\ell +1)(s+1)(s+2)}\biggr )^{1/q}\\&\quad \times \Bigl \{\bigl [(s+1)M_q(a,c,d)+(2^{s+2}-s-3)M_q(b,c,d)\bigl ]^{1/q}\\&\quad +\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigl ]^{1/q}\Bigl \}; \end{aligned}$$
     
  2. 2.
    When \(s=-1\),
    $$\begin{aligned} |\Phi (f;a,b,c,d)|\le & {} \frac{(b-a)(d-c)}{16}\biggl (\frac{q-1}{2q-\ell -1}\biggr )^{1-1/q} \biggl (\frac{4}{\ell +1}\biggr )^{1/q}\Bigl \{\bigl [M_q(a,c,d)\\&+(2\ln 2-1)M_q(b,c,d)\bigl ]^{1/q} +\bigl [(2\ln 2-1)M_q(a,c,d) +M_q(b,c,d)\bigr ]^{1/q}\Bigl \}, \end{aligned}$$
     
where \(M_q(u,c,d)\) is defined by (19).

Proof

If \(s \in (-1,1]\), similar to the proof of the inequality (17), we can acquire
$$\begin{aligned} |\Phi&(f;a,b,c,d)| \\&\le (b-a)(d-c)\biggl \{\biggl (\int _0^{1/2}\int _0^{1/2}t\lambda ^{(q-\ell )/(q-1)}\mathrm{d}t\mathrm{d}\lambda \biggr )^{1-1/q}\\&\quad \times \biggl [\int _0^{1/2}\int _0^{1/2}t\lambda ^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\quad +\biggl ( \int _{1/2}^1\int _0^{1/2}t(1-\lambda )^{(q-\ell )/(q-1)}\mathrm{d}t\mathrm{d}\lambda \biggr )^{1-1/q}\\&\quad \times \biggl [ \int _{1/2}^1\int _0^{1/2}t(1-\lambda )^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\quad +\biggl (\int _0^{1/2}\int _{1/2}^1(1-t)\lambda ^{(q-\ell )/(q-1)}\mathrm{d}t\mathrm{d}\lambda \biggr )^{1-1/q}\\&\quad \times \biggl [\int _0^{1/2}\int _{1/2}^1(1-t)\lambda ^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\quad +\biggl (\int _{1/2}^1\int _{1/2}^1(1-t)(1-\lambda )^{(q-\ell )/(q-1)}\mathrm{d}t\mathrm{d}\lambda \biggr )^{1-1/q}\\&\quad \times \biggl [ \int _{1/2}^1\int _{1/2}^1(1-t)(1-\lambda )^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\biggl \}\\&=(b-a)(d-c)\biggl [\frac{q-1}{8(2q-\ell -1)}\biggl (\frac{1}{2}\biggr )^{(2q-\ell -1)/(q-1)}\biggr ]^{1-1/q}\\&\quad \times \biggl \{ \biggl [\int _0^{1/2}\int _0^{1/2}t\lambda ^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\quad +\biggl [ \int _{1/2}^1\int _0^{1/2}t(1-\lambda )^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\quad +\biggl [\int _0^{1/2}\int _{1/2}^1(1-t)\lambda ^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\quad +\biggl [\int _{1/2}^1\int _{1/2}^1(1-t)(1-\lambda )^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\biggl \}\\&\le (b-a)(d-c)\biggl (\frac{q-1}{2q-\ell -1}\biggr )^{1-1/q} \biggl (\frac{1}{2}\biggr )^{(5q-\ell -4)/q}\\&\quad \times \biggl \{ 2\biggl [\frac{1}{2^{s+\ell +3}(s+1)(s+2)(\ell +1)}\bigl [(s+1)M_q(a,c,d)+(2^{s+2}-s-3)M_q(b,c,d)\biggl ]^{1/q}\\&\quad +2\biggl [\frac{1}{2^{s+\ell +3}(s+1)(s+2)(\ell +1)}\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigr ]\biggl ]^{1/q}\biggl \}\\&=\frac{(b-a)(d-c)}{16}\biggl (\frac{q-1}{2q-\ell -1}\biggr )^{1-1/q} \biggl (\frac{1}{2^{s-1}(\ell +1)(s+1)(s+2)}\biggr )^{1/q}\\&\quad \times \Bigl \{\bigl [(s+1)M_q(a,c,d)+(2^{s+2}-s-3)M_q(b,c,d)\bigl ]^{1/q}\\&\quad +\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigl ]^{1/q}\Bigl \}. \end{aligned}$$
If \(s=-1\), similarly one can see that
$$\begin{aligned}&|\Phi (f;a,b,c,d)| \le (b-a)(d-c)\biggl (\frac{q-1}{2q-\ell -1}\biggr )^{1-1/q} \biggl (\frac{1}{2}\biggr )^{(5q-\ell -4)/q}\\&\qquad \times \biggl \{ \biggl [\int _0^{1/2}\int _0^{1/2}t\lambda ^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\qquad +\biggl [ \int _{1/2}^1\int _0^{1/2}t(1-\lambda )^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\qquad +\biggl [\int _0^{1/2}\int _{1/2}^1(1-t)\lambda ^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\qquad +\biggl [ \int _{1/2}^1\int _{1/2}^1(1-t)(1-\lambda )^\ell \biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\biggl \}\\&\quad \le (b-a)(d-c)\biggl (\frac{q-1}{2q-\ell -1}\biggr )^{1-1/q} \biggl (\frac{1}{2}\biggr )^{(5q-\ell -4)/q}\\&\qquad \times \biggl \{ 2\biggl [\frac{1}{2^{\ell +2}(\ell +1)} \bigl [M_q(a,c,d)+(2\ln 2-1)M_q(b,c,d)\bigl ]\biggl ]^{1/q}\\&\qquad +2\biggl [\frac{1}{2^{\ell +2}(\ell +1)}\bigl [(2\ln 2-1)M_q(a,c,d) +M_q(b,c,d)\bigr ]\biggl ]^{1/q}\biggl \}\\&\quad =\frac{(b-a)(d-c)}{16}\biggl (\frac{q-1}{2q-\ell -1}\biggr )^{1-1/q} \biggl (\frac{4}{\ell +1}\biggr )^{1/q}\\&\qquad \times \Bigl \{\bigl [M_q(a,c,d)+(2\ln 2-1)M_q(b,c,d)\bigl ]^{1/q}\\&\qquad +\bigl [(2\ln 2-1)M_q(a,c,d) +M_q(b,c,d)\bigr ]^{1/q}\Bigl \}. \end{aligned}$$
The proof of Theorem 5 is complete.\(\square\)

