Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity

1.   Introduction and preliminaries

Convexity is very important in the field of mathematical analysis and optimization theory. It is a basic concept in mathematics which has been extended and generalized in different ways by using various techniques. For example one of the generalizations is exponentially $(\alpha, h-m)$-convexity, that contains $(\alpha, h-m)$-convexity, exponentially $(h-m)$-convexity, $(h-m)$-convexity, exponentially $(\alpha, m)$-convexity, $(\alpha, m)$-convexity and several related convexities.

Definition 1. ^[1] Let $J\subseteq\mathbb{R}$ be an interval containing $(0, 1)$ and let $h: J\rightarrow\mathbb{R}$ be a non-negative function. Then a function $\eta: I\rightarrow\mathbb{R}$ (where $I\subseteq\mathbb{R}$ is an interval) is said to be exponentially $(\alpha, h-m)$-convex, if inequality (1.1) must holds for all $\alpha, m\in[0, 1]$, ${a_1}, {a_2} \in I$, $\tau\in(0, 1)$ and $\varsigma\in\mathbb{R}$:

If we put $\alpha = 1$ in (1.1), then we get the following definition of exponentially $(h-m)$-convex functions:

Definition 2. Let $J\subseteq\mathbb{R}$ be an interval containing $(0, 1)$ and let $h: J\rightarrow\mathbb{R}$ be a non-negative function. Then a function $\eta: I\rightarrow\mathbb{R}$ (where $I\subseteq\mathbb{R}$ is an interval) is said to be exponentially $(h-m)$-convex, if inequality (1.2) must holds for all $m\in[0, 1]$, ${a_1}, {a_2} \in I$, $\tau\in(0, 1)$ and $\varsigma\in\mathbb{R}$:

If we put $h(\tau) = \tau$ in (1.1), then we get the following definition of exponentially $(\alpha, m)$-convex functions:

Definition 3. A function $\eta: I\rightarrow\mathbb{R}$ (where $I\subseteq\mathbb{R}$ is an interval) is said to be exponentially $(\alpha, m)$-convex, if inequality (1.3) must holds for all $\alpha, m\in[0, 1]$, ${a_1}, {a_2} \in I$, $\tau\in(0, 1)$ and $\varsigma\in\mathbb{R}$:

Remark 1. 1. If we fix $\alpha = 1$ and $h(\tau) = \tau^s$ in (1.1), we recover the definition of exponentially $(s, m)$-convexity defined by Qiang et al. in ^[2].

2. If we fix $\alpha = m = 1$ and $h(\tau) = \tau^s$ in (1.1), we recover the definition of exponentially $s$-convexity defined by Mehreen et al. in ^[3].

3. If we fix $\alpha = m = 1$ and $h(\tau) = \tau$ in (1.1), we recover the definition of exponentially convexity defined by Awan et al. in ^[4].

4. If we fix $\varsigma = 0$ in (1.1), we recover the definition of $(\alpha, h-m)$-convexity defined by Farid et al. in ^[5].

5. If we fix $\varsigma = \alpha = 0$ and $\alpha = 1$ in (1.1), we recover the definition of $(h-m)$-convexity defined by Özdemir et al. in ^[6].

6. If we fix $\varsigma = 0$ and $h(\tau) = \tau$ in (1.1), we recover the definition of $(\alpha, m)$-convexity defined by Mihesan in ^[7].

7. If we fix $\varsigma = 0$, $\alpha = 1$ and $h(\tau) = \tau^s$ in (1.1), we recover the definition of $(s, m)$-convexity defined by Efthekhari in ^[8].

8. If we fix $\varsigma = 0$, $\alpha = m = 1$ and $h(\tau) = \tau^s$ in (1.1), we recover the definition of $s$-convexity defined by Hudzik and Maligranda in ^[9].

9. If we fix $\varsigma = 0$, $\alpha = 1$ and $h(\tau) = \tau$ in (1.1), we recover the definition of $m$-convexity defined by Toader in ^[10].

10. If we fix $\varsigma = 0$ and $\alpha = m = 1$in (1.1), we recover the definition of $h$-convexity defined by Varosanec in ^[11].

