Some new Hermite–Hadamard type inequalities for differentiable co-ordinated convex functions

Abstract

In the paper, the authors establish some new Hermite–Hadamard type inequalities for differentiable co-ordinated convex functions of two variables.

Keywords:

• Hermite–Hadamard type inequality
• differentiable co-ordinated convex functions

Public Interest Statement

In the paper, the authors establish some new Hermite–Hadamard type inequalities for differentiable co-ordinated convex functions of two variables.

1. Introduction

The following definitions are well known in the literature.

Definition 1.1.

A function $f:I\subseteq \mathbb{R}=(-\infty ,+\infty )\to \mathbb{R}$, if$$f(\mathit{\lambda }x+(1-\mathit{\lambda })y)\le \mathit{\lambda }f(x)+(1-\mathit{\lambda })f(y)$$

is valid for all $x,y\in I$ and $\mathit{\lambda }\in [0,1]$, then we say that f is a convex function on I.

Many important inequalities have been established for the class of convex functions, but the most famous is the Hermite–Hadamard inequality (see for instance Pečarić, Proschan, & Tong, 1991). This double inequality is stated as$$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_a^{b}f(x)\text{d}x\le \frac{f(a)+f(b)}{2},$$

where $f:I\subseteq \mathbb{R}\to \mathbb{R}$ a convex function, a, b ∊ I with a < b.

A modification for convex functions on Δ, which are also known as co-ordinated convex functions, was introduced by Dragomir (2001) and Dragomir and Pearce (2000) as follows.

Definition 1.2.

A function $f:\mathrm{\Delta }\to \mathbb{R}$ is said to be convex on the co-ordinates on Δ = [a, b] × [c, d] $\subseteq {\mathbb{R}}^2$ with a < b and c < d if the partial mappings$$f_y:[a,b]\to \mathbb{R},f_y(u)=f_y(u,y)\text{and}{f}_x:[c,d]\to \mathbb{R},f_x(v)=f_x(x,v)$$

are convex where defined for all (x, y) ∊ Δ.

A formal definition for co-ordinated convex functions may be stated as follows.

Definition 1.3.

A function $f:\mathrm{\Delta }\to \mathbb{R}$ is said to be convex on the co-ordinates on Δ = [a, b] × [c, d] $\subseteq {\mathbb{R}}^2$ with a < b and c < d if$$f(tx+(1-t)z,\mathit{\lambda }y+(1-\mathit{\lambda })w)\le t\mathit{\lambda }f(x,y)+t(1-\mathit{\lambda })f(x,w)+(1-t)\mathit{\lambda }f(z,y)+(1-t)(1-\mathit{\lambda })f(z,w)$$

for all $t,\mathit{\lambda }\in [0,1],(x,y),(z,w)\in \mathrm{\Delta }$.

The following Hermite–Hadamard type inequality for co-ordinated convex functions on the rectangle form the plane ${\mathbb{R}}^2$ was also proved in Dragomir (2001).

Theorem 1.1.

(Dragomir, 2001) Let $f:\mathrm{\Delta }=[a,b]\times [c,d]\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be convex on the co-ordinates on Δ. Then$$f\left(\frac{a+b}{2},\frac{c+d}{2}\right)\le \frac{1}{2}\left[\frac{1}{b-a}{\int }_a^{b}f\left(x,\frac{c+d}{2}\right)\text{d}x+\frac{1}{d-c}{\int }_c^{d}f\left(\frac{a+b}{2},y\right)\text{d}y\right]\le \frac{1}{(b-a)(d-c)}{\int }_a^b{\int }_c^{d}f(x,y)\text{d}y\text{d}x\le \frac{1}{4}\left[\frac{1}{b-a}\left({\int }_a^{b}f(x,c)\text{d}x+{\int }_a^{b}f(x,d)\text{d}x\right)+\frac{1}{d-c}\left({\int }_c^{d}f(a,y)\text{d}y+{\int }_c^{d}f(b,y)\text{d}y\right)\right]\le \frac{1}{4}\left[f(a,c)+f(b,c)+f(a,d)+f(b,d)\right].$$

Theorem 1.2.