Corollary 3

Under the conditions of Theorem 5, when \(\ell =1\),
  1. 1.
    If \(s \in (-1,1]\), then
    $$\begin{aligned} |\Phi (f;a,b,c,d)|&\le \frac{(b-a)(d-c)}{32} \biggl (\frac{1}{2^{s-1}(s+1)(s+2)}\biggr )^{1/q}\\&\quad \times \Bigl \{\bigl [(s+1)M_q(a,c,d)+(2^{s+2}-s-3)M_q(b,c,d)\bigl ]^{1/q}\\&\quad +\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigl ]^{1/q}\Bigl \}; \end{aligned}$$
     
  2. 2.
    if \(s=-1\), then
    $$\begin{aligned} |\Phi (f;a,b,c,d)|\le & {} \frac{(b-a)(d-c)2^{2/q}}{32}\\&\times \Bigl \{\bigl [M_q(a,c,d)+(2\ln 2-1)M_q(b,c,d)\bigl ]^{1/q} +\bigl [(2\ln 2-1)M_q(a,c,d) +M_q(b,c,d)\bigr ]^{1/q}\Bigl \}. \end{aligned}$$
     

Corollary 4

Under the conditions of Theorem 5, when \(\ell =q\),
  1. 1.
    If \(s \in (-1,1]\), then
    $$\begin{aligned} |\Phi (f;a,b,c,d)|&\le \frac{(b-a)(d-c)}{16} \biggl (\frac{1}{2^{s-1}(q+1)(s+1)(s+2)}\biggr )^{1/q}\\&\quad \times \Bigl \{\bigl [(s+1)M_q(a,c,d)+(2^{s+2}-s-3)M_q(b,c,d)\bigl ]^{1/q}\\&\quad +\bigl [(2^{s+2}-s-3)M_q(a,c,d) +(s+1)M_q(b,c,d)\bigl ]^{1/q}\Bigl \}; \end{aligned}$$
     
  2. 2.
    If \(s=-1\), then
    $$\begin{aligned}&|\Phi (f;a,b,c,d)| \le \frac{(b-a)(d-c)}{16} \biggl (\frac{4}{q+1}\biggr )^{1/q}\\&\quad \times \Bigl \{\bigl [M_q(a,c,d)+(2\ln 2-1)M_q(b,c,d)\bigl ]^{1/q} +\bigl [(2\ln 2-1)M_q(a,c,d) +M_q(b,c,d)\bigr ]^{1/q}\Bigl \}. \end{aligned}$$
     

Theorem 6

Let \(f:\Delta =[a,b]\times [c,d]\subseteq \mathbb {R}^2\rightarrow \mathbb {R}\) have partial derivatives and \(\frac{\partial ^2f}{\partial x\partial y}\in L_1(\Delta )\). If \(\bigl | \frac{\partial ^2f}{\partial x\partial y}\bigr |^q\) is an (s, QC)-convex function on the co-ordinates on \(\Delta\) with \(a<b\) and \(c<d\) for some \(s \in (-1,1]\) and \(q>1\), then
$$\begin{aligned} |\Phi (f;a,b,c,d)|\le & {} \frac{(b-a)(d-c)}{4}\biggl (\frac{q-1}{2q-1}\biggl )^{2(1-1/q)} \biggl (\frac{1}{s+1}\biggr )^{1/q}\\&\times \biggl [\max \biggl \{\biggl |\frac{\partial ^2 f(a,c)}{\partial x\partial y}\biggl |^q, \biggl | \frac{\partial ^2 f(a,d)}{\partial x\partial y}\biggl |^q\biggl \} +\max \biggl \{\biggl | \frac{\partial ^2f(b,c)}{\partial x\partial y}\biggl |^q, \biggl | \frac{\partial ^2f(b,d)}{\partial x\partial y}\biggl |^q\biggl \}\biggr ]^{1/q}. \end{aligned}$$

Proof

From Lemma 1, Hölder’s integral inequality, the co-ordinated \((s, \text {QC})\)-convexity of \(\bigl | \frac{\partial ^2f}{\partial x\partial y}\bigr |^q\), and Lemma 2, it follows that
$$\begin{aligned}&|\Phi (f;a,b,c,d)| \\&\quad \le (b-a)(d-c)\biggl (\int _0^1\int _0^1|K(t,\lambda )|^{q/(q-1)}\mathrm{d}t\mathrm{d}\lambda \biggr )^{1-1/q}\\&\qquad \times \biggl [\int _0^1\int _0^1\biggl | \frac{\partial ^2}{\partial x\partial y} f(ta+(1-t)b,\lambda c+(1-\lambda )d)\biggr |^q\mathrm{d}t\mathrm{d}\lambda \biggl ]^{1/q}\\&\quad \le (b-a)(d-c)\biggl [\biggl (\frac{q-1}{2q-1}\biggl )^2\biggl (\frac{1}{4}\biggr )^{q/(q-1)}\biggr ]^{1-1/q}\\&\qquad \times \biggl \{\int _0^1\biggl [t^{s}\max \biggl \{\biggl |\frac{\partial ^2 f(a,c)}{\partial x\partial y}\biggl |^q, \biggl | \frac{\partial ^2 f(a,d)}{\partial x\partial y}\biggl |^q\biggl \} +(1-t)^{s}\max \biggl \{\biggl | \frac{\partial ^2f(b,c)}{\partial x\partial y}\biggl |^q, \biggl | \frac{\partial ^2f(b,d)}{\partial x\partial y}\biggl |^q\biggl \} \biggr ]\mathrm{d}t \biggr \}^{1/q}\\&\quad =\frac{(b-a)(d-c)}{4}\biggl (\frac{q-1}{2q-1}\biggl )^{2(1-1/q)} \biggl (\frac{1}{s+1}\biggr )^{1/q}\\&\qquad \times \biggl [\max \biggl \{\biggl |\frac{\partial ^2 f(a,c)}{\partial x\partial y}\biggl |^q, \biggl | \frac{\partial ^2 f(a,d)}{\partial x\partial y}\biggl |^q\biggl \} +\max \biggl \{\biggl | \frac{\partial ^2f(b,c)}{\partial x\partial y}\biggl |^q, \biggl | \frac{\partial ^2f(b,d)}{\partial x\partial y}\biggl |^q\biggl \} \biggr ]^{1/q}. \end{aligned}$$
Theorem 6 is thus proved.\(\square\)

