A note on generalized convex functions

Abstract

In the article, we provide an example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex and its converse is not true in general, define the coordinate $(\eta _{1}, \eta _{2})$-convex function and establish its Hermite–Hadamard type inequality.

1 Introduction

Let $I\subseteq \mathbb{R}$ be an interval. Then a real-valued function $\varPsi : I\mapsto \mathbb{R}$ is said to be convex on I if the inequality

$$ \varPsi \bigl[\lambda a+(1-\lambda )b\bigr]\leq \lambda \varPsi (a)+(1-\lambda ) \varPsi (b) $$

(1.1)

holds for all $a, b\in I$ and $\lambda \in (0, 1)$. Ψ is said to be concave if inequality (1.1) is reversed.

It is well known that the convexity theory has wide applications in special functions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], differential equations [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61] and bivariate means [62,63,64,65,66,67]. Recently, the extensions, generalizations, refinements and variants for the convexity have attracted the attention of many researchers. For example, Schur convexity [68,69,70], GA-convexity [71], GG-convexity [72], s-convexity [73, 74], preinvexity [75], strong convexity [76,77,78,79] and others [80,81,82,83,84,85].

Dragomir [86] defined the coordinate convex as follows.

Definition 1.1

(See [86])

Let $I_{1}, I_{2}\subseteq \mathbb{R}$ be two interval, $\varPsi : I_{1}\times I_{2}\mapsto \mathbb{R}$ be a real-valued function, and the partial mappings $\varPsi _{y}: I_{1}\mapsto \mathbb{R}$ and $\varPsi _{x}: I_{2}\mapsto \mathbb{R}$ be defined by

$$ \varPsi _{y}(u)=\varPsi (u, y), \qquad \varPsi _{x}(v)=\varPsi (x, v), $$

respectively. Then Ψ is said to be coordinate convex on $I_{1}\times I_{2}$ if $\varPsi _{y}$ is convex on $I_{1}$ for all $y\in I_{2}$ and $\varPsi _{x}$ is convex on $I_{2}$ for all $x\in I_{1}$.

Remark 1.2

Dragomir [86] proved that every convex function is coordinate convex, but not vice versa.

Next, we recall the concept of η-convexity which can be found in the literature [87].

Definition 1.3

(See [87])

Let $I\subseteq \mathbb{R}$ be an interval, and $\varPsi : I\mapsto \mathbb{R}$ and $\eta : \mathbb{R} \times \mathbb{R}\mapsto \mathbb{R}$ be two real-valued functions. Then Ψ is said to be η-convex if the inequality

$$ \varPsi \bigl[\mu x+(1-\mu )y\bigr]\leq \varPsi (y)+\mu \eta \bigl[\varPsi (x), \varPsi (y) \bigr] $$

holds for all $x, y\in I$ and $\mu \in [0, 1]$.

Note that the η-convexity reduces to the usual convexity if $\eta (x, y)=x-y$ in Definition 1.3.

The main purpose of the article is to give a non-trivial example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex but not vice versa, define the coordinate $(\eta _{1}, \eta _{2})$-convex function and establish its Hermite–Hadamard type inequality.

2 Main results

To begin this section, it is interesting to give the definition of η-convex function defined on rectangle, and give a non-trivial example for a η-convex function defined on rectangle is not convex.

Definition 2.1

Let $I_{1}, I_{2}\subseteq \mathbb{R}$ be two intervals, and $\varPsi : I_{1}\times I_{2}\mapsto \mathbb{R}$ and $\eta : \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ be two real-valued functions. Then Ψ is said to be η-convex if the inequality

$$ \varPsi \bigl[\mu x+(1-\mu )z, \mu y+(1-\mu )w\bigr]\leq \varPsi (z, w)+\mu \eta \bigl[ \varPsi (x,y), \varPsi (z, w)\bigr] $$

holds for all $(x, y), (z, w)\in I_{1}\times I_{2}$ and $\mu \in [0, 1]$.

Example 2.2

Let $\varPsi : [1,5]\times [1,5]\mapsto \mathbb{R}$ and $\eta : \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ be defined by

$$ \varPsi (x, y)=x^{2}y^{2}, \qquad \eta (x, y)=104x+103y. $$

Then Ψ is η-convex on $[1,5]\times [1,5]$, but it is not convex.