11. If we fix $\varsigma = 0$, $\alpha = m = 1$ and $h(\tau) = \tau$ in (1.1), we recover the definition of convexity.

A convex function is elegantly interpreted in the coordinate plane by the well known Hermite-Hadamard inequality ^[12], stated as follows:

Theorem 1.1. Let $\eta:[{a_1}, {a_2}]\rightarrow \mathbb{R}$ be a convex function such that ${a_1} < {a_2}$. Then following inequality holds:

The Hermite-Hadamard inequality is generalized in various ways by using different fractional integral operators (see, for example ^{[3,4,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]}). In this paper we will further generalize this inequality by using a new generalized convexity and fractional integral operators containing an extended generalized Mittag-Leffler function. The results of this paper also generalize results of ^{[17,18,19,20,22,24,25,29,30]}.

In ^[31], Andrić et al. defined the generalized fractional integral operators containing generalized Mittag-Leffler function as follows:

Definition 4. Let $\kappa, \theta, \delta, l, \omega, c\in \mathbb{C}$, $\Re(\theta), \Re(\delta), \Re(l) > 0$, $\Re(c) > \Re(\omega) > 0$ with $p\geq0$, $r > 0$ and $0 < q\leq r+\Re(\theta)$. Let $\eta\in L_{1}[{a_1}, {a_2}]$ and $\psi\in[{a_1}, {a_2}].$ Then the generalized fractional integral operators $\Upsilon_{\theta, \delta, l, \kappa, {a_1}^{+}}^{\omega, r, q, c}\eta$ and $\Upsilon_{\theta, \delta, l, \kappa, {a_2}^{-}}^{\omega, r, q, c}\eta$ are defined by:

where $E_{\theta, \delta, l}^{\omega, r, q, c}(\tau; p)$ is the generalized Mittag-Leffler function defined as follows:

In ^[32], Farid defined the following unified integral operators:

Definition 5. Let $\eta, \mu: [{a_1}, {a_2}]\rightarrow \mathbb{R}$, (with $0 < {a_1} < {a_2}$) be two functions such that $\eta$ is positive and integrable on $[{a_1}, {a_2}]$ and $\mu$ is differentiable and strictly increasing on $[{a_1}, {a_2}]$. Also, let $\frac{\gamma}{\psi}$ be an increasing function on $[{a_1}, \infty)$ and $\kappa, \delta, l, \omega, c\in \mathbb{C}$, $\Re(\delta), \Re(l) > 0$, $\Re(c) > \Re(\omega) > 0$ with $p\geq0$, $\theta, r > 0$ and $0 < q\leq r+\theta$. Then for $\psi\in[{a_1}, {a_2}]$ the integral operators $_{\mu}\Upsilon_{\theta, \delta, l, {a_1}^{+}}^{\gamma, \omega, r, q, c}\eta$ and $_{\mu}\Upsilon_{\theta, \delta, l, {a_2}^{-}}^{\gamma, \omega, r, q, c}\eta$ are defined by:

If we put $\gamma(\psi) = \psi^\delta$ in (1.6) and (1.7), then we get the following generalized fractional integral operators containing Mittag-Leffler function:

Definition 6. Let $\eta, \mu: [{a_1}, {a_2}]\rightarrow \mathbb{R}$, (with $0 < {a_1} < {a_2}$) be two functions such that $\eta$ is positive and integrable on $[{a_1}, {a_2}]$ and $\mu$ is differentiable and strictly increasing on $[{a_1}, {a_2}]$. Also, let $\kappa, \delta, l, \omega, c\in \mathbb{C}$, $\Re(\delta), \Re(l) > 0$, $\Re(c) > \Re(\omega) > 0$ with $p\geq0$, $\theta, r > 0$ and $0 < q\leq r+\theta$. Then for $\psi\in[{a_1}, {a_2}]$ the integral operators $_{\mu}\Upsilon_{\theta, \delta, l, \kappa, {a_1}^{+}}^{\omega, r, q, c}\eta$ and $_{\mu}\Upsilon_{\theta, \delta, l, \kappa, {a_2}^{-}}^{ \omega, r, q, c}\eta$ are defined by:

Remark 2. Operators (1.8) and (1.9) are the generalizations of the following fractional integral operators:

1. Choosing $\mu(\psi) = \psi$, we recover the fractional integral operators defined in (1.4) and (1.5).

2. Choosing $\mu(\psi) = \psi$ and $p = 0$, we recover the fractional integral operators defined by Salim-Faraj in ^[33].

