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\title{Generalized $q$-Srivastava-Attiya operator on multivalent functions}
\author{Rizwan Salim Badar}
\address{Comsats University Islamabad, \\ Department of Mathematics, Park Road,\\
Islamabad,Pakistan.}
\email{rizwansbadar@gmail.com}
%
\author{Khalida Inayat Noor}
\address{Comsats University Islamabad, \\ Department of Mathematics, Park Road,\\
Islamabad,Pakistan.}
\email{khalidainayat@comsats.edu.pk}
%
\subjclass{30C45, 30C80, 30H05.}
\keywords{multivalent functions, $q$-difference operator, $q$-Srivastava-Attiya
operator, starlike and convex functions
$q$-generalized Bernardi operator.}
\begin{abstract}
In this article, we define a generalized $q$-integral operator on multivalent
functions. It generalizes many known linear operators in Geometric Function
Theory (GFT). Inclusions results, convolution properties and $q$-Bernardi
integral preservation of the subclasses of analytic functions are discussed.
\end{abstract}
\maketitle

\section{Introduction}
The study of $q$-extension of classical calculus has been point of focus for
various researchers due to its applications. In Physics, $q$-calculus is
amicably used in theories of quantum fields, Newton quantum gravity, special
relativity and many other notable fields. In Mathematics, various branches has
been established due to its applications in basic hypergeometric functions,
combinatorics, calculus of variations, optimal control problems, $q$-transform
analysis. It dates back to great mathematicians of 17th century L. Euler and
C. Jacobi. F.H. Jackson formally defined $q$-difference operator and
$q$-integral operator in \cite{J1,J2}. For comprehensive details of concepts of $q$-calculus, see \cite{ERNST}.
The concepts of GFT has been studied in
context of $q$-calculus and $q$-analogues of various subclasses of analytic
functions are defined by the researchers, see \cite{SRI,Ismail,
AGARWAL,PURO,SAHOO,KS,N,CP,SRI2,SIB}. Using the convolution of normalized
analytic functions, several $q$-operators are defined by the researchers, see
details in \cite{SRI1}. In this paper we define a generalized $q$-Srivastava
Attiya operator and study its application on multivalent functions.

Let $A(p)$ $(p\in%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
=\{0,1,2,..\})$ denote the set of multivalent functions say $f$ given as%
\begin{equation}
f(z)=z^{p}+\sum_{k=2}^{\infty}a_{k+p-1}z^{k+p-1}, \label{1}%
\end{equation}
analytic in the open unit disc $E=\{z:|z|<1\}$.

Let $f(z)$ be given by (\ref{1}) and $g(z)$ defined as \newline%
\[
f(z)=z^{p}+\sum_{k=2}^{\infty}b_{k+p-1}z^{k+p-1}.
\]
Then Hadamard product (or convolution) of $f$ and $g$ is defined by\newline%
\[
(f\ast g)(z)=z^{p}+\sum_{k=2}^{\infty}a_{k+p-1}b_{k+p-1}z^{k+p-1}.
\]


Let $f,g\in A.$ Then $f$ is subordinate to $g$, written as $f\prec g$ or
$f(z)\prec g(z)$, $z\in E$, if there exists a Schwartz function $\omega(z)$
analytic in $E$ with $\omega(0)=0$ and $\left\vert \omega(z)\right\vert <1$
for $z\in E$ such that $f(z)=g(\omega(z))$.

A subset $D\subset%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$ is called $q$-geometric if $zq\in D$ whenever $z\in D$ and it contains all
the geometric sequences $\left\{  zq^{k}\right\}  _{0}^{\infty}.$ In GFT, the
$q$-derivative of $f(z)$ is defined as;%
\begin{equation}
d_{q}f(z)=\frac{f(z)-f(qz)}{(1-q)z},\text{ }q\in(0,1),\quad(z\in
D\setminus\left\{  0\right\}  ), \label{2}%
\end{equation}
and $d_{q}f(0)=f^{\prime}(0)$. For a function $g(z)=z^{k}$, the $q$-derivative
is%
\[
d_{q}g(z)=[k]z^{k-1},
\]
where $\left[  k\right]  =\frac{1-q^{k}}{1-q}=1+q+q^{2}+....+q^{k-1}.$