(Ozdemir, Akdemir, Kavurmaci, & Avci, 2011) Let $f:\mathrm{\Delta }=[a,b]\times [c,d]\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a partial differentiable function on Δ. If $\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|$ is convex on the co-ordinates on Δ, then$$\left|\frac{1}{9}\left[f\left(a,\frac{c+d}{2}\right)+f\left(b,\frac{c+d}{2}\right)+4f\left(\frac{a+b}{2},\frac{c+d}{2}\right)+f\left(\frac{a+b}{2},c\right)+f\left(\frac{a+b}{2},d\right)\right]+\frac{1}{36}\left\{f(a,c)+f(a,d)+f(b,c)+f(b,d)\right\}+\frac{1}{(b-a)(d-c)}{\int }_a^b{\int }_c^{d}f(x,y)\text{d}x\text{d}y-A\right|\le {\left(\frac{5}{72}\right)}^2(b-a)(d-c)\left\{\left|\frac{{\mathit{\partial }}^2}{\mathit{\partial }t\mathit{\partial }\mathit{\lambda }}f(a,c)\right|+\left|\frac{{\mathit{\partial }}^2}{\mathit{\partial }t\mathit{\partial }\mathit{\lambda }}f(a,d)\right|+\left|\frac{{\mathit{\partial }}^2}{\mathit{\partial }t\mathit{\partial }\mathit{\lambda }}f(b,c)\right|+\left|\frac{{\mathit{\partial }}^2}{\mathit{\partial }t\mathit{\partial }\mathit{\lambda }}f(b,d)\right|\right\},$$

where$$A=\frac{1}{b-a}{\int }_a^b\left[\frac{f(x,c)+4f\left(x,\frac{c+d}{2}\right)+f(x,d)}{6}\right]\text{d}x+\frac{1}{d-c}{\int }_c^d\left[\frac{f(a,y)+4f\left(\frac{a+b}{2},y\right)+f(b,y)}{6}\right]\text{d}y.$$

Theorem 1.3.

(Latif & Dragomir, 2012) Let $f:\mathrm{\Delta }=[a,b]\times [c,d]\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a partial differentiable function on Δ. If $\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|$ is convex on the co-ordinates on Δ, then$$\left|\frac{1}{(b-a)(d-c)}{\int }_a^b{\int }_c^{d}f(x,y)\text{d}x\text{d}y+f\left(\frac{a+b}{2},\frac{c+d}{2}\right)-A\right|\le \frac{(b-a)(d-c)}{\text{64}}\left\{\left|\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f(a,c)\right|+\left|\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f(a,d)\right|+\left|\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f(b,c)\right|+\left|\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f(b,d)\right|\right\},$$

where$$A=\frac{1}{b-a}{\int }_a^{b}f\left(x,\frac{c+d}{2}\right)\text{d}x+\frac{1}{d-c}{\int }_c^{d}f\left(\frac{a+b}{2},y\right)\text{d}y.$$

2. Main results

The following lemma is necessary and plays an important role in establishing our main results:

Lemma 2.1.

