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On generalized strongly modified h-convex functions

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Abstract

We derive some properties and results for a new extended class of convex functions, generalized strongly modified h-convex functions. Moreover, we discuss Schur-type, Hermite–Hadamard-type, and Fejér-type inequalities for this class. The crucial fact is that this extended class has awesome properties similar to those of convex functions.

Introduction

Nowadays, in science and modern analysis the convexity plays an important role in economics, statistics, management science, engineering, and optimization theory. For instance, Barani et al. [1] presented the Hermite–Hadamard inequality for functions with preinvex absolute values of derivatives. Characterizations of convexity via Hadamard’s inequality has been studied in [2]. In 2003, Dragomir and Pearce [3] proposed some applications of Hermite–Hadamard inequalities. In 2015, Dragomir [4] presented inequalities of Hermite–Hadamard type for h-convex functions on linear spaces. Some other interesting results can be found in books [5, 6] and research papers [7, 8]. In the recent years, generalizations and extensions were made rapidly for convex functions; for a recent generalization, see [911].

Convexity in the classical sense for a function \(g:L=[a_{1},a_{2}] \subset \mathbb{R} \rightarrow \mathbb{R} \) is defined as

$$\begin{aligned} g\bigl(ta_{1}+(1-t)a_{2}\bigr) \leq t g(a_{1})+(1-t)g(a_{2}), \end{aligned}$$

where \(a_{1},a_{2} \in L\) and \(t \in [0,1]\).

The work on the convexity is extended day by day by using some techniques; see [1214]. The strongly extended convexity is widely used in optimization, economics, and nonlinear programming.

Convex functiosn satisfy several inequalities in which famous inequalities are of Schur type, Hermite–Hadamard-type, and Fejér-type inequalities. The Hermite–Hadamard-type inequality introduced by Jaques Hadamard for classical convex functions \(g:L=[a_{1},a_{2}]\subset \mathbb{R} \rightarrow \mathbb{R} \) as

$$\begin{aligned} g \biggl(\frac{a_{1}+a_{2}}{2} \biggr)\leq \frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}}g(x)\,dx \leq \frac{g(a_{1})+g(a_{2})}{2}. \end{aligned}$$

For extended versions of this inequality, see [12] and [13]. For further reading, see [1519].

Lipot Fejér presented an extended version of the Hermite–Hadamard inequality, known as the Fejér inequality or a weighted version of the Hermite–Hadamard inequality. If \(f:I\rightarrow \mathbb{R} \) is a convex function, then

$$\begin{aligned} g \biggl(\frac{a_{1}+a_{2}}{2} \biggr) \int _{a_{1}}^{a_{2}}w(x)\,dx \leq \frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}}w(x)g(x)\,dx \leq \frac{g(a_{1})+g(a_{2})}{2} \int _{a_{1}}^{a_{2}}w(x)\,dx, \end{aligned}$$

where \(a_{1}\leq a_{2} \), and \(w:I\rightarrow \mathbb{R} \) is nonnegative, integrable, and symmetric about \(\frac{a+b}{2} \). For further extended versions and development, see [20] and [8].

In this paper, we first present some preliminaries and basic results. In the next section, we investigate Schur-type, Hermite–Hadamard-type, and Fejér-type inequalities for the newly introduced class of functions.

Preliminaries

In this section, we investigate a new class of convexity by using a basic result. There is no loss of generality in the extended version of convexity. To get asymptotic results, it is necessary to put some restrictions: L is an interval in \(\mathbb{R}\), and \(\eta : A \times A \rightarrow B \subseteq \mathbb{R}\) is a bifunction.

Definition 1

(h-convex function [21])

Let \(g,h:L\subset \mathbb{R}\rightarrow \mathbb{R}\) be nonnegative functions. Then g is called an h-convex function if

$$\begin{aligned} g \bigl(ta_{1}+(1-t)a_{2} \bigr)\leq h(t)g(a_{1})+h(1-t)g(a_{2}) \end{aligned}$$

for all \(a_{1},a_{2}\in L\) and \(t\in [0,1]\).

Definition 2

(Modified h-convex function [13])

Let \(g,h:L\subset \mathbb{R} \rightarrow \mathbb{R} \) be nonnegative functions. Then g is called a modified h-convex function if

$$\begin{aligned} g \bigl(t a_{1}+(1-t)a_{2} \bigr)\leq h(t)g(a_{1})+\bigl(1-h(t)\bigr)g(a_{2}) \end{aligned}$$
(1)

for all \(a_{1},a_{2}\in L\) and \(t\in [0,1]\).