3. Choosing $\mu(\psi) = \psi$ and $l = r = 1$, we recover the fractional integral operators defined by Rahman et al. in ^[34].

4. Choosing $\mu(\psi) = \psi$, $p = 0$ and $l = r = 1$, we recover the fractional integral operators defined by Srivastava-Tomovski in ^[35].

5. Choosing $\mu(\psi) = \psi$, $p = 0$ and $l = r = q = 1$, we recover the fractional integral operators defined by Prabhakar in ^[36].

6. Choosing $\mu(\psi) = \psi$ and $\kappa = p = 0$, we recover the Riemann-Liouville fractional integral operators.

In ^[26], Mehmood et al. given the following formulas which we will use frequently:

The aim of this paper is to establish the generalized Hermite-Hadamard inequalities for exponentially $(\alpha, h-m)$-convex functions, exponentially $(h-m)$-convex functions and exponentially $(\alpha, m)$-convex functions. These inequalities are produced by using the generalized fractional integral operators (1.8) and (1.9) containing Mittag-Leffler function via a monotone increasing function. These inequalities lead to produce the Hermite-Hadamard inequalities for various kinds of convexities (see Remark 1) and well-known fractional integral operators (see Remark 2).

In the upcoming section we prove the Hermite-Hadamard inequalities for generalized fractional integral operators (1.8) and (1.9) via exponentially $(\alpha, h-m)$-convex functions. Further we present them for generalized fractional integral operators (1.8) and (1.9) via exponentially $(h-m)$-convex functions. Also we give these inequalities for exponentially $(\alpha, m)$-convex functions.

2.   Fractional Hermite-Hadamard inequalities for exponentially $(\alpha, h-m)$-convex functions

First we give the following Hermite-Hadamard inequality for exponentially $(\alpha, h-m)$-convex functions via further generalized fractional integral operators.

Theorem 2.1. Let $\eta: [{a_1}, m{a_2}]\subset[0, \infty)\to\mathbb{R}$, $0 < {a_1} < m{a_2}$ be a positive, integrable and exponentially $(\alpha, h-m)$-convex function. Let $\mu: [{a_1}, m{a_2}]\to\mathbb{R}$ be differentiable and strictly increasing. Then for generalized fractional integral operators, the following inequalities hold:

where $D(\varsigma) = e^{\varsigma\mu(a_2)}$ for $\varsigma < 0$, $D(\varsigma) = e^{\varsigma\mu(a_1)}$ for $\varsigma\geq0$.

Proof. From exponentially $(\alpha, h-m)$-convexity of $\eta$, we have

Multiplying (2.2) by $\tau^{\delta-1}E_{\theta, \delta, l}^{\omega, r, q, c}(\kappa \tau^{\theta}; p)$ and integrating over $[0, 1]$, we have

Putting $\mu(\psi) = \tau\mu(a_1)+m(1-\tau)\mu({a_2})$ and $\mu(\phi) = (1-\tau)\frac{\mu(a_1)}{m}+\tau\mu({a_2})$ in (2.3), then by using (1.8), (1.9) and (1.10), the first inequality of (2.1) can be achieved.

Again from exponentially $(\alpha, h-m)$-convexity of $\eta$, we have the following inequalities:

Multiplying (2.4) by $h\left(\frac{1}{2^\alpha}\right)$ and (2.5) by $mh\left(\frac{2^\alpha-1}{2^\alpha}\right)$, then adding resulting inequalities, we have

Now multiplying (2.6) by $\tau^{\delta-1}E_{\theta, \delta, l}^{\omega, r, q, c}(\kappa \tau^{\theta}; p)$ and integrating over $[0, 1]$, we have

Putting $\mu(\psi) = \tau\mu(a_1)+m(1-\tau)\mu({a_2})$ and $\mu(\phi) = (1-\tau)\frac{\mu(a_1)}{m}+\tau\mu({a_2})$ in (2.7), then by using (1.8) and (1.9), the second inequality of (2.1) can be achieved.