From (\ref{1}) and (\ref{2}) we easily get that%

\[
d_{q}f(z)=\left[  p\right]  z^{p-1}+\sum_{k=2}^{\infty}[k+p-1]\text{
}a_{k+p-1}z^{k+p-2}.
\]


Let $f(z)$ and $g(z)$ be defined on a $q$-geometric set $D\subset%
%TCIMACRO{\U{2102} }%
%
\mathbb{C}
%EndExpansion
$ such that $q$-derivatives of $f(z)$ and $g(z)$ exist $\forall$ $z\in
\mathbb{C}.$ Then
for complex numbers $b,c$ we have :%
\[
d_{q}(bf(z)\pm cg(z))=bd_{q}f(z)\pm cd_{q}g(z).
\]%
\[
d_{q}(f(z)g(z))=f(qz)d_{q}g(z)+g(z)d_{q}f(z).
\]%
\[
d_{q}\left(  \frac{f(z)}{g(z)}\right)  =\frac{g(z)\text{ }d_{q}f(z)-f(z)\text{
}d_{q}g(z)}{g(z)g(qz)},\text{ }g(z)g(qz)\neq0.
\]%
\[
d_{q}\left(  \log f(z)\right)  =\frac{\ln q^{-1}}{1-q}\frac{d_{q}f(z)}%
{f(z)}\text{.}%
\]


Jackson \cite{J1} introduced the q-integral of a function $f$ is given by
\[
\int_{0}^{z}f(t)d_{q}t=z(1-q)\sum_{k=0}^{\infty}q^{k}f(q^{k}z),
\]
provided that series converges.

Consider a $q$-analogue of Lerch-Hurwitz function%
\[
\text{\ }\Phi_{q}(s,b;z)=\sum_{k=0}^{\infty }\frac{z^{k}}{[k+b]^{s}},z\in E,
\]
$ \ \ \ (b\in%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
\setminus%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{0}^{-};s\in%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$ when $\left\vert z\right\vert <1;\operatorname{Re}(s)>1$ when $\left\vert
z\right\vert
=1),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $%

which is a convergent series of radius 1. Now we define the generalized
$q$-Srivastava Attiya operator $J_{q,b}^{s,p}:\ A(p)\rightarrow A(p)$ as%
\begin{equation}
J_{q,b}^{s,p}f(z)=\Psi_{q}(s,b;z)\ast f(z), \label{3}%
\end{equation}
where,%
\begin{equation}
\Psi_{q}(s,b;z)=\left[  1+b\right]  ^{s}\left\{  \Phi_{q}(s,b;z)-[b]^{-s}%
\right\}. \label{4}%
\end{equation}


From (\ref{1}), (\ref{3}) and (\ref{4}), we have%
\begin{equation}
J_{q,b}^{s,p}f(z)=z^{p}+\sum_{k=2}^{\infty}\left(  \frac{\left[  1+b\right]
}{[k+b]}\right)  ^{s}a_{k+p-1}z^{k+p-1}. \label{5}%
\end{equation}
We observe that $J_{q,b}^{0,p}f(z)=f(z)$. The operator $J_{q,b}^{s,p}$ reduces
to known linear operators for different values of parameters $p,b$ and $s$ as:

\textbf{(i)} For $p=1,$ $s=1,b=0$ and $q\rightarrow1^{-}$ $,$ $J_{q,b}^{s,p}$
reduces to Alexander operator \cite{ALX}.

\textbf{(ii)} If $p=1,$ it is $q$-Srivastava Attiya operator discussed in
\cite{SAK}.

\textbf{(iii)} For $p=1,$ $s=1,b>0$ it reduces to $q$-Choi-Saigo-Srivastava
operator discussed in \cite{WANG}.

\textbf{(iv)} For $s=\alpha$ $(\alpha>0),$ $b=p$ and $q\rightarrow1^{-},$ it
is operator discussed in \cite{JAHAN}.

For any complex number $s,$ the operator $I_{q,s}^{b,p}:A(p)\rightarrow A(p)$
is defined as;%
\begin{equation}
I_{q,b}^{s,p}f(z)=z^{p}+\sum_{k=2}^{\infty}\left(  \frac{\left[  k+b\right]
}{[1+b]}\right)  ^{s}a_{k+p-1}z^{k+p-1}. \label{6}%
\end{equation}
The operator $I_{q,b}^{s,p}$ also reduces to known linear operators as:

\textbf{(i)} For $p=1,$ $s\in%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
_{0},$ $b=0,$ it is $q$-S\~{a}l\~{a}gean differential operator \cite{GS}.

