Skip to main content

Strongly \((\eta ,\omega )\)-convex functions with nonnegative modulus

Abstract

We introduce a new class of functions called strongly \((\eta,\omega)\)-convex functions. This class of functions generalizes some recently introduced notions of convexity, namely, the η-convex functions and strongly η-convex functions. We also establish inequalities of the Hermite–Hadamard–Fejér’s type, which generalize results of Delavar and Dragomir (Math. Inequal. Appl. 20(1):203–216, 2017) and Awan et al. (Filomat 31(18):5783–5790, 2017). In addition, we obtain some new results for this class of functions. Finally, we apply our results to the k-Riemann–Liouville fractional integral operators to obtain more results in this direction.

Introduction

The field of mathematical inequalities, derived from different families of convexity, has been a booming area in recent times. The literature is replete with plethora of such results. The theory of inequalities, especially integral inequalities, has found its place in many areas of mathematical sciences. For instance, it is generally known that there are functions whose integrals cannot be computed analytically, but estimates of such integrals would suffice. Hence an inequality is desired in this case. With the help of convexity the Jensen, Jensen–Steffensen, Slater, Favard, Berwald, Fejér, Hermite–Hadamard inequalities, and their generalizations have all been established. In this work, we concern ourselves with the Fejér and Hermite–Hadamard inequalities.

We start our discussion by collating the following foundational definition and results.

Definition 1

([3])

A function \(F:J\rightarrow \mathbb{R}\) is said to be η-convex with respect to \(\eta :\mathbb{R}\times \mathbb{R}\to \mathbb{R}\) if the following inequality holds:

$$ F\bigl(\tau x+(1-\tau )y\bigr)\leq F(y)+\tau \eta \bigl(F(x), F(y)\bigr) $$

for all \(x, y\in J\) and \(\tau \in [0,1]\).

We recapture the classical definition of convexity if the bifunction \(\eta (x,y)=x-y\). Recently, Delavar and Dragomir [1] obtained the following theorems for the class of η-convex functions.

Theorem 2

([1])

Let\(F:[\alpha ,\beta ]\to \mathbb{R}\)with\(\alpha <\beta \). Suppose that the functionFsatisfies the following conditions:

  1. (a)

    Fisη-convex andηbounded above on\(F([\alpha ,\beta ])\times F([\alpha ,\beta ])\);

  2. (b)

    \(F\in L_{\infty }([\alpha , \beta ])\).

Then we have the following inequalities:

$$ \begin{aligned}[b] &F \biggl(\frac{\alpha +\beta }{2} \biggr)- \frac{K_{\eta }}{2} \\ &\quad \leq \frac{1}{\beta -\alpha } \int ^{\beta }_{\alpha } F(u)\,du\\ &\quad \leq \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{4}+ \frac{F(\alpha )+F(\beta )}{2} \\ &\quad \leq \frac{F(\alpha )+F(\beta )}{2}+\frac{K_{\eta }}{2}, \end{aligned} $$
(1)

where\(K_{\eta }\)is an upper bound of η.

Theorem 3

([1])

Let\(F:[\alpha ,\beta ]\to \mathbb{R}\)and\(G:(\alpha ,\beta )\to [0,\infty )\)with\(\alpha <\beta \). Suppose that the functionsFandGsatisfy the following conditions:

  1. (a).

    Fisη-convex;

  2. (b).

    \(F\in L_{\infty }([\alpha , \beta ])\);

  3. (c).

    \(G\in L_{1}((\alpha , \beta ))\);

  4. (d).

    \(G(\alpha +\beta -u)=G(u)\)for all\(u\in (\alpha ,\beta )\).

Then we have the following inequality:

$$ \begin{aligned}[b] & \int ^{\beta }_{\alpha } F(u)G(u)\,du\\ &\quad \leq \biggl( \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{2(\beta -\alpha )} \biggr) \int _{\alpha }^{\beta }(\beta -u)G(u)\,du\\ &\quad\quad{} + \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{\beta }G(u) \,du. \end{aligned} $$
(2)

Definition 1 was further generalized by Awan et al.

Definition 4

([2])

A function \(F:J\subset \mathbb{R}\to \mathbb{R}\) is said to be strongly η-convex with respect to \(\eta :\mathbb{R}\times \mathbb{R}\to \mathbb{R}\) and modulus \(\mu \geq 0\) if

$$\begin{aligned} F\bigl(\tau x+(1-\tau )y\bigr)\leq F(y)+\tau \eta \bigl(F(x), F(y)\bigr)-\mu \tau (1- \tau ) (x-y)^{2} \end{aligned}$$

for all \(x, y\in J\) and \(\tau \in [0,1]\).

Example 5

The function \(F(x)=x^{2}\) is strongly η-convex with respect to the function \(\eta (x,y)=2x+y\) and modulus \(\mu =1\).

For related and recent results associated with the η-convex functions, we refer the interested reader to the papers [410] and the references therein.

The authors in [2] proved the following result.

Theorem 6

([2])

Let\(F:[\alpha ,\beta ]\to \mathbb{R}\)with\(\alpha <\beta \). Suppose that the functionFsatisfies the following conditions:

  1. (a)

    Fis stronglyη-convex with respect to modulus\(\mu \geq 0\)andηbounded above on\(F([\alpha ,\beta ])\times F([\alpha ,\beta ])\);

  2. (b)

    \(F\in L_{\infty }([\alpha , \beta ])\).

Then we have the following inequalities:

$$ \begin{aligned}[b] &F \biggl(\frac{\alpha +\beta }{2} \biggr)- \frac{K_{\eta }}{2}+ \frac{\mu }{12}(\beta -\alpha )^{2} \\ &\quad \leq \frac{1}{\beta -\alpha } \int ^{\beta }_{\alpha } F(u)\,du\\ &\quad \leq \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{4}+ \frac{F(\alpha )+F(\beta )}{2}-\frac{\mu }{6}(\beta - \alpha )^{2} \\ &\quad \leq \frac{F(\alpha )+F(\beta )}{2}+\frac{K_{\eta }}{2}- \frac{\mu }{6}(\beta - \alpha )^{2}, \end{aligned} $$
(3)

where\(K_{\eta }\)is an upper bound of η.

Stimulated by the above-mentioned work, we aim to achieve the following goals:

  1. 1.

    to introduce a new class of functions in Sect. 2, which generalizes preexisting notions of convexity;

  2. 2.

    to extend Theorems 3 and 6 to this new class of functions (see Sect. 3) and then apply the results obtained thereafter to the k-Riemann–Liouville fractional integrals;

  3. 3.

    finally, to establish many new integral inequalities in this direction.

A new class of convexity

We now introduce a new definition as a generalization of Definition 4.

Definition 7

A function \(F:J\subset \mathbb{R}\to \mathbb{R}\) is said to be strongly \((\eta ,\omega )\)-convex with respect to \(\eta :\mathbb{R}\times \mathbb{R}\to \mathbb{R}\), \(\omega :[0,1]\rightarrow [0,\infty )\), and modulus \(\mu \geq 0\) if

$$\begin{aligned} F\bigl(\tau x+(1-\tau )y\bigr)\leq F(y)+\omega (\tau )\eta \bigl(F(x), F(y)\bigr)-\mu \tau (1-\tau ) (x-y)^{2} \end{aligned}$$
(4)

for all \(x, y\in J\) and \(\tau \in [0,1]\).

Evidently, by taking \(\omega (\tau )=\tau \) we get Definition 4. Substituting \(\tau =0\) into (4), we obtain

$$ \omega (0)\eta \bigl(F(x), F(y)\bigr)\geq 0. $$

For \(\tau =1\), we get

$$ \omega (1)\eta \bigl(F(x), F(y)\bigr)\geq F(x)-F(y). $$

If, in addition, we set \(x=y\) in (4), then we obtain

$$ \omega (\tau )\eta \bigl(F(x), F(x)\bigr)\geq 0. $$

We now present an example of a strongly \((\eta ,\omega )\)-convex function.

