1. Introduction
The classical calculus of derivatives and integrals which involves integer
orders is extended with fractional orders that belong to the real numbers.
In last few decades, the fractional calculus theory receives more attention
due to its significant applications in several scopes such as physics, fluid
dynamics, computer networking, image processing, biology, signal processing,
control theory and other scopes. Because of the importance of fractional
calculus, many researchers have shown their intense interest. One of the
prevalent approaches among researchers is the use of fractional derivatives
and integral operators. As a consequence, several distinct kinds of
fractional integrals and derivatives operators have been realized, such as
the Liouville, Riemann-Liouville, Katugampola, Weyl types, Hadamard and some
other types which can be found in Kilbas et al., [1].
Hilfer [2] in (2000), through his contribution to improve the
fractional calculus, established a new fractional derivative operator for
any real order \(\delta \), which gives the Caputo derivative and the
Riemann-Liouville fractional operator. The primary concept and properties
and more information of \(\psi \)-Riemann-Liouville fractional derivative and
integral can be found in [1]. In (2017), Almeida [3], introduced \(%
\psi \)-Caputo fractional derivative and investigated its significant
properties. Recently, in 2018, Sousa and Oliveira [4], introduced a
generalization of many existing fractional derivative operators called \(\psi
\)-Hilfer derivative.
The mathematical inequalities play a very reliable role in classical
integral and differential equations as well as in the past few years, many
of useful mathematical inequalities have been originated by many authors,
see [5,6,7,8]. One of the most significant
integral inequalities is that discovered by Hermite [9] and Hadamard
[10] for convex function \(f\) as follows
\begin{equation}
f\left( \frac{a+b}{2}\right) \leq \frac{1}{a-b}\int_{a}^{b}f\left( x\right)
dx\leq \frac{f\left( a\right) +f\left( b\right) }{2}. \label{inqq1}
\end{equation}
(1)
If \(f\) is a concave function then both inequalities in (1) are
held in a reversed direction. For some historical of Hermite-Hadamard
inequalities [
11] and the references therein. In the last few
decades, these inequalities have been received a considerable attention by
many authors and several articles have appeared in the literature, see [
12,
13,
14]. In 2010, Dahmani [
15], studied the
Hermite-Hadamard type inequalities for concave functions by means of
Riemann-Liouville fractional integral. Sarikaya
et al., in 2013 [
16],
gave the Hermite-Hadamard type inequalities for convex function using
Riemann-Liouville fractional integral. In 2014, Set
et al., established
Hermite-Hadamard type inequalities for s-convex functions in the second
sense proved by Dragomir
et al., [
17] and \(m\)-convex functions via
fractional integrals. In (2015), Noor
et al., [
18], derived some quantum
estimates for Hermite-Hadamard inequalities for \(q\)-differentiable quasi
convex functions and \(q\)-differentiable convex functions. Liu
et al., [
19]
in 2016, introduced some inequalities of Hermite-Hadamard type for
MT-convex functions using classical integrals and Riemann-Liouville
fractional integrals. In 2017, Agarwal
et al., [
20], obtained some
Hermite-Hadamard type inequalities for convex functions via
\((k,s)-\)Riemann-Liouville fractional integrals. Muhammad A. Khan [
21] in
2018, proved new Hermite-Hadamard inequalities for convex function,
\(s-\)convex and coordinate convex functions by using conformable fractional
integrals. Recently in 2019, a lots of researchers studied
Hermite-Hadamard inequalities for several kinds of convexity of the
functions, for more details we refer the readers to see [
22,
23]. Very recently, in 2020, Chudziak and Ołdak introduced notion of a
co-ordinated \((F,G)\)-convex function defined on an interval in \(
\mathbb{R}
^{2}.\)
The main objective of this paper is to establish some new fractional
integral Hermite-Hadamard inequalities for concave functions by using \(\psi
– \)Riemann-Liouville fractional integral operator. Moreover, we introduce
some new fractional integral inequalities related to the Hermite-Hadamard
inequalities via \(\psi -\)Riemann-Liouville fractional integral operator. The
paper is organized as follows: In Section 2, we collect some
notations, definitions, results and preliminary facts which are used
throughout this paper. In Section 3, we present the reverse
Hermite-Hadamard’s inequalities for concave functions. In Section 4, we
give some other related results of Hermite-Hadamard type inequalities which
involving \(\psi -\)Riemann-Liouville fractional integral operator.
2. Basic definitions and tools
This section is dedicated for some basic definitions and properties of
fractional integrals used to obtain and discuss our new results. We also
outline some basic results related to this work.
Let \(\delta >0,\) \(m\in
\mathbb{N}
,\) with \(\Upsilon =\left[ a,b\right] \) \(\left( -\infty \leq a< t< b\leq \infty
\right) ,\) be a finite or infinite interval. Assume that \(f\) be an
integrable function defined on \(\Upsilon \) and \(\psi :\Upsilon \rightarrow
\mathbb{R}
\) be an increasing function for all \(t\in \Upsilon ,\) which belong to \(
C^{1}\left( \Upsilon ,
\mathbb{R}
\right) \) with condition that \(\psi ^{\prime }\left( t\right) \) must be
nonzero along the interval \(\Upsilon \). The \(\psi -\)Riemann-Liouville
fractional derivative of order \(\delta \) of a function \(f\) are defined by [1,24]:
\begin{eqnarray}
D_{a^{+}}^{\delta ;\psi }f\left( t\right) &=&\left( \frac{1}{\psi ^{\prime
}\left( t\right) }\frac{d}{dt}\right) ^{m}\mathcal{I}_{a^{+}}^{m-\delta
;\psi }f\left( t\right) \notag \\
&=&\frac{1}{\Gamma \left( m-\delta \right) }\left( \frac{1}{\psi ^{\prime
}\left( t\right) }\frac{d}{dt}\right) ^{m}\int_{a}^{t}\psi ^{\prime }\left(
\xi \right) \left[ \psi \left( t\right) -\psi \left( \xi \right) \right]
^{m-\delta -1}f\left( \xi \right) d\xi, \label{ID3}
\end{eqnarray}
(2)
and
\begin{eqnarray}
D_{b^{-}}^{\delta ;\psi }f\left( t\right) &=&\left( -\frac{1}{\psi ^{\prime
}\left( t\right) }\frac{d}{dt}\right) ^{m}\mathcal{I}_{b^{-}}^{m-\delta
;\psi }f\left( t\right) \notag \\
&=&\frac{1}{\Gamma \left( m-\delta \right) }\left( -\frac{1}{\psi ^{\prime
}\left( t\right) }\frac{d}{dt}\right) ^{m}\int_{t}^{b}\psi ^{\prime }\left(
\xi \right) \left[ \psi \left( \xi \right) -\psi \left( t\right) \right]
^{m-\delta -1}f\left( \xi \right) d\xi . \label{ID4}
\end{eqnarray}
(3)
Definition 1.
