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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.

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.

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 \(\frac{1}{4} [M_{\eta _{1}}+M_{\eta _{2}} ]\). □

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.

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.

References

  1. 1.

    Guessab, A., Schmeisser, G.: Sharp integral inequalities of the Hermite–Hadamard type. J. Approx. Theory 115(2), 260–288 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Shi, H.-P., Zhang, H.-Q.: Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J. Math. Anal. Appl. 361(2), 411–419 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Zhou, W.-J., Zhang, L.: Convergence of a regularized factorized quasi-Newton method for nonlinear least squares problems. Comput. Appl. Math. 29(2), 195–214 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Yang, X.-S., Zhu, Q.-X., Huang, C.-X.: Generalized lag-synchronization of chaotic mix-delayed systems with uncertain parameters and unknown perturbations. Nonlinear Anal., Real World Appl. 12(1), 93–105 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Zhu, Q.-X., Huang, C.-X., Yang, X.-S.: Exponential stability for stochastic jumping BAM neural networks with time-varying and distributed delays. Nonlinear Anal. Hybrid Syst. 5(1), 52–77 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Dai, Z.-F., Wen, F.-H.: A modified CG-DESCENT method for unconstrained optimization. J. Comput. Appl. Math. 235(11), 3332–3341 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Gou, K., Sun, B.: Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions. Appl. Math. Comput. 217(21), 8765–8777 (2011)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Lin, L., Liu, Z.-Y.: An alternating projected gradient algorithm for nonnegative matrix factorization. Appl. Math. Comput. 217(24), 9997–10002 (2011)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Zhang, L., Li, J.-L.: A new globalization technique for nonlinear conjugate gradient methods for nonconvex minimization. Appl. Math. Comput. 217(24), 10295–10304 (2011)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Xiao, C.-E., Liu, J.-B., Liu, Y.-L.: An inverse pollution problem in porous media. Appl. Math. Comput. 218(7), 3649–3653 (2011)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Huang, C.-X., Liu, L.-Z.: Sharp function inequalities and boundness for Toeplitz type operator related to general fractional singular integral operator. Publ. Inst. Math. 92(106), 165–176 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Zhou, W.-J.: On the convergence of the modified Levenberg–Marquardt method with a nonmonotone second order Armijo type line search. J. Comput. Appl. Math. 239, 152–161 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Zhang, L., Jian, S.-Y.: Further studies on the Wei–Yao–Liu nonlinear conjugate gradient method. Appl. Math. Comput. 219(14), 7616–7621 (2013)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Li, X.-F., Tang, G.-J., Tang, B.-Q.: Stress field around a strike-slip fault in orthotropic elastic layers via a hypersingular integral equation. Comput. Math. Appl. 66(11), 2317–2326 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Qin, G.-X., Huang, C.-X., Xie, Y.-Q., Wen, F.-H.: Asymptotic behavior for third-order quasi-linear differential equations. Adv. Differ. Equ. 2013, Article ID 305 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Zhou, W.-J., Chen, X.-L.: On the convergence of a modified regularized Newton method for convex optimization with singular solutions. J. Comput. Appl. Math. 239, 179–188 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Wang, M.-K., Chu, Y.-M.: Asymptotical bounds for complete elliptic integrals of the second kind. J. Math. Anal. Appl. 402(1), 119–126 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A power mean inequality involving the complete elliptic integrals. Rocky Mt. J. Math. 44(5), 1661–1667 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Wang, M.-K., Chu, Y.-M., Jiang, Y.-P.: Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mt. J. Math. 46(2), 679–691 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl. 2017, Article ID 106 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Article ID 210 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler–Mascheroni constant. J. Inequal. Appl. 2018, Article ID 118 (2018)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Huang, T.-R., Tan, S.-Y., Ma, X.-Y., Chu, Y.-M.: Monotonicity properties and bounds for the complete p-elliptic integrals. J. Inequal. Appl. 2018, Article ID 239 (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Yang, Z.-H., Qian, W.-M., Chu, Y.-M.: Monotonicity properties and bounds involving the complete elliptic integrals of the first kind. Math. Inequal. Appl. 21(4), 1185–1199 (2018)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Qiu, S.-L., Ma, X.-Y., Chu, Y.-M.: Sharp Landen transformation inequalities for hypergeometric functions, with applications. J. Math. Anal. Appl. 474(2), 1306–1337 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Wang, M.-K., Chu, Y.-M., Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 22(2), 601–617 (2019)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Wang, M.-K., Chu, Y.-M., Zhang, W.: Precise estimates for the solution of Ramanujan’s generalized modular equation. Ramanujan J. 49(3), 653–668 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Wang, M.-K., Zhang, W., Chu, Y.-M.: Monotonicity, convexity and inequalities involving the generalized elliptic integrals. Acta Math. Sci. 39B(5), 1440–1450 (2019)

    Google Scholar 

  30. 30.