\textbf{(ii)} For $p=1,$ $s=-1$ and $q\rightarrow1^{-}$, it reduces to
Owa-Srivastava Integral Operator \cite{OWA}.

The following identities holds for the two operators $J_{q,b}^{s,p}(z)$and
$I_{q,b}^{s,p}(z)$,%
\begin{align}
zd_{q}(J_{q,b}^{s+1,p}f(z))  &  =q^{p-1}\left(  1+\frac{\left[  b\right]
}{q^{b}}\right)  J_{q,b}^{s,p}f(z)+\left(  \frac{\left[  p-1\right]  -\left[
b\right]  }{q^{b}}\right)  J_{q,b}^{s+1,p}f(z).\label{7}\\
zd_{q}(I_{q,s}^{b,p}f(z))  &  =q^{p-1}\left(  1+\frac{\left[  b\right]
}{q^{b}}\right)  I_{q,s+1}^{b,p}f(z)+\left(  \frac{\left[  p-1\right]  -\left[
b\right]  }{q^{b}}\right)  I_{q,s}^{b,p}f(z). \label{8}%
\end{align}

Here we prove the identity (\ref{7}) as;%
\begin{equation*}
  zd_{q}(J_{q,b}^{s+1,p}f(z)) =\left[ p\right] z^{p}+\sum_{k=2}^{\infty
}\left( \frac{\left[ 1+b\right] }{[k+b]}\right) ^{s+1}a_{k+p-1}\left[ k+p-1%
\right] z^{k+p-1}, 
\end{equation*}
or equivalently,
\begin{multline*}
  zd_{q}(J_{q,b}^{s+1,p}f(z))  = \left[ p\right] z^{p}+\sum_{k=2}^{\infty }\left( \tfrac{\left[ 1+b\right]
}{[k+b]}\right) ^{s+1} \times \\
    a_{k+p-1}\left[ (k+b)+(p-b-1)\right] z^{k+p-1}.
\end{multline*}
Using the property $\left[ a+b\right] =q^{b}[a]+[b]$ we have:%
\begin{multline*}
  zd_{q}(J_{q,b}^{s+1,p}f(z))=\left[ p\right] z^{p}+\sum_{k=2}^{\infty }\left(
\frac{\left[ 1+b\right] }{[k+b]}\right) ^{s+1} \times \\
  a_{k+p-1}\left\{q^{p-b-1}[k+b]+[p-b-1]\right\} z^{k+p-1}.
\end{multline*}
By adding and subtracting the terms $q^{p-b-1}[1+b]z^{p}$ and $[p-b-1]z^{p}$%
, using the property $\left[ a+b\right] =q^{b}[a]+[b]$ and rearranging the
terms: we get%
\begin{equation*}
zd_{q}(J_{q,b}^{s+1,p}f(z))=q^{p-1}\left( 1+\frac{\left[ b\right] }{q^{b}}%
\right) J_{q,b}^{s,p}f(z)+\left( \frac{\left[ p-1\right] -\left[ b\right] }{%
q^{b}}\right) J_{q,b}^{s+1,p}f(z).
\end{equation*}%
On same lines, we can prove the identity (\ref{8}) as well.

\bigskip

\begin{definition}
A function $f\in A(p)$ is said to be in the class $ST_{q}^{p}(\varphi)$ if and
only if
\begin{equation}
\frac{zd_{q}f(z)}{\left[  p\right]  f(z)}\prec\varphi(z); \label{9}%
\end{equation}
where $\varphi\in\Omega,$ the class of analytic and convex multivalent
functions in $E.$
\end{definition}

\begin{definition}
A function $f\in A(p)$ is said to be in the class $CV_{q}^{p}(\varphi)$ if and
only if
\[
\frac{d_{q}(zd_{q}f(z))}{\left[  p\right]  d_{q}f(z)}\prec\varphi(z);
\]
where $\varphi\in\Omega,$ the class of analytic and convex multivalent
functions in $E.$
\end{definition}

