1 Introduction, definitions, and preliminaries

Convex functions are very important in the field of integral inequalities. A lot of fractional integral inequalities and novel results have been established due to convex functions (for more details, see [1, 8, 13, 14]).

Definition 1

A function \(f: I\rightarrow\mathbb{R}\), where I is an interval in \(\mathbb{R}\), is said to be a convex function if

$$ f\bigl(tx+(1-t)y\bigr)\leq tf(x)+(1-t)f(y) $$
(1)

holds for \(t\in[0,1]\) and \(x,y\in I\).

A convex function \(f: I\rightarrow\mathbb{R}\) is also equivalently defined by the Hadamard inequality

$$ f \biggl(\frac{a+b}{2} \biggr)\leq\frac{1}{b-a} \int_{a}^{b}f(x)\,dx \leq\frac{f(a)+f(b)}{2}, $$

where \(a,b\in I\), \(a< b\).

The concept of m-convexity was introduced in [17] and since then many properties, especially inequalities, have been obtained for this class of functions (see [3, 6, 7, 18]).

Definition 2

A function \(f:[0,b]\rightarrow\mathbb{R}\), \(b>0\) is called m-convex, where \(m\in[0,1]\), if for every \(x,y\in[0,b]\) and \(t\in[0,1]\), we have

$$ f \bigl(tx+m(1-t)y \bigr)\leq tf(x)+m(1-t)f(y). $$

For \(m=1\), we recapture the definition of convex functions, and for \(m=0\), the definition of star-shaped functions defined on \([0,b]\). We recall that a function \(f:[0,b]\to\mathbb{R}\) is called star-shaped if

$$f(tx)\leq tf(x)\quad \mbox{for all } t\in[0,1] \mbox{ and } x\in[0,b]. $$

If we denote by \(K_{m}(b)\) the set of m-convex functions defined on \([0,b]\) for which \(f(0)<0\), then

$$K_{1}(b) \subset K_{m}(b) \subset K_{0}(b), $$

whenever \(m\in(0,1)\). Note that in the class \(K_{1}(b)\) there are only convex functions \(f:[0,b]\to\mathbb{R}\) for which \(f(0)\leq0\) (see [4]), while \(k_{0}(b)\) contains star-shaped functions.

Example 1.1

([6])

The function \(f:[0,\infty)\to\mathbb{R}\), given by

$$f(x)=\frac{1}{12} \bigl(x^{4}-5x^{3}+9x^{2}-5x \bigr), $$

is a \(\frac{16}{17}\)-convex function but it is not m-convex for any \(m\in(\frac{16}{17},1]\).

For more results and inequalities related to m-convex functions, one can consult, for example, [3, 6, 7] along with the references therein.

Recently in [2] Andrić et al. defined an extended generalized Mittag-Leffler function \(E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(\cdot;p)\) as follows.

Definition 3

([2])

Let \(\mu,\alpha,l,\gamma,c\in\mathbb{C}\), \(\Re(\mu),\Re(\alpha ),\Re(l)>0\), \(\Re(c)>\Re(\gamma)>0\) with \(p\geq0\), \(\delta>0\), and \(0< k\leq\delta+\Re(\mu)\). Then the extended generalized Mittag-Leffler function \(E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p)\) is defined by

$$ E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p)= \sum _{n=0}^{\infty }\frac{\beta_{p}(\gamma+nk,c-\gamma)}{\beta(\gamma,c-\gamma)} \frac{(c)_{nk}}{\Gamma(\mu n +\alpha)} \frac{t^{n}}{(l)_{n \delta}}, $$
(2)

where \(\beta_{p}\) is the generalized beta function defined by

$$ \beta_{p}(x,y) = \int_{0}^{1}t^{x-1}(1-t)^{y-1}e^{-\frac {p}{t(1-t)}}\,dt $$

and \((c)_{nk}\) is the Pochhammer symbol defined as \((c)_{nk}=\frac {\Gamma(c+nk)}{\Gamma(c)}\).