Definition 3

(Generalized modified h-convex function)

Let functions g, h: \(J\subset \mathbb{R}\rightarrow \mathbb{R} \) be nonnegative functions. Then \(g:I\subset \mathbb{R}\rightarrow \mathbb{R}\) is called a generalized modified h-convex function if

$$\begin{aligned} g \bigl(t a_{1}+(1-t)a_{2} \bigr)\leq g(a_{2})+h(t) \eta \bigl(g(a_{1}),g(a _{2}) \bigr) \end{aligned}$$
(2)

for all \(a_{1},a_{2}\in I\) and \(t\in [0,1]\).

Definition 4

(Wright-convex function [20])

A function \(g:L\subset \mathbb{R}\rightarrow \mathbb{R} \) is said to be Wright-convex if

$$\begin{aligned} g \bigl((1-t)a_{1} +t a_{2} \bigr)+g \bigl(t a_{1}+(1-t)a_{2} \bigr) \leq g(a_{1})+g(a_{2}) \end{aligned}$$

for all \(a_{1},a_{2}\in L\) and \(t\in [0,1]\).

Definition 5

(Additivity)

A function η is said to be additive if \(\eta (x_{1},y_{1})+\eta (x_{2},y_{2}) =\eta (x_{1}+x_{2},y_{1}+y _{2})\) for all \(x_{1},x_{2},y_{1},y_{2} \in \mathbb{R} \); see [22] for more detail.

Definition 6

(Nonnegative homogeneity)

A function η is said to be nonnegatively homogeneous if \(\eta ( \lambda a_{1},\lambda a_{2} )= \lambda \eta (a_{1},a_{2})\) for all \(a_{1},a_{2} \in \mathbb{R}\) and \(\lambda \geq 0 \).

Definition 7

(Supermultiplicativity [23])

A function \(g:L\subset \mathbb{R}\rightarrow \mathbb{R_{+}} \) is said to be a supermultiplicative function if \(g(a_{1}a_{2})\geq g(a_{1}) g(a_{2})\) for all \(a_{1},a_{2} \in L \), \(t \in [0,1]\).

Definition 8

(Similar-order functions [24])

Functions f and g are said to be of similar order on \(L\subseteq \mathbb{R}\) if \(\langle f(x)-f(y),g(x)-g(y)\rangle \geqslant 0\) for all \(x,y \in L\).

Now we are going to introduce a new extended definition of convexity.

Definition 9

(Generalized strongly modified h-convex function)

Let \(g,h:L\subset \mathbb{R}\rightarrow \mathbb{R} \) be nonnegative functions. Then g is called a generalized strongly modified h-convex function if

$$\begin{aligned} g \bigl(t a_{1}+(1-t)a_{2} \bigr)\leq g(a_{2})+h(t) \eta \bigl(g(a_{1}),g(a _{2}) \bigr)-\mu t(1-t) (a_{1}-a_{2})^{2} \end{aligned}$$
(3)

for all \(a_{1},a_{2}\in L\) and \(t\in [0,1]\).

Remark 1

  1. 1.

    Inequality (3) reduces to inequality (1) if \(\mu =0\) and \(\eta (x,y)=x-y\).

  2. 2.

    Definition (9) becomes the definition of a classical convex function when \(\mu =0\), \(\eta (x,y)=x-y \), and \(h(t)=t\).

  3. 3.

    Inequality(3) reduces to inequality (2) when \(\mu =0\).

  4. 4.

    If \(h(t)=t \), then definition (9) reduces to the definition of a strongly generalized convex function [12].

Example 1

A function \(g:L=[a_{1},a_{2}] \subset \mathbb{R}\rightarrow \mathbb{R}\) is defined by \(g(x)=x^{2} \), \(\eta (a_{1},a_{2})=2a_{1}+a _{2} \), and \(h(t)\geq t \), then g is a generalized strongly modified h-convex function.

Main results

This section contains some basic and straightforward results. The following proposition shows the linearity of the class of generalized strongly modified h-convex functions.

Proposition 1

Letfandgbe generalized strongly modifiedh-convex functions whereηis additive and nonnegatively homogeneous. Then for all \(a,b \in \mathbb{R} \), \(af+bg\)is also a generalized strongly modifiedh-convex function.

Proposition 2

Let \(h_{1} \), \(h_{2} \)be nonnegative functions onLsuch that \(h_{2}(t)\leqslant h_{1}(t)\). Ifgis a generalized strongly modified \(h_{2} \)-convex function, thengis also a generalized strongly modified \(h_{1} \)-convex function.