Proof

Let $\mu \in [0, 1]$. Then for any $(x, y), (z, w)\in [1,5]$ we have

$$\begin{aligned}& \varPsi \bigl[\mu x+(1-\mu )z,\mu y+(1-\mu )w\bigr] \\& \quad =\bigl[\mu x+(1-\mu )z\bigr]^{2}\bigl[\mu y+(1-\mu )w \bigr]^{2} \\& \quad =\bigl[z^{2}+\mu \bigl(\mu x^{2}+\mu z^{2}-2z^{2}\bigr)+2\mu (1-\mu )xz\bigr] \\& \qquad {}\times \bigl[w^{2}+\mu \bigl(\mu y^{2}+\mu w^{2}-2w^{2}\bigr)+2\mu (1-\mu )yw\bigr] \\& \quad \leq \bigl[z^{2}+\mu x^{2}+2\mu (1-\mu )xz\bigr] \bigl[w^{2}+\mu y^{2}+2\mu (1-\mu )yw\bigr] \\& \quad \leq \bigl[z^{2}+\mu x^{2}+\mu (1-\mu ) \bigl(x^{2}+z^{2}\bigr)\bigr] \bigl[w^{2}+\mu y^{2}+ \mu (1-\mu ) \bigl(y^{2}+w^{2}\bigr) \bigr] \\& \quad \leq \bigl[z^{2}+\mu \bigl(x^{2}+x^{2}+z^{2} \bigr)\bigr] \bigl[w^{2}+\mu \bigl(y^{2}+y^{2}+w^{2} \bigr)\bigr] \\& \quad =z^{2}w^{2}+\mu \bigl[2y^{2}z^{2}+z^{2}w^{2}+2x^{2}w^{2}+w^{2}z^{2} \bigr]+\mu ^{2}\bigl[4x^{2}y^{2}+2x^{2}w^{2}+2y^{2}z^{2}+z^{2}w^{2} \bigr] \\& \quad \leq \varPsi (z,w)+\mu \bigl[2y^{2}z^{2}+z^{2}w^{2}+2x^{2}w^{2}+w^{2}z^{2} \bigr]+ \mu \bigl[4x^{2}y^{2}+2x^{2}w^{2}+2y^{2}z^{2}+z^{2}w^{2} \bigr] \\& \quad =\varPsi (z,w)+\mu \bigl[4x^{2}y^{2}+3z^{2}w^{2}+4 \bigl(z^{2}y^{2}+x^{2}w^{2}\bigr) \bigr]. \end{aligned}$$

(2.1)

Note that

$$ z^{2}\leq 25x^{2}, \qquad x^{2}\leq 25z^{2}. $$

(2.2)

It follows from (2.1) and (2.2) that

$$\begin{aligned}& \varPsi \bigl[\mu x+(1-\mu )z,\mu y+(1-\mu )w\bigr] \\& \quad \leq \varPsi (z,w)+\mu \bigl[104x^{2}y^{2}+103z^{2}w^{2} \bigr] \\& \quad =\varPsi (z,w)+\mu \eta \bigl[\varPsi (x,y),\varPsi (z,w)\bigr], \end{aligned}$$

which shows that Ψ is η-convex on $[1,5]\times [1,5]$. It is easily to verify that Ψ is not convex on $[1,5]\times [1,5]$, for details see [79]. □

Next, we introduce the definition of coordinate $(\eta _{1}, \eta _{2})$-convexity.

Definition 2.3

Let $I_{1}, I_{2}\subseteq \mathbb{R}$ be two intervals, $\varPsi : I_{1}\times I_{2}\mapsto \mathbb{R}$, $\eta _{1}, \eta _{2}: \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ be three real-valued functions, and the partial mappings $\varPsi _{y}: I_{1} \mapsto \mathbb{R}$ and $\varPsi _{x}: I_{2}\mapsto \mathbb{R}$ be defined by

$$ \varPsi _{y}(u)=\varPsi (u, y), \qquad \varPsi _{x}(v)=\varPsi (x, v). $$

Then Ψ is said to be coordinate $(\eta _{1}, \eta _{2})$-convex on $I_{1}\times I_{2}$ if $\varPsi _{y}$ is $\eta _{1}$-convex on $I_{1}$ and $\varPsi _{x}$ is $\eta _{2}$-convex on $I_{2}$. In particular, if $\eta _{1}=\eta _{2}=\eta$, then Ψ is said to be coordinate η-convex.