Example 8

Let \(F(x)=x^{2}\). The function F is strongly \((\eta ,\omega )\)-convex with respect to the bifunction \(\eta (x, y)=2x+y\), \(\omega (\tau )=\tau \), and modulus \(\mu =1\). To see this, let \(\tau \in [0, 1]\). Then

$$\begin{aligned} &F(y)+\omega (\tau )\eta \bigl(F(x), F(y)\bigr)-\mu \tau (1-\tau ) (x-y)^{2} \\ &\quad = y^{2}+\tau \bigl(2x^{2}+y^{2}\bigr)- \tau (1- \tau ) (y-x)^{2} \\ &\quad =\tau ^{2}x^{2}+2xy\tau (1-\tau )+(1-\tau )^{2}y^{2}+ \tau \bigl(x^{2}+2y^{2}\bigr) \\ &\quad \geq \tau ^{2}x^{2}+2xy\tau (1-\tau )+(1-\tau )^{2}y^{2} \\ &\quad =F\bigl(\tau x+(1-\tau )y\bigr). \end{aligned}$$

We wrap up this section by showing, by means of the next example, that the class of strongly \((\eta ,\omega )\)-convex functions is wider than the class of strongly η-convex functions.

Example 9

The function \(F(x)=\sqrt{x}\) defined on \([0, 1]\) is strongly \((\eta ,\omega )\)-convex with respect to \(\eta (p,q)=\sqrt{ \vert p^{2}-q^{2} \vert }\) (\(p,q\in \mathbb{R}\)), \(\omega (\tau )=\sqrt{\tau }\) (\(\tau \in [0, 1]\)), and \(\mu =0\). To prove this claim, let \(x,y,\tau \in [0, 1]\). Then

$$\begin{aligned} &F(y)+\omega (\tau )\eta \bigl(F(x), F(y)\bigr)-\mu \tau (1-\tau ) (x-y)^{2} \\ &\quad = \sqrt{y}+\sqrt{\tau }\sqrt{ \vert x-y \vert } \\ &\quad \geq \sqrt{y+\tau (x-y)} \\ &\quad =F\bigl(\tau x+(1-\tau )y\bigr). \end{aligned}$$

Next, we argue that there are no \(\eta :[0,1]\times [0, 1]\to \mathbb{R}\) and \(\mu \geq 0\) for which F is strongly η-convex. We prove this by contradiction. Suppose there are \(\eta :[0,1]\times [0, 1]\to \mathbb{R}\) and \(\mu \geq 0\) such that F is strongly η-convex. Then for all \(x,y\in [0, 1]\),

$$ \sqrt{\tau x+(1-\tau )y}\leq \sqrt{y}+\tau \eta (\sqrt{x},\sqrt{y})- \mu \tau (1-\tau ) (x-y)^{2},\quad \tau \in [0, 1]. $$

Let \(x>0\) and \(y=0\). We get

$$ \sqrt{\tau x}\leq \tau \eta (\sqrt{x},0)-\mu \tau (1-\tau )x^{2}, \quad \tau \in [0, 1]. $$

This implies

$$ \sqrt{x}\leq \sqrt{\tau }\eta (\sqrt{x},0)-\mu \sqrt{\tau }(1-\tau )x^{2}, \quad \tau \in (0, 1]. $$

Taking limit as \(\tau \to 0^{+}\), we obtain \(x=0\), contradicting the fact that \(x>0\). Therefore our claim is justified.

Main results

We break this section into three subsections. We start by presenting Hermite–Hadamard–Fejér-type results and give an application to the k-Riemann–Liouville fractional integral. Thereafter, we conclude by establishing three more theorems for the class of \((\eta ,\omega )\)-convex functions.

Inequalities of the Hermite–Hadamard–Fejér type

Theorem 10

Let\(F:[\alpha ,\beta ]\to \mathbb{R}\)and\(G:(\alpha ,\beta )\to [0,\infty )\)with\(\alpha <\beta \). Suppose that the functionsFandGsatisfy the following conditions:

  1. (a)

    Fis strongly\((\eta ,\omega )\)-convex with modulus\(\mu \geq 0\), ηbounded above on\(F([\alpha ,\beta ])\times F([\alpha ,\beta ])\), and\(\omega \in L_{\infty }([0, 1])\);

  2. (b)

    \(F\in L_{\infty }([\alpha , \beta ])\);

  3. (c)

    \(G\in L_{1}((\alpha , \beta ))\);

  4. (d)

    \(G(\alpha +\beta -u)=G(u)\)for all\(u\in (\alpha ,\beta )\).

Then we have the following inequalities:

$$ \begin{aligned}[b] & \biggl(F \biggl(\frac{\alpha +\beta }{2} \biggr)-\omega \biggl( \frac{1}{2} \biggr)K_{\eta } \biggr) \int ^{\beta }_{\alpha } G(u)\,du+ \frac{\mu }{4} \int ^{\beta }_{\alpha } (2u-\alpha -\beta )^{2}G(u) \,du\\ &\quad \leq \int ^{\beta }_{\alpha } F(u)G(u)\,du\\ &\quad \leq \biggl( \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{2} \biggr) \int _{\alpha }^{\beta }\omega \biggl( \frac{\beta -u}{\beta -\alpha } \biggr)G(u)\,du\\ &\quad\quad{} + \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{\beta }G(u) \,du-\mu \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u)G(u)\,du\\ &\quad \leq \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{\beta }G(u) \,du+K_{\eta } \int _{\alpha }^{\beta }\omega \biggl( \frac{\beta -u}{\beta -\alpha } \biggr)G(u)\,du\\ &\quad\quad{} -\mu \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u)G(u)\,du, \end{aligned} $$
(5)

where\(K_{\eta }\)is an upper bound of η.

Proof

For all \(\tau \in [0, 1]\), we have

$$ F \biggl(\frac{\alpha +\beta }{2} \biggr)=F \biggl[\frac{1}{2} \biggl( \frac{\alpha +\beta +\tau (\beta -\alpha )}{2} \biggr)+ \biggl(1- \frac{1}{2} \biggr) \biggl( \frac{\alpha +\beta -\tau (\beta -\alpha )}{2} \biggr) \biggr]. $$

Since F is strongly \((\eta ,\omega )\)-convex, we obtain

$$ \begin{aligned}[b] F \biggl(\frac{\alpha +\beta }{2} \biggr)&\leq \omega \biggl( \frac{1}{2} \biggr)\eta \biggl(F \biggl( \frac{\alpha +\beta +\tau (\beta -\alpha )}{2} \biggr), F \biggl( \frac{\alpha +\beta -\tau (\beta -\alpha )}{2} \biggr) \biggr) \\ & \quad{} +F \biggl(\frac{\alpha +\beta -\tau (\beta -\alpha )}{2} \biggr)- \frac{\mu \tau ^{2}}{4}(\beta -\alpha )^{2} \end{aligned} $$
(6)

for all \(\tau \in [0, 1]\). Since \(K_{\eta }\) is an upper bound of η, we get

$$ F \biggl(\frac{\alpha +\beta }{2} \biggr)\leq \omega \biggl( \frac{1}{2} \biggr)K_{\eta }+F \biggl( \frac{\alpha +\beta -\tau (\beta -\alpha )}{2} \biggr)- \frac{\mu \tau ^{2}}{4}( \beta -\alpha )^{2}, \quad \tau \in [0, 1], $$

that is,

$$ F \biggl(\frac{\alpha +\beta }{2} \biggr)-\omega \biggl( \frac{1}{2} \biggr)K_{\eta }\leq F \biggl( \frac{\alpha +\beta -\tau (\beta -\alpha )}{2} \biggr)- \frac{\mu \tau ^{2}}{4}(\beta -\alpha )^{2} $$
(7)

for all \(\tau \in [0, 1]\). Similarly, we can also write

$$ F \biggl(\frac{\alpha +\beta }{2} \biggr)=F \biggl[\frac{1}{2} \biggl( \frac{\alpha +\beta -\tau (\beta -\alpha )}{2} \biggr)+ \biggl(1- \frac{1}{2} \biggr) \biggl( \frac{\alpha +\beta +\tau (\beta -\alpha )}{2} \biggr) \biggr]. $$

