Here we recall some basic definitions and properties of elementary fractional integral operators.
Definition 2.1
A function \(\varphi: ( [ a,b ] \subseteq \mathbb{R} ) \to \mathbb{R} \) is said to be a convex function if
$$\begin{aligned} \varphi \bigl( \lambda r+ ( 1-\lambda ) s \bigr) \leq \lambda \varphi ( r ) + ( 1-\lambda ) \varphi ( s ) \end{aligned}$$
(2.1)
for all \(r,s\in [ a,b ] \) and \(\lambda \in [ 0,1 ] \). We say that φ is concave if inequality (2.1) is reversed.
Definition 2.2
([2])
For an integrable function φ on \([ a,b ] \) and \(a\geq 0\), we have, for all \(\mu >0\),
$$\begin{aligned} \mathcal{I}_{a^{+}}^{\mu }\varphi ( t ) = \frac{1}{\Gamma ( \mu ) } \int _{a}^{t} ( t-u ) ^{\mu -1}\varphi ( u ) \,du,\quad u>a \end{aligned}$$
(2.2)
and
$$\begin{aligned} \mathcal{I}_{b^{-}}^{\mu }\varphi ( t ) = \frac{1}{\Gamma ( \mu ) } \int _{t}^{b} ( u-t ) ^{\mu -1}\varphi ( u ) \,du,\quad t< b, \end{aligned}$$
(2.3)
where \(\Gamma ( \mu ) =\int _{0}^{\infty }e^{-x}\) \(x^{\mu -1}\,dx\) is the gamma function, and \(\mathcal{I}_{a^{+}}^{0}\varphi ( t ) =\mathcal{I}_{b^{-}}^{0}\varphi ( t ) =\varphi ( t ) \). The functions \(\mathcal{I}_{a^{+}}^{\mu }\varphi ( t ) \) and \(\mathcal{I}_{b^{-}}^{\mu }\varphi ( t ) \) are called the left- and right-sided Riemann–Liouville fractional integrals, respectively, of the function φ for the order μ.
Definition 2.3
([2, 3])
For an integrable function φ on the interval Λ and for an increasing function \(\psi \in C^{1} ( \Lambda,\mathbb{R} ) \) such that \(\psi ^{\prime } ( t ) \neq 0\), \(t\in \Lambda \), we have, for all \(\mu >0\),
$$\begin{aligned} ^{\psi }\mathcal{I}_{a^{+}}^{\mu }\varphi ( t ) = \frac{1}{\Gamma ( \mu ) } \int _{a}^{t}\psi ^{\prime } ( u ) \bigl[ \psi ( t ) -\psi ( u ) \bigr] ^{\mu -1}\varphi ( u ) \,du \end{aligned}$$
(2.4)
and
$$\begin{aligned} ^{\psi }\mathcal{I}_{b^{-}}^{\mu }\varphi ( t ) = \frac{1}{\Gamma ( \mu ) } \int _{t}^{b}\psi ^{\prime } ( u ) \bigl[ \psi ( u ) -\psi ( t ) \bigr] ^{\mu -1}\varphi ( u ) \,du. \end{aligned}$$
(2.5)
The functions \({}^{\psi }\mathcal{I}_{a^{+}}^{\mu }\varphi ( t ) \) and \({}^{\psi }\mathcal{I}_{b^{-}}^{\mu }\varphi ( t ) \) are called the left- and right-sided ψ-Riemann–Liouville fractional integrals, respectively, of the function φ for the order μ.