Let \(\delta >0\) and \(f\) be an integrable function defined on \(\Upsilon \) and
\(\psi \left( t\right) \in C^{1}\left( \Upsilon ,
\mathbb{R}
\right) \) be an increasing function such that \(\psi ^{\prime }\left(
t\right) \neq 0\) for all \(t\in \Upsilon .\) The left and right \(\psi -\)
Riemann-Liouville fractional integral of order \(\delta \) with respect to the
function \(\psi \) of a function \(f\) are respectively defined by [1,24]:
\begin{equation}
\mathcal{I}_{a^{+}}^{\delta ;\psi }f\left( t\right) =\frac{1}{\Gamma \left(
\delta \right) }\int_{a}^{t}\psi ^{\prime }\left( \xi \right) \left[ \psi
\left( t\right) -\psi \left( \xi \right) \right] ^{\delta -1}f\left( \xi
\right) d\xi, \label{ID1}
\end{equation}
(4)
and
\begin{equation}
\mathcal{I}_{b^{-}}^{\delta ;\psi }f\left( t\right) =\frac{1}{\Gamma \left(
\delta \right) }\int_{t}^{b}\psi ^{\prime }\left( \xi \right) \left[ \psi
\left( \xi \right) -\psi \left( t\right) \right] ^{\delta -1}f\left( \xi
\right) d\xi . \label{ID2}
\end{equation}
(5)
Lemma 1.
[1] Let \(\delta >0\) and \(\mu >0.\) If \(f\left( t\right) =\left[ \psi
\left( t\right) -\psi \left( \xi \right) \right] ^{\mu -1},\) then
\begin{equation*}
\mathcal{I}_{a^{+}}^{\delta ;\psi }f\left( t\right) =\frac{\Gamma \left( \mu
\right) }{\Gamma \left( \delta +\mu \right) }\left[ \psi \left( t\right)
-\psi \left( \xi \right) \right] ^{\delta +\mu -1}.
\end{equation*}
Definition 2.
The function \(f\) \(:\left( \Lambda \subseteq
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) is said to be concave function if the following inequality holds
\begin{equation}
f\left( \lambda x+\left( 1-\lambda \right) y\right) \geq \lambda f\left(
x\right) +\left( 1-\lambda \right) f\left( y\right) , \label{cod}
\end{equation}
(6)
for all \(x,y\in \Lambda \) and \(\lambda \in \left[ 0,1\right] .\) We say that \(
f\) is convex if the inequality (6) is reversed.
Theorem 3.
[16] Let \(f:\left[ a,b\right] \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) be a convex positive function on \(\left[ a,b\right] \) with \(0\leq a0,\) the following inequality holds:
\begin{equation}
\frac{f\left( a\right) +f\left( b\right) }{2}\leq \frac{\Gamma \left( \delta
+1\right) }{2\left( a+b\right) ^{\delta }}\left[ \mathcal{I}_{a^{+}}^{\delta
}f\left( b\right) +\mathcal{I}_{b^{-}}^{\delta }f\left( a\right) \right]
\leq f\left( \frac{a+b}{2}\right) . \label{ineq1}
\end{equation}
(7)
Lemma 2.
[15] Let \(h:\left[ a,b\right] \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) be a concave. Then the following inequality holds:
\begin{equation*}
h\left( a\right) +h\left( b\right) \leq h\left[ a+b-t\right] +h\left(
t\right) \leq 2h\left( \frac{a+b}{2}\right) .
\end{equation*}
Theorem 4.
[15] Let \(f\) and \(g\) be two positive functions on \(\left[
0,\infty \right) \). If \(f\) and \(g\) are a concave functions on \(\left[
0,\infty \right) \), then for all \(p>1\), \(q>1\) and \(\delta >0,\) the following
inequality holds:
\begin{eqnarray*}
2^{-p-q}\left[ f\left( 0\right) +f\left( x\right) \right] ^{p}\left[
g\left( 0\right) +g\left( x\right) \right] ^{q}\left( \mathcal{I}^{\delta
}x^{\delta -1}\right) ^{2} \left. \leq \right. \mathcal{I}^{\delta }\left[ x^{\delta -1}f^{p}\left(
x\right) \right] \mathcal{I}^{\delta }\left( x^{\delta -1}g^{q}\left(
x\right) \right) .
\end{eqnarray*}
Theorem 5.
[15] Let \(f\) and \(g\) be two positive functions on \(\left[
0,\infty \right) \). If \(f\) and \(g\) are a concave functions on \(\left[
0,\infty \right) \), then for all \(p>1\), \(q>1\) and \(\delta >0,\) \(\sigma >0\)
the following inequality holds:
\begin{eqnarray*}
&&2^{2-p-q}\left[ f\left( 0\right) +f\left( x\right) \right] ^{p}\left[
g\left( 0\right) +g\left( x\right) \right] ^{q}\left( \mathcal{I}\left(
^{\delta }x^{\sigma -1}\right) \right) ^{2} \\
&&\left. \leq \right. \frac{\Gamma \left( \sigma \right) }{\Gamma \left(
\delta \right) }\mathcal{I}^{\sigma }\left[ x^{\sigma -1}f^{p}\left(
x\right) \right] +\mathcal{I}^{\delta }\left[ x^{\sigma -1}f^{p}\left(
x\right) \right] \frac{\Gamma \left( \sigma \right) }{\Gamma \left(
\delta \right) }\mathcal{I}^{\sigma }\left[ x^{\sigma -1}g^{q}\left(
x\right) \right] +\mathcal{I}^{\delta }\left[ x^{\sigma -1}g^{q}\left(
x\right) \right] .
\end{eqnarray*}
3. The reverse Hermite-Hadamard’s inequalities for fractional integral
Now, we give the reverse Hermite-Hadamard’s inequalities involving concave
functions for \(\psi \)-Riemann-Liouville fractional integral operators.
Theorem 6.
Let \(\psi :\left[ a,b\right] \longrightarrow \Lambda \subseteq
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) with \(0\leq a< b\) be an increasing and bijective function having a
continuous derivative \(\psi ^{\prime }\left( x\right) \neq 0\,\) \(\forall \) \(%
x\in \left[ a,b\right] ,\) \(\psi \left( 0\right) =0,\) \(\psi \left( 1\right)
=1\) and \(f:\Lambda \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) be an increasing and differentiable function on \(\Lambda ^{\circ }\) such
that \(\left( f\circ \psi \right) :\left[ a,b\right] \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) be an integrable mapping on \(\left[ a,b\right] \). If \(\psi \) and \(f\) are
concave and positive functions, then the following inequality holds:
\begin{eqnarray}
f\left( \frac{\psi \left( a\right) +\psi \left( b\right) }{2}\right) \geq %
\frac{\Gamma \left( \delta +1\right) }{2\left[ \psi \left( a\right) +\psi
\left( b\right) \right] ^{\delta }}\left[ \mathcal{I}_{a^{+}}^{\delta ;\psi
}\left( f\circ \psi \right) \left( b\right) +\mathcal{I}_{b^{-}}^{\delta
;\psi }\left( f\circ \psi \right) \left( a\right) \right] \geq \frac{\left( f\circ \psi \right) \left( a\right) +\left( f\circ \psi
\right) \left( b\right) }{2}. \label{inq1}
\end{eqnarray}
(8)
Proof.