In [2] properties of the generalized Mittag-Leffler function are discussed, and it is given that \(E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p)\) is absolutely convergent for \(k<\delta+\Re(\mu)\). Let S be the sum of series of absolute terms of the Mittag-Leffler function \(E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p)\), then we have \(\vert E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p) \vert \leq S\). We use this property of Mittag-Leffler function in our results where we need.

The corresponding left and right sided extended generalized fractional integral operators are defined as follows.

Definition 4

([2])

Let \(\omega,\mu,\alpha,l,\gamma,c\in\mathbb{C}\), \(\Re(\mu),\Re(\alpha),\Re(l)>0\), \(\Re(c)>\Re(\gamma)>0\) with \(p\geq0\), \(\delta>0\) and \(0< k\leq \delta+\Re(\mu)\). Let \(f\in L_{1}[a,b]\) and \(x\in[a,b]\). Then the extended generalized fractional integral operators \(\epsilon_{\mu,\alpha,l,\omega,a^{+}}^{\gamma,\delta,k,c}f \) and \(\epsilon_{\mu,\alpha,l,\omega,b^{-}}^{\gamma,\delta,k,c}f\) are defined by

$$ \bigl( \epsilon_{\mu,\alpha,l,\omega,a^{+}}^{\gamma,\delta,k,c}f \bigr) (x;p)= \int_{a}^{x}(x-t)^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega(x-t)^{\mu};p\bigr)f(t)\,dt $$
(3)

and

$$ \bigl( \epsilon_{\mu,\alpha,l,\omega,b^{-}}^{\gamma,\delta,k,c}f \bigr) (x;p)= \int_{x}^{b}(t-x)^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega(t-x)^{\mu};p\bigr)f(t)\,dt. $$
(4)

From extended generalized fractional integral operators, we have

$$\begin{aligned} & (\epsilon_{\mu,\alpha,l,\omega,a^{+}}^{\gamma,\delta,k,c}1 )(x;p) \\ &\quad = \int_{a}^{x}(x-t)^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(w(x-t)^{\mu};p)\,dt \\ &\quad = \int_{a}^{x}(x-t)^{\alpha-1}\sum_{n=0}^{\infty}\frac{\mathrm{B}_{p}(\gamma+nk,c-\gamma)}{\mathrm{B}(\gamma,c-\gamma)}\frac {(c)_{nk}}{\Gamma(\mu n+\alpha)}\frac{w^{n}(x-t)^{\mu n}}{(l)_{n\delta}}\,dt \\ &\quad =\sum_{n=0}^{\infty}\frac{\mathrm{B}_{p}(\gamma+nk,c-\gamma )}{\mathrm{B}(\gamma,c-\gamma)}\frac{(c)_{nk}}{\Gamma(\mu n+\alpha)}\frac{w^{n} }{(l)_{n\delta}} \int_{a}^{x}(x-t)^{\mu n+\alpha-1}\,dt \\ &\quad = (x-a )^{\alpha}\sum_{n=0}^{\infty}\frac{\mathrm{B}_{p}(\gamma+nk,c-\gamma)}{\mathrm{B}(\gamma,c-\gamma)}\frac {(c)_{nk}}{\Gamma(\mu n+\alpha)}\frac{w^{n} }{(l)_{n\delta}} (x-a )^{\mu n}\frac{1}{\mu n + \alpha}. \end{aligned}$$

Hence

$$ \bigl(\epsilon_{\mu,\alpha,l,\omega,a^{+}}^{\gamma,\delta,k,c}1 \bigr) (x;p)= (x-a )^{\alpha} E_{\mu,\alpha +1,l}^{\gamma,\delta,k,c}\bigl(w(x-a)^{\mu};p \bigr), $$

and similarly

$$ \bigl(\epsilon_{\mu,\alpha,l,\omega,b^{-}}^{\gamma,\delta,k,c}1 \bigr) (x;p)= (b-x )^{\alpha} E_{\mu,\alpha +1,l}^{\gamma,\delta,k,c}\bigl(w(b-x)^{\mu};p \bigr). $$