From this inequality we get

$$ F \biggl(\frac{\alpha +\beta }{2} \biggr)-\omega \biggl( \frac{1}{2} \biggr)K_{\eta }\leq F \biggl( \frac{\alpha +\beta +\tau (\beta -\alpha )}{2} \biggr)- \frac{\mu \tau ^{2}}{4}(\beta -\alpha )^{2} $$
(8)

for all \(\tau \in [0, 1]\). Adding (7) and (8), we obtain the following inequality for \(\tau \in [0, 1]\):

$$ \begin{aligned}[b] F \biggl(\frac{\alpha +\beta }{2} \biggr)- \omega \biggl(\frac{1}{2} \biggr)K_{\eta }&\leq \frac{1}{2}F \biggl( \frac{\alpha +\beta -\tau (\beta -\alpha )}{2} \biggr)+\frac{1}{2}F \biggl(\frac{\alpha +\beta +\tau (\beta -\alpha )}{2} \biggr) \\ & \quad{} -\frac{\mu \tau ^{2}}{4}(\beta -\alpha )^{2}. \end{aligned} $$
(9)

Multiplying (9) by \(G (\frac{\alpha +\beta +\tau (\beta -\alpha )}{2} )\), integrating over \((0, 1)\) with respect to the variable τ, and using item (d) and a change of variable, we get

$$ \begin{aligned} & \biggl(F \biggl(\frac{\alpha +\beta }{2} \biggr)-\omega \biggl( \frac{1}{2} \biggr)K_{\eta } \biggr)\frac{2}{\beta -\alpha } \int _{ \frac{\alpha +\beta }{2}}^{\beta } G(u)\,du\\ &\quad \leq \frac{1}{\beta -\alpha } \int ^{\frac{\alpha +\beta }{2}}_{ \alpha } F(u)G(u)\,du+ \frac{1}{\beta -\alpha } \int _{ \frac{\alpha +\beta }{2}}^{\beta } F(u)G(u)\,du\\ & \quad\quad{} -\frac{\mu }{2}\frac{1}{\beta -\alpha } \int _{ \frac{\alpha +\beta }{2}}^{\beta } (2u-\alpha -\beta )^{2}G(u) \,du. \end{aligned} $$

This implies that

$$ \begin{aligned}[b] &2 \biggl(F \biggl( \frac{\alpha +\beta }{2} \biggr)-\omega \biggl( \frac{1}{2} \biggr)K_{\eta } \biggr) \int _{\frac{\alpha +\beta }{2}}^{ \beta } G(u)\,du\\ &\quad \leq \int ^{\beta }_{\alpha } F(u)G(u)\,du-\frac{\mu }{2} \int _{ \frac{\alpha +\beta }{2}}^{\beta } (2u-\alpha -\beta )^{2}G(u) \,du. \end{aligned} $$
(10)

Multiplying again (9) by \(G (\frac{\alpha +\beta -\tau (\beta -\alpha )}{2} )\) and proceeding as before, we get

$$ \begin{aligned}[b] &2 \biggl(F \biggl( \frac{\alpha +\beta }{2} \biggr)-\omega \biggl( \frac{1}{2} \biggr)K_{\eta } \biggr) \int ^{\frac{\alpha +\beta }{2}}_{ \alpha } G(u)\,du\\ &\quad \leq \int ^{\beta }_{\alpha } F(u)G(u)\,du-\frac{\mu }{2} \int ^{ \frac{\alpha +\beta }{2}}_{\alpha } (2u-\alpha -\beta )^{2}G(u) \,du. \end{aligned} $$
(11)

Adding (10) and (11) gives

$$ \begin{aligned}[b] & \biggl(F \biggl(\frac{\alpha +\beta }{2} \biggr)-\omega \biggl( \frac{1}{2} \biggr)K_{\eta } \biggr) \int ^{\beta }_{\alpha } G(u)\,du\\ &\quad \leq \int ^{\beta }_{\alpha } F(u)G(u)\,du-\frac{\mu }{4} \int ^{ \beta }_{\alpha } (2u-\alpha -\beta )^{2}G(u) \,du, \end{aligned} $$
(12)

which gives the first inequality.

Next, we prove the second inequality. For this, let u be any element in \([\alpha ,\beta ]\). Then u can be expressed as

$$ u=\tau \alpha + (1-\tau )\beta \quad \text{with } \tau = \frac{\beta -u}{\beta -\alpha }. $$

Using the strong \((\eta ,\omega )\)-convexity of F, we obtain

$$ F(u)\leq F(\beta )+\omega \biggl(\frac{\beta -u}{\beta -\alpha } \biggr)\eta \bigl(F(\alpha ),F(\beta ) \bigr)-\mu (u-\alpha ) ( \beta -u). $$

Multiplying this inequality by \(G(u)\) and integrating over \((\alpha ,\beta )\) with respect to the variable u, we get

$$ \begin{aligned}[b] \int _{\alpha }^{\beta }F(u)G(u)\,du&\leq F(\beta ) \int _{\alpha }^{ \beta }G(u)\,du+\eta \bigl(F(\alpha ),F( \beta ) \bigr) \int _{ \alpha }^{\beta }\omega \biggl(\frac{\beta -u}{\beta -\alpha } \biggr)G(u) \,du\\ &\quad{} -\mu \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u)G(u)\,du. \end{aligned} $$
(13)

Similarly, we can also write

$$ u=\tau \beta + (1-\tau )\alpha \quad \text{with }\tau = \frac{u-\alpha }{\beta -\alpha }. $$

Applying again the strong \((\eta ,\omega )\)-convexity of F gives

$$ F(u)\leq F(\alpha )+\omega \biggl(\frac{u-\alpha }{\beta -\alpha } \biggr)\eta \bigl(F(\beta ),F(\alpha ) \bigr)-\mu (u-\alpha ) ( \beta -u). $$

Multiplying this inequality by \(G(u)\), proceeding as outlined before, and noting that \(G(\alpha +u(\beta -\alpha ))=G(\beta -u(\beta -\alpha ))\), we get

$$ \begin{aligned}[b] \int _{\alpha }^{\beta }F(u)G(u)\,du&\leq F(\alpha ) \int _{\alpha }^{ \beta }G(u)\,du+\eta \bigl(F(\beta ),F( \alpha ) \bigr) \int _{ \alpha }^{\beta }\omega \biggl(\frac{\beta -u}{\beta -\alpha } \biggr)G(u) \,du\\ & \quad{} -\mu \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u)G(u)\,du, \end{aligned} $$
(14)

where we have used the fact that

$$ \int _{\alpha }^{\beta }\omega \biggl( \frac{u-\alpha }{\beta -\alpha } \biggr)G(u)\,du= \int _{\alpha }^{ \beta }\omega \biggl(\frac{\beta -u}{\beta -\alpha } \biggr)G(u)\,du. $$

Now adding (13) and (14) gives

$$ \begin{aligned}[b] \int _{\alpha }^{\beta }F(u)G(u)\,du&\leq \biggl( \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{2} \biggr) \int _{\alpha }^{\beta }\omega \biggl( \frac{\beta -u}{\beta -\alpha } \biggr)G(u)\,du\\ &\quad{} + \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{\beta }G(u) \,du-\mu \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u)G(u)\,du. \end{aligned} $$
(15)

The last inequality follows by using the upper bound \(K_{\eta }\) in (15):

$$ \begin{aligned} & \biggl( \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{2} \biggr) \int _{\alpha }^{\beta }\omega \biggl( \frac{\beta -u}{\beta -\alpha } \biggr)G(u)\,du\\ & \quad\quad{} + \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{\beta }G(u) \,du-\mu \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u)G(u)\,du\\ &\quad \leq \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{\beta }G(u) \,du+K_{\eta } \int _{\alpha }^{\beta }\omega \biggl( \frac{\beta -u}{\beta -\alpha } \biggr)G(u)\,du\\ &\quad\quad{} -\mu \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u)G(u)\,du. \end{aligned} $$

This completes the proof. □

Corollary 11

Let\(F:[\alpha ,\beta ]\to \mathbb{R}\)and\(G:(\alpha ,\beta )\to [0,\infty )\)with\(\alpha <\beta \). Suppose that the functionsFandGsatisfy the following conditions:

  1. (a)

    Fis stronglyη-convex with modulus\(\mu \geq 0\)andηbounded above on\(F([\alpha ,\beta ])\times F([\alpha ,\beta ])\);

  2. (b)

    \(F\in L_{\infty }([\alpha , \beta ])\);

  3. (c)

    \(G\in L_{1}((\alpha , \beta ))\);

  4. (d)

    \(G(\alpha +\beta -u)=G(u)\)for all\(u\in (\alpha ,\beta )\).