    Wang, M.-K., Chu, H.-H., Chu, Y.-M.: Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals. J. Math. Anal. Appl. https://doi.org/10.1016/j.jmaa.2019.123388

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Huang, C.-X., Yang, Z.-C., Yi, T.-S., Zou, X.-F.: On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differ. Equ. 256(7), 2101–2114 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Tang, W.-S., Sun, Y.-J.: Construction of Runge–Kutta type methods for solving ordinary differential equations. Appl. Math. Comput. 234, 179–191 (2014)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Huang, C.-X., Guo, S., Liu, L.-Z.: Boundedness on Morrey space for Toeplitz type operator associated to singular integral operator with variable Calderón–Zygmund kernel. J. Math. Inequal. 8(3), 453–464 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Xie, D.-X., Li, J.: A new analysis of electrostatic free energy minimization and Poisson–Boltzmann equation for protein in ionic solvent. Nonlinear Anal., Real World Appl. 21, 185–196 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Zhou, W.-J., Wang, F.: A PRP-based residual method for large-scale monotone nonlinear equations. Appl. Math. Comput. 261, 1–7 (2015)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Dai, Z.-F., Chen, X.-H., Wen, F.-H.: A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations. Appl. Math. Comput. 270, 378–386 (2015)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Liu, Y.-C., Wu, J.: Multiple solutions of ordinary differential systems with min-max terms and applications to the fuzzy differential equations. Adv. Differ. Equ. 2015, Article ID 379 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Fang, X.-P., Deng, Y.-J., Li, J.: Plasmon resonance and heat generation in nanostructures. Math. Methods Appl. Sci. 38(18), 4663–4672 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Dai, Z.-F.: Comments on a new class of nonlinear conjugate gradient coefficients with global convergence properties. Appl. Math. Comput. 276, 297–300 (2016)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Li, J.-L., Sun, G.-Y., Zhang, R.-M.: The numerical solution of scattering by infinite rough interfaces based on the integral equation method. Comput. Math. Appl. 71(7), 1491–1502 (2016)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Tan, Y.-X., Jing, K.: Existence and global exponential stability of almost periodic solution for delayed competitive neural networks with discontinuous activations. Math. Methods Appl. Sci. 39(11), 2821–2839 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Duan, L., Huang, C.-X.: Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model. Math. Methods Appl. Sci. 40(3), 814–822 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Duan, L., Huang, L.-H., Guo, Z.-Y., Fang, X.-W.: Periodic attractor for reaction-diffusion high-order Hopfield neural networks with time-varying delays. Comput. Math. Appl. 73(2), 233–245 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Wang, W.-S., Chen, Y.-Z.: Fast numerical valuation of options with jump under Merton’s model. J. Comput. Appl. Math. 318, 79–92 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Huang, C.-X., Liu, L.-Z.: Boundedness of multilinear singular integral operator with a non-smooth kernel and mean oscillation. Quaest. Math. 40(3), 295–312 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Hu, H.-J., Liu, L.-Z.: Weighted inequalities for a general commutator associated to a singular integral operator satisfying a variant of Hörmander’s condition. Math. Notes 101(5–6), 830–840 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Generalized Lyapunov–Razumikhin method for retarded differential inclusions: applications to discontinuous neural networks. Discrete Contin. Dyn. Syst. 22B(9), 3591–3614 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Wang, W.-S.: On A-stable one-leg methods for solving nonlinear Volterra functional differential equations. Appl. Math. Comput. 314, 380–390 (2017)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Hu, H.-J., Zou, X.-F.: Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc. Am. Math. Soc. 145(11), 4763–4771 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Tan, Y.-X., Huang, C.-X., Sun, B., Wang, T.: Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition. J. Math. Anal. Appl. 458(2), 1115–1130 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Tang, W.-S., Zhang, J.-J.: Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods. Appl. Math. Comput. 323, 204–219 (2018)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Duan, L., Fang, X.-W., Huang, C.-X.: Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting. Math. Methods Appl. Sci. 41(5), 1954–1965 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Liu, Z.-Y., Wu, N.-C., Qin, X.-R., Zhang, Y.-L.: Trigonometric transform splitting methods for real symmetric Toeplitz systems. Comput. Math. Appl. 75(8), 2782–2794 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Huang, C.-X., Qiao, Y.-C., Huang, L.-H., Agarwal, R.P.: Dynamical behaviors of a food-chain model with stage structure and time delays. Adv. Differ. Equ. 2018, Article ID 186 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Periodic orbit analysis for the delayed Filippov system. Proc. Am. Math. Soc. 146(11), 4667–4682 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Wang, J.-F., Chen, X.-Y., Huang, L.-H.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J. Math. Anal. Appl. 469(1), 405–427 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Wang, J.-F., Huang, C.-X., Huang, L.-H.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear Anal. Hybrid Syst. 33, 162–178 (2019)