We use the following notations in our results:

$$ C_{\alpha,a^{+}}(x;p)= \bigl(\epsilon_{\mu,\alpha,l,\omega,a^{+}}^{\gamma,\delta,k,c}1 \bigr) (x;p) $$
(5)

and

$$ C_{\alpha,b^{-}}(x;p)= \bigl(\epsilon_{\mu,\alpha,l,\omega,b^{-}}^{\gamma,\delta,k,c}1 \bigr) (x;p). $$
(6)

For more information related to Mittag-Leffler functions and corresponding fractional integral operators, the readers are referred to [9–12, 15, 16, 19].

In this paper we give general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function and deduce some results already published in [1, 5, 6, 8, 13]. Also we give a Hadamard type inequality for convex and m-convex functions by involving an extended Mittag-Leffler function.

2 Main results

Here we give some fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function and corresponding fractional integral operators given in (3) and (4). The following lemma is useful to establish the results.

Lemma 2.1

Let \(f:[a,mb]\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,mb]\) with \(0\leq a< mb\). Also let \(g:[a,mb]\rightarrow\mathbb{R}\) be a continuous function on \([a,mb]\), then the following identity for extended generalized fractional integral operators holds:

$$\begin{aligned} & \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl[f(a)+f(mb) \bigr] \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\quad = \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}f'(t)\,dt \\ &\qquad{} - \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha }f'(t)\,dt. \end{aligned}$$
(7)

Proof

On integrating by parts one can have

$$\begin{aligned} & \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha}f'(t)\,dt \\ &\quad = \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha}f(mb) \\ &\qquad {}-\alpha \int _{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k,c}\bigl(\omega t^{\mu };p \bigr)f(t)\,dt \end{aligned}$$
(8)

and

$$\begin{aligned} & \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha}f'(t)\,dt \\ &\quad =- \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha}f(a) \\ & \qquad {}+\alpha \int _{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha-1}g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k,c}\bigl(\omega t^{\mu };p \bigr)f(t)\,dt. \end{aligned}$$
(9)

Subtracting (9) from (8), we get (7) which is the required identity. □

If we take \(m=1\) in (7), then we get the following identity for a convex function.

Corollary 2.2

Let \(f:[a,b]\subseteq[0,\infty)\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,b]\) with \(a< b\). Also let \(g:[a,b]\rightarrow\mathbb{R}\) be continuous on \([a,b]\), then the following identity for extended generalized fractional integral operators holds:

$$\begin{aligned} & \biggl( \int_{a}^{b}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha}\bigl[f(a)+f(b) \bigr] \\ &\qquad{} -\alpha \int _{a}^{b} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k,c}\bigl(\omega t^{\mu };p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{b} \biggl( \int_{t}^{b}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c}\bigl( \omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha -1}g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega t^{\mu };p\bigr)f(t)\,dt \\ &\quad = \int_{a}^{b} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha}f'(t)\,dt \\ &\qquad{} - \int_{a}^{b} \biggl( \int_{t}^{b}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c}\bigl( \omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha}f'(t)\,dt. \end{aligned}$$
(10)

We use identity (7) to establish the following fractional integral inequality.

Theorem 2.3

Let \(f:[a,mb]\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,mb]\) with \(0\leq a< mb\). Also let \(g:[a,mb]\rightarrow\mathbb{R}\) be a continuous function on \([a,mb]\). If \(|f'|\) is an m-convex function on \([a,mb]\), then the following inequality for extended generalized fractional integral operators holds:

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ &\quad \leq\frac{(mb-a)^{\alpha+1} \Vert g \Vert _{\infty}^{\alpha}S^{\alpha }}{(\alpha+1)}\bigl( \bigl\vert f'(a) \bigr\vert +m \bigl\vert f'(b) \bigr\vert \bigr) \end{aligned}$$
(11)

for \(k<\delta+\Re(\mu)\) and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,mb]}|g(t)|\).