Example 2.4

Let $\varPsi : [0, \infty )\times [0, \infty ) \mapsto \mathbb{R}$ be defined by $\varPsi (x, y)=-|x|-y^{2}$, $\eta _{1}(x, y)=-x-y$ and $\eta _{2}(x, y)=-x-2y$. Then Ψ is coordinate $(\eta _{1}, \eta _{2})$-convex on $[0, \infty )\times [0, \infty )$.

Proof

Let $x_{1}, y_{1}\in [0, \infty )$ and $\mu \in [0, 1]$. Then for any $(x, y)\in [0, \infty )$ we clearly see that

$$\begin{aligned}& \varPsi _{y}\bigl(\mu x_{1}+(1-\mu )x_{2}\bigr)=- \bigl\vert \mu x_{1}+(1-\mu )x_{2} \bigr\vert -y^{2}, \end{aligned}$$

(2.3)

$$\begin{aligned}& \varPsi _{y}(x_{2})+\mu \eta _{1}\bigl(\varPsi _{y}(x_{1}), \varPsi _{y}(x_{2}) \bigr) \\& \quad =- \vert x_{2} \vert -y^{2}+\mu \eta _{1}\bigl(- \vert x_{1} \vert -y^{2}, - \vert x_{2} \vert -y^{2}\bigr) \\& \quad =- \vert x_{2} \vert -y^{2}+\mu \bigl( \vert x_{1} \vert + \vert x_{2} \vert +2y^{2} \bigr), \end{aligned}$$

(2.4)

$$\begin{aligned}& \varPsi _{x}\bigl(\mu y_{1}+(1-\mu )y_{2} \bigr)=- \vert x \vert -\bigl(\mu y_{1}+(1-\mu )y_{2} \bigr)^{2}, \end{aligned}$$

(2.5)

$$\begin{aligned}& \varPsi _{x}(y_{2})+\mu \eta _{2}\bigl(\varPsi _{x}(y_{1}), \varPsi _{x}(y_{2}) \bigr) \\& \quad =- \vert x \vert -y_{2}^{2}+\mu \eta _{2}\bigl(- \vert x \vert -y_{1}^{2}, - \vert x \vert -y_{2}^{2}\bigr) \\& \quad=- \vert x \vert -y_{2}^{2}+\mu \bigl(y_{1}^{2}+2y_{2}^{2}+3 \vert x \vert \bigr). \end{aligned}$$

(2.6)

It follows from (2.3)–(2.6) that

$$\begin{aligned}& \varPsi _{y}(x_{2})+\mu \eta _{1}\bigl(\varPsi _{y}(x_{1}), \varPsi _{y}(x_{2}) \bigr)-\varPsi _{y}\bigl(\mu x_{1}+(1-\mu )x_{2} \bigr) \\& \quad =\mu \bigl( \vert x_{1} \vert + \vert x_{2} \vert +2y^{2}\bigr)+ \bigl\vert \mu x_{1}+(1-\mu )x_{2} \bigr\vert - \vert x_{2} \vert \\& \quad \geq 2\mu y^{2}+\mu \vert x_{1} \vert +\mu \vert x_{2} \vert +(1-\mu ) \vert x_{2} \vert -\mu \vert x_{1} \vert - \vert x _{2} \vert \\& \quad =2\mu y^{2}\geq 0, \end{aligned}$$

(2.7)

$$\begin{aligned}& \varPsi _{x}(y_{2})+\mu \eta _{2}\bigl(\varPsi _{x}(y_{1}), \varPsi _{x}(y_{2}) \bigr)-\varPsi _{x}\bigl(\mu y_{1}+(1-\mu )y_{2} \bigr) \\& \quad =3\mu \vert x \vert +2\mu (1-\mu )y_{1}y_{2}+ \mu (1+\mu )y_{1}^{2}+\mu ^{2}y_{2} ^{2}\geq 0. \end{aligned}$$

(2.8)

Therefore, Ψ is coordinate $(\eta _{1}, \eta _{2})$-convex on $[0, \infty )\times [0, \infty )$ follows from (2.7) and (2.8). □

Theorem 2.5

Let $I_{1}, I_{2}\subseteq \mathbb{R}$be two interval and $\eta : \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$be a real-valued function. ThenΨis coordinateη-convex on $I_{1}\times I_{2}$ifΨisη-convex on $I_{1}\times I _{2}$.