By using operators given by (\ref{5}) and (\ref{6}), we define the classes%
\[
ST_{q,b}^{s,p}\left(  \varphi\right)  =\left\{  f\in A(p):J_{q,b}^{s,p}f(z)\in
ST_{q}^{p}(\varphi)\right\}
\]


and%
\[
\widetilde{ST}_{q,s}^{b,p}\left(  \varphi\right)  =\left\{  f\in
A(p):I_{q,s}^{b,p}f(z)\in ST_{q}^{p}(\varphi)\right\}.
\]


Similarly;%
\[
CV_{q,b}^{s,p}\left(  \varphi\right)  =\left\{  f\in A(p):J_{q,b}^{s,p}f(z)\in
CV_{q}^{p}(\varphi)\right\}
\]


and%
\[
\widetilde{CV}_{q,s}^{b,p}\left(  \varphi\right)  =\left\{  f\in
A(p):I_{q,s}^{b,p}f(z)\in CV_{q}^{p}(\varphi)\right\}  .
\]
It is noted%
\[
f\in CV_{q,b}^{s,p}\left(  \varphi\right)  \Leftrightarrow\frac{zd_{q}%
f(z)}{\left[  p\right]  }\in ST_{q,b}^{s,p}\left(  \varphi\right)
\]


and%
\[
f\in\widetilde{CV}_{q,s}^{b,p}\left(  \varphi\right)  \Leftrightarrow
\frac{zd_{q}f(z)}{\left[  p\right]  }\in\widetilde{ST}_{q,s}^{b,p}\left(
\varphi\right).
\]
We need the following Lemma to obtain our results;

\begin{lemma}
\label{lem:2.1}\cite{HSL} Let $\beta$ and $\gamma$ be complex numbers with
$\beta\neq0,$ and let $h(z)$ be a regular in $E$ with $h(0)=1$ and
$\operatorname{Re}[\beta h(z)+\gamma]>0.$ If $p(z)=1+p_{1}z+p_{2}z^{2}+...$is
analytic in $E,$ then $p(z)+\frac{zd_{q}p(z)}{\beta p(z)+\gamma}\prec
h(z)\Rightarrow p(z)\prec h(z).$
\end{lemma}

\section{Main Results}

\subsection{Inclusions Results}

In this section, we proved the inclusions results of the classes with respect
to parameter $s.$

\begin{theorem}
\label{theorem1}Let $\varphi(z)$ be analytic and convex multivalent function
with $\varphi(0)=1$ and $\operatorname{Re}(\varphi(z))>0$ for $z\in E.$ Then
$ST_{q,b}^{s,p}\left(  \varphi\right)  \subset ST_{q,b}^{s+1,p}\left(
\varphi\right)  $ if $\ \operatorname{Re}(q^{b-p+1})>1$.
\end{theorem}

\begin{proof}
Let $f\in ST_{q,b}^{s,p}\left(  \varphi\right)  $ and we set%
\begin{equation}
\frac{zd_{q}(J_{q,b}^{s+1,p}f(z))}{[p]J_{q,b}^{s+1,p}f(z)}=Q(z),\text{ }z\in
E, \label{10}%
\end{equation}
where $Q(z)$ is analytic in $E$ with $Q(0)=1.$

Using identity (\ref{7}), we get%
\begin{equation}
\frac{zd_{q}(J_{q,b}^{s+1,p}f(z))}{J_{q,b}^{s+1,p}f(z)}=M_{q}\frac
{J_{q,b}^{s,p}f(z)}{J_{q,b}^{s+1,p}f(z)}-\gamma_{q}, \label{11}%
\end{equation}
where $M_{q}=q^{p-1}\left(  1+\frac{\left[  b\right]  }{q^{b}}\right)  $ and
$\gamma_{q}=\frac{\left[  p-1\right]  -\left[  b\right]  }{q^{b}}.$

From (\ref{10}) and (\ref{11}), we have
\[
\lbrack p]Q(z)+\gamma_{q}=M_{q}\frac{J_{q,b}^{s,p}f(z)}{J_{q,b}^{s+1,p}f(z)}.
\]