Proof

From Lemma 2.1, we have

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int _{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ & \quad \leq \int_{a}^{mb} \biggl\vert \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr\vert ^{\alpha } \bigl\vert f'(t) \bigr\vert \,dt \\ &\qquad{} + \int_{a}^{mb} \biggl\vert \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr\vert ^{\alpha} \bigl\vert f'(t) \bigr\vert \,dt. \end{aligned}$$
(12)

Using absolute convergence of the Mittag-Leffler function and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,b]}|g(t)|\), we have

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ & \quad \leq \Vert g \Vert _{\infty}^{\alpha}S^{\alpha} \biggl( \int _{a}^{mb}(t-a)^{\alpha} \bigl\vert f'(t) \bigr\vert \,dt+ \int_{a}^{mb}(mb-t)^{\alpha } \bigl\vert f'(t) \bigr\vert \,dt \biggr). \end{aligned}$$
(13)

Since \(|f'|\) is an m-convex function, we have

$$ \bigl\vert f'(t) \bigr\vert \leq \frac{mb-t}{mb-a} \bigl\vert f'(a) \bigr\vert +m \frac{t-a}{mb-a} \bigl\vert f'(b) \bigr\vert $$
(14)

for \(t\in[a,mb]\).

Using (14) in (13), we have

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ & \quad \leq \Vert g \Vert _{\infty}^{\alpha}S^{\alpha} \biggl( \int _{a}^{mb}(t-a)^{\alpha} \biggl( \frac{mb-t}{mb-a} \bigl\vert f'(a) \bigr\vert +m \frac {t-a}{mb-a} \bigl\vert f'(b) \bigr\vert \biggr)\,dt \\ &\qquad{} + \int_{a}^{mb}(mb-t)^{\alpha} \biggl( \frac {mb-t}{mb-a} \bigl\vert f'(a) \bigr\vert +m \frac{t-a}{mb-a} \bigl\vert f'(b) \bigr\vert \biggr)\,dt \biggr) . \end{aligned}$$
(15)

After simple calculation of the above inequality, we get (11) which is required. □

If we take \(m=1\) in (11), then we get the following result for a convex function.

Corollary 2.4

Let \(f:[a,b]\subseteq[0,\infty)\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,b]\) with \(a< b\). Also let \(g:[a,b]\rightarrow\mathbb{R}\) be a continuous function on \([a,b]\). If \(|f'|\) is a convex function on \([a,b]\), then the following inequality for extended generalized fractional integral operators holds:

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{b}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha}\bigl[f(a)+f(b) \bigr] \\ &\quad{} -\alpha \int_{a}^{b} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k,c}\bigl(\omega t^{\mu };p\bigr)f(t)\,dt \\ &\quad{} -\alpha \int_{a}^{b} \biggl( \int_{t}^{b}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c}\bigl( \omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha -1}g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega t^{\mu };p\bigr)f(t)\,dt \biggr\vert \\ & \leq\frac{(b-a)^{\alpha+1} \Vert g \Vert _{\infty}^{\alpha}S^{\alpha }}{(\alpha+1)}\bigl[ \bigl\vert f'(a) \bigr\vert + \bigl\vert f'(b) \bigr\vert \bigr] \end{aligned}$$
(16)

for \(k<\delta+\Re(\mu)\) and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,b]}|g(t)|\).

Remark 2.5

In Theorem 2.3.

  1. (i)

    If we put \(p=0\), then we get [6, Theorem 3.2].

  2. (ii)

    If we put \(\omega=p=0\) and \(m=1\), then we get [13, Theorem 6].

  3. (iii)

    If we take \(\omega=p=0\), \(m=1\), \(\alpha=\frac{\mu }{k}\), and \(g(s)=1\), then we get [8, Corollary 2.3].

  4. (iv)

    For \(g(s)=1\) along with \(\omega=p=0\), \(m=1\), and \(\alpha =\mu\), we get [13, Corollary 2].