Proof

Let $(x,y)\in I_{1}\times I_{2}$, $u, v\in I_{2}$ and $z, w\in I_{1}$. Then it follows from the η-convexity of the function Ψ on $I_{1}\times I_{2}$ that

$$\begin{aligned} \varPsi _{x}\bigl(\mu v+(1-\mu )u\bigr) =&\varPsi \bigl(x, \mu v+(1-\mu )u\bigr) \\ =&\varPsi \bigl(\mu x+(1-\mu )x, \mu v+(1-\mu )u\bigr) \\ \leq& \varPsi (x, u)+\mu \eta \bigl(\varPsi (x, v), \varPsi (x, u)\bigr) \\ =&\varPsi _{x}(u)+\mu \eta \bigl(\varPsi _{x}(v), \varPsi _{x}(u)\bigr) \end{aligned}$$

(2.9)

and

$$\begin{aligned} \varPsi _{y}\bigl(\mu z+(1-\mu )w\bigr) =&\varPsi \bigl(\mu z+(1-\mu )w, y \bigr) \\ =&\varPsi \bigl(\mu z+(1-\mu )w, \mu y+(1-\mu )y\bigr) \\ \leq &\varPsi (w, y)+\mu \eta \bigl(\varPsi (z, y), \varPsi (w, y)\bigr) \\ =&\varPsi _{y}(w)+\mu \eta \bigl(\varPsi _{y}(z), \varPsi _{y}(w)\bigr). \end{aligned}$$

(2.10)

Inequalities (2.9) and (2.10) imply that $\varPsi _{x}$ is η-convex on $I_{2}$ and $\varPsi _{y}$ is η-convex on $I_{1}$. Therefore, Ψ is coordinate η-convex on $I_{1}\times I_{2}$. □

Example 2.6

Let $I_{1}=I_{2}=[0, \infty )$, $\varPsi , \eta : I_{1}\times I_{2}\mapsto [0, \infty )$ be defined by

$$ \varPsi (x, y)=xy, \qquad \eta (x, y)=x+y. $$

(2.11)

Then Ψ is coordinate η-convex on $I_{1}\times I_{2}$ but it is not η-convex on $I_{1}\times I_{2}$.

Proof

Let $x, y, u, v, z, w\in [0, \infty )$ and $\mu \in [0, 1]$. Then it follows from (2.11) that

$$\begin{aligned}& \varPsi _{x}\bigl(\mu u+(1-\mu )v\bigr)=\varPsi \bigl(x, \mu u+(1-\mu )v \bigr) \\& \hphantom{\varPsi _{x}\bigl(\mu u+(1-\mu )v\bigr)} =x\bigl(\mu u+(1-\mu )v\bigr)=-\mu xv+x(\mu u+v), \end{aligned}$$

(2.12)

$$\begin{aligned}& \varPsi (x, v)+\mu \eta \bigl(\varPsi (x, u), \varPsi (x, v)\bigr)=xv+\mu \eta (xu, xv) \\& \hphantom{\varPsi (x, v)+\mu \eta \bigl(\varPsi (x, u), \varPsi (x, v)\bigr)} =xv+\mu (xu+xv)=\mu xv+x(\mu u+v), \end{aligned}$$

(2.13)

$$\begin{aligned}& \varPsi _{y}\bigl(\mu z+(1-\mu )w\bigr)=\varPsi \bigl(\mu z+(1-\mu )w, y \bigr) \\& \hphantom{\varPsi _{y}\bigl(\mu z+(1-\mu )w\bigr)} =y\bigl(\mu z+(1-\mu )w\bigr)=-\mu yw+y(\mu z+w), \end{aligned}$$

(2.14)

$$\begin{aligned}& \varPsi (w, y)+\mu \eta \bigl(\varPsi (z, y), \varPsi (w, y)\bigr)=wy+\mu \eta (zy, wy) \\& \hphantom{\varPsi (w, y)+\mu \eta \bigl(\varPsi (z, y), \varPsi (w, y)\bigr)} =wy+\mu (zy+wy)=\mu yw+y(\mu z+w). \end{aligned}$$

(2.15)

Inequalities (2.12)–(2.15) imply that

$$ \varPsi _{x}\bigl(\mu u+(1-\mu )v\bigr)\leq \varPsi (x, v)+\mu \eta \bigl(\varPsi (x, u), \varPsi (x, v)\bigr) $$

(2.16)

and

$$ \varPsi _{y}\bigl(\mu z+(1-\mu )w\bigr)\leq \varPsi (w, y)+\mu \eta \bigl(\varPsi (z, y), \varPsi (w, y)\bigr). $$

(2.17)

Note that

$$ \varPsi _{x}\bigl(\mu u+(1-\mu )v\bigr)=\varPsi \bigl(\mu x+(1-\mu )x, \mu u+(1-\mu )v\bigr) $$

(2.18)

and

$$ \varPsi _{y}\bigl(\mu z+(1-\mu )w\bigr)=\varPsi \bigl(\mu z+(1-\mu )w, \mu y+(1-\mu )y\bigr). $$

(2.19)

Therefore, Ψ is coordinate η-convex on $I_{1}\times I_{2}$ follows from (2.16)–(2.19).