For any \(x,y\in \left[ a,b\right] ,\) using the concavity of \(f\) and \(\psi ,\)
we have
\begin{eqnarray}
\left( f\circ \psi \right) \left[ \lambda x+\left( 1-\lambda \right) y\right]
&=&f\left[ \psi \left[ \lambda x+\left( 1-\lambda \right) y\right] \right]
\notag \\
&\geq &f\left[ \lambda \psi \left( x\right) +\left( 1-\lambda \right) \psi
\left( y\right) \right] \notag \\
&\geq &\lambda f\left[ \psi \left( x\right) \right] +\left( 1-\lambda
\right) f\left[ \psi \left( y\right) \right] \notag \\
&=&\lambda \left( f\circ \psi \right) \left( x\right) +\left( 1-\lambda
\right) \left( f\circ \psi \right) \left( y\right) . \label{inq2}
\end{eqnarray}
(9)
Putting \(\lambda =\frac{1}{2}\) and using (9), we can write
\begin{equation}
f\left( \frac{\psi \left( x\right) +\psi \left( y\right) }{2}\right) \geq
\frac{\left( f\circ \psi \right) \left( x\right) +\left( f\circ \psi \right)
\left( y\right) }{2}. \label{inq3}
\end{equation}
(10)
Let
\begin{equation}
\psi \left( x\right) =\psi \left( t\right) \psi \left( a\right) +\left[
1-\psi \left( t\right) \right] \psi \left( b\right), \label{inq4}
\end{equation}
(11)
and
\begin{equation}
\psi \left( y\right) =\left[ 1-\psi \left( t\right) \right] \psi \left(
a\right) +\psi \left( t\right) \psi \left( b\right) , \label{inq5}
\end{equation}
(12)
where \(x,y\) are variables containing \(t.\) By substituting (11) and (12) in (10), we get
\begin{eqnarray}
2f\left( \frac{\psi \left( a\right) +\psi \left( b\right) }{2}\right) \geq
f\left[ \psi \left( t\right) \psi \left( a\right) +\left[ 1-\psi \left(
t\right) \right] \psi \left( b\right) \right]+f\left[ \left[ 1-\psi \left( t\right) \right] \psi \left( a\right) +\psi
\left( t\right) \psi \left( b\right) \right] . \label{inq6}
\end{eqnarray}
(13)
Now, multiplying both sides of (13) by \(\psi ^{\prime }\left(
t\right) \psi ^{\delta -1}\left( t\right) ,\) then integrating the resulting
inequality with respect to \(t\) over \(\left[ 0,1\right] ,\) we obtain
\begin{eqnarray}
\frac{2}{\delta }f\left( \frac{\psi \left( a\right) +\psi \left( b\right) }{2%
}\right) &\geq &\int_{0}^{1}\psi ^{\prime }\left( t\right) \psi ^{\delta
-1}\left( t\right) f\left[ \psi \left( t\right) \psi \left( a\right) +\left[
1-\psi \left( t\right) \right] \psi \left( b\right) \right] dt \notag \\
&&+\int_{0}^{1}\psi ^{\prime }\left( t\right) \psi ^{\delta -1}\left(
t\right) f\left[ \left[ 1-\psi \left( t\right) \right] \psi \left( a\right)
+\psi \left( t\right) \psi \left( b\right) \right] dt. \notag \\
&=&I_{1}+I_{2}. \label{inq7}
\end{eqnarray}
(14)
From (11), we have
\begin{equation*}
\frac{d}{dt}\psi \left( x\right) =\frac{d}{dt}\left[ \psi \left( t\right)
\psi \left( a\right) +\left[ 1-\psi \left( t\right) \right] \psi \left(
b\right) \right] \Longrightarrow \frac{\psi ^{\prime }\left( x\right) dx}{%
\psi \left( a\right) -\psi \left( b\right) }=\psi ^{\prime }\left( t\right)
dt,
\end{equation*}
and
\begin{equation*}
\psi \left( t\right) =\frac{\psi \left( x\right) -\psi \left( b\right) }{%
\psi \left( a\right) -\psi \left( b\right) }.
\end{equation*}
So, we have
\begin{eqnarray}
I_{1} &=&\int_{b}^{a}\left( \frac{\psi \left( x\right) -\psi \left( b\right)
}{\psi \left( a\right) -\psi \left( b\right) }\right) ^{\delta -1}f\left(
\psi \left( x\right) \right) \frac{\psi ^{\prime }\left( x\right) dx}{\psi
\left( a\right) -\psi \left( b\right) } \notag \\
&=&\int_{a}^{b}\psi ^{\prime }\left( x\right) \left( \frac{\psi \left(
b\right) -\psi \left( x\right) }{\psi \left( b\right) -\psi \left( a\right) }%
\right) ^{\delta -1}\left( f\circ \psi \right) \left( x\right) \frac{dx}{%
\psi \left( b\right) -\psi \left( a\right) } \notag \\
&=&\frac{\Gamma \left( \delta \right) }{\left[ \psi \left( b\right) -\psi
\left( a\right) \right] ^{\delta }}\mathcal{I}_{a^{+}}^{\delta ;\psi }\left(
f\circ \psi \right) \left( b\right) . \label{inq8}
\end{eqnarray}
(15)
Also from (12), we have
\begin{equation*}
\frac{d}{dt}\psi \left( y\right) =\frac{d}{dt}\left[ \left[ 1-\psi \left(
t\right) \right] \psi \left( a\right) +\psi \left( t\right) \psi \left(
b\right) \right] \Longrightarrow \frac{\psi ^{\prime }\left( y\right) dy}{%
\left[ \psi \left( b\right) -\psi \left( a\right) \right] }=\psi ^{\prime
}\left( t\right) dt,
\end{equation*}
and
\begin{equation*}
\psi \left( t\right) =\frac{\psi \left( y\right) -\psi \left( a\right) }{%
\psi \left( b\right) -\psi \left( a\right) }.
