1 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 [9–11].

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 [12–14]. 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 [15–19].

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.

2 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.

3 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.