Next, we prove that Ψ is not η-convex on $I_{1}\times I _{2}$.

Let $\mu \in (0, 1)$, $x=w=1$ and $y=z=0$. Then (2.11) leads to

$$\begin{aligned}& \varPsi \bigl(\mu x+(1-\mu )z, \mu y+(1-\mu )w\bigr) \\& \quad =\varPsi (\mu , 1-\mu )=\mu (1-\mu )>0, \end{aligned}$$

(2.20)

$$\begin{aligned}& \varPsi (z, w)+\mu \eta \bigl(\varPsi (x, y), \varPsi (z, w)\bigr) \\& \quad =\varPsi (0, 1)+\mu \eta \bigl(\varPsi (1, 0), \varPsi (0, 1)\bigr)=0. \end{aligned}$$

(2.21)

From (2.20) and (2.21) we clearly see that Ψ is not η-convex on $I_{1}\times I_{2}$. □

Next, we establish a Hermite–Hadamard type inequality for the coordinate $(\eta _{1}, \eta _{2})$-convex function.

Theorem 2.7

Let $a, b, c, d\in \mathbb{R}$with $a< b$and $c< d$, $\varPsi : [a, b]\times [c, d]\mapsto \mathbb{R}$, $\eta _{1}, \eta _{2}: \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$be three real-valued functions such thatΨis coordinate $(\eta _{1}, \eta _{2})$-convex on $[a, b]\times [c, d]$and

$$ \eta _{1}(x, y)\leq M_{\eta _{1}}, \qquad \eta _{2}(x, y)\leq M_{\eta _{2}} $$

for all $x, y\in \mathbb{R}$, where $M_{\eta _{1}}$and $M_{\eta _{2}}$are two positive constants. Then

$$\begin{aligned}& \begin{gathered}[b] \varPsi \biggl(\frac{a+b}{2}, \frac{c+d}{2} \biggr)- \frac{M_{\eta _{1}}+M _{\eta _{2}}}{2} \\ \quad \leq \frac{1}{2} \biggl[\frac{1}{b-a} \int _{a}^{b}\varPsi \biggl(x, \frac{c+d}{2} \biggr)\,dx +\frac{1}{d-c} \int _{c}^{d}\varPsi \biggl(\frac{a+b}{2}, y \biggr)\,dy \biggr]-\frac{M _{\eta _{1}}+M_{\eta _{2}}}{4} \\ \quad \leq \frac{1}{(b-a)(d-c)} \int _{c}^{d} \int _{a}^{b}\varPsi (x, y)\,dx\,dy \\ \quad \leq \frac{1}{4} \biggl[\frac{1}{b-a} \int _{a}^{b} \bigl(\varPsi (x, c)+ \varPsi (x, d) \bigr)\,dx+ \frac{1}{d-c} \int _{c}^{d} \bigl(\varPsi (a, y)+ \varPsi (b, y) \bigr)\,dy \biggr]\\ \qquad {}+\frac{M_{\eta _{1}}+M_{\eta _{2}}}{4} \\ \quad \leq \frac{1}{4} \bigl[\varPsi (a, c)+\varPsi (b, c)+\varPsi (a, d)+\varPsi (b, d) \bigr]+ \frac{5}{4} [M_{\eta _{1}}+M_{\eta _{2}} ]. \end{gathered} \end{aligned}$$

(2.22)

Proof

For any fixed $x\in [a, b]$, $\varPsi _{x}(y)=\varPsi (x, y)$ is $\eta _{2}$-convex on $[c, d]$ due to Ψ is coordinate $(\eta _{1}, \eta _{2})$-convex on $[a, b]\times [c, d]$. It follows from [77, Theorem 5] that

$$ \varPsi \biggl(x, \frac{c+d}{2} \biggr)- \frac{M_{\eta _{2}}}{2}\leq \frac{1}{d-c} \int _{c}^{d}\varPsi (x, y)\,dy \leq \frac{\varPsi (x, c)+\varPsi (x, d)}{2}+\frac{M_{\eta _{2}}}{2}. $$