Let $f:\mathrm{\Omega }\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a twice partial differentiable mapping on Ω^° (the interior of Ω) and let Δ: = [a, b] × [c, d] ⊆ Ω^° with a < b and c < d. If $\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\in L_1(\mathrm{\Delta })$, where L₁(Δ) denotes the set of all Lebesgue integrable functions on Δ, then$$I(f):=\frac{16}{(d-c)(b-a)}\left[f\left(\frac{a+b}{2},\frac{c+d}{2}\right)-\frac{1}{d-c}{\int }_c^{d}f\left(\frac{a+b}{2},y\right)\text{d}y-\frac{1}{b-a}{\int }_a^{b}f\left(x,\frac{c+d}{2}\right)\text{d}x+\frac{1}{(d-c)(b-a)}{\int }_c^d{\int }_a^{b}f(x,y)\text{d}x\text{d}y\right]={\int }_0^1{\int }_0^{1}t\mathit{\lambda }\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f\left(\frac{t}{2}a+\left(1-\frac{t}{2}\right)b,\frac{\mathit{\lambda }}{2}c+\left(1-\frac{\mathit{\lambda }}{2}\right)d\right)\text{d}t\text{d}\mathit{\lambda }+{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f\left(\left(1-\frac{t}{2}\right)a+\frac{t}{2}b,\left(1-\frac{\mathit{\lambda }}{2}\right)c+\frac{\mathit{\lambda }}{2}d\right)\text{d}t\text{d}\mathit{\lambda }-{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f\left(\frac{t}{2}a+\left(1-\frac{t}{2}\right)b,\left(1-\frac{\mathit{\lambda }}{2}\right)c+\frac{\mathit{\lambda }}{2}d\right)\text{d}t\text{d}\mathit{\lambda }-{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f\left(\left(1-\frac{t}{2}\right)a+\frac{t}{2}b,\frac{\mathit{\lambda }}{2}c+\left(1-\frac{\mathit{\lambda }}{2}\right)d\right)\text{d}t\text{d}\mathit{\lambda }.$$

Proof

By integration by parts, we have$${\int }_0^1{\int }_0^{1}t\mathit{\lambda }\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f\left(\frac{t}{2}a+\left(1-\frac{t}{2}\right)b,\frac{\mathit{\lambda }}{2}c+\left(1-\frac{\mathit{\lambda }}{2}\right)d\right)\text{d}t\text{d}\mathit{\lambda }=\frac{4}{(b-a)(d-c)}\left[f\left(\frac{a+b}{2},\frac{c+d}{2}\right)-{\int }_0^{1}f\left(\frac{a+b}{2},\frac{\mathit{\lambda }}{2}c+\left(1-\frac{\mathit{\lambda }}{2}\right)d\right)\text{d}\mathit{\lambda }-{\int }_0^{1}f\left(\frac{t}{2}a+\left(1-\frac{t}{2}\right)b,\frac{c+d}{2}\right)\text{d}t+{\int }_0^1{\int }_0^{1}f\left(\frac{t}{2}a+\left(1-\frac{t}{2}\right)b,\frac{\mathit{\lambda }}{2}c+\left(1-\frac{\mathit{\lambda }}{2}\right)d\right)\text{d}t\text{d}\mathit{\lambda }\right]=\frac{4}{(b-a)(d-c)}\left[f\left(\frac{a+b}{2},\frac{c+d}{2}\right)-\frac{2}{d-c}{\int }_{\frac{c+d}{2}}^{d}f\left(\frac{a+b}{2},y\right)\text{d}y-\frac{2}{b-a}{\int }_{\frac{a+b}{2}}^{b}f\left(x,\frac{c+d}{2}\right)\text{d}x+\frac{4}{(b-a)(d-c)}{\int }_{\frac{c+d}{2}}^d{\int }_{\frac{a+b}{2}}^{b}f(x,y)\text{d}x\text{d}y\right].$$