Proof

As g is generalized strongly modified h-convex function, for all \(a_{1} ,a_{2} \in L\) and \(t \in [0,1] \), we have

$$\begin{aligned} g \bigl(t a_{1}+(1-t)a_{2} \bigr) \leq& g(a_{2})+h_{2}(t) \eta \bigl(g(a _{1}),g(a_{2}) \bigr)-\mu t(1-t) (a_{1}-a_{2})^{2} \\ \leq& g(a_{2})+h_{1}(t) \eta \bigl(g(a_{1}),g(a_{2}) \bigr)-\mu t(1-t) (a_{1}-a _{2})^{2}. \end{aligned}$$

This completes the proof. □

Remark 2

If g is a generalized strongly modified \(h_{1} \)-convex and \(h_{1} (t)\leqslant h_{2}(t)\), then g is a generalized strongly modified \(h_{2} \)-convex function.

Proposition 3

Letfbe a linear function such that \(f(x)-f(y)=x-y \), and letgbe a generalized strongly modifiedh-convex function. Then \(g\circ f \)is also a generalized strongly modifiedh-convex function.

Proof

As f is a linear function such that \(f(x)-f(y)=x-y \) and g is a generalized strongly modified h-convex function, for all \(a_{1} ,a_{2} \in L \) and \(t \in [0,1] \), we get

$$\begin{aligned} (g\circ f) \bigl(t a_{1}+(1-t)a_{2}\bigr) =&g(t f(a_{1})+(1-t)f(a_{2}) \\ \leq &(g\circ f) (a_{2})+h(t)\eta \bigl((g\circ f) (a_{1}),(g \circ f) (a _{2})\bigr) \\ &{}-\mu t(1-t) \bigl(f(a_{1})-f(a_{2}) \bigr)^{2} \\ =&(g\circ f) (a_{2})+h(t)\eta \bigl((g\circ f) (a_{1}),(g \circ f) (a_{2})\bigr) \\ &{}-\mu t(1-t) (a_{1}-a_{2})^{2} . \end{aligned}$$

This shows that \(g\circ f \) is a generalized strongly modified h-convex function. □

Proposition 4

Let functions \(g_{j} :L\subset \mathbb{R}\rightarrow \mathbb{R}\)be generalized strongly modifiedh-convex functions, \(\sum_{j=1}^{m} \lambda _{j} =1\), and letηbe additive non-negatively homogeneous function. Then their linear combination \(f:\mathbb{R}\rightarrow \mathbb{R} \)is also a generalized strongly modifiedh-convex function.

Proof

As \(g_{j}:L \subset \mathbb{R}\rightarrow \mathbb{R} \) be generalized strongly modified h-convex functions, for \(a_{1},a_{2} \in L \) and \(t\in [0,1] \), let

$$\begin{aligned} f(x)= \sum_{j=1}^{m}\lambda _{j}g_{j}(x). \end{aligned}$$

Set \(x=(t a_{1}+(1-t)a_{2})\). Then

$$\begin{aligned} f \bigl(t a_{1}+(1-t)a_{2} \bigr) =& \sum _{j=1}^{m}\lambda _{j}g_{j} \bigl(t a_{1}+(1-t)a_{2} \bigr) \\ \leq & \sum_{j=1}^{m} \lambda _{j}g_{j}(a_{2})+h(t) \sum _{j=1}^{m} \lambda _{j} \eta \bigl(g_{i}(a_{1}),g_{i}(a_{2}) \bigr) \\ &{}-\mu t(1-t) (a_{1}-a_{2})^{2} \sum _{j=1}^{m} \lambda _{j} \\ = &f(a_{2})+h(t) \eta \Biggl( \sum_{j=1}^{m} \lambda _{j}g_{i}(a_{1}), \sum _{j=1}^{m} \lambda _{j}g_{i}(a_{2}) \Biggr) \\ &{}-\mu t(1-t) (a_{1}-a_{2})^{2} \\ = &f(a_{2})+h(t) \eta \bigl( f(a_{1}),f(a_{2}) \bigr)-\mu t(1-t) (a _{1}-a_{2})^{2}. \end{aligned}$$

This completes the proof. □

Corollary 1

Every generalized strongly modifiedh-convex function is a generalized modified convex function.