Conclusions

Our main results in this paper are Definitions  11 to 12 and those integral inequalities of Hermite–Hadamard type in Theorems 4 to 6.

Declarations

Authors’ contributions

Both authors contributed equally to the manuscript. Both authors read and approved the final manuscript.

Acknowledgements

The authors thank the anonymous referees for their careful corrections to and valuable comments on the original version of this paper. This work was partially supported by the National Natural Science Foundation of China under Grant No. 11361038 and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, China.

Competing interests

The authors declare that they have no competing interests.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics, Inner Mongolia University for Nationalities
(2)
Department of Mathematics, School of Science, Tianjin Polytechnic University
(3)
Institute of Mathematics, Henan Polytechnic University

References

  1. Alomari M, Darus M (2008) Hadamard-type inequalities for s-convex functions. Int Math Forum 40(3):1965–1975Google Scholar
  2. Bai S-P, Qi F, Wang S-H (2016) Some new integral inequalities of Hermite C–Hadamard type for \((\alpha, m; P)\)-convex functions on co-ordinates. J Appl Anal Comput 6(1):171–178. doi:https://doi.org/10.11948/2016014 Google Scholar
  3. Dragomir SS (2001) On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwan J Math 5(4):775–788Google Scholar
  4. Dragomir SS, Pearce CEM (1998) Quasi-convex functions and Hadamard’s inequality. Bull Aust Math Soc 57(3):377–385. doi:https://doi.org/10.1017/S0004972700031786 View ArticleGoogle Scholar
  5. Dragomir SS, Pearce CEM (2000) Selected topics on Hermite–Hadamard type inequalities and applications, RGMIA monographs. Victoria University, Melbourne, Australia. http://rgmia.org/monographs/hermite_hadamard.html
  6. Hudzik H, Maligranda L (1994) Some remarks on \(s\)-convex functions. Aequ Math 48(1):100–111. doi:https://doi.org/10.1007/BF01837981 View ArticleGoogle Scholar
  7. Hwang D-Y, Tseng K-L, Yang G-S (2007) Some Hadamard’s inequalities for co-ordinated convex functions in a rectangle from the plane. Taiwan J Math 11:63–73Google Scholar
  8. Latif MA, Dragomir SS (2012) On some new inequalities for differentiable co-ordinated convex functions. J Inequal Appl 2012:28. doi:https://doi.org/10.1186/1029-242X-2012-28 View ArticleGoogle Scholar
  9. Özdemir ME, Set E, Sarikaya MZ (2011) Some new Hadamard-type inequalities for co-ordinated \(m\)-convex and \((\alpha, m)\)-convex functions. Hacet J Math Stat 40(2):219–229Google Scholar
  10. Özdemir ME, Akdemir AO, Yildiz Ç (2012a) On co-ordinated quasi-convex functions. Czechoslov Math J 62(4):889–900. doi:https://doi.org/10.1007/s10587-012-0072-z View ArticleGoogle Scholar
  11. Özdemir ME, Latif MA, Akdemir AO (2012b) On some Hadamard-type inequalities for product of two \(s\)-convex functions on the co-ordinates. J Inequal Appl 2012:21. doi:https://doi.org/10.1186/1029-242X-2012-21 View ArticleGoogle Scholar