Similarly, we have$${\int }_0^1{\int }_0^{1}t\mathit{\lambda }\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f\left(\left(1-\frac{t}{2}\right)a+\frac{t}{2}b,\left(1-\frac{\mathit{\lambda }}{2}\right)c+\frac{\mathit{\lambda }}{2}d\right)\text{d}t\text{d}\mathit{\lambda }=\frac{4}{(b-a)(d-c)}\left[f\left(\frac{a+b}{2},\frac{c+d}{2}\right)-\frac{2}{d-c}{\int }_c^{\frac{c+d}{2}}f\left(\frac{a+b}{2},y\right)\text{d}y-\frac{2}{b-a}{\int }_a^{\frac{a+b}{2}}f\left(x,\frac{c+d}{2}\right)\text{d}x+\frac{4}{(b-a)(d-c)}{\int }_c^{\frac{c+d}{2}}{\int }_a^{\frac{a+b}{2}}f(x,y)\text{d}x\text{d}y\right],$$$${\int }_0^1{\int }_0^{1}t\mathit{\lambda }\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f\left(\frac{t}{2}a+\left(1-\frac{t}{2}\right)b,\left(1-\frac{\mathit{\lambda }}{2}\right)c+\frac{\mathit{\lambda }}{2}d\right)\text{d}t\text{d}\mathit{\lambda }=-\frac{4}{(b-a)(d-c)}\left[f\left(\frac{a+b}{2},\frac{c+d}{2}\right)-\frac{2}{d-c}{\int }_c^{\frac{c+d}{2}}f\left(\frac{a+b}{2},y\right)\text{d}y-\frac{2}{b-a}{\int }_{\frac{a+b}{2}}^{b}f\left(x,\frac{c+d}{2}\right)\text{d}x+\frac{4}{(b-a)(d-c)}{\int }_c^{\frac{c+d}{2}}{\int }_{\frac{a+b}{2}}^{b}f(x,y)\text{d}x\text{d}y\right],$$

and$${\int }_0^1{\int }_0^{1}t\mathit{\lambda }\frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }y}f\left(\left(1-\frac{t}{2}\right)a+\frac{t}{2}b,\frac{\mathit{\lambda }}{2}c+\left(1-\frac{\mathit{\lambda }}{2}\right)d\right)\text{d}t\text{d}\mathit{\lambda }=-\frac{4}{(b-a)(d-c)}\left[f\left(\frac{a+b}{2},\frac{c+d}{2}\right)-\frac{2}{d-c}{\int }_{\frac{c+d}{2}}^{d}f\left(\frac{a+b}{2},y\right)\text{d}y-\frac{2}{b-a}{\int }_a^{\frac{a+b}{2}}f\left(x,\frac{c+d}{2}\right)\text{d}x+\frac{4}{(b-a)(d-c)}{\int }_{\frac{c+d}{2}}^d{\int }_a^{\frac{a+b}{2}}f(x,y)\text{d}x\text{d}y\right].$$

The proof of Lemma 2.1 is complete.

Theorem 2.1.

Let $f:\mathrm{\Omega }\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a twice partial differentiable mapping on Ω^° (the interior of Ω) and let Δ: = [a, b] × [c, d] ⊆ Ω^° with a < b, c < d and $\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\in L_1(\mathrm{\Delta })$, where L₁(Δ) denotes the set of all Lebesgue integrable functions on Δ. If ${\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|}^q$ is convex on the co-ordinates on Δ and q ≥ 1, then the following inequality holds:$$\left|I(f)\right|\le \frac{1}{4}{\left(\frac{\text{1}}{\text{9}}\right)}^{1/q}\left\{g_q(1,2,2,4)+g_q(4,2,2,1)+g_q(2,1,4,2)+g_q(2,4,1,2)\right\},$$

where $f_{xy}(x,y)=\frac{{\mathit{\partial }}^{2}f\left(x,y\right)}{\mathit{\partial }x\mathit{\partial }y}$ and$$g_q(r_1,r_2,r_3,r_4)={\left[r_1{\left|f_{xy}(a,c)\right|}^q+r_2{\left|f_{xy}(a,d)\right|}^q+r_3{\left|f_{xy}(b,c)\right|}^q+r_4{\left|f_{xy}(b,d)\right|}^q\right]}^{1/q}.$$

Proof.