Then we have the following inequalities:

$$ \begin{aligned} & \biggl(F \biggl(\frac{\alpha +\beta }{2} \biggr)- \frac{K_{\eta }}{2} \biggr) \int ^{\beta }_{\alpha } G(u)\,du+\frac{\mu }{4} \int ^{\beta }_{ \alpha } (2u-\alpha -\beta )^{2}G(u) \,du\\ &\quad \leq \int ^{\beta }_{\alpha } F(u)G(u)\,du\\ &\quad \leq \biggl( \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{2(\beta -\alpha )} \biggr) \int _{\alpha }^{\beta }(\beta -u)G(u)\,du\\ &\quad\quad{} + \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{\beta }G(u) \,du-\mu \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u)G(u)\,du\\ &\quad \leq \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{\beta }G(u) \,du+\frac{K_{\eta }}{\beta -\alpha } \int _{\alpha }^{\beta }(\beta -u)G(u) \,du\\ & \quad\quad{} -\mu \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u)G(u)\,du, \end{aligned} $$

where\(K_{\eta }\)is an upper bound of η.

Proof

The desired result follows by setting \(\omega (\tau )=\tau \) in Theorem 10. □

Remark 12

If we take \(\mu =0\) in Corollary 11, then we recapture Theorem 3 due to Delavar and Dragomir. By taking \(G(\tau )=1\) for all \(\tau \in (\alpha ,\beta )\) in Corollary 11 we recover Theorem 6 due to Awan et al. Also, Corollary 11 reduces to Theorem 2 by taking \(G\equiv 1\) and \(\mu =0\).

Corollary 13

Let\(F:[\alpha ,\beta ]\to \mathbb{R}\)and\(H:(\alpha ,\beta )\to [0,\infty )\)with\(\alpha <\beta \). Suppose that the functionsFandHsatisfy the following conditions:

  1. (a)

    Fis strongly\((\eta ,\omega )\)-convex with modulus\(\mu \geq 0\), ηbounded above on\(F([\alpha ,\beta ])\times F([\alpha ,\beta ])\), and\(\omega \in L_{\infty }([0, 1])\);

  2. (b)

    \(F\in L_{\infty }([\alpha , \beta ])\);

  3. (c)

    \(H\in L_{1}((\alpha , \beta ))\).

Then we have the following inequalities:

$$ \begin{aligned} & \biggl(F \biggl(\frac{\alpha +\beta }{2} \biggr)-\omega \biggl( \frac{1}{2} \biggr)K_{\eta } \biggr) \int ^{\beta }_{\alpha } H(u)\,du\\ &\quad\quad{} +\frac{\mu }{8} \int ^{\beta }_{\alpha } (2u-\alpha -\beta )^{2} \bigl(H(u)+H(\alpha +\beta -u) \bigr)\,du\\ &\quad \leq \frac{1}{2} \int ^{\beta }_{\alpha } F(u) \bigl(H(u)+H(\alpha + \beta -u) \bigr)\,du\\ &\quad \leq \biggl( \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{4} \biggr) \int _{\alpha }^{\beta } \biggl[\omega \biggl( \frac{\beta -u}{\beta -\alpha } \biggr)+\omega \biggl( \frac{u-\alpha }{\beta -\alpha } \biggr) \biggr]H(u) \,du\\ &\quad\quad{} + \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{ \beta }H(u)\,du-\frac{\mu }{2} \int _{\alpha }^{\beta }(u-\alpha ) ( \beta -u) \bigl(H(u)+H( \alpha +\beta -u) \bigr)\,du\\ &\quad \leq \biggl(\frac{F(\alpha )+F(\beta )}{2} \biggr) \int _{\alpha }^{\beta }H(u) \,du+\frac{K_{\eta }}{2} \int _{\alpha }^{\beta } \biggl[\omega \biggl( \frac{\beta -u}{\beta -\alpha } \biggr)+\omega \biggl( \frac{u-\alpha }{\beta -\alpha } \biggr) \biggr]H(u) \,du\\ &\quad\quad{} -\frac{\mu }{2} \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u) \bigl(H(u)+H( \alpha +\beta -u) \bigr)\,du, \end{aligned} $$

where\(K_{\eta }\)is an upper bound of η.

Proof

Let \(G:(\alpha ,\beta )\to \mathbb{R}\) be the function defined by

$$ G(u)=H(u)+H(\alpha +\beta -u), \quad u\in (\alpha ,\beta ). $$

Since \(H\in L_{1}((\alpha , \beta ))\), it follows also that \(G\in L_{1}((\alpha , \beta ))\). Also, by the definition of the function G we have that for \(u\in (\alpha ,\beta )\),

$$\begin{aligned} G(\alpha +\beta -u)&=H(\alpha +\beta -u)+H(u)=G(u). \end{aligned}$$

Hence, items (c) and (d) of Theorem 10 are satisfied. Therefore, applying Theorem 10 to the function G, we get the desired inequalities. □

Application to the k-Riemann–Liouville fractional operators

We start by recalling the definition of the k-Riemann–Liouville fractional integrals: the left- and right-sided k-Riemann–Liouville fractional integral operators \({}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon }\) and \({}_{k}{\mathcal{J}}_{\beta ^{-}}^{\epsilon }\) of order \(\epsilon >0\) for a real-valued continuous function \(F(x)\) are defined as

$$ {}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon }F(x) = \frac{1}{k\varGamma _{k}(\epsilon )} \int _{\alpha }^{x}(x-t)^{ \frac{\epsilon }{k}-1}F(t)\,dt, \quad x> \alpha , $$
(16)

and

$$ {}_{k}{\mathcal{J}}_{\beta ^{-}}^{\epsilon }F(x) = \frac{1}{k\varGamma _{k}(\epsilon )} \int _{x}^{\beta } (t-x )^{ \frac{\epsilon }{k}-1}F(t)\,dt, \quad x< \beta , $$
(17)

where \(k>0\), and \(\varGamma _{k}\) is the k-gamma function given by

$$ \varGamma _{k}(x):= \int _{0}^{\infty }t^{x-1}e^{-\frac{t^{k}}{k}}\,dt, \quad \operatorname{Re}(x)>0, $$

with the properties \(\varGamma _{k}(x+k)=x\varGamma _{k}(x)\) and \(\varGamma _{k}(k)=1\).

In what follows, we will need the following functions \(\mathcal{U}, \mathcal{V}, \mathcal{W}:[\alpha ,\beta ]\to \mathbb{R}\) defined by

$$\begin{aligned}& \mathcal{U}(x)=(2x-\alpha -\beta )^{2}, \\ & \mathcal{V}(x)=(x-\alpha ) (\beta -x), \end{aligned}$$

and

$$ \mathcal{W}(x)=\omega \biggl(\frac{\beta -x}{\beta -\alpha } \biggr)+ \omega \biggl( \frac{x-\alpha }{\beta -\alpha } \biggr). $$

Applying Corollary 13, we get the following result.

Corollary 14

Let\(F:[\alpha ,\beta ]\to \mathbb{R}\)with\(\alpha <\beta \). Suppose that the functionFsatisfies the following conditions:

  1. (a)

    Fis strongly\((\eta ,\omega )\)-convex with modulus\(\mu \geq 0\), ηbounded above on\(F([\alpha ,\beta ])\times F([\alpha ,\beta ])\), and\(\omega \in L_{\infty }([0, 1])\);

  2. (b)

    \(F\in L_{\infty }([\alpha , \beta ])\).