Applying logarithmic $q$-differentiation,
\begin{equation}
\frac{zd_{q}(J_{q,b}^{s,p}f(z))}{J_{q,b}^{s,p}f(z)}=[p]Q(z)+\frac
{[p]zd_{q}Q(z)}{Q(z)+\gamma_{q}}. \label{12}%
\end{equation}

As $f\in ST_{q,b}^{s,p}\left(  \varphi\right)  $ so,%
\begin{equation}
\frac{zd_{q}(J_{q,b}^{s,p}f(z))}{J_{q,b}^{s,p}f(z)}\prec\varphi(z). \label{13}%
\end{equation}


From (\ref{12}) and (\ref{13}), we have%

\[
\lbrack p]Q(z)+\frac{[p]zd_{q}Q(z)}{Q(z)+\gamma_{q}}\prec\varphi(z).
\]

As $\operatorname{Re}(q^{b-p+1})>1$ and $\operatorname{Re}(\varphi)>0$ then by
Lemma \ref{lem:2.1}, we have $Q(z)\prec\varphi(z)$ which implies $f\in
ST_{q,b}^{s+1,p}\left(  \varphi\right)  .$ So $ST_{q,b}^{s,p}\left(
\varphi\right)  \subset ST_{q,b}^{s+1,p}\left(  \varphi\right)  .$
\end{proof}

\begin{theorem}
Let $\varphi(z)$ be same as in Theorem \ref{theorem1}. Then $CV_{q,b}%
^{s,p}\left(  \varphi\right)  \subset CV_{q,b}^{s+1,p}\left(  \varphi\right)
$ if $\operatorname{Re}(q^{b-p+1})>1$.

\begin{proof}
It is evident from the fact $f\in CV_{q,b}^{s,p}\left(  \varphi\right)
\Leftrightarrow\frac{zd_{q}f(z)}{\left[  p\right]  }\in ST_{q,b}^{s,p}\left(
\varphi\right)  .$
\end{proof}
\end{theorem}

We can easily prove the following result by using Lemma \ref{lem:2.1} and
identity relation (\ref{8}).

\begin{theorem}
Let $\varphi(z)$ be analytic and convex multivalent function with
$\varphi(0)=1$ and $\operatorname{Re}(\varphi(z))>0$ for $z\in E.$Then%
\[
\widetilde{ST}_{q,s}^{b,p}\left(  \varphi\right)  \subset\widetilde
{ST}_{q,s+1}^{b,p}\left(  \varphi\right)
\]


and%
\[
\widetilde{CV}_{q,s+1}^{b,p}\left(  \varphi\right)  \subset\widetilde
{CV}_{q,s}^{b,p}\left(  \varphi\right)  .
\]

\end{theorem}

\subsection{Integral Preservation under Generalized $q$-Bernardi Integral Operator}

In \cite{SQA}, the $q$-Bernardi integral operator $L_{c,p}f(z)$ for
multivalent functions is defined as;
\begin{equation}
L_{c,p}f(z)=\frac{\left[  p+c\right]  }{z^{c}}\int_{0}^{z}t^{c-1}f(t)d_{q}t,
\label{14}%
\end{equation}

where $f\in A(p)$ given by (\ref{1}) with $c>-p.$

\begin{theorem}
If $f\in$ $ST_{q,b}^{s,p}\left(  \varphi\right)  $ then $L_{c,p}f(z)\in
ST_{q,b}^{s,p}\left(  \varphi\right)  .$

\begin{proof}
Let $f(z)\in ST_{q,b}^{s,p}\left(  \varphi\right).$ Consider
\begin{equation}
\frac{zd_{q}(J_{q,b}^{s,p}(L_{c,p}f(z)))}{\left[  p\right]  J_{q,b}%
^{s,p}(L_{c,p}f(z))}=Q(z),%
%TCIMACRO{\U{b8} }%
\label{15}%
\end{equation}

where $Q(z)$ is analytic in $E$ with $Q(0)=1.$ From (\ref{14}), after some
calculations, we can write%
\[
zd_{q}(L_{b,p}f(z))=\left[  p+c\right]  f(z)-\left[  c\right]  L_{c,p}f(z).
\]