Using Lemma 2.1, since ${\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|}^q$ is convex on the co-ordinates on Δ and Hölder inequality, then$$\left|I(f)\right|\le {\int }_0^1{\int }_0^{1}t\mathit{\lambda }\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\left(\frac{t}{2}a+\left(1-\frac{t}{2}\right)b,\frac{\mathit{\lambda }}{2}c+\left(1-\frac{\mathit{\lambda }}{2}\right)d\right)\right|\text{d}t\text{d}\mathit{\lambda }+{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\left(\left(1-\frac{t}{2}\right)a+\frac{t}{2}b,\left(1-\frac{\mathit{\lambda }}{2}\right)c+\frac{\mathit{\lambda }}{2}d\right)\right|\text{d}t\text{d}\mathit{\lambda }+{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\left(\frac{t}{2}a+\left(1-\frac{t}{2}\right)b,\left(1-\frac{\mathit{\lambda }}{2}\right)c+\frac{\mathit{\lambda }}{2}d\right)\right|\text{d}t\text{d}\mathit{\lambda }+{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\left(\left(1-\frac{t}{2}\right)a+\frac{t}{2}b,\frac{\mathit{\lambda }}{2}c+\left(1-\frac{\mathit{\lambda }}{2}\right)d\right)\right|\text{d}t\text{d}\mathit{\lambda }\le {\left({\int }_0^1{\int }_0^{1}t\mathit{\lambda }\text{d}t\text{d}\mathit{\lambda }\right)}^{1-1/q}\left\{{\left[{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\left(\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(a,c)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,d)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(b,c)\right|}^q+\left(1-\frac{2}{t}\right)\left(1-\frac{2}{\mathit{\lambda }}\right){\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\left(\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,c)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(a,d)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,c)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\left(\left(\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,c)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(a,d)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,c)\right|}^q\right)+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^{1}t\mathit{\lambda }\left(\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(a,c)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,d)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(b,c)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}\right\}\le \frac{1}{4}{\left(\frac{1}{9}\right)}^{1/q}\left\{g_q(1,2,2,4)+g_q(4,2,2,1)+g_q(2,1,4,2)+g_q(2,4,1,2)\right\}$$

Theorem 2.1 is proved.

If taking q = 1 in Theorem 2.1, we can derive the following corollary.

Corollary 2.1.1.

Under the conditions of Theorem 2.1, when q = 1, we have$$\left|I(f)\right|\le \frac{1}{4}\left[\left|f_{xy}(a,c)\right|+\left|f_{xy}(a,d)\right|+\left|f_{xy}(b,c)\right|+\left|f_{xy}(b,d)\right|\right].$$

Theorem 2.2.

Let $f:\mathrm{\Omega }\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a twice partial differentiable mapping on Ω^° (the interior of Ω) and let Δ: = [a, b] × [c, d] ⊆ Ω^° with a < b, c < d and $\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\in L_1(\mathrm{\Delta })$, where L₁(Δ) denotes the set of all Lebesgue integrable functions on Δ. If ${\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|}^q$ is convex on the co-ordinates on Δ and q > 1, then the following inequality holds:$$\left|I(f)\right|\le \frac{1}{16}{\left(\frac{4(q-1)}{2q-1}\right)}^{2(1-1/q)}\left\{g_q(1,3,3,9)+g_q(9,3,3,1)+g_q(3,1,9,3)+g_q(3,9,1,3)\right\},$$

where g(r₁, r₂, r₃, r₄) is defined in Theorem 2.1.

Proof.

Using Lemma 2.1, and ${\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|}^q$ is convex on the co-ordinates on Δ and Hölder’s inequality, we have$$\left|I(f)\right|\le {\left({\int }_0^1{\int }_0^1{(t\mathit{\lambda })}^{q/(q-1)}\text{d}t\text{d}\mathit{\lambda }\right)}^{1-1/q}\left\{{\left[{\int }_0^1{\int }_0^1\left(\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(a,c)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,d)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(b,c)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^1\left(\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,c)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(a,d)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,c)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^1\left(\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,c)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(a,d)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,c)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^1\left(\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(a,c)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,d)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(b,c)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}\right\}=\frac{1}{16}{\left(\frac{4(q-1)}{2q-1}\right)}^{2(1-1/q)}\left\{g_q(1,3,3,9)+g_q(9,3,3,1)+g_q(3,1,9,3)+g_q(3,9,1,3)\right\}.$$

Theorem 2.2 is proved.