Then we have the following inequalities:

$$ \begin{aligned}[b] &F \biggl(\frac{\alpha +\beta }{2} \biggr)-\omega \biggl( \frac{1}{2} \biggr)K_{\eta } \\ &\quad\quad{} +\frac{\mu }{8} \frac{\varGamma _{k}(\epsilon +k)}{(\beta -\alpha )^{\frac{\epsilon }{k}}} \bigl[{}_{k}{ \mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{U}(\beta )+{}_{k}{ \mathcal{J}}_{\beta ^{-}}^{\epsilon }\mathcal{U}(\alpha ) \bigr] \\ &\quad \leq \frac{\varGamma _{k}(\epsilon +k)}{2(\beta -\alpha )^{\frac{\epsilon }{k}}} \bigl[{}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon }F( \beta )+{}_{k}{ \mathcal{J}}_{\beta ^{-}}^{\epsilon }F(\alpha ) \bigr] \\ &\quad \leq \frac{\varGamma _{k}(\epsilon +k)}{(\beta -\alpha )^{\frac{\epsilon }{k}}} \biggl( \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{4} \biggr){ {}_{k}{ \mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{W}( \beta )} \\ &\quad\quad{} +\frac{F(\alpha )+F(\beta )}{2}-\frac{\mu }{2} \frac{\varGamma _{k}(\epsilon +k)}{(\beta -\alpha )^{\frac{\epsilon }{k}}} \bigl[{}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{V}(\beta )+{}_{k}{ \mathcal{J}}_{\beta ^{-}}^{\epsilon }\mathcal{V}(\alpha ) \bigr] \\ &\quad \leq \frac{K_{\eta }}{2} \frac{\varGamma _{k}(\epsilon +k)}{(\beta -\alpha )^{\frac{\epsilon }{k}}}{{}_{k}{ \mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{W}(\beta )} \\ &\quad\quad{} +\frac{F(\alpha )+F(\beta )}{2}-\frac{\mu }{2} \frac{\varGamma _{k}(\epsilon +k)}{(\beta -\alpha )^{\frac{\epsilon }{k}}} \bigl[{}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{V}(\beta )+{}_{k}{ \mathcal{J}}_{\beta ^{-}}^{\epsilon }\mathcal{V}(\alpha ) \bigr], \end{aligned} $$
(18)

where\(K_{\eta }\)is an upper bound of η.

Proof

Let

$$ H(x)= \frac{(\beta -x)^{\frac{\epsilon }{k}-1}}{k\varGamma _{k}(\epsilon )}, \quad x\in (\alpha ,\beta ), $$

where \(\epsilon ,k>0\). The function H clearly satisfies the conditions of Corollary 13 since

$$ \begin{aligned} \int _{\alpha }^{\beta }H(x)\,dx&=\frac{1}{k\varGamma _{k}(\epsilon )} \int _{\alpha }^{\beta }(\beta -x)^{\frac{\epsilon }{k}-1}\,dx \\ &= \frac{(\beta -\alpha )^{\frac{\epsilon }{k}}}{\varGamma _{k}(\epsilon +k)}< \infty . \end{aligned} $$

We obtain the intended inequalities by applying Corollary 13 to the function H and the following identities:

$$\begin{aligned}& \int ^{\beta }_{\alpha } (2u-\alpha -\beta )^{2} \bigl(H(u)+H(\alpha + \beta -u) \bigr)\,du={}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon } \mathcal{U}(\beta )+{}_{k}{\mathcal{J}}_{\beta ^{-}}^{\epsilon } \mathcal{U}(\alpha ), \\ & \int ^{\beta }_{\alpha } F(u) \bigl(H(u)+H(\alpha +\beta -u) \bigr)\,du={}_{k}{ \mathcal{J}}_{\alpha ^{+}}^{\epsilon }F(\beta )+{}_{k}{\mathcal{J}}_{ \beta ^{-}}^{\epsilon }F(\alpha ), \\ & \int _{\alpha }^{\beta } \biggl[\omega \biggl( \frac{\beta -u}{\beta -\alpha } \biggr)+\omega \biggl( \frac{u-\alpha }{\beta -\alpha } \biggr) \biggr]H(u) \,du={}_{k}{ \mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{W}(\beta ), \end{aligned}$$

and

$$ \int _{\alpha }^{\beta }(u-\alpha ) (\beta -u) \bigl(H(u)+H( \alpha + \beta -u) \bigr)\,du={}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon } \mathcal{V}(\beta )+{}_{k}{\mathcal{J}}_{\beta ^{-}}^{\epsilon } \mathcal{V}(\alpha ). $$

 □

Corollary 15

Let\(F:[\alpha ,\beta ]\to \mathbb{R}\)with\(\alpha <\beta \). Suppose that the functionFsatisfies the following conditions:

  1. (a)

    Fis stronglyη-convex with modulus\(\mu \geq 0\)andηbounded above on\(F([\alpha ,\beta ])\times F([\alpha ,\beta ])\);

  2. (b)

    \(F\in L_{\infty }([\alpha , \beta ])\).

Then we have the following inequalities:

$$\begin{aligned}& F \biggl(\frac{\alpha +\beta }{2} \biggr)-\frac{K_{\eta }}{2} \\ & \quad \quad{} +\frac{\mu }{8} \frac{\varGamma _{k}(\epsilon +k)}{(\beta -\alpha )^{\frac{\epsilon }{k}}} \bigl[ {}_{k}{ \mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{U}(\beta )+{}_{k}{ \mathcal{J}}_{\beta ^{-}}^{\epsilon }\mathcal{U}(\alpha ) \bigr] \\ & \quad \leq \frac{\varGamma _{k}(\epsilon +k)}{2(\beta -\alpha )^{\frac{\epsilon }{k}}} \bigl[{}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon }F( \beta )+{}_{k}{ \mathcal{J}}_{\beta ^{-}}^{\epsilon }F(\alpha ) \bigr] \\ & \quad \leq \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{4} \\ & \quad\quad{} +\frac{F(\alpha )+F(\beta )}{2}-\frac{\mu }{2} \frac{\varGamma _{k}(\epsilon +k)}{(\beta -\alpha )^{\frac{\epsilon }{k}}} \bigl[{}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{V}(\beta )+{}_{k}{ \mathcal{J}}_{\beta ^{-}}^{\epsilon }\mathcal{V}(\alpha ) \bigr] \\ & \quad \leq \frac{K_{\eta }}{2}+\frac{F(\alpha )+F(\beta )}{2} \\ & \quad\quad{} -\frac{\mu }{2} \frac{\varGamma _{k}(\epsilon +k)}{(\beta -\alpha )^{\frac{\epsilon }{k}}} \bigl[{}_{k}{ \mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{V}(\beta )+{}_{k}{ \mathcal{J}}_{\beta ^{-}}^{\epsilon }\mathcal{V}(\alpha ) \bigr], \end{aligned}$$
(19)

where\(K_{\eta }\)is an upper bound of η.

Proof

The proof follows by setting \(\omega (\tau )=\tau \), \(\tau \in [0,1]\), in Corollary 14. For this, we notice that

$$ \mathcal{W}(x)=1,\quad x\in [\alpha ,\beta ], $$

and thus

$$ {}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon }\mathcal{W}(\beta )= \frac{(\beta -\alpha )^{\frac{\epsilon }{k}}}{\varGamma _{k}(\epsilon +k)}. $$

 □

By substituting \(\mu =0\) in the corollary, we obtain the following result for the η-convex functions.

Corollary 16

Let\(F:[\alpha ,\beta ]\to \mathbb{R}\)with\(\alpha <\beta \). Suppose that the functionFsatisfies the following conditions:

  1. (a)

    Fisη-convex andηbounded above on\(F([\alpha ,\beta ])\times F([\alpha ,\beta ])\);

  2. (b)

    \(F\in L_{\infty }([\alpha , \beta ])\).