If we choose $\alpha = 1$ in (2.1), then we get following Hermite-Hadamard inequality for exponentially $(h-m)$-convex functions.

Corollary 2.2. Let $\eta: [{a_1}, m{a_2}]\subset[0, \infty)\to\mathbb{R}$, $0 < {a_1} < m{a_2}$ be a positive, integrable and exponentially $(h-m)$-convex functions. Let $\mu: [{a_1}, m{a_2}]\to\mathbb{R}$ be differentiable and strictly increasing. Then for generalized fractional integral operators, the following inequalities hold:

where $D(\varsigma) = e^{\varsigma\mu(a_2)}$ for $\varsigma < 0$, $D(\varsigma) = e^{\varsigma\mu(a_1)}$ for $\varsigma\geq0$.

If we choose $h(\tau) = \tau$ in (2.1), then we get following Hermite-Hadamard inequality for exponentially $(\alpha, m)$-convex functions.

Corollary 2.3. Let $\eta: [{a_1}, m{a_2}]\subset[0, \infty)\to\mathbb{R}$, $0 < {a_1} < m{a_2}$, be a positive, integrable and exponentially $(\alpha, m)$-convex function. Let $\mu: [{a_1}, m{a_2}]\to\mathbb{R}$ be differentiable and strictly increasing. Then for generalized fractional integral operators, the following inequalities hold:

where $D(\varsigma) = e^{\varsigma\mu(a_2)}$ for $\varsigma < 0$, $D(\varsigma) = e^{\varsigma\mu(a_1)}$ for $\varsigma\geq0$.

Remark 3. 1. If we choose $\varsigma = p = 0$, $\alpha = m = 1$, $\mu(\psi) = \psi$ and $h(\tau) = \tau$ in (2.1), we recover the result in [17,Theorem 2.1].

2. If we choose $\varsigma = p = 0$, $\alpha = 1$, $\mu(\psi) = \psi$ and $h(\tau) = \tau$ in (2.1), we recover the result in [18,Theorem 3].

3. If we choose $\varsigma = 0$, $\alpha = 1$ and $\mu(\psi) = \psi$ in (2.1), we recover the result in [24,Theorem 2.1].

4. If we choose $\varsigma = 0$, $\alpha = m = 1$, $\mu(\psi) = \psi$ and $h(\tau) = \tau$ in (2.1), we recover the result in [25,Theorem 2.1].

5. If we choose $\varsigma = 0$, $\alpha = 1$, $\mu(\psi) = \psi$ and $h(\tau) = \tau$ in (2.1), we recover the result in [25,Theorem 3.1].

6. If we choose $\varsigma = p = \kappa = 0$, $\alpha = m = 1$, $\mu(\psi) = \psi$ and $h(\tau) = \tau$ in (2.1), we recover the result in [29,Theorem 2].

In the following we give another version of the Hermite-Hadamard inequality for exponentially $(\alpha, h-m)$-convex functions via further generalized fractional integral operators.

Theorem 2.4. Let $\eta: [{a_1}, m{a_2}]\subset[0, \infty)\to\mathbb{R}$, $0 < {a_1} < m{a_2}$ be a positive, integrable and exponentially $(\alpha, h-m)$-convex functions. Let $\mu: [{a_1}, m{a_2}]\to\mathbb{R}$ be differentiable and strictly increasing. Then for generalized fractional integral operators, the following inequalities hold:

where $D(\varsigma) = e^{\varsigma\mu(a_2)}$ for $\varsigma < 0$, $D(\varsigma) = e^{\varsigma\mu(a_1)}$ for $\varsigma\geq0$.

Proof. From exponentially $(\alpha, h-m)$-convexity of $\eta$, we have

Multiplying (2.11) by $\tau^{\delta-1}E_{\theta, \delta, l}^{\omega, r, q, c}(\kappa \tau^{\theta}; p)$ and integrating over $[0, 1]$, we have

Putting $\mu(\psi) = \frac{\tau}{2}\mu(a_1)+m\frac{(2-\tau)}{2}\mu({a_2})$ and $\mu(\phi) = \frac{(2-\tau)}{2}\frac{\mu({a_1})}{m}+\frac{\tau}{2}\mu({a_2})$ in (2.12), then by using (1.8), (1.9) and (1.10), the first inequality of (2.9) can be achieved.