Definition 2.4
([35])
For an integrable function φ and \(\omega >0\), we have, for all \(\mu \in \mathbb{C} \), \(\operatorname{Re} ( \mu ) \geq 0\),
$$\begin{aligned} \bigl( D_{a^{+}}^{\mu,\omega }\varphi \bigr) ( t ) &=D^{m, \omega } \mathcal{I}_{a^{+}}^{m-\mu,\omega } \varphi ( t ) \\ &=\frac{D_{t}^{m,\omega }}{\omega ^{m-\mu }\Gamma ( m-\mu ) }\int _{a}^{t}\exp \biggl[ \frac{\omega -1}{\omega } ( t-u ) \biggr] ( t-u ) ^{m-\mu -1}\varphi ( u ) \,du \end{aligned}$$
(2.6)
and
$$\begin{aligned} \bigl( D_{b^{-}}^{\mu,\omega }\varphi \bigr) ( t ) &= _{\gamma }D^{m,\omega } \mathcal{I}_{b^{-}}^{m- \mu,\omega } \varphi ( t ) \\ &=\frac{_{\gamma }D_{t}^{m,\omega }}{\omega ^{m-\mu }\Gamma ( m-\mu ) } \int _{t}^{b}\exp \biggl[ \frac{\omega -1}{\omega } ( u-t ) \biggr] ( u-t ) ^{m-\mu -1}\varphi ( u ) \,du, \end{aligned}$$
(2.7)
where
$$\begin{aligned} D^{m,\omega }= \underset{m\text{-times}}{\underbrace{D^{\omega }D^{\omega }\cdots D^{\omega }}},\quad m= \bigl[ \operatorname{Re} ( \mu ) \bigr] +1 \end{aligned}$$
and
$$\begin{aligned} \bigl({}_{\gamma }D^{\omega }\varphi \bigr) ( t ) = ( 1-\omega ) \varphi ( t ) -\omega \varphi ^{ \prime } ( t ),\qquad {} _{\gamma }D^{m,\omega } = \underset{m\text{-times}}{\underbrace{_{\gamma }D_{\gamma }^{\omega }D^{\omega }\cdots{}_{\gamma }D^{\omega }}}. \end{aligned}$$
The functions \(( D_{a^{+}}^{\mu,\omega }\varphi ) ( t ) \) and \(( D_{b^{-}}^{\mu,\omega }\varphi ) ( t ) \) are called the left- and right-sided proportional fractional derivatives, respectively, of the function φ for the order μ.
Definition 2.5
([35])
For an integrable function φ and \(\omega >0\), we have, for all \(\mu \in \mathbb{C} \), \(\operatorname{Re} ( \mu ) \geq 0\),
$$\begin{aligned} \bigl( \mathcal{I}_{a^{+}}^{\mu,\omega } \varphi \bigr) ( t ) = \frac{1}{\omega ^{\mu }\Gamma ( \mu ) }\int _{a}^{t}\exp \biggl[ \frac{\omega -1}{\omega } ( t-u ) \biggr] ( t-u ) ^{\mu -1}\varphi ( u ) \,du \end{aligned}$$
(2.8)
and
$$\begin{aligned} \bigl( \mathcal{I}_{b^{-}}^{\mu,\omega } \varphi \bigr) ( t ) = \frac{1}{\omega ^{\mu }\Gamma ( \mu ) }\int _{t}^{b}\exp \biggl[ \frac{\omega -1}{\omega } ( u-t ) \biggr] ( u-t ) ^{\mu -1}\varphi ( u ) \,du \end{aligned}$$
(2.9)
The functions \(( \mathcal{I}_{a^{+}}^{\mu,\omega } \varphi ) ( t ) \) and \(( \mathcal{I}_{b^{-}}^{\mu,\omega } \varphi ) ( t ) \) are called the left- and right-sided proportional fractional integrals, respectively, of the function φ for the order μ.