Then we have the following inequalities:

$$ \begin{aligned}[b] &F \biggl(\frac{\alpha +\beta }{2} \biggr)- \frac{K_{\eta }}{2} \\ &\quad \leq \frac{\varGamma _{k}(\epsilon +k)}{2(\beta -\alpha )^{\frac{\epsilon }{k}}} \bigl[{}_{k}{\mathcal{J}}_{\alpha ^{+}}^{\epsilon }F( \beta )+{}_{k}{ \mathcal{J}}_{\beta ^{-}}^{\epsilon }F(\alpha ) \bigr] \\ &\quad \leq \frac{\eta (F(\alpha ),F(\beta ) )+\eta (F(\beta ),F(\alpha ) )}{4}+ \frac{F(\alpha )+F(\beta )}{2} \\ &\quad \leq \frac{K_{\eta }}{2}+\frac{F(\alpha )+F(\beta )}{2}, \end{aligned} $$
(20)

where\(K_{\eta }\)is an upper bound of η.

More integral inequalities

We now proceed to obtain more results associated with this new class of functions. For this, we will need the following lemma.

Lemma 17

([11])

Let\(F:J\subset \mathbb{R}\to \mathbb{R}\), and let\(\alpha ,\beta \in J\)with\(\alpha <\beta \). SupposeFsatisfies the following conditions:

  1. (a)

    Fis differentiable in the interior ofJdenoted by\(J^{\circ }\);

  2. (b)

    \(F'\in L_{1}([\alpha ,\beta ])\).

Then, for any\(\lambda \in \mathbb{R}\), we have the identity

$$ \lambda F(\alpha )+(1-\lambda )F(\beta )-\frac{1}{\beta -\alpha } \int _{\alpha }^{\beta }F(u)\,du=(\beta -\alpha ) \int _{0}^{1} (\tau - \lambda )F' \bigl((1-\tau )\alpha +\tau \beta \bigr)\,d\tau . $$

Theorem 18

Assume that a functionFsatisfies the conditions of Lemma 17. If, in addition, \(\vert F' \vert \)is strongly\((\eta ,\omega )\)-convex on\([\alpha ,\beta ]\)with modulus\(\mu \geq 0\)and\(\omega \in L_{\infty }([0, 1])\), then for any\(\lambda \in \mathbb{R}\), we have the following inequalities:

$$ \begin{aligned}[b] & \biggl\vert \lambda F(\alpha )+(1-\lambda )F( \beta )- \frac{1}{\beta -\alpha } \int _{\alpha }^{\beta }F(u)\,du\biggr\vert \\ &\quad \leq \textstyle\begin{cases} h (\lambda -\frac{1}{2} ) \vert F'(\alpha ) \vert + h\eta ( \vert F'( \beta ) \vert , \vert F'(\alpha ) \vert )\mathcal{W}_{1}(\lambda )+ \frac{\mu h^{3}(1-2\lambda )}{12},&\lambda \geq 1, \\ h (\lambda ^{2}-\lambda +\frac{1}{2} ) \vert F'(\alpha ) \vert + h \eta ( \vert F'(\beta ) \vert , \vert F'(\alpha ) \vert )\mathcal{W}_{2}(\lambda ) \\ \quad{} + \frac{\mu h^{3}(2\lambda ^{4}-4\lambda ^{3}+2\lambda -1)}{12}, &0< \lambda < 1, \\ h (\frac{1}{2}-\lambda ) \vert F'(\alpha ) \vert + h\eta ( \vert F'( \beta ) \vert , \vert F'(\alpha ) \vert )\mathcal{W}_{3}(\lambda )+ \frac{\mu h^{3}(2\lambda -1)}{12}, &\lambda \leq 0, \end{cases}\displaystyle \end{aligned} $$
(21)

where\(h=\beta -\alpha \),

$$\begin{aligned}& \mathcal{W}_{1}(\lambda )= \int _{0}^{1}(\lambda -\tau )\omega (\tau ) \,d\tau , \\& \mathcal{W}_{2}(\lambda )= \int _{0}^{\lambda }(\lambda -\tau )\omega ( \tau )\,d\tau + \int _{\lambda }^{1}(\tau -\lambda )\omega (\tau )\,d \tau , \end{aligned}$$

and

$$ \mathcal{W}_{3}(\lambda )= \int _{0}^{1}(\tau -\lambda )\omega (\tau ) \,d\tau . $$

Proof

We start by observing that

$$ \int _{0}^{1} \vert \tau -\lambda \vert \,d\tau = \textstyle\begin{cases} \lambda -\frac{1}{2},&\lambda \geq 1, \\ \lambda ^{2}-\lambda +\frac{1}{2}, &0< \lambda < 1, \\ \frac{1}{2}-\lambda ,&\lambda \leq 0, \end{cases} $$
(22)

and

$$ \int _{0}^{1} \vert \tau -\lambda \vert \tau (1-\tau )\,d\tau = \textstyle\begin{cases} \frac{2\lambda -1}{12},&\lambda \geq 1, \\ \frac{4\lambda ^{3}-2\lambda ^{4}-2\lambda +1}{12}, &0< \lambda < 1, \\ \frac{1-2\lambda }{12},&\lambda \leq 0. \end{cases} $$
(23)

Now using Lemma 17 and the strong \((\eta ,\omega )\)-convexity of \(\vert F' \vert \), we get

$$\begin{aligned}& \biggl\vert \lambda F(\alpha )+(1-\lambda )F( \beta )- \frac{1}{\beta -\alpha } \int _{\alpha }^{\beta }F(u)\,du\biggr\vert \\& \quad = \biggl\vert (\beta -\alpha ) \int _{0}^{1} (\tau -\lambda )F' \bigl((1- \tau )\alpha +\tau \beta \bigr)\,d\tau \biggr\vert \\& \quad \leq (\beta -\alpha ) \int _{0}^{1} \vert \tau -\lambda \vert \bigl\vert F' \bigl((1- \tau )\alpha +\tau \beta \bigr) \bigr\vert \,d \tau \\& \quad \leq (\beta -\alpha ) \int _{0}^{1} \vert \tau -\lambda \vert \bigl[ \bigl\vert F'(\alpha ) \bigr\vert + \omega (\tau )\eta \bigl( \bigl\vert F'(\beta ) \bigr\vert , \bigl\vert F'( \alpha ) \bigr\vert \bigr)-\mu \tau (1-\tau ) (\beta -\alpha )^{2} \bigr]\,d\tau \\& \quad =(\beta -\alpha ) \bigl\vert F'(\alpha ) \bigr\vert \int _{0}^{1} \vert \tau -\lambda \vert \,d\tau + (\beta -\alpha )\eta \bigl( \bigl\vert F'(\beta ) \bigr\vert , \bigl\vert F'(\alpha ) \bigr\vert \bigr) \int _{0}^{1} \vert \tau -\lambda \vert \omega (\tau )\,d\tau \\& \quad\quad{} -\mu (\beta -\alpha )^{3} \int _{0}^{1} \vert \tau -\lambda \vert \tau (1-\tau )\,d\tau . \end{aligned}$$

Hence the desired result is obtained by using (22) and (23). □

Theorem 19

Assume that a functionFsatisfies the conditions of Lemma 17. If, in addition, \(\vert F' \vert ^{q} \) (\(q>1\)) is strongly\((\eta ,\omega )\)-convex on\([\alpha ,\beta ]\)with modulus\(\mu \geq 0\)and\(\omega \in L_{\infty }([0, 1])\), then for any\(\lambda \in \mathbb{R}\), we have the following inequalities:

$$ \begin{aligned}[b] & \biggl\vert \lambda F(\alpha )+(1-\lambda )F( \beta )- \frac{1}{\beta -\alpha } \int _{\alpha }^{\beta }F(u)\,du\biggr\vert \\ &\quad \leq \textstyle\begin{cases} (\beta -\alpha ) (\frac{\lambda ^{p+1}-(\lambda -1)^{p+1}}{p+1} )^{\frac{1}{p}}{\mathcal{A}}_{q}(\eta ,\omega ;\mu ),& \lambda \geq 1, \\ (\beta -\alpha ) (\frac{\lambda ^{p+1}-(1-\lambda )^{p+1}}{p+1} )^{\frac{1}{p}}{\mathcal{A}}_{q}(\eta ,\omega ;\mu ), &0< \lambda < 1, \\ (\beta -\alpha ) ( \frac{(1-\lambda )^{p+1}-(-\lambda )^{p+1}}{p+1} )^{\frac{1}{p}}{ \mathcal{A}}_{q}(\eta ,\omega ;\mu ), &\lambda \leq 0, \end{cases}\displaystyle \end{aligned} $$
(24)

where\(\frac{1}{p}+\frac{1}{q}=1\), and

$$ {\mathcal{A}}_{q}(\eta ,\omega ;\mu )= \biggl( \bigl\vert F'(\alpha ) \bigr\vert ^{q}+\eta \bigl( \bigl\vert F'(\beta ) \bigr\vert ^{q}, \bigl\vert F'(\alpha ) \bigr\vert ^{q} \bigr) \int _{0}^{1} \omega (\tau )\,d\tau - \frac{\mu (\beta -\alpha )^{2}}{6} \biggr)^{ \frac{1}{q}}. $$