Theorem 2.3.

Let $f:\mathrm{\Omega }\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a twice partial differentiable mapping on Ω^° (the interior of Ω) and let Δ: = [a, b] × [c, d] ⊆ Ω^° with a < b, c < d and $\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\in L_1(\mathrm{\Delta })$, where L₁(Δ) denotes the set of all Lebesgue integrable functions on Δ. If ${\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|}^q$ is convex on the co-ordinates on Δ and q > 1, then the following inequality holds:$$\left|I(f)\right|\le \frac{q-1}{2(2q-1)}{\left(\frac{2q-1}{12(q-1)}\right)}^{1/q}\left\{g_q(1,3,2,6)+g_q(6,2,3,1)+g_q(3,1,6,2)+g_q(2,6,1,3)\right\},$$

where g(r₁, r₂, r₃, r₄) is defined in Theorem 2.1.

Proof.

By Lemma 2.1, since ${\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|}^q$ is convex on the co-ordinates on Δ and Hölder’s inequality, we get$$\left|I(f)\right|\le {\left({\int }_0^1{\int }_0^{1}t{\mathit{\lambda }}^{q/(q-1)}\text{d}t\text{d}\mathit{\lambda }\right)}^{1-1/q}\left\{{\left[{\int }_0^1{\int }_0^{1}t\left(\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(a,c)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,d)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(b,c)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^{1}t\left(\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,c)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(a,d)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,c)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^{1}t\left(\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,c)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(a,d)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,c)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^{1}t\left(\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(a,c)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,d)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(b,c)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}\right\}=\frac{q-1}{2(2q-1)}{\left(\frac{2q-1}{12(q-1)}\right)}^{1/q}\left\{g_q(1,3,2,6)+g_q(6,2,3,1)+g_q(3,1,6,2)+g_q(2,6,1,3)\right\}.$$

Theorem 2.3 is proved.

Theorem 2.4.

Let $f:\mathrm{\Omega }\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a twice partial differentiable mapping on Ω^° (the interior of Ω) and let Δ: = [a, b] × [c, d] ⊆ Ω^° with a < b, c < d and $\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\in L_1(\mathrm{\Delta })$, where L₁(Δ) denotes the set of all Lebesgue integrable functions on Δ. If ${\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|}^q$ is convex on the co-ordinates on Δ and q ≥ 1 and q ≥ r, s > 0 then the following inequality holds:$$\left|I(f)\right|\le {\left(\frac{1}{4(r+2)(s+2)}\right)}^{1/q}{\left(\frac{{(q-1)}^2}{(2q-r-1)(2q-s-1)}\right)}^{1-1/q}\times \left\{g_q\left(1,\frac{s+3}{s+1},\frac{r+3}{r+1},\frac{(r+3)(s+3)}{(r+1)(s+1)}\right)+g_q\left(\frac{(r+3)(s+3)}{(r+1)(s+1)},\frac{r+3}{r+1},\frac{s+3}{s+1},1\right)+g_q\left(\frac{s+3}{s+1},1,\frac{(r+3)(s+3)}{(r+1)(s+1)},\frac{r+3}{r+1}\right)+g_q\left(\frac{r+3}{r+1},\frac{(r+3)(s+3)}{(r+1)(s+1)},1,\frac{s+3}{s+1}\right)\right\}.$$

where g(r₁, r₂, r₃, r₄) is defined in Theorem 2.1.

Proof.