Definition 2.6
([36])
For an integrable function φ, a strictly increasing continuous function ψ on \([ a,b ] \), and \(\omega \in ( 0,1 ] \), we have, for all \(\mu \in \mathbb{C} \), \(\operatorname{Re} ( \mu ) \geq 0\),
$$\begin{aligned} \bigl( ^{\psi }D_{a^{+}}^{\mu,\omega }\varphi \bigr) ( t ) ={}& {}^{\psi }D^{m,\omega } {}^{\psi } \mathcal{I}_{a^{+}}^{m-\mu,\omega } \varphi ( t ) \\ ={}&\frac{^{\psi }D_{t}^{m,\omega }}{\omega ^{m-\mu }\Gamma ( m-\mu ) } \int _{a}^{t}\exp \biggl[ \frac{\omega -1}{\omega } \bigl( \psi ( t ) -\psi ( u ) \bigr) \biggr] \\ &{}\times \bigl( \psi ( t ) -\psi ( u ) \bigr) ^{m-\mu -1} \psi ^{\prime } ( u ) \varphi ( u ) \,du \end{aligned}$$
(2.10)
and
$$\begin{aligned} \bigl( ^{\psi }D_{b^{-}}^{\mu,\omega }\varphi \bigr) ( t ) ={}& {} _{\gamma }^{\psi }D^{m,\omega } {}^{ \psi }\mathcal{I}_{b^{-}}^{m-\mu,\omega } \varphi ( t ) \\ ={}&\frac{_{\gamma }^{\psi }D_{t}^{m,\omega }}{\omega ^{m-\mu }\Gamma ( m-\mu ) } \int _{t}^{b}\exp \biggl[ \frac{\omega -1}{\omega } \bigl( \psi ( u ) -\psi ( t ) \bigr) \biggr] \\ &{}\times \bigl( \psi ( u ) -\psi ( t ) \bigr) ^{m-\mu -1}\psi ^{\prime } ( u ) \varphi ( u ) \,du, \end{aligned}$$
(2.11)
where
$$\begin{aligned} {}^{\psi }D^{m,\omega }= \underset{m\text{-times}}{\underbrace{ {}^{\psi }D^{\omega } {}^{\psi }D^{\omega } \cdots {}^{\psi }D^{\omega } }},\quad m= \bigl[ \operatorname{Re} ( \mu ) \bigr] +1 \end{aligned}$$
and
$$\begin{aligned} \bigl({}_{\gamma }^{\psi }D^{\omega }\varphi \bigr) ( t ) = ( 1- \omega ) \varphi ( t ) - \omega \frac{\varphi ^{\prime } ( t ) }{\psi ^{\prime } ( t ) },\qquad {} _{\gamma }^{\psi }D^{m,\omega } = \underset{m\text{-times}}{\underbrace{{} _{\gamma }^{\psi }D^{\omega } {} _{\gamma }^{\psi }D^{\omega } \cdots {} _{\gamma }^{\psi }D^{\omega } }}. \end{aligned}$$
The functions \(( ^{\psi }D_{a^{+}}^{\mu,\omega }\varphi ) ( t ) \) and \(( ^{\psi }D_{b^{-}}^{\mu,\omega }\varphi ) ( t ) \) are called, respectively, the left- and right-sided proportional fractional derivatives of the function φ with respect to the function ψ for the order μ.
Definition 2.7
([36])
For an integrable function φ, a strictly increasing continuous function ψ on \([ a,b ] \), and \(\omega \in ( 0,1 ] \), we have, for all \(\mu \in \mathbb{C} \), \(\operatorname{Re} ( \mu ) \geq 0\),
$$\begin{aligned} &\bigl( {}^{\psi }\mathcal{I}_{a^{+}}^{\mu,\omega } \varphi \bigr) ( t ) \\ &\quad = \frac{1}{\omega ^{\mu }\Gamma ( \mu ) }\int _{a}^{t}\exp \biggl[ \frac{\omega -1}{\omega } \bigl( \psi ( t ) -\psi ( u ) \bigr) \biggr] \bigl( \psi ( t ) -\psi ( u ) \bigr) ^{\mu -1} \psi ^{\prime } ( u ) \varphi ( u ) \,du \end{aligned}$$
(2.12)
and
$$\begin{aligned} &\bigl( {}^{\psi }\mathcal{I}_{b^{-}}^{\mu,\omega } \varphi \bigr) ( t ) \\ &\quad = \frac{1}{\omega ^{\mu }\Gamma ( \mu ) }\int _{t}^{b}\exp \biggl[ \frac{\omega -1}{\omega } \bigl( \psi ( u ) -\psi ( t ) \bigr) \biggr] \bigl( \psi ( u ) -\psi ( t ) \bigr) ^{\mu -1} \psi ^{\prime } ( u ) \varphi ( u ) \,du. \end{aligned}$$
(2.13)
The functions \(( {}^{\psi }\mathcal{I}_{a^{+}}^{\mu,\omega } \varphi ) ( t ) \) and \(( {}^{\psi }\mathcal{I}_{b^{-}}^{\mu,\omega } \varphi ) ( t ) \) are called, respectively, the left- and right-sided proportional fractional integrals of the function φ with respect to the function ψ for the order μ.
