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A note on generalized convex functions

  • Syed Zaheer Ullah
  • Muhammad Adil Khan
  • Yu-Ming ChuEmail author
Open Access
Research

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.

Keywords

Convex function Coordinate convex function η-convex function Coordinate \((\eta _{1}, \eta _{2})\)-convex function 

MSC

26D15 26A51 39B62 

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.

Notes

Availability of data and materials

Not applicable.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Funding

This work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485).

Competing interests

The authors declare that they have no competing interests.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PeshawarPeshawarPakistan
  2. 2.Department of MathematicsHuzhou UniversityHuzhouChina
  3. 3.School of Mathematics and StatisticsChangsha University of Science and TechnologyChangshaChina

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