Proof

Let g be a generalized modified h-convex function. Then

$$\begin{aligned} g \bigl(t a_{1}+(1-t)a_{2} \bigr) \leq & g(a_{2})+h(t) \eta \bigl(g(a_{1}),g(a_{2}) \bigr)-\mu t(1-t) (a_{1}-a_{2})^{2} \\ \leq & g(a_{2})+h(t) \eta \bigl(g(a_{1}),g(a_{2}) \bigr) \end{aligned}$$

for all \(a_{1} ,a_{2} \in L\subset \mathbb{R}\). □

Corollary 2

Ifgis generalized strongly convex function and \(t\leq h(t) \), thengis a generalized strongly modifiedh-convex function.

Theorem 1

(Schur-type inequality)

Let \(g :L\rightarrow \mathbb{R}\)be a generalized strongly modifiedh-convex function, lethbe a supermultiplicative function, and let \(\eta : N\times N\rightarrow M\)be a bifunction for appropriate \(A,B\subseteq \mathbb{R}\). Then for \(a_{1},a_{2},a_{3} \in L \)such that \(a_{1}< a_{2}< a_{3} \)and \(a_{3}-a_{1},a_{3} -a_{2},a_{2}-a_{1} \in L\), we have the inequality

$$\begin{aligned} h(a_{3}-a_{1})g(a_{2}) \leq & h(a_{3}-a_{1})g(a_{3})+h(a_{3}-a_{2}) \eta \bigl(g(a_{1}),g(a_{2})\bigr) \\ &{}-\mu (a_{3}-a_{2}) (a_{2}-a_{1})h(a_{3}-a_{1}) \end{aligned}$$
(4)

if and only ifgis a generalized strongly modifiedh-convex function.

Proof

Let \(a_{1},a_{2},a_{3} \in L\subset \mathbb{R} \) be such that \(\frac{(a_{3}-a_{2})}{(a_{3}-a_{1})} \in (0,1)\subseteq L \), \(\frac{(a _{2}-a_{1})}{(a_{3}-a_{1})} \in (0,1)\subseteq L\), and \(\frac{(a_{3}-a _{2})}{(a_{3}-a_{1})}+\frac{(a_{2}-a_{1})}{(a_{3}-a_{1})}=1\). Then

$$\begin{aligned} h(a_{3}-a_{1})=h \biggl(\frac{a_{3}-a_{1}}{a_{3}-a_{2}}({a_{3}-a_{2}}) \biggr) \geq h \biggl(\frac{a_{3}-a_{1}}{a_{3}-a_{2}} \biggr)h({a_{3}-a_{2}}) \end{aligned}$$

as h is supermultiplicative.

Suppose \(h({a_{3}-a_{2}})\geq 0\). Then by the definition of g we have

$$\begin{aligned} g \bigl(t x+(1-t)y \bigr)\leq g(y)+h(t) \eta \bigl(g(x),g(y)\bigr)-\mu t(1-t) (x-y)^{2}. \end{aligned}$$
(5)

Inserting \(\frac{(a_{3}-a_{2})}{(a_{3}-a_{1})}=t\), \(x=a_{1}\), and \(y= a_{3} \) into inequality (5), we obtain

$$\begin{aligned} \begin{aligned}&\begin{aligned} g \biggl(\frac{(a_{3}-a_{2})}{(a_{3}-a_{1})} a_{1}+ \biggl(1-\frac{(a_{3}-a _{2})}{(a_{3}-a_{1})}\biggr)a_{3} \biggr) \leq {}& g(a_{3})+h \biggl(\frac{(a _{3}-a_{2})}{(a_{3}-a_{1})} \biggr) \eta \bigl(g(a_{1}),g(a_{3})\bigr) \\ &{}- \mu (a_{3}-a_{2}) (a_{2}-a_{1}) \\ \leq{} & g(a_{3})+\frac{h(a_{3}-a_{2})}{h(a_{3}-a_{1})} \eta \bigl(g(a_{1}),g(a _{3})\bigr) \\ &{}-\mu (a_{3}-a_{2}) (a_{2}-a_{1}), \end{aligned} \\ &\begin{aligned} g(a_{2}) h(a_{3}-a_{1}) \leq {}&h(a_{3}-a_{1})g(a_{3}) \\ &{}+h(a_{3}-a_{2}) \eta \bigl(g(a_{1}),g(a_{3}) \bigr) \\ &{}- \mu (a_{3}-a_{2}) (a_{2}-a_{1})h(a_{3}-a_{1}). \end{aligned} \end{aligned} \end{aligned}$$
(6)