\end{equation*}
So, we have
\begin{eqnarray}
I_{2} &=&\int_{a}^{b}\left( \frac{\psi \left( y\right) -\psi \left( a\right)
}{\psi \left( b\right) -\psi \left( a\right) }\right) ^{\delta -1}f\left(
\psi \left( y\right) \right) \frac{\psi ^{\prime }\left( y\right) dy}{\left[
\psi \left( b\right) -\psi \left( a\right) \right] } \notag \\
&=&\frac{1}{\left[ \psi \left( b\right) -\psi \left( a\right) \right]
^{\delta }}\int_{a}^{b}\psi ^{\prime }\left( y\right) \left[ \psi \left(
y\right) -\psi \left( a\right) \right] ^{\delta -1}\left( f\circ \psi
\right) \left( y\right) dy \notag \\
&=&\frac{\Gamma \left( \delta \right) }{\left[ \psi \left( b\right) -\psi
\left( a\right) \right] ^{\delta }}\mathcal{I}_{b^{-}}^{\delta ;\psi }\left(
f\circ \psi \right) \left( a\right) . \label{inq9}
\end{eqnarray}
(16)
Using (14), (15) and (16), we get
\begin{equation}
f\left( \frac{\psi \left( a\right) +\psi \left( b\right) }{2}\right) \geq
\frac{\Gamma \left( \delta +1\right) }{2\left[ \psi \left( b\right) -\psi
\left( a\right) \right] ^{\delta }}\left[ \mathcal{I}_{a^{+}}^{\delta ;\psi
}\left( f\circ \psi \right) \left( b\right) +\mathcal{I}_{b^{-}}^{\delta
;\psi }\left( f\circ \psi \right) \left( a\right) \right] , \label{inq10}
\end{equation}
(17)
which is the first inequality in (8). To prove the second
inequality and using the concavity of \(f\) and \(\psi \), we can write for \(
\lambda \in \left[ 0,1\right]\)
\begin{equation}
f\left[ \psi \left( t\right) \psi \left( a\right) +\left[ 1-\psi \left(
t\right) \right] \psi \left( b\right) \right] \geq \psi \left( t\right)
\left( f\circ \psi \right) \left( a\right) +\left[ 1-\psi \left( t\right) %
\right] \left( f\circ \psi \right) \left( b\right), \label{inq11}
\end{equation}
(18)
and
\begin{equation}
f\left[ \left[ 1-\psi \left( t\right) \right] \psi \left( a\right) +\psi
\left( t\right) \psi \left( b\right) \right] \geq \left[ 1-\psi \left(
t\right) \right] \left( f\circ \psi \right) \left( a\right) +\psi \left(
t\right) \left( f\circ \psi \right) \left( b\right) . \label{inq12}
\end{equation}
(19)
Adding (18) and (19), we obtain
\begin{eqnarray}
f\left[ \psi \left( t\right) \psi \left( a\right) +\left[ 1-\psi \left(
t\right) \right] \psi \left( b\right) \right] +f\left[ \left[ 1-\psi \left(
t\right) \right] \psi \left( a\right) +\psi \left( t\right) \psi \left(
b\right) \right]\left. \geq \left( f\circ \psi \right) \left( b\right) +\left( f\circ \psi
\right) \left( a\right) .\right. \label{inq13}
\end{eqnarray}
(20)
Now, multiplying both sides of (20) by \(\psi ^{\prime }\left(
t\right) \psi ^{\delta -1}\left( t\right) ,\) then integrating the resulting
inequality with respect to \(t\) over \(\left[ 0,1\right] ,\) we obtain
\begin{equation}
\frac{\Gamma \left( \delta \right) }{\left[ \psi \left( b\right) -\psi
\left( a\right) \right] ^{\delta }}\left[ \mathcal{I}_{a^{+}}^{\delta ;\psi
}\left( f\circ \psi \right) \left( b\right) +\mathcal{I}_{b^{-}}^{\delta
;\psi }\left( f\circ \psi \right) \left( a\right) \right] \geq \frac{\left(
f\circ \psi \right) \left( b\right) +\left( f\circ \psi \right) \left(
a\right) }{\delta }. \label{inq14}
\end{equation}
(21)
Hence, by combining the inequalities (17) and (20), we get
the desired inequality (8).
Remark 1.
- (i) If we put \(\psi \left( x\right) =x,\) \(\forall x\in \left[ a,b\right] ,\)
for \(f\) a convex function, then both inequalities (8) reversed and
Theorem 6 reduce to Theorem 3 obtained by Sarikaya
et al., in [16] for classical Riemann-Liouville fractional integral.
- (ii) Applying inequalities (8) for \(\psi \left( x\right) =x\), \(%
\forall x\in \left[ a,b\right] \) and \(\delta =1,\) for \(f\) be a convex
function, we obtain the classical Hermite-Hadamard inequalities (1).
4. Hermite-Hadamard type inequalities for fractional integral
In this section, we generalize some Hermite-Hadamard type inequalities
involving concave functions introduced by Dahmani [
15] using the
Riemann-Liouville fractional integral with respect to other monotone and
bijective function. In present part, we use only the left-sided fractional
integrals (4). Moreover, we consider \(a=0\) to obtain and discuss our
results. We first prove the following lemma:
Lemma 3.
Let \(\psi :\left[ 0,\infty \right) \longrightarrow \Lambda \)
be an increasing and bijective function having a continuous derivative \(\psi
^{\prime }\left( t\right) \neq 0\,\) \(\forall \) \(t\in \left[ 0,\infty \right)
, \) \(\psi \left( 0\right) =0,\) \(\psi \left( 1\right) =1\) and \(h:\Lambda
\longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) be an increasing and differentiable function on \(\Lambda ^{\circ }\) such
that \(\left( h\circ \psi \right) :\left[ 0,\infty \right) \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) be an integrable mapping on \(\left[ 0,\infty \right) \). If \(h\) is a
concave functions on \(\Lambda \), then we have
\begin{eqnarray}
\left( h\circ \psi \right) \left( c\right) +\left( h\circ \psi \right)
\left( d\right) \leq h\left[ \psi \left( c\right) +\psi \left( d\right)
-\psi \left( t\right) \right] +\left( h\circ \psi \right) \left( t\right)
\leq 2h\left( \frac{\psi \left( c\right) +\psi \left( d\right) }{2}\right)
. \label{ieeq}
\end{eqnarray}
(22)
Proof.
Since \(h\) be a concave function on \(\Lambda ,\) so for any \(c,d\in \left[
0,\infty \right) ,\) we can write
\begin{eqnarray}
h\left( \frac{\psi \left( c\right) +\psi \left( d\right) }{2}\right)
=h\left( \frac{\psi \left( c\right) +\psi \left( d\right) +\psi \left(
t\right) -\psi \left( t\right) }{2}\right) \geq \frac{h\left[ \psi \left( c\right) +\psi \left( d\right) -\psi \left(
t\right) \right] +h\left[ \psi \left( t\right) \right] }{2}. \label{inq19}
\end{eqnarray}
(23)
If we choose \(\psi \left( t\right) =\lambda \psi \left( c\right) +\left[
1-\lambda \right] \psi \left( d\right) ,\) then we have
\begin{eqnarray*}
&&\frac{1}{2}\left[ h\left[ \psi \left( c\right) +\psi \left( d\right)
-\lambda \psi \left( c\right) -\left[ 1-\lambda \right] \psi \left( d\right) %
\right] +h\left[ \lambda \psi \left( c\right) +\left[ 1-\lambda \right] \psi
\left( d\right) \right] \right] \\
&&\left. =\frac{1}{2}\left[ h\left[ \lambda \psi \left( d\right) +\left[
1-\lambda \right] \psi \left( c\right) \right] +h\left[ \lambda \psi \left(
c\right) +\left[ 1-\lambda \right] \psi \left( d\right) \right] \right]
\right. .