Remark 2.6

In Corollary 2.4.

  1. (i)

    If we put \(p=0\), then we get [1, Theorem 3.2].

  2. (ii)

    If we put \(\omega=p=0\), then we get [13, Theorem 6].

  3. (iii)

    For \(\omega=p=0\), \(\alpha=\frac{\mu}{k}\), and \(g(s)=1\), we get [8, Corollary 2.3].

  4. (iv)

    For \(g(s)=1\) along with \(\omega=p=0\), we get [13, Corollary 2].

Next we give the following fractional integral inequality.

Theorem 2.7

Let \(f:[a,mb]\rightarrow\mathbb{R}\) be a differentiable function such that \(f\in L_{1}[a,mb]\) with \(0\leq a< mb\). Also let \(g:[a,mb]\rightarrow\mathbb{R}\) be a continuous function on \([a,mb]\). If \(|f'|^{q}\) is a convex function on \([a,mb]\), then for \(q>0\) the following inequality for extended generalized fractional integral operators holds:

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ &\quad \leq\frac{2(mb-a)^{\alpha+1} \Vert g \Vert _{\infty}^{\alpha}S^{\alpha }}{(\alpha p+1)^{\frac{1}{q}}} \biggl(\frac { \vert f'(a) \vert ^{q}+m \vert f'(b) \vert ^{q}}{2} \biggr)^{\frac{1}{q}} \end{aligned}$$
(17)

for \(k<\delta+\Re(\mu)\) and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,mb]}|g(t)|\) and \(\frac{1}{p}+\frac{1}{q}=1\).

Proof

From Lemma 2.1 and by using Hölder’s inequality, we have

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ & \quad \leq \biggl( \int_{a}^{mb} \biggl\vert \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr\vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{mb} \bigl\vert f'(t) \bigr\vert ^{q}\,dt \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \int_{a}^{mb} \biggl\vert \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr\vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{mb} \bigl\vert f'(t) \bigr\vert ^{q}\,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$
(18)

Using absolute convergence of the Mittag-Leffler function and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,b]}|g(t)|\), we have

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int _{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ & \quad \leq \Vert g \Vert _{\infty}^{\alpha}S^{\alpha} \biggl( \biggl( \int _{a}^{mb} \vert t-a \vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \\ &\qquad {}+ \biggl( \int _{a}^{mb} \vert mb-t \vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \biggr) \biggl( \int _{a}^{mb} \bigl\vert f'(t) \bigr\vert ^{q}\,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$
(19)

Since \(|f'(t)|^{q}\) is an m-convex function, we have

$$ \bigl\vert f'(t) \bigr\vert ^{q}\leq \frac{mb-t}{mb-a} \bigl\vert f'(a) \bigr\vert ^{q}+m \frac{t-a}{mb-a} \bigl\vert f'(b) \bigr\vert ^{q}. $$
(20)

Using (20) in (19), we have

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ & \quad \leq \Vert g \Vert _{\infty}^{\alpha}S^{\alpha} \biggl( \biggl( \int _{a}^{mb} \vert t-a \vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}}+ \biggl( \int _{a}^{mb} \vert mb-t \vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \biggr) \\ & \qquad {}\times \biggl( \int_{a}^{mb}\frac{mb-t}{mb-a} \bigl\vert f'(a) \bigr\vert ^{q}+m\frac {t-a}{mb-a} \bigl\vert f'(b) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} . \end{aligned}$$
(21)

After simple calculation of the above inequality, we get (17) which is required. □

If we take \(m=1\) in (17), then we get the following result for a convex function.