Conversely, suppose inequality (4) holds and insert \(a_{1}=x \), \(a_{2}=tx+(1-t)y \), and \(a_{3}=y \) into inequality (4). Then we get

$$\begin{aligned} &\begin{aligned} h(y-x)g \bigl(t x+(1-t)y \bigr) \leq {}& h(y-x)g(y)+h(y-x)h(t) \eta \bigl(g(x),g(y)\bigr) \\ &{}-\mu h(y-x)t(y-x) (1-t) (y-x), \end{aligned} \\ &g \bigl(t x+(1-t)y \bigr) \leq g(y)+h(t) \eta \bigl(g(x),g(y)\bigr)-\mu t(1-t) (x-y)^{2}. \end{aligned}$$

This completes the proof. □

Remark 3

  1. 1.

    By taking \(h(t)=t\) in (4) it is reduced to aSchur-type inequality for generalized strongly convex functions.

  2. 2.

    If \(\mu =0\) and \(\eta (x,y)=x-y\), then (4) is reduced to a Schur-type inequality for modified h-convex functions; see [13].

Further, we will discuss the Hermite–Hadamard-type inequality for generalized strongly modified h-convex functions.

Theorem 2

(Hermit–Hadamard-type inequality)

Let function \(g:L\rightarrow \mathbb{R} \)be a generalized strongly modifiedh-convex function on \([a_{1},a_{2}] \)with \(a_{1}< a_{2} \). Then

$$\begin{aligned} g \biggl(\frac{a_{1}+a_{2}}{2} \biggr)-h\biggl(\frac{1}{2}\biggr) M_{\eta }+ \frac{ \mu }{12}(a_{2}-a_{1})^{2} \leq & \frac{1}{a_{2}-a_{1}} \int _{a_{1}} ^{a_{2}}g((x)\,dx \\ \leq& g(a_{2})+N_{\eta } -\frac{\mu }{6}(a_{2}-a_{1})^{2}. \end{aligned}$$
(7)

Proof

Choosing \(w=ta_{1} +(1-t)a_{2}\) and \(z=(1-t)a_{1} +ta_{2}\), we have

$$\begin{aligned} g \biggl(\frac{a_{1}+a_{2}}{2} \biggr) =&g \biggl(\frac{w+z}{2} \biggr) \\ =& g \biggl(\frac{ta_{1} +(1-t)a_{2}+(1-t)a_{1} +ta_{2}}{2} \biggr). \end{aligned}$$

Now by the definition of g we have

$$\begin{aligned} g \biggl(\frac{a_{1}+a_{2}}{2} \biggr) \leq & g\bigl((1-t)a_{1} +ta_{2}\bigr)+h\biggl( \frac{1}{2}\biggr) \eta \bigl(g \bigl(ta_{1} +(1-t)a_{2}\bigr),g\bigl((1-t)a_{1} +ta_{2}\bigr)\bigr) \\ &{}-\mu \frac{1}{2}\biggl(1-\frac{1}{2}\biggr) (a_{2}-a_{1})^{2}(2t-1)^{2}. \end{aligned}$$

Integrating with respect to t on [0,1], we get

$$\begin{aligned} g \biggl(\frac{a_{1}+a_{2}}{2} \biggr) \leq & \int _{0}^{1}g\bigl((1-t)a _{1} +ta_{2}\bigr)\,dt \\ &{}+h\biggl(\frac{1}{2}\biggr) \int _{0}^{1}\eta \bigl(g\bigl(ta_{1} +(1-t)a_{2}\bigr),g\bigl((1-t)a _{1} +ta_{2} \bigr)\bigr)\,dt \\ &{}- \frac{\mu }{4}(a_{2}-a_{1})^{2} \int _{0}^{1}(2t-1)^{2}\,dt. \end{aligned}$$

Putting \(x=(1-t)a_{1} +ta_{2}\), we get

$$ \begin{aligned}&g \biggl(\frac{a_{1}+a_{2}}{2} \biggr) \leq \frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}}g(x)\,dx+h\biggl( \frac{1}{2}\biggr) M_{\eta } -\frac{\mu }{12}(a_{2}-a_{1})^{2}, \\ &g \biggl(\frac{a_{1}+a_{2}}{2} \biggr)-h\biggl(\frac{1}{2}\biggr) M_{\eta }+ \frac{ \mu }{12}(a_{2}-a_{1})^{2} \leq \frac{1}{a_{2}-a_{1}} \int _{a_{1}} ^{a_{2}}g((x)\,dx. \end{aligned} $$
(8)

