Fractional inequalities of the Hermite–Hadamard type for $ m $-polynomial convex and harmonically convex functions

1.   Introduction

The sets $ {\bf T} $ and $ {\bf S}\subseteq \mathbb{R}\setminus\{0\} $ are called convex and harmonically convex, respectively if

Whenever used, we shall always consider $ {\bf T} $ as a convex set and $ {\bf S} $ as a harmonically convex set. Let $ m\in\mathbb{N} $. Recall that a function $ \varphi:{\bf T}\to\mathbb{R} $ is said to be $ m $-polynomial convex ^[31] on $ {\bf T} $ if

for all $ q, z\in{\bf S} $ and $ \varsigma\in [0, 1] $. For this class of functions, Toplu et al. established the following double inequality of the Hermite–Hadamard type.

Theorem 1 (^[31]). Let $ \varphi:{\bf T}\to\mathbb{R} $ be an $ m $-polynomial convex function. If $ \xi, \delta\in {\bf T} $ with $ \xi < \delta $, and $ \varphi $ is Lebesgue integrable on $ [\xi, \delta] $, then the following Hermite–Hadamard type inequality holds:

The inequality (1.1) boils down to the classical Hermite–Hadamard inequality for convex functions if we take $ m = 1 $. Recently, Awan et al. ^[2] introduced the notion of $ m $-polynomial harmonically convex functions as follows: a real valued function $ \varphi:{\bf S}\to{\mathbb{R}}^+: = [0, \infty) $ is $ m $-harmonically convex if

for all $ q, z\in{\bf S} $ and $ \varsigma\in [0, 1] $. In the same paper, the authors established the following Hermite–Hadamard type inequality for this class of functions:

Theorem 2 (^[2]). Let $ \varphi:{\bf S}\to{\mathbb{R}}^{+} $ be an $ m $-polynomial harmonically convex function. If $ \xi, \delta\in {\bf S} $ with $ 0 < \xi < \delta $, and $ \varphi $ is Lebesgue integrable on $ [\xi, \delta] $, then the following Hermite–Hadamard type inequality holds:

In the sequel, we will denote the sets of all $ m $-polynomial convex and $ m $-polynomial harmonically convex functions from A into B by $ {\bf XP}_m\left(A, B\right) $ and $ {\bf HXP}_m\left(A, B\right) $, respectively. The classical Hermite–Hadamard inequality has generated load of generalizations and extensions to other class of convexity. There are dozens of articles in this direction. We invite the interested reader to see the following articles ^{[3,4,5,6,8,10,11,12,13,14,15,16,17,18,19,20,22,23,24,25,26,27,28,29,30,32,33,34]} and the references cited therein.

Now, recall that the left- and right-sided $ \zeta $-Riemann–Liouville fractional integral operators $ _{ \zeta}{\mathcal{J}}_{ \xi^+}^{\epsilon} $ and $ _{ \zeta}{\mathcal{J}}_{ \delta^-}^{\epsilon} $ of order $ \epsilon > 0 $, for a real valued continuous function $ \varphi(r) $, are defined as (^[21]):

and

where $ \zeta > 0 $, and $ \Gamma_ \zeta $ is the $ \zeta $-gamma function given by

with the properties $ \Gamma_ \zeta(r+ \zeta) = r \Gamma_ \zeta(r) $ and $ \Gamma_ \zeta(\zeta) = 1 $. If $ \zeta = 1 $, we simply write

The beta function $ \mathcal{B} $ is defined by

Another fractional integral operators of interest is the Caputo–Fabrizio operators ^[1]: let $ L^2(\xi, \delta) $ be the space of square integrable functions on the interval $ (\xi, \delta) $ and

If $ \varphi\in H^1(\xi, \delta) $, $ \xi < \delta $ and $ \mu\in[0, 1] $, then the left- and right-sided Caputo–Fabrizio fractional integral operators $ ^{cf}{\mathcal{I}}_{ \xi}^{\mu} $ and $ ^{cf}{\mathcal{I}}_{ \delta}^{\mu} $ are defined by

and

where $ B:[0, 1]\to (0, \infty) $ is a normalization function satisfying $ B(0) = B(1) = 1 $.