\end{eqnarray*}
Using the concavity of \(h\), we obtain
\begin{eqnarray}
\frac{1}{2}\left[ h\left[ \lambda \psi \left( d\right) +\left[ 1-\lambda %
\right] \psi \left( c\right) \right] +h\left[ \lambda \psi \left( c\right) +%
\left[ 1-\lambda \right] \psi \left( d\right) \right] \right]\left. \geq \frac{1}{2}\left[ \left( h\circ \psi \right) \left( c\right)
+\left( h\circ \psi \right) \left( d\right) \right] \right. . \label{inq20}
\end{eqnarray}
(24)
By (23) and (24), we get
\begin{eqnarray*}
\left( h\circ \psi \right) \left( c\right) +\left( h\circ \psi \right)
\left( d\right) \leq h\left[ \psi \left( c\right) +\psi \left( d\right)
-\psi \left( t\right) \right] +\left( h\circ \psi \right) \left( t\right) \leq 2h\left( \frac{\psi \left( c\right) +\psi \left( d\right) }{2}\right)
,
\end{eqnarray*}
which is the required inequality (22).
Theorem 7.
Let \(\psi :\left[ 0,\infty \right) \longrightarrow \Lambda \) be
an increasing positive and bijective function having a continuous derivative
\(\psi ^{\prime }\left( x\right) \neq 0\,\) \(\forall \) \(x\in \left[ 0,\infty
\right) ,\) \(\psi \left( 0\right) =0,\) \(\psi \left( 1\right) =1\) and \(%
f,g:\Lambda \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) be an increasing and differentiable functions on \(\Lambda ^{\circ }\) such
that \(\left( f\circ \psi \right) ,\left( g\circ \psi \right) :\left[
0,\infty \right) \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) are two integrable mappings on \(\left[ 0,\infty \right) \). If \(f\) and \(g\)
are a concave functions on \(\Lambda \). Then for all \(p>1\), \(q>1\) and \(\delta
>0,\) the following inequality holds:
\begin{eqnarray}
&&2^{-p-q}\left[ f\left( 0\right) +\left( f\circ \psi \right) \left(
x\right) \right] ^{p}\left[ g\left( 0\right) +\left( g\circ \psi \right)
\left( x\right) \right] ^{q}\left( \mathcal{I}^{\delta ;\psi }\psi ^{\delta
-1}\left( x\right) \right) ^{2} \notag \\
&&\left. \leq \right. \mathcal{I}^{\delta ;\psi }\left[ \psi ^{\delta
-1}\left( x\right) \left( f\circ \psi \right) ^{p}\left( x\right) \right]
\mathcal{I}^{\delta ;\psi }\left[ \psi ^{\delta -1}\left( x\right) \left(
g\circ \psi \right) ^{q}\left( x\right) \right] . \label{inq31}
\end{eqnarray}
(25)
Proof.
Since \(f^{p}\) and \(g^{q}\) are a concave functions on \(\Lambda ,\) so by
Lemma (3), for any \(x,y>0,\) we have
\begin{equation}
f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left( x\right) \leq
f^{p}\left[ \psi \left( x\right) -\psi \left( y\right) \right] +\left(
f\circ \psi \right) ^{p}\left( y\right) \leq 2\left( f\circ \psi \right)
^{p}\left( \frac{x}{2}\right), \label{inq15}
\end{equation}
(26)
and
\begin{equation}
g^{q}\left( 0\right) +\left( g\circ \psi \right) ^{q}\left( x\right) \leq
g^{q}\left[ \psi \left( x\right) -\psi \left( y\right) \right] +\left(
g\circ \psi \right) ^{q}\left( y\right) \leq 2\left( g\circ \psi \right)
^{q}\left( \frac{x}{2}\right) . \label{inq16}
\end{equation}
(27)
Multiplying both sides of (26) and (27) by \(\frac{\psi
^{\prime }\left( y\right) }{\Gamma \left( \delta \right) }\left[ \psi \left(
x\right) -\psi \left( y\right) \right] ^{\delta -1}\psi ^{\delta -1}\left(
y\right) ,\) \(y\in \left( 0,x\right) \) and integrating the resulting
inequalities with respect to \(y\) over \(\left( 0,x\right) ,\) we obtain
\begin{eqnarray}
&&\frac{f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left( x\right)
}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left( y\right) %
\left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta -1}\psi
^{\delta -1}\left( y\right) dy \notag \\
&&\left. \leq \frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi
^{\prime }\left( y\right) \left[ \psi \left( x\right) -\psi \left( y\right) %
\right] ^{\delta -1}\psi ^{\delta -1}\left( y\right) f^{p}\left[ \psi \left(
x\right) -\psi \left( y\right) \right] dy\right. \notag \\
&&\;\;\;\left. +\frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime
}\left( y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right]
^{\delta -1}\psi ^{\delta -1}\left( y\right) \left( f\circ \psi \right)
^{p}\left( y\right) dy\right. \notag \\
&&\left. \leq \right. \frac{2\left( f\circ \psi \right) ^{p}\left( \frac{x}{2%
}\right) }{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta
-1}\psi ^{\delta -1}\left( y\right) dy, \label{inq18}
\end{eqnarray}
(28)
and
\begin{eqnarray}
&&\frac{g^{q}\left( 0\right) +\left( g\circ \psi \right) ^{q}\left( x\right)
}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left( y\right) %
\left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta -1}\psi
^{\delta -1}\left( y\right) dy \notag \\
&&\left. \leq \frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi
^{\prime }\left( y\right) \left[ \psi \left( x\right) -\psi \left( y\right) %
\right] ^{\delta -1}\psi ^{\delta -1}\left( y\right) g^{q}\left[ \psi \left(
x\right) -\psi \left( y\right) \right] dy\right. \notag \\
&&\;\;\;\left. +\frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime
}\left( y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right]
^{\delta -1}\psi ^{\delta -1}\left( y\right) \left( g\circ \psi \right)
^{q}\left( y\right) dy\right. \notag \\
&&\left. \leq \frac{2\left( g\circ \psi \right) ^{q}\left( \frac{x}{2}%
\right) }{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta
-1}\psi ^{\delta -1}\left( y\right) dy\right. . \label{inq17}
\end{eqnarray}
(29)
Using the change of variable \(\psi \left( u\right) =\psi \left( x\right)
-\psi \left( y\right) ,\) where \(u\in \left[ 0,\infty \right) \) is a variable
containing \(y,\) we have
\begin{equation*}
\frac{d}{dy}\left[ \psi \left( u\right) \right] =\frac{d}{dy}\left[ \psi
\left( x\right) -\psi \left( y\right) \right] \Longrightarrow \psi ^{\prime
}\left( u\right) du=-\psi ^{\prime }\left( y\right) dy.