In the right-hand side of inequality (8), we set \(x=ta_{1} +(1-t)a_{2} \), and using the definition of g, we get

$$ \begin{aligned}& \int _{a_{1}}^{a_{2}}g(x)\,dx \leq (a_{2}-a_{1})g(a_{2})+(a_{2}-a_{1}) \int _{0}^{1}h(t) \eta (g\bigl(a_{1},g(a_{2}) \bigr)\,dt -\frac{\mu }{6}(a_{2}-a_{1})^{2}, \\ &\frac{1}{(a_{2}-a_{1})} \int _{a_{1}}^{a_{2}}g(x)\,dx \leq g(a_{2})+N _{\eta } -\frac{\mu }{6}(a_{2}-a_{1})^{2}. \end{aligned} $$
(9)

Now from inequalities (8) and (9) we get

$$\begin{aligned} g \biggl(\frac{a_{1}+a_{2}}{2} \biggr)-h\biggl( \frac{1}{2}\biggr) M_{\eta }+ \frac{ \mu }{12}(a_{2}-a_{1})^{2} \leq &\frac{1}{a_{2}-a_{1}} \int _{a_{1}} ^{a_{2}}g((x)\,dx \\ \leq& g(a_{2})+N_{\eta }-\frac{\mu }{6}(a_{2}-a_{1})^{2}. \end{aligned}$$
(10)

This completes the proof. □

Remark 4

  1. 1.

    If we take \(\mu =0\) and \(\eta (x,y)=x-y\), then the Hermite–Hadamard-type inequality (10) is reduced to Hermite–Hadamard-type inequality for modified h-convex functions; for details, see [13].

  2. 2.

    If we put \(h(t)=t \) in (10), then we get a Hermite–Hadamard-type inequality for generalized strongly convex functions; see [12].

  3. 3.

    If we take \(\mu =0 \), \(\eta (x,y)=x-y \) and \(h(t)=t \), then inequality (10) is reduced to a Hemite–Hadard-type inequality for classical convex functions.

Now we prove the following lemma by using technique of [25]. This lemma has the crucial fact that generalized strongly modified h-convex functions behave like classic convex functions.

Lemma 1

Letgbe a generalized modifiedh-convex function, and suppose that \(\eta (x,y)=-\eta (y,x)\). Then

$$\begin{aligned} g(a_{1}+a_{2}-x)\leq g(a_{1})+g(a_{2})-g(x) \quad \forall x\in [a_{1},a_{2}], \end{aligned}$$

where \(x=ta_{1}+(1-t)a_{2} \)and \(t\in [0,1]\).

Proof

As g is generalized modified h-convex function, for \(x=ta_{1}+(1-t)a_{2}\), we get

$$\begin{aligned} g(a_{1}+a_{2}-x) =&g\bigl((1-t)a_{1}+ta_{2} \bigr) \\ \leq & g(a_{1})+h(t)\eta \bigl(g(a_{2}),g(a_{1}) \bigr) \\ =&g(a_{1})+g(a_{2})-g(a_{2})-h(t)\eta (g(a_{1}),g(a_{2}) \\ =&g(a_{1})+g(a_{2})-\bigl[g(a_{2})+h(t)\eta (g(a_{1}),g(a_{2})\bigr] \\ \leq &g(a_{1})+g(a_{2})-g(x). \end{aligned}$$

This completes the proof. □

Lemma 2

Letgbe q the generalized strongly modifiedh-convex function, and suppose that \(\eta (x,y)=-\eta (y,x)\). Then

$$\begin{aligned} g(a_{1}+a_{2}-x)\leq g(a_{1})+g(a_{2})-g(x) \quad \forall x\in [a_{1},a_{2}], \end{aligned}$$
(11)

where \(x=ta_{1}+(1-t)a_{2} \)and \(t\in [0,1]\).

Proof

Let g be a generalized strongly modified h-convex function. Then for \(x=ta_{1}+(1-t)a_{2}\), we get

$$\begin{aligned} g(a_{1}+a_{2}-x) =&g\bigl((1-t)a_{1}+ta_{2} \bigr) \\ \leq & g(a_{1})+h(t)\eta \bigl(g(a_{2}),g(a_{1}) \bigr)-\mu t(1-t) (a_{1}-a_{2})^{2} \\ \leq &g(a_{1})+g(a_{2})-g(a_{2})-h(t)\eta (g(a_{1}),g(a_{2}) \\ &{}-\mu t(1-t) (a_{1}-a_{2})^{2}+2\mu t(1-t) (a_{1}-a_{2})^{2} \\ \leq &g(a_{1})+g(a_{2})-\bigl[g(a_{2})+h(t) \eta (g(a_{1}),g(a_{2})-\mu t(1-t) (a _{1}-a_{2})^{2} \bigr] \\ \leq &g(a_{1})+g(a_{2})-g(x). \end{aligned}$$

This completes the proof. □

It is very interesting that when g is a modified h-convex function [13], generalized modified h-convex, or generalized strongly modified h-convex function, then inequality (11) holds.