    MathSciNet  Article  Google Scholar 

  58. 58.

    Jiang, Y.-J., Xu, X.-J.: A monotone finite volume method for time fractional Fokker–Planck equations. Sci. China Math. 62(4), 783–794 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Peng, J., Zhang, Y.: Heron triangles with figurate number sides. Acta Math. Hung. 157(2), 478–488 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Tian, Z.-L., Liu, Y., Zhang, Y., Liu, Z.-Y., Tian, M.-Y.: The general inner-outer iteration method based on regular splittings for the PageRank problem. Appl. Math. Comput. 356, 479–501 (2019)

    MathSciNet  Google Scholar 

  61. 61.

    Wang, W.-S., Chen, Y.-Z., Fang, H.: On the variable two-step IMEX BDF method for parabolic integro-differential equations with nonsmooth initial data arising in finance. SIAM J. Numer. Anal. 57(3), 1289–1317 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  62. 62.

    Chu, Y.-M., Wang, M.-K., Qiu, S.-L.: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41–51 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  63. 63.

    Zhao, T.-H., Zhou, B.-C., Wang, M.-K., Chu, Y.-M.: On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, Article ID 42 (2019)

    MathSciNet  Article  Google Scholar 

  64. 64.

    Wang, J.-L., Qian, W.-M., He, Z.-Y., Chu, Y.-M.: On approximating the Toader mean by other bivariate means. J. Funct. Spaces 2019, Article ID 6082413 (2019)

    MathSciNet  MATH  Google Scholar 

  65. 65.

    Qian, W.-M., Xu, H.-Z., Chu, Y.-M.: Improvements of bounds for the Sándor–Yang means. J. Inequal. Appl. 2019, Article ID 73 (2019)

    Article  Google Scholar 

  66. 66.

    He, X.-H., Qian, W.-M., Xu, H.-Z., Chu, Y.-M.: Sharp power mean bounds for two Sándor–Yang means. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2627–2638 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  67. 67.

    Qian, W.-M., He, Z.-Y., Zhang, H.-W., Chu, Y.-M.: Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean. J. Inequal. Appl. 2019, Article ID 168 (2019)

    MathSciNet  Article  Google Scholar 

  68. 68.

    Chu, Y.-M., Wang, G.-D., Zhang, X.-H.: The Schur multiplicative and harmonic convexities of the complete symmetric function. Math. Nachr. 284(5–6), 653–663 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  69. 69.

    Chu, Y.-M., Xia, W.-F., Zhang, X.-H.: The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications. J. Multivar. Anal. 105, 412–421 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  70. 70.

    Wu, S.-H., Chu, Y.-M.: Schur m-power convexity of generalized geometric Bonferroni mean involving three parameters. J. Inequal. Appl. 2019, Article ID 57 (2019)

    MathSciNet  Article  Google Scholar 

  71. 71.

    Zhang, X.-M., Chu, Y.-M., Zhang, X.-H.: The Hermite–Hadamard type inequality of GA-convex functions and its applications. J. Inequal. Appl. 2010, Article ID 507560 (2010)

    MathSciNet  MATH  Google Scholar 

  72. 72.

    Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Conformable integral inequalities of the Hermite–Hadamard type in terms of GG- and GA-convexities. J. Funct. Spaces 2019, Article ID 6926107 (2019)

    MathSciNet  MATH  Google Scholar 

  73. 73.

    Adil Khan, M., Chu, Y.-M., Khan, T.U., Khan, J.: Some new inequalities of Hermite–Hadamard type for s-convex functions with applications. Open Math. 15(1), 1414–1430 (2017)

    MathSciNet  MATH  Google Scholar 

  74. 74.

    Adil Khan, M., Hanif, M., Khan, Z.A., Ahmad, K., Chu, Y.-M.: Association of Jensen’s inequality for s-convex function with Csiszár divergence. J. Inequal. Appl. 2019, Article ID 162 (2019)

    Article  Google Scholar 

  75. 75.

    Khurshid, Y., Adil Khan, M., Chu, Y.-M., Khan, Z.A.: Hermite–Hadamard–Fejér inequalities for conformable fractional integrals via preinvex functions. J. Funct. Spaces 2019, Article ID 3146210 (2019)

    MATH  Google Scholar 

  76. 76.

    Song, Y.-Q., Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: Integral inequalities involving strongly convex functions. J. Funct. Spaces 2018, Article ID 6595921 (2018)

    MathSciNet  MATH  Google Scholar 

  77. 77.

    Zaheer Ullah, S., Adil Khan, M., Chu, Y.-M.: Majorization theorems for strongly convex functions. J. Inequal. Appl. 2019, Article ID 58 (2019)

    MathSciNet  Article  Google Scholar 

  78. 78.

    Zaheer Ullah, S., Adil Khan, M., Khan, Z.A., Chu, Y.-M.: Integral majorization type inequalities for the functions in the sense of strong convexity. J. Funct. Spaces 2019, Article ID 9487823 (2019)

    MathSciNet  MATH  Google Scholar 

  79. 79.

    Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: The concept of coordinate strongly convex functions and related inequalities. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2235–2251 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  80. 80.

    Chu, Y.-M., Adil Khan, M., Ali, T., Dragomir, S.S.: Inequalities for α-fractional differentiable functions. J. Inequal. Appl. 2017, Article ID 93 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  81. 81.

    Adil Khan, M., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, Article ID 70 (2018)

    MathSciNet  Article  Google Scholar 

  82. 82.

    Adil Khan, M., Chu, Y.-M., Kashuri, A., Liko, R., Ali, G.: Conformable fractional integrals versions of Hermite–Hadamard inequalities and their generalizations. J. Funct. Spaces 2018, Article ID 6928130 (2018)

    MathSciNet  MATH  Google Scholar 

  83. 83.

    Adil Khan, M., Iqbal, A., Suleman, M., Chu, Y.-M.: Hermite–Hadamard type inequalities for fractional integrals via Green’s function. J. Inequal. Appl. 2018, Article ID 161 (2018)

    MathSciNet  Article  Google Scholar 

  84. 84.

    Adil Khan, M., Khurshid, Y., Du, T.-S., Chu, Y.-M.: Generalization of Hermite–Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces 2018, Article ID 5357463 (2018)

    MathSciNet  MATH  Google Scholar 

  85. 85.

    Adik Khan, M., Wu, S.-H., Ullah, H., Chu, Y.-M.: Discrete majorization type inequalities for convex functions on rectangles. J. Inequal. Appl. 2019, Article ID 16 (2019)

    MathSciNet  Article  Google Scholar 

  86. 86.

    Dragomir, S.S.: On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwan. J. Math. 5(4), 775–788 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  87. 87.

    Delavar, M.R., Dragomir, S.S.: On η-convexity. Math. Inequal. Appl. 20(1), 203–216 (2017)

    MathSciNet  MATH  Google Scholar 

  88. 88.

    Eshaghi Gordji, M., Rostamian Delavar, M., Dragomir, S.S.: Some inequalities related to η-convex functions. Available at http://www.ajmaa.org/RGMIA/papers/v18/v18a08.pdf

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

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Zaheer Ullah, S., Adil Khan, M. & Chu, Y. A note on generalized convex functions. J Inequal Appl 2019, 291 (2019). https://doi.org/10.1186/s13660-019-2242-0

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MSC

  • 26D15
  • 26A51
  • 39B62

Keywords

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