Now applying the operator $J_{q,b}^{s,p}$ on both sides, we have
\begin{equation}
\frac{zd_{q}(J_{q,b}^{s,p}(L_{c,p}f(z)))}{J_{q,b}^{s,p}(L_{c,p}f(z))}=\left[
p+c\right]  \frac{J_{q,b}^{s,p}f(z)}{J_{q,b}^{s,p}(L_{c,p}f(z))}-\left[
c\right]  . \label{16}%
\end{equation}


Now applying logarithmic $q$-differentiation on both sides of (\ref{16}), after
some calculations and using (\ref{15}), we get%
\begin{equation}
\frac{zd_{q}(J_{q,b}^{s,p}f(z))}{\left[  p\right]  J_{q,b}^{s,p}%
f(z)}=Q(z)+\frac{z[d_{q}Q(z)]}{\left[  p\right]  Q(z)+\left[  c\right]  }.
\label{17}%
\end{equation}


As $f\in$ $ST_{q,b}^{s,p}\left(  \varphi\right),$ so from (\ref{16}) and
(\ref{17}), we have
\[
Q(z)+\frac{z[d_{q}Q(z)]}{\left[  p\right]  Q(z)+\left[  b\right]  }%
\prec\varphi(z).
\]


As $\ \operatorname{Re}([p]\varphi(z)+[c])>0$ so by Lemma \ref{lem:2.1}, we
have $Q(z)\prec\varphi(z)$ which implies $L_{c,p}f(z)\in ST_{q,b}^{s,p}\left(
\varphi\right).$
\end{proof}
\end{theorem}

\begin{theorem}
If $f(z)\in CV_{q,b}^{s,p}\left(  \varphi\right)$ then $L_{c,p}f(z)\in CV_{q,b}^{s,p}\left(  \varphi\right)$.
\end{theorem}

\begin{proof}
It is immediate consequence of the fact $f\in CV_{q,b}^{s,p}\left(
\varphi\right)  \Leftrightarrow\frac{zd_{q}f(z)}{\left[  p\right]  }\in
ST_{q,b}^{s,p}\left(  \varphi\right).$
\end{proof}

\subsection{Convolution Property of $ST_{q,b}^{s,p}\left(\varphi\right)  $}

We now obtain convolution property for the class $ST_{q,b}%
^{s,p}\left(  \varphi\right).  $

\begin{theorem}
Let $f\in ST_{q,b}^{s,p}\left(  \varphi\right).$ Then%
\begin{multline}\label{18}
f(z)=z^{\left[  p\right]  }.\exp\left(  \frac{\ln q^{-1}}{1-q}\left[
p\right]  \int_{0}^{z}\frac{\varphi(\omega(\varsigma))-1}{\varsigma}%
d_{q}\varsigma\right) \\
  \ast\left(  z^{p}+\sum_{k=2}^{\infty}\left(
\frac{\left[  k+b\right]  }{[1+b]}\right)  ^{s}a_{k+p-1}z^{k+p-1}\right),
\end{multline}
where $\omega$ is Schwartz function.

\begin{proof}
Suppose that $f\in ST_{q,b}^{s,p}\left(  \varphi\right).$  The subordination
condition (\ref{9}) can be written as:%
\begin{equation}
\frac{zd_{q}(J_{q,b}^{s,p}f(z))}{J_{q,b}^{s,p}f(z)}=\left[  p\right]
\varphi(\omega(z)), \label{19}%
\end{equation}
where $\omega$ is Schwartz function$.$

From (\ref{19}), after $q$-integrating we get%
\begin{equation}
\log\left(  \frac{J_{q,b}^{s,p}f(z)}{z^{[p]}}\right)  =\frac{\ln q^{-1}}%
{1-q}\left[  p\right]  \int_{0}^{z}\frac{\varphi(\omega(\varsigma
))-1}{\varsigma}d_{q}\varsigma. \label{20}%
\end{equation}

It follows from (\ref{20}) that%
\begin{equation}
J_{q,b}^{s,p}f(z)=z^{\left[  p\right]  }.\exp\left(  \frac{\ln q^{-1}}%
{1-q}\left[  p\right]  \int_{0}^{z}\frac{\varphi(\omega(\varsigma
))-1}{\varsigma}d_{q}\varsigma\right). \label{21}%
\end{equation}

The assertion can be obtained easily from (\ref{5}) and (\ref{21}).
\end{proof}
\end{theorem}

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\end{document}