Corollary 2.8

Let \(f:[a,b]\subseteq[0,\infty)\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,b]\) with \(a< b\). Also let \(g:[a,b]\rightarrow\mathbb{R}\) be a continuous function on \([a,b]\). If \(|f'|^{q}\) is a convex function on \([a,b]\), then for \(q>0\) the following inequality for extended generalized fractional integral operators holds:

$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{b}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha}\bigl[f(a)+f(b) \bigr] \\ &\qquad{} -\alpha \int_{a}^{b} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha -1}g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega t^{\mu};p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{b} \biggl( \int_{t}^{b}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k,c}\bigl( \omega s^{\mu};p\bigr)\,ds \biggr)^{\alpha -1}g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k,c} \bigl(\omega t^{\mu };p\bigr)f(t)\,dt \biggr\vert \\ &\quad \leq\frac{2(b-a)^{\alpha+1} \Vert g \Vert _{\infty}^{\alpha}S^{\alpha }}{(\alpha p+1)^{\frac{1}{q}}} \biggl[\frac { \vert f'(a) \vert ^{q}+ \vert f'(b) \vert ^{q}}{2} \biggr]^{\frac{1}{q}} \end{aligned}$$
(22)

for \(k<\delta+\Re(\mu)\) and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,b]}|g(t)|\) and \(\frac{1}{p}+\frac{1}{q}=1\).

Remark 2.9

In Theorem 2.7.

  1. (i)

    If we put \(p=0\), then we get [6, Theorem 3.6].

  2. (ii)

    If we put \(\omega=p=0\) and \(m=1\), then we get [13, Theorem 7].

  3. (iii)

    If we take \(\omega=p=0\), \(m=1\) along with \(\alpha=\frac {\mu}{k}\), then we get [8, Theorem 2.5].

  4. (iv)

    If we take \(g(s)=1\), \(m=1\), and \(\omega=p=0\), then we get [5, Theorem 2.3].

  5. (v)

    If we put \(\omega=p=0\), \(m=1\), and \(\alpha=1\), then we get [5, Corollary 3].

Remark 2.10

In Corollary 2.8.

  1. (i)

    If we put \(p=0\), then we get [1, Theorem 3.5].

  2. (ii)

    If we put \(\omega=p=0\), then we get [13, Theorem 7].

  3. (iii)

    If we put \(\omega=p=0\), \(\alpha=1\), then we get [13, Corollary 3].

  4. (iv)

    If we take \(\omega=p=0\) along with \(\alpha=\frac{\mu }{k}\), then we get [8, Theorem 2.5].

  5. (v)

    If we take \(g(s)=1\) and \(\omega=p=0\), then we get [5, Theorem 2.3].

In the next result we give Hadamard type inequalities for m-convex functions via an extended Mittag-Leffler function.

Theorem 2.11

Let \(f:[a,mb]\rightarrow\mathbb{R}\) be a function such that \(f\in L_{1}[a,mb]\) with \(0\leq a< mb\). If f is m-convex on \([a,mb]\), then the following inequalities for extended generalized fractional integral operators hold:

$$\begin{aligned} &2f \biggl(\frac{a+mb}{2} \biggr)C_{\alpha, (\frac {a+mb}{2} )^{+}}(mb;p) \\ &\quad \leq \bigl( \epsilon_{\mu,\alpha,l,\omega',(\frac {a+mb}{2})^{+}}^{\gamma,\delta,k, c}f \bigr) (mb;p) + m^{\alpha+1} \bigl( \epsilon_{\mu,\alpha,l,m^{\mu}\omega',(\frac {a+mb}{2m})^{-}}^{\gamma,\delta,k, c}f \bigr) \biggl( \frac {a}{m};p \biggr) \\ &\quad \leq\frac{1}{mb-a} \biggl( f(a)-m^{2}f \biggl(\frac{a}{m^{2}} \biggr) \biggr)C_{\alpha+1, (\frac{a+mb}{2} )^{+}}(mb;p) \\ &\qquad{} +m^{\alpha+1} \biggl( f(b)+mf \biggl(\frac{a}{m^{2}} \biggr) \biggr)C_{\alpha, (\frac{a+mb}{2m} )^{-}} \biggl(\frac {a}{m};p \biggr), \end{aligned}$$
(23)

where \(\omega'=\frac{2^{\mu}\omega}{(mb-a)^{\mu}}\).