(2.23)

Integrating each side of inequality (2.23) with respect to the variable x on $[a, b]$ leads to

$$\begin{aligned}& \frac{1}{b-a} \int _{a}^{b}\varPsi \biggl(x, \frac{c+d}{2} \biggr)\,dx-\frac{M _{\eta _{2}}}{2} \\& \quad \leq \frac{1}{(b-a)(d-c)} \int _{c}^{d} \int _{a}^{b}\varPsi (x, y)\,dx\,dy \\& \quad \leq \frac{1}{2(b-a)} \int _{a}^{b} \bigl[\varPsi (x, c)+\varPsi (x, d) \bigr]\,dx+\frac{M _{\eta _{2}}}{2}. \end{aligned}$$

(2.24)

By similar arguments we have

$$\begin{aligned}& \frac{1}{d-c} \int _{c}^{d}\varPsi \biggl(\frac{a+b}{2}, y \biggr)\,dy-\frac{M _{\eta _{1}}}{2} \\& \quad \leq \frac{1}{(b-a)(d-c)} \int _{c}^{d} \int _{a}^{b}\varPsi (x, y)\,dx\,dy \\& \quad \leq \frac{1}{2(d-c)} \int _{c}^{d} \bigl[\varPsi (a, y)+\varPsi (b, y) \bigr]\,dy+\frac{M _{\eta _{1}}}{2}. \end{aligned}$$

(2.25)

Adding (2.24) and (2.25) we get the second and third inequalities of (2.22).

Making use of the $(\eta _{1}, \eta _{2})$-convexity of the function Ψ on $[a, b]\times [c, d]$ and [88, Theorem 5] again we get

$$\begin{aligned}& \varPsi \biggl(\frac{a+b}{2}, \frac{c+d}{2} \biggr)- \frac{M_{\eta _{2}}}{2}\leq \frac{1}{b-a} \int _{a}^{b}\varPsi \biggl(x, \frac{c+d}{2} \biggr)\,dx, \end{aligned}$$

(2.26)

$$\begin{aligned}& \varPsi \biggl(\frac{a+b}{2}, \frac{c+d}{2} \biggr)- \frac{M_{\eta _{1}}}{2}\leq \frac{1}{d-c} \int _{c}^{d}\varPsi \biggl(\frac{a+b}{2}, y \biggr)\,dy, \end{aligned}$$

(2.27)

$$\begin{aligned}& \frac{1}{b-a} \int _{a}^{b}\varPsi (x, c)\,dx\leq \frac{\varPsi (a, c)+\varPsi (b, c)}{2}+\frac{M_{\eta _{2}}}{2}, \end{aligned}$$

(2.28)

$$\begin{aligned}& \frac{1}{b-a} \int _{a}^{b}\varPsi (x, d)\,dx\leq \frac{\varPsi (a, d)+\varPsi (b, d)}{2}+\frac{M_{\eta _{2}}}{2}, \end{aligned}$$

(2.29)

$$\begin{aligned}& \frac{1}{d-c} \int _{c}^{d}\varPsi (a, y)\,dy\leq \frac{\varPsi (a, c)+\varPsi (a, d)}{2}+\frac{M_{\eta _{1}}}{2}, \end{aligned}$$

(2.30)

$$\begin{aligned}& \frac{1}{d-c} \int _{c}^{d}\varPsi (b, y)\,dy\leq \frac{\varPsi (b, c)+\varPsi (b, d)}{2}+\frac{M_{\eta _{1}}}{2}. \end{aligned}$$

(2.31)

Therefore, the first inequality of (2.22) follows from (2.26) and (2.27) with adding $-\frac{1}{2}M_{\eta _{2}}$ and $-\frac{1}{2}M_{\eta _{1}}$ respectively, and the last inequality in (2.22) can be derived from (2.28)–(2.31) immediately, with adding $\frac{1}{4} [M_{\eta _{1}}+M_{\eta _{2}} ]$. □

3 Results and discussion

In the article, we establish a non-trivial example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex and its converse is not true in general. Furthermore, we define a new class of function which is coordinate $(\eta _{1}, \eta _{2})$-convex function and prove its well-known Hermite–Hadamard type inequality.

4 Conclusion

We find an example for η-convex function defined on rectangle is not convex. The authors define a coordinate $(\eta _{1}, \eta _{2})$-convex function and prove its results. Our approach may have further applications in the theory of η-convexity.