Lemma 2.1
([36])
Let \(\omega \in ( 0,1 ] \), \(\operatorname{Re} ( \alpha )>0\) and \(\operatorname{Re} ( \mu ) >0\). Then, if ψ be continuous function and defined for \(t\geq a\) or \(t\leq b\), we have
$$\begin{aligned} & {}^{\psi }\mathcal{I}_{a^{+}}^{\mu,\omega } \bigl( {}^{\psi }\mathcal{I}_{a^{+}}^{\alpha,\omega } \varphi \bigr) ( t ) = {}^{\psi }\mathcal{I}_{a^{+}}^{ \alpha,\omega } \bigl( {}^{\psi }\mathcal{I}_{a^{+}}^{ \mu,\omega } \varphi \bigr) ( t ) = \bigl( {}^{\psi } \mathcal{I}_{a^{+}}^{\mu +\alpha,\omega } \varphi \bigr) ( t ), \end{aligned}$$
(2.14)
$$\begin{aligned} & {}^{\psi }\mathcal{I}_{b^{-}}^{\mu,\omega } \bigl( {}^{\psi }\mathcal{I}_{b^{-}}^{\alpha,\omega } \varphi \bigr) ( t ) = {}^{\psi }\mathcal{I}_{b^{-}}^{ \mu,\upsilon } \bigl( {}^{\psi }\mathcal{I}_{b^{-}}^{ \alpha,\omega } \varphi \bigr) ( t ) = \bigl( {}^{\psi } \mathcal{I}_{b^{-}}^{\mu +\alpha,\omega } \varphi \bigr) ( t ). \end{aligned}$$
(2.15)
Lemma 2.2
([36])
Let \(0\leq m< [ \operatorname{Re} ( \mu ) ] +1\). Then, we have
$$\begin{aligned} &{}^{\psi }D^{m,\omega } \bigl( {}^{\psi } \mathcal{I}_{a^{+}}^{ \mu,\omega } \varphi \bigr) ( t ) = \bigl( {}^{\psi }\mathcal{I}_{a^{+}}^{\mu -m,\omega } \varphi \bigr) ( t ), \end{aligned}$$
(2.16)
$$\begin{aligned} &{}_{\gamma }^{\psi }D^{m,\omega } \bigl( {}^{\psi } \mathcal{I}_{b^{-}}^{\mu,\omega } \varphi \bigr) ( t ) = \bigl( {}^{\psi }\mathcal{I}_{b^{-}}^{\mu -m,\omega } \varphi \bigr) ( t ). \end{aligned}$$
(2.17)
Along this paper, we need the following identity from [37]:
Let \(\omega \in ( 0,1 ] \), \(\mu \in \mathbb{C} \), \(\operatorname{Re} ( \mu ) \geq 0\), and let ψ be a strictly increasing continuous function. Then for any constant k, we have
$$\begin{aligned} \bigl( ^{\psi }\mathcal{I}_{a^{+}}^{\mu,\omega }k \bigr) ( b ) =\frac{ ( \psi ( b ) -\psi ( a ) ) ^{\mu }}{\omega ^{\mu }\Gamma ( \mu +1 ) }k. \end{aligned}$$
(2.18)