Using these fractional integral operators in (1.4) and (1.5), Gürbüz et al. established the following fractional version of the Hermite–Hadamard inequality:

Theorem 3 (^[7]). Let $ \varphi:{\bf T}\to\mathbb{R} $ be a convex function on $ {\bf T} $. If $ \xi, \delta\in {\bf T} $ with $ \xi < \delta $, and $ \varphi $ is Lebesgue integrable on $ [\xi, \delta] $, then the following double inequality holds:

where $ \mu\in[0, 1] $, $ s\in [\xi, \delta] $ and $ B(\mu) > 0 $ is a normalization function.

Since the classes of convexity introduced here are new, much work have not been done in this sense. This work is geared towards further development around inequalities for these classes. In view of this, we aim to achieve the following objectives:

1. To establish new Hermite–Hadamard type inequalities for the class of $ m $-polynomial convex functions involving the Caputo–Fabrizio integral operators. Our first result in this direction generalizes and extends Theorem 3.

2. To obtain inequalities of the Hermite–Hadamard type for functions that are $ m $-polynomial harmonically convex functions via the $ \zeta $-Riemann–Liouville fractional integral operators. This, in turn, also complement and generalize some existing results in the literature.

2.   Main results

2.1.   Inequalities for $ m $-polynomial convex functions

Inequalities of the Hermite–Hadamard type, for $ m $-polynomial convex functions, are hereby presented. The results, presented herein, involve the Caputo–Fabrizio operators.

Theorem 4. Let $ \varphi:{\bf T}\to \mathbb{R} $ be a Lebesgue integrable function on $ [\xi, \delta] $ with $ \xi < \delta $ and $ \xi, \delta\in {\bf T} $. If $ \varphi\in{\bf XP}_m\left({\bf T}, \mathbb{R}\right) $, then

where $ \mu\in(0, 1] $, $ s\in [\xi, \delta] $ and $ B(\mu) > 0 $ is a normalization function.

Proof. Given that $ \varphi\in{\bf XP}_m\left({\bf T}, \mathbb{R}\right) $, it follows from (1.1) that

Multiplying both sides of (2.1) by $ \frac{\mu(\delta- \xi)}{2B(\mu)} $ gives:

By adding $ \frac{2(1-\mu)}{B(\mu)} \varphi(s) $ to both sides of (2.2), we get:

This implies that

On the other hand, we also get from (1.1) the following inequality:

If we multiply (2.4) by $ \frac{\mu(\delta- \xi)}{2B(\mu)} $ and then add $ \frac{2(1-\mu)}{B(\mu)} \varphi(s) $ to the resulting inequality, we obtain:

Hence, the desired result is obtained by combining (2.3) and (2.5).

Remark 1. By taking $ m = 1 $, Theorem 4 becomes Theorem 3.

Theorem 5. Let $ \varphi, \upsilon:{\bf T}\to \mathbb{R} $ be two functions such that $ \varphi \upsilon $ is Lebesgue integrable function on $ [\xi, \delta] $ with $ \xi < \delta $ and $ \xi, \delta\in {\bf T} $. If $ \varphi\in{\bf XP}_{m_1}\left({\bf S}, \mathbb{R}\right) $, $ \upsilon\in{\bf XP}_{m_2}\left({\bf T}, \mathbb{R}\right) $, then

where $ \mu\in(0, 1] $, $ s\in [\xi, \delta] $ and $ B(\mu) > 0 $ is a normalization function, and

Proof. Let $ \varphi\in{\bf XP}_{m_1}\left({\bf T}, \mathbb{R}\right) $ and $ \upsilon\in{\bf XP}_{m_2}\left({\bf T}, \mathbb{R}\right) $. Then for $ \varsigma\in[0, 1] $, we have:

and

Multiplying (2.6) and (2.7) gives:

This implies that

Integrating both sides of (2.8) with respect to $ \varsigma $ over $ [0, 1] $ results to:

That is,

Now, multiplying (2.9) by $ \frac{\mu(\delta- \xi)}{2B(\mu)} $ and then adding $ \frac{2(1-\mu)}{B(\mu)} \varphi(s) \upsilon(s) $ to the result to obtain:

Hence,

from which we get the intended inequality.