Proof

Since f is an m-convex function, we have

$$ 2f \biggl(\frac{a+mb}{2} \biggr)\leq f \biggl( \frac{t}{2}a+\frac {2-t}{2}mb \biggr)+mf \biggl(\frac{2-t}{2m}a+ \frac{t}{2}b \biggr). $$
(24)

Also from m-convexity of f, we have

$$\begin{aligned} &f \biggl(\frac{t}{2}a+m\frac{2-t}{2}b \biggr)+mf \biggl( \frac {2-t}{2m}a+\frac{t}{2}b \biggr) \\ &\quad \leq\frac{t}{2} \biggl(f(a)-m^{2}f \biggl(\frac{a}{m^{2}} \biggr) \biggr)+m \biggl(f(b)+mf \biggl(\frac{a}{m^{2}} \biggr) \biggr). \end{aligned}$$
(25)

Multiplying (24) by \(t^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c}(\omega t^{\mu}; p)\) on both sides and then integrating over \([0,1]\), we have

$$\begin{aligned} &2f \biggl(\frac{a+mb}{2} \biggr) \int_{0}^{1}t^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega t^{\mu}; p\bigr)\,dt \\ &\quad \leq \int_{0}^{1}t^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega t^{\mu}; p\bigr)f \biggl(\frac{t}{2}a+ \frac{2-t}{2}mb \biggr)\,dt \\ &\qquad{} +m \int_{0}^{1}t^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega t^{\mu}; p\bigr)f \biggl(\frac{2-t}{2m}a+ \frac{t}{2}b \biggr)\,dt. \end{aligned}$$
(26)

Putting \(u=\frac{t}{2}a+\frac{2-t}{2}mb\) and \(v=\frac {2-t}{2m}a+\frac{t}{2}b\) in (26), we have

$$\begin{aligned} &2f \biggl(\frac{a+mb}{2} \biggr) \int_{\frac {a+mb}{2}}^{mb}(mb-u)^{\alpha-1} E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega' (mb-u)^{\mu}; p\bigr)\,du \\ & \quad \leq \int_{\frac{a+mb}{2}}^{mb}(mb-u)^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega' (mb-u)^{\mu}; p\bigr)f(u)\,du \\ &\qquad{} +m^{\alpha+1} \int_{\frac{a}{m}}^{\frac{a+mb}{2m}} \biggl(v-\frac {a}{m} \biggr)^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \biggl(m^{\mu} \omega'\biggl(v-\frac{a}{m}\biggr)^{\mu}:p \biggr)f(v)\,dv . \end{aligned}$$

By using (3), (4), and (5) we get the first inequality of (23).

Now multiplying (25) by \(t^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c}(\omega t^{\mu}; p)\) on both sides and then integrating over \([0,1]\), we have

$$\begin{aligned} & \int_{0}^{1}t^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega t^{\mu}; p\bigr)f \biggl(\frac{t}{2}a+m \frac{2-t}{2}b \biggr)\,dt \\ &\qquad{} +m \int_{0}^{1}t^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega t^{\mu}; p\bigr)f \biggl(\frac{2-t}{2m}a+ \frac{t}{2}b \biggr) \\ & \quad \leq\frac{1}{2} \biggl(f(a)-m^{2}f\biggl(\frac{a}{m^{2}} \biggr) \biggr) \int _{0}^{1}t^{\alpha}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega t^{\mu}; p\bigr)\,dt \\ &\qquad{} +m \biggl(f(b)+mf\biggl(\frac{a}{m^{2}}\biggr) \biggr) \int_{0}^{1}t^{\alpha -1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega t^{\mu}; p\bigr)\,dt. \end{aligned}$$
(27)