\end{equation*}
Then, we can write
\begin{eqnarray}
&&\frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta
-1}\psi ^{\delta -1}\left( y\right) f^{p}\left[ \psi \left( x\right) -\psi
\left( y\right) \right] dy \notag \\
&&\left. =\right. \frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi
^{\prime }\left( u\right) \left[ \psi \left( x\right) -\psi \left( u\right) %
\right] ^{\delta -1}\psi ^{\delta -1}\left( u\right) \left( f\circ \psi
\right) ^{p}\left( u\right) du\left. =\right. \mathcal{I}^{\delta ;\psi }\left[ \psi ^{\delta -1}\left(
x\right) \left( f\circ \psi \right) ^{p}\left( x\right) \right],
\label{inq21}
\end{eqnarray}
(30)
and
\begin{eqnarray}
&&\frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta
-1}\psi ^{\delta -1}\left( y\right) g^{q}\left[ \psi \left( x\right) -\psi
\left( y\right) \right] dy \notag \\
&&\left. =\right. \frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi
^{\prime }\left( u\right) \left[ \psi \left( x\right) -\psi \left( u\right) %
\right] ^{\delta -1}\psi ^{\delta -1}\left( u\right) \left( g\circ \psi
\right) ^{q}\left( u\right) du \left. =\right. \mathcal{I}^{\delta ;\psi }\left[ \psi ^{\delta -1}\left(
x\right) \left( g\circ \psi \right) ^{q}\left( x\right) \right] .
\label{inq22}
\end{eqnarray}
(31)
Now, by using (28) and (30), we get
\begin{eqnarray}
\left[ f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left( x\right) %
\right] \mathcal{I}^{\delta ;\psi }\psi ^{\delta -1}\left( x\right) \leq 2%
\mathcal{I}^{\delta ;\psi }\left[ \psi ^{\delta -1}\left( x\right) \left(
f\circ \psi \right) ^{p}\left( x\right) \right] \leq 2\left( f\circ \psi \right) ^{p}\left( \frac{x}{2}\right) \mathcal{I}%
^{\delta ;\psi }\psi ^{\delta -1}\left( x\right), \label{inq23}
\end{eqnarray}
(32)
and using (29) and (31), we get
\begin{eqnarray}
\left[ g^{q}\left( 0\right) +\left( g\circ \psi \right) ^{q}\left( x\right) %
\right] \mathcal{I}^{\delta ;\psi }\psi ^{\delta -1}\left( x\right) \leq 2%
\mathcal{I}^{\delta ;\psi }\left[ \psi ^{\delta -1}\left( x\right) \left(
g\circ \psi \right) ^{q}\left( x\right) \right]
\leq 2\left( g\circ \psi \right) ^{q}\left( \frac{x}{2}\right) \mathcal{I}%
^{\delta ;\psi }\psi ^{\delta -1}\left( x\right) . \label{inq24}
\end{eqnarray}
(33)
The inequalities (32) and (33) yields
\begin{eqnarray}
&&\left[ f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left(
x\right) \right] \left[ g^{q}\left( 0\right) +\left( g\circ \psi \right)
^{q}\left( x\right) \right] \left( \mathcal{I}^{\delta ;\psi }\psi ^{\delta
-1}\left( x\right) \right) ^{2} \notag \\
&&\left. \leq 4\mathcal{I}^{\delta ;\psi }\left[ \psi ^{\delta -1}\left(
x\right) \left( f\circ \psi \right) ^{p}\left( x\right) \right] \mathcal{I}%
^{\delta ;\psi }\left[ \psi ^{\delta -1}\left( x\right) \left( g\circ \psi
\right) ^{q}\left( x\right) \right] .\right. \label{inq25}
\end{eqnarray}
(34)
On the other hand, we have \(f\) and \(g\) are positive functions and \(\psi \) is
increasing function on \(\left[ 0,\infty \right) \). Then for any \(x>0,\) \(
p\geq 1,\) \(q\geq 1\), we can write
\begin{equation}
\left[ \frac{f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left(
x\right) }{2}\right] ^{\frac{1}{p}}\geq \frac{f\left( 0\right) +\left(
f\circ \psi \right) \left( x\right) }{2}, \label{inq26}
\end{equation}
(35)
and
\begin{equation}
\left[ \frac{g^{q}\left( 0\right) +\left( g\circ \psi \right) ^{q}\left(
x\right) }{2}\right] ^{\frac{1}{q}}\geq \frac{g\left( 0\right) +\left(
g\circ \psi \right) \left( x\right) }{2}. \label{inq27}
\end{equation}
(36)
Multiplying both sides of (35) and (36) by \(\frac{\psi
^{\prime }\left( y\right) }{\Gamma \left( \delta \right) }\left[ \psi \left(
x\right) -\psi \left( y\right) \right] ^{\delta -1}\psi ^{\delta -1}\left(
y\right) ,\) \(y\in \left( 0,x\right) \), then integrating the resulting
inequalities with respect to \(y\) over \(\left( 0,x\right) ,\) we get
\begin{equation}
\frac{f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left( x\right) }{%
2}\mathcal{I}^{\delta ;\psi }\psi ^{\delta -1}\left( x\right) \geq 2^{-p}%
\left[ f\left( 0\right) +\left( f\circ \psi \right) \left( x\right) \right]
^{p}\mathcal{I}^{\delta ;\psi }\psi ^{\delta -1}\left( x\right),
\label{inq28}
\end{equation}
(37)
and
\begin{equation}
\frac{g^{q}\left( 0\right) +\left( g\circ \psi \right) ^{q}\left( x\right) }{%
2}\mathcal{I}^{\delta ;\psi }\psi ^{\delta -1}\left( x\right) \geq 2^{-q}%
\left[ g\left( 0\right) +\left( g\circ \psi \right) \left( x\right) \right]
^{q}\mathcal{I}^{\delta ;\psi }\psi ^{\delta -1}\left( x\right) .
\label{inq29}
\end{equation}
(38)
The inequalities (37) and (38) yields
\begin{eqnarray}
&&\frac{1}{4}\left[ f^{p}\left( 0\right) +\left( f\circ \psi \right)
^{p}\left( x\right) \right] \left[ g^{q}\left( 0\right) +\left( g\circ \psi
\right) ^{q}\left( x\right) \right] \left( \mathcal{I}^{\delta ;\psi }\psi
^{\delta -1}\left( x\right) \right) ^{2} \notag \\
&&\left. \geq \right. 2^{-p-q}\left[ f\left( 0\right) +\left( f\circ \psi
\right) \left( x\right) \right] ^{p}\left[ g\left( 0\right) +\left( g\circ
\psi \right) \left( x\right) \right] ^{q}\left( \mathcal{I}^{\delta ;\psi
}\psi ^{\delta -1}\left( x\right) \right) ^{2}. \label{inq30}
\end{eqnarray}
(39)
Combining the inequalities (34) and (39), we obtain the
desired inequality (25).
Remark 2.
- (i) If we put \(\psi \left( x\right) =x\) for all \(x\in \left[ 0,\infty
\right) ,\) then Lemma 3 reduce to Lemma 2 and
Theorem 7 reduce to Theorem 4 obtained by Dahmani in
[15].
- (ii) Applying Theorem 7 for \(\psi \left( x\right) =x\)
for all \(x\in \left[ 0,\infty \right) ,\) \(\delta =1,\) we obtain Theorem
5 obtained by Set et al., in [25].