Remark 2. Set $ m_1 = m_2 = 1 $ in Theorem 5. Then we recover [7,Theorem 3] .

2.2.   Inequalities for $ m $-polynomial harmonically convex functions

In this subsection, we present some new Hermite–Hadamard type results involving the $ \zeta $-Riemann–Liouville fractional integral operators.

Theorem 6. Let $ \varphi:{\bf S}\to {\mathbb{R}}^{+} $ be a Lebesgue integrable function on $ [\xi, \delta] $ with $ 0 < \xi < \delta $ and $ \xi, \delta\in {\bf S} $. If $ \varphi\in{\bf HXP}_m\left({\bf S}, {\mathbb{R}}^{+}\right) $ and $ \zeta, \epsilon > 0 $, then

where $ \tilde{ \varphi}(r) = \frac{1}{r} $ and $ \mathcal{B} $ is the beta function defined by (1.3).

Proof. Given that $ \varphi\in{\bf HXP}_m\left({\bf S}, {\mathbb{R}}^{+}\right) $, we get the following relation:

This implies that for all $ q, z\in {\bf S} $:

Now, let $ q = \frac{ \xi \delta}{ \varsigma \xi+(1- \varsigma) \delta} $ and $ z = \frac{ \xi \delta}{ \varsigma \delta+(1- \varsigma) \xi} $. Then (2.10) becomes:

Multiplying both sides of (2.11) by $ \varsigma^{\frac{ \epsilon}{ \zeta}-1} $ and integrating with respect to $ \varsigma $ over $ [0, 1] $, we get:

where $ \tilde{ \varphi}(r) = \frac{1}{r} $. This implies that

Next, substituting $ q = \xi $ and $ z = \delta $ in (1.2) gives

Reversing the role of $ \xi $ and $ \delta $ in (2.13) produces:

If we now add (2.13) and (2.15), multiply the resulting inequality by $ \varsigma^{\frac{ \epsilon}{ \zeta}-1} $ and integrate with respect to $ \varsigma\in [0, 1] $, then we obtain:

From (2.15), we get:

Combining (2.12) and (2.16), we get the desired result.

Remark 3. If we take $ \epsilon = \zeta = 1 $, then Theorem 6 reduces to Theorem 2 . If, on the other hand, we let $ m = 1 $, then we get from Theorem 6 the following corollary:

Corollary 1. Let $ \varphi:{\bf S}\to {\mathbb{R}}^{+} $ be a Lebesgue integrable function on $ [\xi, \delta] $ with $ \xi < \delta $ and $ \xi, \delta\in {\bf S} $. If $ \varphi $ is harmonically convex and $ \zeta, \epsilon > 0 $, then

Theorem 7. Let $ \varphi, \upsilon:{\bf S}\to {\mathbb{R}}^{+} $ be two functions such that $ \varphi \upsilon $ is Lebesgue integrable function on $ [\xi, \delta] $ with $ 0 < \xi < \delta $ and $ \xi, \delta\in {\bf S} $. If $ \varphi\in{\bf HXP}_{m_1}\left({\bf S}, {\mathbb{R}}^{+}\right) $, $ \upsilon\in{\bf HXP}_{m_2}\left({\bf S}, {\mathbb{R}}^{+}\right) $ and $ \zeta, \epsilon > 0 $, then

where $ \mathcal{D}(\xi, \delta): = \varphi(\xi) \upsilon(\xi)+ \varphi(\delta) \upsilon(\delta) $, $ \mathcal{F}(\xi, \delta): = \varphi(\xi) \upsilon(\delta)+ \varphi(\delta) \upsilon(\xi) $, $ \tilde{ \varphi} $ is as defined in Theorem 6, and $ \Delta _j(\varsigma) $, $ j = \overline{1, 4} $, as defined in Theorem 5.