  12. Özdemir ME, Yildiz Ç, Akdemir AO (2012c) On some new Hadamard-type inequalities for co-ordinated quasi-convex functions. Hacettepe J Math Stat 41(5):697–707Google Scholar
  13. Özdemir ME, Kavurmaci H, Akdemir AO, Avci M (2012d) Inequalities for convex and \(s\)-convex functions on \(\Delta =[a, b]\times [c, d]\). J Inequal Appl 2012:20. doi:https://doi.org/10.1186/1029-242X-2012-20 View ArticleGoogle Scholar
  14. Özdemir ME, Akdemir AO, Kavurmaci H (2014) On the Simpson’s inequality for co-ordinated convex functions. Turkish J Anal Number Theory 2(5):165–169. doi:https://doi.org/10.12691/tjant-2-5-2 View ArticleGoogle Scholar
  15. Özdemir ME, Yildiz C, Akdemir AO (2014) On the co-ordinated convex functions. Appl Math Inf Sci 8(3):1085–1091. doi:https://doi.org/10.12785/amis/080318 View ArticleGoogle Scholar
  16. Pečarić J, Proschan F, Tong YL (1992) Convex functions, partial orderings, and statistical applications. Mathematics in science and engineering, vol 187. Academic Press, New YorkGoogle Scholar
  17. Qi F, Xi B-Y (2013) Some integral inequalities of Simpson type for GA-\(\varepsilon\)-convex functions. Georgian Math J 20(4):775–788. doi:https://doi.org/10.1515/gmj-2013-0043 View ArticleGoogle Scholar
  18. Roberts AW, Varberg DE (1973) Convex functions. Academic Press, New YorkGoogle Scholar
  19. Sarikaya MZ, Set E, Özdemir ME, Dragomir SS (2012) New some Hadamard’s type inequalities for co-ordinated convex functions. Tamsui Oxf J Math Sci 28(2):137–152Google Scholar
  20. Wu Y, Qi F, Pei Z-L, Bai S-P (2016) Hermite–Hadamard type integral inequalities via \((s, m)-P\)-convexity on co-ordinates. J Nonlinear Sci Appl 9(3):876–884Google Scholar
  21. Xi B-Y, Qi F (2012) Some integral inequalities of Hermite–Hadamard type for convex functions with applications to means. J Funct Spaces Appl 2012(980438):14. doi:https://doi.org/10.1155/2012/980438 Google Scholar
  22. Xi B-Y, Bai R-F, Qi F (2012) Hermite–Hadamard type inequalities for the \(m\)- and \((\alpha, m)\)-geometrically convex functions. Aequ Math 84(3):261–269. doi:https://doi.org/10.1007/s00010-011-0114-x View ArticleGoogle Scholar
  23. Xi B-Y, Qi F (2013) Some Hermite–Hadamard type inequalities for differentiable convex functions and applications. Hacet J Math Stat 42(3):243–257Google Scholar
  24. Xi B-Y, Qi F (2015a) Inequalities of Hermite–Hadamard type for extended \(s\)-convex functions and applications to means. J Nonlinear Convex Anal 16(5):873–890Google Scholar
  25. Xi B-Y, Qi F (2015b) Integral inequalities of Hermite–Hadamard type for \(((\alpha, m), \log )\)-convex functions on co-ordinates. Probl Anal Issues Anal 4(2):72–91. doi:https://doi.org/10.15393/j3.art.2015.2829 Google Scholar
  26. Xi B-Y, Qi F (2015c) Some new integral inequalities of Hermite-Hadamard type for \((\log, (\alpha, m))\)-convex functions on co-ordinates. Stud Univ Babeş-Bolyai Math 60(4):509–525Google Scholar
  27. Xi B-Y, Bai S-P, Qi F (2015) Some new inequalities of Hermite–Hadamard type for \((\alpha ,m_1)\)-\((s,m_2)\)-convex functions on co-ordinates. ResearchGate dataset. doi:https://doi.org/10.13140/2.1.2919.7126

Copyright

© Wu and Qi. 2016