No, we give the following version of Theorem 7 with two
parameters for \(\psi \)-Riemann-Liouville fractional integral operator.
Theorem 8.
Let \(\psi :\left[ 0,\infty \right) \longrightarrow \Lambda \) be
an increasing positive and bijective function having a continuous derivative
\(\psi ^{\prime }\left( x\right) \neq 0\,\) \(\forall \) \(x\in \left[ 0,\infty
\right) ,\) \(\psi \left( 0\right) =0,\) \(\psi \left( 1\right) =1\) and \(%
f,g:\Lambda \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) be an increasing and differentiable functions on \(\Lambda ^{\circ }\) such
that \(\left( f\circ \psi \right) ,\left( g\circ \psi \right) :\left[
0,\infty \right) \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\) are two integrable mappings on \(\left[ 0,\infty \right) \). If \(f\) and \(g\)
are a concave functions on \(\Lambda \). Then for all \(p>1\), \(q>1\) and \(\delta
>0,\) \(\sigma >0,\) the following inequality holds:
\begin{eqnarray}
&&2^{2-p-q}\left[ f\left( 0\right) +\left( f\circ \psi \right) \left(
x\right) \right] ^{p}\left[ g\left( 0\right) +\left( g\circ \psi \right)
\left( x\right) \right] ^{q}\left( \mathcal{I}^{\delta ;\psi }\psi ^{\sigma
-1}\left( x\right) \right) ^{2} \notag \\
&&\left. \leq \right. \frac{\Gamma \left( \sigma \right) }{\Gamma \left(
\delta \right) }\mathcal{I}^{\sigma ;\psi }\left[ \psi ^{\delta -1}\left(
x\right) \left( f\circ \psi \right) ^{p}\left( x\right) \right] +\mathcal{I}%
^{\delta ;\psi }\left[ \psi ^{\sigma -1}\left( x\right) \left( f\circ \psi
\right) ^{p}\left( x\right) \right] \notag \\
&&\;\;\;\left. \times \right. \frac{\Gamma \left( \sigma \right) }{\Gamma \left(
\delta \right) }\mathcal{I}^{\sigma ;\psi }\left[ \psi ^{\delta -1}\left(
x\right) \left( g\circ \psi \right) ^{q}\left( x\right) \right] +\mathcal{I}%
^{\delta ;\psi }\left[ \psi ^{\sigma -1}\left( x\right) \left( g\circ \psi
\right) ^{q}\left( x\right) \right] . \label{inq44}
\end{eqnarray}
(40)
Proof.
By using Lemma 3 and as \(f^{p}\) and \(g^{q}\) are concave
functions on \(\Lambda ,\) then we have for any \(x,y>0\)
\begin{equation}
f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left( x\right) \leq
f^{p}\left[ \psi \left( x\right) -\psi \left( y\right) \right] +\left(
f\circ \psi \right) ^{p}\left( y\right) \leq 2\left( f\circ \psi \right)
^{p}\left( \frac{x}{2}\right), \label{inq32}
\end{equation}
(41)
and
\begin{equation}
g^{q}\left( 0\right) +\left( g\circ \psi \right) ^{q}\left( x\right) \leq
g^{q}\left[ \psi \left( x\right) -\psi \left( y\right) \right] +\left(
g\circ \psi \right) ^{q}\left( y\right) \leq 2\left( g\circ \psi \right)
^{q}\left( \frac{x}{2}\right) . \label{inq33}
\end{equation}
(42)
Now, multiplying both sides of (41) and (42) by \(\frac{\psi ^{\prime }\left( y\right) }{\Gamma \left(
\delta \right) }\left[ \psi \left( x\right) -\psi \left( y\right) \right]
^{\delta -1}\psi ^{\sigma -1}\left( y\right) ,\) \(y\in \left( 0,x\right) \), then integrating the
resulting inequalities with respect to \(y\) over \(\left( 0,x\right) ,\) we
obtain
\begin{eqnarray}
&&\frac{f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left( x\right)
}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left( y\right) %
\left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta -1}\psi
^{\sigma -1}\left( y\right) dy \notag \\
&&\left. \leq \frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi
^{\prime }\left( y\right) \left[ \psi \left( x\right) -\psi \left( y\right) %
\right] ^{\delta -1}\psi ^{\sigma -1}\left( y\right) f^{p}\left[ \psi \left(
x\right) -\psi \left( y\right) \right] dy\right. \notag \\
&&\;\;\;\left. +\frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime
}\left( y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right]
^{\delta -1}\psi ^{\sigma -1}\left( y\right) \left( f\circ \psi \right)
^{p}\left( y\right) dy\right. \notag \\
&&\left. \leq \right. \frac{2\left( f\circ \psi \right) ^{p}\left( \frac{x}{2%
}\right) }{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta
-1}\psi ^{\sigma -1}\left( y\right) dy, \label{inq34}
\end{eqnarray}
(43)
and
\begin{eqnarray}
&&\frac{g^{q}\left( 0\right) +\left( g\circ \psi \right) ^{q}\left( x\right)
}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left( y\right) %
\left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta -1}\psi
^{\sigma -1}\left( y\right) dy \notag \\
&&\left. \leq \frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi
^{\prime }\left( y\right) \left[ \psi \left( x\right) -\psi \left( y\right) %
\right] ^{\delta -1}\psi ^{\sigma -1}\left( y\right) g^{q}\left[ \psi \left(
x\right) -\psi \left( y\right) \right] dy\right. \notag \\
&&\;\;\;\left. +\frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime
}\left( y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right]
^{\delta -1}\psi ^{\sigma -1}\left( y\right) \left( g\circ \psi \right)
^{q}\left( y\right) dy\right. \notag \\
&&\left. \leq \frac{2\left( g\circ \psi \right) ^{q}\left( \frac{x}{2}%
\right) }{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta
-1}\psi ^{\sigma -1}\left( y\right) dy\right. . \label{inq35}
\end{eqnarray}
(44)
Using the change of variable \(\psi \left( u\right) =\psi \left( x\right)
-\psi \left( y\right) ,\) where \(u\in \left[ 0,\infty \right) \) is a variable
containing \(y,\) we have
\begin{eqnarray}
&&\frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta
-1}\psi ^{\delta -1}\left( y\right) f^{p}\left[ \psi \left( x\right) -\psi
\left( y\right) \right] dy \notag \\
&&\left. =\right. \frac{\Gamma \left( \sigma \right) }{\Gamma \left( \sigma