Theorem 3

(Fejér-type inequality)

Let \(g:[a_{1},a _{2}]\rightarrow \mathbb{R} \)be a generalized strongly modifiedh-convex, and let \(w:[a_{1},a_{2}]\rightarrow \mathbb{R}\)be nonnegative, integrable, and symmetric with respect to \(\frac{a_{1}+a _{2}}{2}\). Then

$$\begin{aligned}& g \biggl(\frac{a_{1}+a_{2}}{2} \biggr) \int _{a_{1}}^{a_{2}}w(x)\,dx +\frac{ \mu }{4} \int _{a_{1}}^{a_{2}}(a_{1}+a_{2}-2x)w(x) \,dx -N_{\eta }(a_{1},a _{2}) \\& \quad \leq \int _{a_{1}}^{a_{2}}g(x)w(x)\,dx \\& \quad \leq \frac{g(a_{1})+g(a_{2})}{2} \int _{a_{1}}^{a_{2}}w(x)\,dx+T_{\eta }(a _{1},a_{2})-\mu \int _{a_{1}}^{a_{2}}(x-a_{2}) (a_{1}-x)w(x)\,dx, \end{aligned}$$
(12)

where

$$\begin{aligned}& N_{\eta }(a_{1},a_{2})=h\biggl( \frac{1}{2}\biggr) \int _{a_{1}}^{a_{2}}\eta \bigl(g(a _{1}+a_{2}-x),g(x) \bigr)w(x)\,dx, \\& T_{\eta }(a_{1},a_{2})=\frac{\eta (g(a_{1}),g(a_{2}))}{2} \int _{a_{1}} ^{a_{2}} h \biggl(\frac{x-a_{2}}{a_{1}-a_{2}} \biggr) w(x)\,dx. \end{aligned}$$

Proof

Let g be a generalized strongly modified h-convex function. Then

$$\begin{aligned}& \begin{aligned}&\begin{aligned} g \biggl(\frac{a_{1}+a_{2}}{2} \biggr) \int _{a_{1}}^{a_{2}}w(x)\,dx ={}& \int _{a_{1}}^{a_{2}}g \biggl(\frac{a_{1}+a_{2}-x+x}{2} \biggr)w(x)\,dx \\ \leq {}& \int _{a_{1}}^{a_{2}} g(x)w(x)\,dx \\ &{}+h\biggl(\frac{1}{2}\biggr) \int _{a_{1}}^{a_{2}} \eta \bigl(g(a_{1}+a_{2}-x),g(x) \bigr)w(x)\,dx \\ &{}- \int _{a_{1}}^{a_{2}} \mu \frac{1}{2}\biggl(1- \frac{1}{2}\biggr) (2x-a_{1}-a_{2})^{2}w(x) \,dx, \end{aligned} \\ &g \biggl(\frac{a_{1}+a_{2}}{2} \biggr) \int _{a_{1}}^{a_{2}}w(x)\,dx +\frac{ \mu }{4} \int _{a_{1}}^{a_{2}}(a_{1}+a_{2}-2x)^{2}w(x) \,dx-N_{\eta }(a _{1},a_{2}) \\ &\quad \leq \int _{a_{1}}^{a_{2}}g(x)w(x)\,dx. \end{aligned} \end{aligned}$$
(13)

In the right hand-side of inequality (13), put \(x=ta_{1}+(1-t)a _{2}\). Then

$$\begin{aligned}& \begin{aligned}& \int _{a_{1}}^{a_{2}}g(x)w(x)\,dx=(a_{2}-a_{1}) \int _{0}^{1}g\bigl(ta_{1}+(1-t)a _{2}\bigr)w\bigl(ta_{1}+(1-t)a_{2}\bigr)\,dt, \\ &\begin{aligned} \frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}}g(x)w(x)\,dx \leq {}& \int _{0} ^{1}g(a_{2})w \bigl(ta_{1}+(1-t)a_{2}\bigr)\,dt \\ &{} +\eta \bigl(g(a_{1}),g(a_{2})\bigr) \int _{0}^{1}h(t)w\bigl(ta_{1}+(1-t)a_{2} \bigr)\,dt \\ &{}-\mu (a_{2}-a_{1})^{2} \int _{0}^{1}t(1-t)w\bigl(ta_{1}+(1-t)a_{2} \bigr)\,dt. \end{aligned} \end{aligned} \end{aligned}$$
(14)