Proof

From identity (22) we get

$$ \int _{0}^{1} \vert \tau -\lambda \vert ^{p}\,d\tau = \textstyle\begin{cases} \frac{\lambda ^{p+1}-(\lambda -1)^{p+1}}{p+1},&\lambda \geq 1, \\ \frac{\lambda ^{p+1}-(1-\lambda )^{p+1}}{p+1}, &0< \lambda < 1, \\ \frac{(1-\lambda )^{p+1}-(-\lambda )^{p+1}}{p+1},&\lambda \leq 0, \end{cases} $$
(25)

where \(p>1\).

Using again Lemma 17, the strong \((\eta ,\omega )\)-convexity of \(\vert F' \vert ^{q}\), and the Hölder inequality, we obtain

$$\begin{aligned}& \biggl\vert \lambda F(\alpha )+(1-\lambda )F( \beta )- \frac{1}{\beta -\alpha } \int _{\alpha }^{\beta }F(u)\,du\biggr\vert \\& \quad = \biggl\vert (\beta -\alpha ) \int _{0}^{1} (\tau -\lambda )F' \bigl((1- \tau )\alpha +\tau \beta \bigr)\,d\tau \biggr\vert \\& \quad \leq (\beta -\alpha ) \int _{0}^{1} \vert \tau -\lambda \vert \bigl\vert F' \bigl((1- \tau )\alpha +\tau \beta \bigr) \bigr\vert \,d \tau \\& \quad \leq (\beta -\alpha ) \biggl( \int _{0}^{1} \vert \tau -\lambda \vert ^{p}\,d \tau \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} \bigl\vert F' \bigl((1-\tau ) \alpha +\tau \beta \bigr) \bigr\vert ^{q}\,d\tau \biggr)^{\frac{1}{q}} \\& \quad \leq (\beta -\alpha ) \biggl( \int _{0}^{1} \vert \tau -\lambda \vert ^{p}\,d \tau \biggr)^{\frac{1}{p}} \\& \quad\quad{}\times \biggl( \int _{0}^{1} \bigl[ \bigl\vert F'( \alpha ) \bigr\vert ^{q}+ \omega (\tau )\eta \bigl( \bigl\vert F'(\beta ) \bigr\vert ^{q}, \bigl\vert F'(\alpha ) \bigr\vert ^{q} \bigr)- \mu \tau (1-\tau ) (\beta -\alpha )^{2} \bigr]\,d\tau \biggr)^{ \frac{1}{q}} \\& \quad =(\beta -\alpha ) \biggl( \int _{0}^{1} \vert \tau -\lambda \vert ^{p}\,d\tau \biggr)^{\frac{1}{p}} \\& \quad\quad{}\times\biggl( \bigl\vert F'(\alpha ) \bigr\vert ^{q}+\eta \bigl( \bigl\vert F'(\beta ) \bigr\vert ^{q}, \bigl\vert F'(\alpha ) \bigr\vert ^{q} \bigr) \int _{0}^{1}\omega (\tau )\,d\tau - \frac{\mu (\beta -\alpha )^{2}}{6} \biggr)^{\frac{1}{q}}. \end{aligned}$$

The desired result is obtained by employing identity (25). □

Theorem 20

Assume that a functionFsatisfies the conditions of Lemma 17. If, in addition, \(\vert F' \vert ^{q} \) (\(q\geq 1\)) is strongly\((\eta ,\omega )\)-convex on\([\alpha ,\beta ]\)with modulus\(\mu \geq 0\)and\(\omega \in L_{\infty }([0, 1])\), then for any\(\lambda \in \mathbb{R}\), we have the following inequalities:

$$ \begin{aligned} & \biggl\vert \lambda F(\alpha )+(1-\lambda )F( \beta )- \frac{1}{\beta -\alpha } \int _{\alpha }^{\beta }F(u)\,du\biggr\vert \\ &\quad \leq \textstyle\begin{cases} h (\lambda -\frac{1}{2} )^{1-\frac{1}{q}} [ \vert F'( \alpha ) \vert ^{q} (\lambda -\frac{1}{2} ) + \eta ( \vert F'( \beta ) \vert ^{q}, \vert F'(\alpha ) \vert ^{q} )\mathcal{W}_{1}(\lambda ) \\ \quad {}+ \frac{\mu h^{2}(1-2\lambda )}{12} ]^{\frac{1}{q}},&\lambda \geq 1, \\ h (\lambda ^{2}-\lambda +\frac{1}{2} )^{1-\frac{1}{q}} [ \vert F'(\alpha ) \vert (\lambda ^{2}-\lambda +\frac{1}{2} ) + \eta ( \vert F'(\beta ) \vert ^{q}, \vert F'(\alpha ) \vert ^{q} )\mathcal{W}_{2}( \lambda ) \\ \quad {}+ \frac{\mu h^{2}(2\lambda ^{4}-4\lambda ^{3}+2\lambda -1)}{12} ]^{ \frac{1}{q}}, &0< \lambda < 1, \\ h (\frac{1}{2}-\lambda )^{1-\frac{1}{q}} [ \vert F'( \alpha ) \vert (\frac{1}{2}-\lambda ) + \eta ( \vert F'(\beta ) \vert , \vert F'(\alpha ) \vert )\mathcal{W}_{3}(\lambda )+ \frac{\mu h^{2}(2\lambda -1)}{12} ]^{\frac{1}{q}}, &\lambda \leq 0, \end{cases}\displaystyle \end{aligned} $$

whereh, \(\mathcal{W}_{1}(\lambda )\), \(\mathcal{W}_{2}(\lambda )\), and\(\mathcal{W}_{3}(\lambda )\)are defined in Theorem 18.

Proof

Applying Lemma 17, the strong \((\eta ,\omega )\)-convexity of \(\vert F' \vert ^{q}\), and the Hölder inequality, we get