\right) \Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
u\right) \left[ \psi \left( x\right) -\psi \left( u\right) \right] ^{\sigma
-1}\psi ^{\delta -1}\left( u\right) \left( f\circ \psi \right) ^{p}\left(
u\right) du \left. =\right. \frac{\Gamma \left( \sigma \right) }{\Gamma \left( \delta
\right) }\mathcal{I}^{\sigma ;\psi }\left[ \psi ^{\delta -1}\left( x\right)
\left( f\circ \psi \right) ^{p}\left( x\right) \right],\notag\\ \label{inq36}
\end{eqnarray}
(45)
and
\begin{eqnarray}
&&\frac{1}{\Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
y\right) \left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta
-1}\psi ^{\delta -1}\left( y\right) g^{q}\left[ \psi \left( x\right) -\psi
\left( y\right) \right] dy \notag \\
&&\left. =\right. \frac{\Gamma \left( \sigma \right) }{\Gamma \left( \sigma
\right) \Gamma \left( \delta \right) }\int_{0}^{x}\psi ^{\prime }\left(
u\right) \left[ \psi \left( x\right) -\psi \left( u\right) \right] ^{\sigma
-1}\psi ^{\delta -1}\left( u\right) \left( g\circ \psi \right) ^{q}\left(
u\right) du \left. =\right. \frac{\Gamma \left( \sigma \right) }{\Gamma \left( \delta
\right) }\mathcal{I}^{\sigma ;\psi }\left[ \psi ^{\delta -1}\left( x\right)
\left( g\circ \psi \right) ^{q}\left( x\right) \right] .\notag\\ \label{37}
\end{eqnarray}
(46)
By using (43) and (45), we obtain
\begin{eqnarray}
\left[ f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left(
x\right) \right] \mathcal{I}^{\delta ;\psi }\psi ^{\sigma -1}\left( x\right)
&\leq &\frac{\Gamma \left( \sigma \right) }{\Gamma \left( \delta \right) }%
\mathcal{I}^{\sigma ;\psi }\left[ \psi ^{\delta -1}\left( x\right) \left(
f\circ \psi \right) ^{p}\left( x\right) \right] +\mathcal{I}^{\delta ;\psi }%
\left[ \psi ^{\sigma -1}\left( x\right) \left( f\circ \psi \right)
^{p}\left( x\right) \right] \notag \\
&\leq &2\left( f\circ \psi \right) ^{p}\left( \frac{x}{2}\right) \mathcal{I}%
^{\delta ;\psi }\psi ^{\sigma -1}\left( x\right), \label{inq38}
\end{eqnarray}
(47)
and using (44) and (46), we get
\begin{eqnarray}
\left[ g^{q}\left( 0\right) +\left( g\circ \psi \right) ^{q}\left(
x\right) \right] \mathcal{I}^{\delta ;\psi }\psi ^{\sigma -1}\left( x\right)
&\leq &\frac{\Gamma \left( \sigma \right) }{\Gamma \left( \delta \right) }%
\mathcal{I}^{\sigma ;\psi }\left[ \psi ^{\delta -1}\left( x\right) \left(
g\circ \psi \right) ^{q}\left( x\right) \right] +\mathcal{I}^{\delta ;\psi }%
\left[ \psi ^{\sigma -1}\left( x\right) \left( g\circ \psi \right)
^{q}\left( x\right) \right] \notag \\
&\leq &2\left( g\circ \psi \right) ^{q}\left( \frac{x}{2}\right) \mathcal{I}%
^{\delta ;\psi }\psi ^{\sigma -1}\left( x\right) . \label{inq39}
\end{eqnarray}
(48)
The inequalities (47) and (48) imply that
\begin{eqnarray}
&&\left[ f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left(
x\right) \right] \left[ g^{q}\left( 0\right) +\left( g\circ \psi \right)
^{q}\left( x\right) \right] \left( \mathcal{I}^{\delta ;\psi }\psi ^{\sigma
-1}\left( x\right) \right) ^{2} \notag \\
&&\left. \leq \frac{\Gamma \left( \sigma \right) }{\Gamma \left( \delta
\right) }\mathcal{I}^{\sigma ;\psi }\left[ \psi ^{\delta -1}\left( x\right)
\left( f\circ \psi \right) ^{p}\left( x\right) \right] +\mathcal{I}^{\delta
;\psi }\left[ \psi ^{\sigma -1}\left( x\right) \left( f\circ \psi \right)
^{p}\left( x\right) \right] \right. \notag \\
&&\;\;\;\left.\times \right. \frac{\Gamma \left( \sigma \right) }{\Gamma \left(
\delta \right) }\mathcal{I}^{\sigma ;\psi }\left[ \psi ^{\delta -1}\left(
x\right) \left( g\circ \psi \right) ^{q}\left( x\right) \right] +\mathcal{I}%
^{\delta ;\psi }\left[ \psi ^{\sigma -1}\left( x\right) \left( g\circ \psi
\right) ^{q}\left( x\right) \right] . \label{inq40}
\end{eqnarray}
(49)
Similarly as before, we have \(f\) and \(g\) are positive functions and \(\psi \)
is increasing function on \(\left[ 0,\infty \right) \). Then for any \(x>0,\) \(%
p\geq 1,\) \(q\geq 1\), we can write
\begin{equation}
\frac{f^{p}\left( 0\right) +\left( f\circ \psi \right) ^{p}\left( x\right) }{%
2}\mathcal{I}^{\delta ;\psi }\psi ^{\sigma -1}\left( x\right) \geq 2^{-p}%
\left[ f\left( 0\right) +\left( f\circ \psi \right) \left( x\right) \right]
^{p}\mathcal{I}^{\delta ;\psi }\psi ^{\sigma -1}\left( x\right),
\label{inq41}
\end{equation}
(50)
and
\begin{equation}
\frac{g^{q}\left( 0\right) +\left( g\circ \psi \right) ^{q}\left( x\right) }{%
2}\mathcal{I}^{\delta ;\psi }\psi ^{\sigma -1}\left( x\right) \geq 2^{-q}%
\left[ g\left( 0\right) +\left( g\circ \psi \right) \left( x\right) \right]
^{q}\mathcal{I}^{\delta ;\psi }\psi ^{\sigma -1}\left( x\right) .
\label{inq42}
\end{equation}
(51)
The inequalities (50) and (51) imply that
\begin{eqnarray}
&&\frac{1}{4}\left[ f^{p}\left( 0\right) +\left( f\circ \psi \right)
^{p}\left( x\right) \right] \left[ g^{q}\left( 0\right) +\left( g\circ \psi
\right) ^{q}\left( x\right) \right] \left( \mathcal{I}^{\delta ;\psi }\psi
^{\sigma -1}\left( x\right) \right) ^{2} \notag \\
&&\left. \geq \right. 2^{-p-q}\left[ f\left( 0\right) +\left( f\circ \psi
\right) \left( x\right) \right] ^{p}\left[ g\left( 0\right) +\left( g\circ
\psi \right) \left( x\right) \right] ^{q}\left( \mathcal{I}^{\delta ;\psi
}\psi ^{\sigma -1}\left( x\right) \right) ^{2}. \label{inq43}
\end{eqnarray}
(52)
Combining the inequalities (49) and (52), we obtain the
desired inequality (40).
Remark 3.
If we put \(\psi \left( x\right) =x\) for all \(x\in \left[ 0,\infty \right) ,\)
then Theorem 7 reduce to Theorem 5 obtained by
Dahmani in [15].
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflicts of interest
The authors declare no conflict of interest.