Proof. Given that $ \varphi\in{\bf HXP}_{m_1}\left({\bf S}, {\mathbb{R}}^{+}\right) $ and $ \upsilon\in{\bf HXP}_{m_2}\left({\bf S}, {\mathbb{R}}^{+}\right) $, we get:

and

This implies:

This gives:

Similarly, we also have

Adding (2.17) and (2.18), we get

Now, multiplying both sides of (2.19) by $ \varsigma^{\frac{ \epsilon}{ \zeta}-1} $ and integrating with respect to $ \varsigma $ over $ [0, 1] $, gives:

Hence, this completes the proof.

Corollary 2. Let $ \varphi, \upsilon:{\bf S}\to {\mathbb{R}}^{+} $ be two functions such that $ \varphi \upsilon $ is Lebesgue integrable function on $ [\xi, \delta] $ with $ 0 < \xi < \delta $ and $ \xi, \delta\in {\bf S} $. If $ \varphi $ and $ \upsilon $ are harmonically convex and $ \zeta, \epsilon > 0 $, then

Proof. Let $ m_1 = m_2 = 1 $. Then, $ \Delta _1(\varsigma) = \varsigma^2 $, $ \Delta _2(\varsigma) = \Delta _3(\varsigma) = \varsigma- \varsigma^2 $ and $ \Delta _4(\varsigma) = 1-2 \varsigma+ \varsigma^2 $. The intended result follows by using Theorem 7.

Theorem 8. Let $ \varphi, \upsilon:{\bf S}\to {\mathbb{R}}^{+} $ be two functions such that $ \varphi \upsilon $ is Lebesgue integrable function on $ [\xi, \delta] $ with $ 0 < \xi < \delta $ and $ \xi, \delta\in {\bf S} $. If $ \varphi\in{\bf HXP}_{m_1}\left({\bf S}, {\mathbb{R}}^{+}\right) $, $ \upsilon\in{\bf HXP}_{m_2}\left({\bf S}, {\mathbb{R}}^{+}\right) $ and $ \zeta, \epsilon > 0 $, then

where $ \tilde{ \varphi} $ is defined in Theorem 6, $ \Lambda_{m}(\varsigma) = \frac{1}{m}\sum_{\vartheta = 1}^m\left[1-(1- \varsigma)^\vartheta\right] $ and $ \tilde{\Lambda}_{m}(\varsigma) = \frac{1}{m}\sum_{\vartheta = 1}^m\left[1- \varsigma^\vartheta\right] $.

Proof. We start by noticing that:

Now, let $ \varsigma\in[0, 1] $. Hence, from (2.11), one gets:

and

Now,

This implies that

Multiplying both sides of (2.20) by $ \varsigma^{\frac{ \epsilon}{ \zeta}-1} $ and integrating with respect to $ \varsigma $ over $ [0, 1] $ to get:

The required result follows.

Corollary 3. Let $ \varphi, \upsilon:{\bf S}\to {\mathbb{R}}^{+} $ be two functions such that $ \varphi \upsilon $ is Lebesgue integrable function on $ [\xi, \delta] $ with $ 0 < \xi < \delta $ and $ \xi, \delta\in {\bf S} $. If $ \varphi $ and $ \upsilon $ are harmonically convex and $ \zeta, \epsilon > 0 $, then

Proof. Let $ m_1 = m_2 = 1 $. Then, $ \Lambda_{m_1}(\varsigma) = \Lambda_{m_2}(\varsigma) = \varsigma $ and $ \tilde{\Lambda}_{m_1}(\varsigma) = \tilde{\Lambda}_{m_2}(\varsigma) = 1- \varsigma $. The intended result follows by using Theorem 8.

3.   Conclusion

Utilizing the Caputo–Fabrizio and generalized Riemann–Liouville fractional integral operators, we proved some inequalities of the Hermite–Hadamard kinds for $ m $-polynomial convex and harmonically convex functions. Our results generalize, extend and complement results in ^[7,9,31].

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.