Using Lemma 2.1, and ${\left|\frac{{\mathit{\partial }}^{2}f}{\mathit{\partial }x\mathit{\partial }y}\right|}^q$ is convex on the co-ordinates on Δ and Hölder inequality, we have$$\left|I(f)\right|\le {\left({\int }_0^1{\int }_0^1{t}^{(q-r)/(q-1)}{\mathit{\lambda }}^{(q-s)/(q-1)}\text{d}t\text{d}\mathit{\lambda }\right)}^{1-1/q}\times \left\{{\left[{\int }_0^1{\int }_0^1{t}^r{\mathit{\lambda }}^s\left(\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(a,c)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,d)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(b,c)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^1{t}^r{\mathit{\lambda }}^s\left(\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,c)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(a,d)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,c)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^1{t}^r{\mathit{\lambda }}^s\left(\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,c)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(a,d)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,c)\right|}^q+\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}+{\left[{\int }_0^1{\int }_0^1{t}^r{\mathit{\lambda }}^s\left(\left(1-\frac{t}{2}\right)\frac{\mathit{\lambda }}{2}{\left|f_{xy}(a,c)\right|}^q+\left(1-\frac{t}{2}\right)\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(a,d)\right|}^q+\frac{t\mathit{\lambda }}{4}{\left|f_{xy}(b,c)\right|}^q+\frac{t}{2}\left(1-\frac{\mathit{\lambda }}{2}\right){\left|f_{xy}(b,d)\right|}^q\right)\text{d}t\text{d}\mathit{\lambda }\right]}^{1/q}\right\}\le {\left(\frac{1}{4(r+2)(s+2)}\right)}^{1/q}{\left(\frac{{(q-1)}^2}{(2q-r-1)(2q-s-1)}\right)}^{1-1/q}\times \left\{g_q\left(1,\frac{s+3}{s+1},\frac{r+3}{r+1},\frac{(r+3)(s+3)}{(r+1)(s+1)}\right)+g_q\left(\frac{(r+3)(s+3)}{(r+1)(s+1)},\frac{r+3}{r+1},\frac{s+3}{s+1},1\right)+g_q\left(\frac{s+3}{s+1},1,\frac{(r+3)(s+3)}{(r+1)(s+1)},\frac{r+3}{r+1}\right)+g_q\left(\frac{r+3}{r+1},\frac{(r+3)(s+3)}{(r+1)(s+1)},1,\frac{s+3}{s+1}\right)\right\}.$$

Theorem 2.4 is proved.

If taking r = s = q in Theorem 2.4, we can derive the following corollary.

Corollary 2.4.1.

Under the conditions of Theorem2.4, when r = s = q, we have$$\left|I(f)\right|\le {\left(\frac{1}{4(q+2)(q+2)}\right)}^{1/q}\left\{g_q\left(1,\frac{q+3}{q+1},\frac{q+3}{q+1},\frac{{(q+3)}^2}{{(q+1)}^2}\right)+g_q\left(\frac{{(q+3)}^2}{{(q+1)}^2},\frac{q+3}{q+1},\frac{q+3}{q+1},1\right)+g_q\left(\frac{q+3}{q+1},1,\frac{{(q+3)}^2}{{(q+1)}^2},\frac{q+3}{q+1}\right)+g_q\left(\frac{q+3}{q+1},\frac{{(q+3)}^2}{{(q+1)}^2},1,\frac{q+3}{q+1}\right)\right\}.$$

where g(r₁, r₂, r₃, r₄) is defined in Theorem 2.1.

Acknowledgements

The authors appreciate the anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation (NNSF) [grant number 11361038] of China; the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region [grant number NJZY14192]; NSF of Inner Mongolia Autonomous Region [grant number 2014BS0106], [grant number 2015BS0123]; the Scientific Innovation Project for Graduates at the Inner Mongolia University for Nationalities [grant number NMDSS1419] China.

Notes on contributors

Xu-Yang Guo

Xu-Yang Guo is being a graduate for master degree of science in applied mathematics at Inner Mongolia University for Nationalities. Her supervisor is the third author, Professor Bo-Yan Xi. Currently, her research interests are in the areas of mathematical inequalities and convex analysis.