Again from exponentially $(\alpha, h-m)$-convexity of $\eta$, we have the following inequalities:

Multiplying (2.13) by $h\left(\frac{1}{2^\alpha}\right)$ and (2.14) by $mh\left(\frac{2^\alpha-1}{2^\alpha}\right)$, then adding resulting inequalities, we have

Now multiplying (2.15) by $\tau^{\delta-1}E_{\theta, \delta, l}^{\omega, r, q, c}(\kappa \tau^{\theta}; p)$ and integrating over $[0, 1]$, we have

Putting $\mu(\psi) = \frac{\tau}{2}\mu(a_1)+m\frac{(2-\tau)}{2}\mu({a_2})$ and $\mu(\phi) = \frac{\tau}{2}\mu({a_2})+\frac{(2-\tau)}{2}\frac{\mu(a_1)}{m}$ in (2.16), then by using (1.8) and (1.9), the second inequality of (2.10) can be achieved.

If we choose $\alpha = 1$ in (2.9), then we get following Hermite-Hadamard inequality for exponentially $(h-m)$-convex functions.

Corollary 2.5. Let $\eta: [{a_1}, m{a_2}]\subset[0, \infty)\to\mathbb{R}$, $0 < {a_1} < m{a_2}$ be a positive, integrable and exponentially $(h-m)$-convex functions. Let $\mu: [{a_1}, m{a_2}]\to\mathbb{R}$ be differentiable and strictly increasing. Then for generalized fractional integral operators, the following inequalities hold:

where $D(\varsigma) = e^{\varsigma\mu(a_2)}$ for $\varsigma < 0$, $D(\varsigma) = e^{\varsigma\mu(a_1)}$ for $\varsigma\geq0$.

If we choose $h(\tau) = \tau$ in 2.9, then we get following Hermite-Hadamard inequality for exponentially $(\alpha, m)$-convex functions.

Corollary 2.6. Let $\eta: [{a_1}, m{a_2}]\subset[0, \infty)\to\mathbb{R}$, $0 < {a_1} < m{a_2}$ be a positive, integrable and exponentially $(\alpha, m)$-convex functions. Let $\mu: [{a_1}, m{a_2}]\to\mathbb{R}$ be differentiable and strictly increasing. Then for generalized fractional integral operators, the following inequalities hold:

where $D(\varsigma) = e^{\varsigma\mu(a_2)}$ for $\varsigma < 0$, $D(\varsigma) = e^{\varsigma\mu(a_1)}$ for $\varsigma\geq0$.

Remark 4. 1. If we choose $\varsigma = p = 0$, $\alpha = 1$ and $\mu(\psi) = \psi$ in (2.9), we recover the result in [19,Theorem 3.10].

2. f we choose $\varsigma = p = \kappa = 0$, $\alpha = 1$ and $\mu(\psi) = \psi$ in (2.9), we recover the result in [20,Theorem 2.1].

3. If we choose $\varsigma = 0$, $\alpha = 1$ and $\mu(\psi) = \psi$ in (2.9), we recover the result in [22,Theorem 2.11].

4. f we choose $\varsigma = p = \kappa = 0$, $\alpha = m = 1$ and $\mu(\psi) = \psi$ in (2.9), we recover the result in [30,Theorem 4].

3.   Conclusions

In this article, we have proposed the generalized fractional Hermite-Hadamard inequalities for a generalized convexity. The results are applicable for fractional integral operators containing Mittag-Leffler functions in their kernels. Also they hold for exponentially $(\alpha, h-m)$-convex functions, exponentially $(h-m)$-convex functions and exponentially $(\alpha, m)$-convex functions which are further linked with several known classes of convex functions. The readers can deduce a plenty of fractional integral inequalities of their choice of fractional integral operators from Remark 2 and convex function of any kind from Remark 1.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, 11601485).

Conflict of interest

The authors do not have any competing interest