Similarly, if we put \(x=ta_{2}+(1-t)a_{1} \) in the right-hand side of inequality (13), then we get the inequality

$$\begin{aligned} \frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}}g(x)w(x)\,dx \leq & \int _{0} ^{1}g(a_{1})w \bigl(ta_{2}+(1-t)a_{1}\bigr)\,dt \\ &{}+\eta \bigl(g(a_{2}),g(a_{1})\bigr) \int _{0}^{1}h(t)w\bigl(ta_{2}+(1-t)a_{1} \bigr)\,dt \\ &{}-\mu (a_{2}-a_{1})^{2} \int _{0}^{1}t(1-t)w\bigl(ta_{2}+(1-t)a_{1} \bigr)\,dt. \end{aligned}$$
(15)

Adding inequalities (14) and (15), where w is symmetric, we get

$$\begin{aligned}& \frac{2}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}}g(x)w(x)\,dx \\& \quad \leq \bigl(g(a _{1})+g(a_{2}) \bigr) \int _{0}^{1}w\bigl(ta_{1}+(1-t)a_{2} \bigr)\,dt \\& \qquad{} + \bigl[\eta \bigl(g(a_{1}),g(a_{2})\bigr)+\eta \bigl(g(a_{2}),g(a_{1})\bigr) \bigr] \int _{0}^{1}h(t)w\bigl(ta_{1}+(1-t)a_{2} \bigr)\,dt \\& \qquad {}-2\mu (a_{2}-a_{1})^{2} \int _{0}^{1}t(1-t)w\bigl(ta_{1}+(1-t)a_{2} \bigr)\,dt. \end{aligned}$$
(16)

Putting \(x=ta_{1}+(1-t)a_{2}\) in the right-hand side of inequality (16), we have

$$\begin{aligned}& \begin{aligned}&\begin{aligned} \int _{a_{1}}^{a_{2}}g(x)w(x)\,dx \leq {}& \frac{ (g(a_{1})+g(a_{2}) )}{2} \int _{a_{1}}^{a_{2}}w(x)\,dx \\ & {}+ \frac{ [\eta (g(a_{1}),g(a_{2}))+\eta (g(a_{2}),g(a_{1})) ]}{2} \int _{a_{1}}^{a_{2}}h(\frac{x-a_{2}}{a_{1}-a_{2}}w(x)\,dx \\ &{}-\mu \int _{a_{1}}^{a_{2}}(x-a_{2}) (a_{1}-x)w(x)\,dx, \end{aligned} \\ &\begin{aligned} \int _{a_{1}}^{a_{2}}g(x)w(x)\,dx \leq {}& \frac{ (g(a_{1})+g(a_{2}) )}{2} \int _{a_{1}}^{a_{2}}w(x)\,dx+ T_{\eta }(a_{1},a_{2}) \\ -{}&\mu \int _{a_{1}}^{a_{2}}(x-a_{2}) (a_{1}-x)w(x)\,dx. \end{aligned} \end{aligned} \end{aligned}$$
(17)

Now from inequalities (13) and (17) we get Fejér-type inequality (12) for generalized strongly modified h-convex functions. □

Remark 5

  1. 1.

    If \(h(t)=t \), then inequality (12) reduced to Fejér type inequality for generalized strongly convex functions, see [12].

  2. 2.

    If we put \(\mu =0 \) and \(\eta (x,y)=x-y\) then inequality (12) becomes a Fejér-type inequality for modified h-convex functions; see [13].

  3. 3.

    If we put \(\mu =0\), \(\eta (x,y)=x-y\), and \(h(t)=t \), then inequality (12) is reduced to a Fejér-type inequality for classical convex functions.

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Zhao, T., Saleem, M.S., Nazeer, W. et al. On generalized strongly modified h-convex functions. J Inequal Appl 2020, 11 (2020). https://doi.org/10.1186/s13660-020-2281-6

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Keywords

  • h-convex function
  • Modified h-convex function
  • Schur-type inequality
  • Hermite–Hadamard inequality
  • Fejér-type inequality