$$ \begin{aligned} & \biggl\vert \lambda F(\alpha )+(1-\lambda )F( \beta )- \frac{1}{\beta -\alpha } \int _{\alpha }^{\beta }F(u)\,du\biggr\vert \\ &\quad = \biggl\vert (\beta -\alpha ) \int _{0}^{1} (\tau -\lambda )F' \bigl((1- \tau )\alpha +\tau \beta \bigr)\,d\tau \biggr\vert \\ &\quad \leq (\beta -\alpha ) \int _{0}^{1} \vert \tau -\lambda \vert \bigl\vert F' \bigl((1- \tau )\alpha +\tau \beta \bigr) \bigr\vert \,d \tau \\ &\quad \leq (\beta -\alpha ) \biggl( \int _{0}^{1} \vert \tau -\lambda \vert \,d\tau \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \vert \tau -\lambda \vert \bigl\vert F' \bigl((1-\tau )\alpha +\tau \beta \bigr) \bigr\vert ^{q}\,d\tau \biggr)^{ \frac{1}{q}} \\ &\quad \leq (\beta -\alpha ) \biggl( \int _{0}^{1} \vert \tau -\lambda \vert \,d\tau \biggr)^{1-\frac{1}{q}} \\ &\quad\quad{} \times \biggl( \int _{0}^{1} \vert \tau -\lambda \vert \bigl[ \bigl\vert F'(\alpha ) \bigr\vert ^{q}+ \omega (\tau ) \eta \bigl( \bigl\vert F'(\beta ) \bigr\vert ^{q}, \bigl\vert F'(\alpha ) \bigr\vert ^{q} \bigr)- \mu \tau (1-\tau ) (\beta -\alpha )^{2} \bigr]\,d\tau \biggr)^{ \frac{1}{q}} \\ &\quad =(\beta -\alpha ) \biggl( \int _{0}^{1} \vert \tau -\lambda \vert \,d\tau \biggr)^{1-\frac{1}{q}} \\ &\quad\quad{} \times \biggl( \bigl\vert F'(\alpha ) \bigr\vert ^{q} \int _{0}^{1} \vert \tau -\lambda \vert \,d \tau +\eta \bigl( \bigl\vert F'(\beta ) \bigr\vert ^{q}, \bigl\vert F'(\alpha ) \bigr\vert ^{q} \bigr) \int _{0}^{1} \vert \tau -\lambda \vert \omega (\tau )\,d\tau \\ &\quad \quad {}-\mu (\beta -\alpha )^{2} \int _{0}^{1} \vert \tau -\lambda \vert \tau (1-\tau )\,d\tau \biggr)^{\frac{1}{q}}. \end{aligned} $$

The intended result is reached by employing identities (22) and (23). □

Conclusion

We introduced the notion of \((\eta ,\omega )\)-convexity. We established inequalities of the Hermite–Hadamard–Fejér type and many novel results for the class of \((\eta ,\omega )\)-convex functions. Applications are also provided by employing Corollary 13 to the k-Riemann–Liouville fractional integral operators. We anticipate that this new class of functions will inspire further investigation in this direction. Some further work in this direction can be found in [1229].

References

  1. 1.

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

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Awan, M.U., Noorb, M.A., Noorb, K.I., Safdarb, F.: On strongly generalized convex functions. Filomat 31(18), 5783–5790 (2017)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Gordji, M.E., Delavar, M.R., Sen, M.D.L.: On φ-convex functions. J. Math. Inequal. 10(1), 173–183 (2016)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Gordji, M.E., Dragomir, S.S., Delavar, M.R.: An inequality related to η-convex functions (II). Int. J. Nonlinear Anal. Appl. 6(2), 26–32 (2015)

    Google Scholar 

  5. 5.

    Khan, M.A., Khurshid, Y., Ali, T.: Hermite–Hadamard inequality for fractional integrals via η-convex functions. Acta Math. Univ. Comen. LXXXVI(1), 153–164 (2017)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Delavar, M.R., De La Sen, M.: Some generalizations of Hermite–Hadamard type inequalities. SpringerPlus 5, 1661 (2016)

    Article  Google Scholar 

  7. 7.

    Nwaeze, E.R.: Inequalities of the Hermite–Hadamard type for quasi-convex functions via the \((k,s)\)-Riemann–Liouville fractional integrals. Fract. Differ. Calc. 8(2), 327–336 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Nwaeze, E.R., Torres, D.F.M.: Novel results on the Hermite–Hadamard kind inequality for η-convex functions by means of the \((k,r)\)-fractional integral operators. In: Dragomir, S.S., Agarwal, P., Jleli, M., Samet, B. (eds.) Advances in Mathematical Inequalities and Applications (AMIA). Trends in Mathematics, pp. 311–321. Birkhäuser, Singapore (2018)

    Google Scholar 

  9. 9.

    Nwaeze, E.R., Kermausuor, S., Tameru, A.M.: Some new k-Riemann–Liouville fractional integral inequalities associated with the strongly η-quasiconvex functions with modulus \(\mu \geq 0\). J. Inequal. Appl. 2018, 139 (2018)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Sarikaya, M.Z., Aktan, N.: On the generalization of some integral inequalities and their applications. Math. Comput. Model. 54, 2175–2182 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Sun, J., Xi, B.-Y., Qi, F.: Some new inequalities of the Hermite–Hadamard type for extended s-convex functions. J. Comput. Anal. Appl. 26(6), 985–996 (2019)

    Google Scholar 

  12. 12.

    Chu, Y.-M., Adil Khan, M., Khan, T.U., Ali, T.: Generalizations of Hermite–Hadamard type inequalities for MT-convex functions. J. Nonlinear Sci. Appl. 9, 4305–4316 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

    Guessab, A., Schmeisser, G.: Sharp integral inequalities of the Hermite–Hadamard type. J. Approx. Theory 1152, 260–288 (2002)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Iqbal, A., Adil Khan, M., Ullah, S., Chu, Y.-M.: Some new Hermite–Hadamard type inequalities associated with conformable fractional integrals and their applications. J. Funct. Spaces 2020, Article ID 9845407 (2020)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Kermausuor, S., Nwaeze, E.R.: Some new inequalities involving the Katugampola fractional integrals for strongly η-convex functions. Tbil. Math. J. 12(1), 117–130 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kermausuor, S., Nwaeze, E.R., Tameru, A.M.: New integral inequalities via the Katugampola fractional integrals for functions whose second derivatives are strongly η-convex. Mathematics 7(2), Article ID 183 (2019)

    Article  Google Scholar 

  18. 18.

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

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Khurshid, Y., Adil Khan, M., Chu, Y.M.: Generalized inequalities via GG-convexity and GA-convexity. J. Funct. Spaces 2019, Article ID 6926107 (2019)

    MATH  Google Scholar 

  20. 20.

    Adil Khan, M., Ali, T., Dragomir, S.S.: Hermite–Hadamard type inequalities for conformable fractional integrals. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112(4), 1033–1048 (2018)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Adil Khan, M., Ali, T., Khan, T.U.: Hermite–Hadamard type inequalities with applications. Fasc. Math. 59, 57–74 (2017)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Adil Khan, M., Chu, Y.-M., Kashuri, A., Liko, R., Ali, G.: New Hermite–Hadamard inequalities for conformable fractional integrals. J. Funct. Spaces 2018, Article ID 6928130 (2018)

    MATH  Google Scholar 

  23. 23.

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

    MathSciNet  Article  Google Scholar 

  24. 24.

    Adil Khan, M., Khurshid, Y., Du, T., 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 

  25. 25.

    Adil Khan, M., Mohammad, N., Nwaeze, E.R., Chu, Y.-M.: Quantum Hermite–Hadamard inequality by means of a Green function. Adv. Differ. Equ. 2020, 99 (2020)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Alabdali, O.Y.M., Guessab, A.: On the approximation of strongly convex functions by an upper or lower operator. Appl. Math. Comput. 247, 1129–1138 (2014)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Nwaeze, E.R., Tameru, A.M.: New parameterized quantum integral inequalities via η-quasiconvexity. Adv. Differ. Equ. 2019, 425 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Nwaeze, E.R.: Generalized fractional integral inequalities by means of quasiconvexity. Adv. Differ. Equ. 2019, 262 (2019)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Nwaeze, E.R.: Integral inequalities via generalized quasiconvexity with applications. J. Inequal. Appl. 2019, 236 (2019)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Many thanks to the referees for their valuable comments and suggestions.

Availability of data and materials

Not applicable.

Funding

This research was supported by the National Science Foundation grant AGEP-1820981.

Author information

Affiliations

Authors

Contributions

The authors read and approved the final manuscript.

Corresponding author

Correspondence to Ana M. Tameru.

Ethics declarations

Competing interests

The authors declare that there are no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tameru, A.M., Nwaeze, E.R. & Kermausuor, S. Strongly \((\eta ,\omega )\)-convex functions with nonnegative modulus. J Inequal Appl 2020, 165 (2020). https://doi.org/10.1186/s13660-020-02436-3

Download citation

MSC

  • 26A51
  • 26D15

Keywords

  • Strongly η-convex function
  • Hölder’s inequality
  • Convex function
  • Hermite–Hadamard–Fejér inequality