Putting \(u=\frac{t}{2}a+m\frac{2-t}{2}b\) and \(v=\frac {2-t}{2m}a+\frac{t}{2}b\) in (27), we have

$$\begin{aligned} & \int_{\frac{a+mb}{2}}^{mb}(mb-u)^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega' (mb-u)^{\mu};p\bigr)f(u)\,du \\ &\qquad{} + \int_{\frac{a}{m}}^{\frac{a+mb}{2m}} \biggl(v-\frac{a}{m} \biggr)^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\biggl(m^{\mu} \omega' \biggl(v-\frac{a}{m} \biggr)^{\mu};p \biggr)f(v)\,dv \\ &\quad \leq\frac{1}{2} \biggl( f(a)-m^{2}f\biggl(\frac{a}{m^{2}} \biggr) \biggr) \int _{\frac{a+mb}{2}}^{mb}(mb-u)^{\alpha}E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega' (mb-u)^{\mu};p \bigr)\,dt \\ &\qquad{} +m^{\alpha+1} \biggl( f(b)+mf\biggl(\frac{a}{m^{2}}\biggr) \biggr) \\ &\qquad {}\times \int_{\frac {a}{m}}^{\frac{a+mb}{2m}} \biggl(v-\frac{a}{m} \biggr)^{\alpha -1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\biggl(m^{\mu} \omega' \biggl(v-\frac{a}{m} \biggr)^{\mu};p \biggr)\,dt. \end{aligned}$$
(28)

By using (3), (4), and (6), we get the second inequality of (23). □

If we take \(m=1\) in (23), then we get the following Hadamard type inequality for a convex function.

Corollary 2.12

Let \(f:[a,b]\subseteq[0,\infty)\rightarrow\mathbb{R}\) be a function such that \(f\in L_{1}[a,b]\) with \(a< b\). If f is convex on \([a,b]\), then the following inequalities for extended generalized fractional integral operators hold:

$$\begin{aligned} &f \biggl(\frac{a+b}{2} \biggr)C_{\alpha, (\frac{a+b}{2} )^{+}}(b;p) \\ &\quad \leq \bigl[ \bigl( \epsilon_{\mu,\alpha,l,\omega ',(\frac{a+b}{2})^{+}}^{\gamma,\delta,k,c}f \bigr) (b;p) + \bigl( \epsilon_{\mu,\alpha,l,\omega',(\frac {a+b}{2})_{-}}^{\gamma,\delta,k,c}f \bigr) (a;p) \bigr] \\ &\quad \leq\frac{f(a)+f(b)}{2}C_{\alpha, (\frac{a+b}{2} )^{-}}(a;p), \end{aligned}$$
(29)

where \(\omega'=\frac{2^{\mu}\omega}{(b-a)^{\mu}}\).

Remark 2.13

In Theorem 2.11.

  1. (i)

    If we put \(p=0\), then we get [6, Theorem 3.10].

  2. (ii)

    If we put \(\omega=p=0\), \(m=1\), and \(\alpha=1\), then we get the classical Hadamard inequality.

Remark 2.14

In Corollary 2.12.

  1. (i)

    If we put \(p=0\), then we get [1, Theorem 3.9].

  2. (ii)

    If we put \(\omega=p=0\) and \(\alpha=1\), then we get the classical Hadamard inequality.

  3. (iii)

    If we take \(\omega=p=0\), then we get [14, Theorem 4].

3 Concluding remarks

We have investigated more general fractional integral inequalities. By selecting specific values of parameters quite interesting results can be obtained. For example selecting \(p=0\), fractional integral inequalities for fractional integral operators defined by Salim and Faraj in [12], selecting \(l=\delta=1\), fractional integral inequalities for fractional integral operators defined by Rahman et al. in [11], selecting \(p=0\) and \(l=\delta=1\), fractional integral inequalities for fractional integral operators defined by Shukla and Prajapati in [15] (see also [16]), selecting \(p=0\) and \(l=\delta=k=1\), fractional integral inequalities for fractional integral operators defined by Prabhakar in [10], selecting \(p=\omega=0\), fractional integral inequalities for Riemann–Liouville fractional integral operators.