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\title{Geometric properties of mixed operator involving Ruscheweyh derivative and S\`{a}l\`{a}gean operator }
\author{Rabha W. Ibrahim}
\address{IEEE:  94086547,  Kuala Lumpur, 59200, \\Malaysia  }
\email{rabhaibrahim@yahoo.com}
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\author{Mayada T. Wazi}
\address{University of Technology, \\ Department of Electromechanical Engineering, \\
Iraq}
\email{mayada.t.wazi@uotechnology.edu.iq}
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\author{Nadia Al-Saidi}
\address{University of Technology, \\ Department of Applied Sciences,\\
Iraq}
\email{nadiamg08@gmail.com}
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\subjclass{30C45}
\keywords{Differential operator; conformable operator; fractional calculus;  unit disk; univalent function; analytic function; subordination and superordination}

\begin{abstract}
Operator theory is a magnificent tool for studying the geometric behaviors of holomorphic functions in the open unit disk. Recently, a combination between two well known differential operators, Ruscheweyh derivative and S\`{a}l\`{a}gean operator are suggested by Lupas in \cite{r1}. In this effort, we shall follow the same principle, to formulate a generalized differential-difference operator.  We deliver a new class of analytic functions containing the generalized operator. Applications are illustrated in the sequel concerning some  differential subordinations of the operator.
\end{abstract}


\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Differential operators in a complex domain play a significant role in functions theory and its information. They have used to describe the geometric interpolation  
of analytic functions in a complex domain. Also, they have utilized to generate new formulas of holomorphic functions. Lately, Lupas \cite{r1} presented a amalgamation of two well-known differential operators prearranged by
Ruscheweyh \cite{r2} and S\`{a}l\`{a}gean \cite{r3}. Later, these operators are investigated by researchers considering different classes and formulas of analytic functions \cite{rr3,rr4}.

\medskip \noindent In this note, we consider a special class of functions in the open unit disk $\sqcup=\{\xi \in \mathbb{C}|\,  |\xi|<1\}$ denoting by  $\Sigma$ and having the series 
\[\varphi (\xi)= \xi+\sum^\infty_{n=2} \varphi_n \xi^n, \quad \xi \in \sqcup.\]
Let $\varphi \in \Sigma,$ then the Ruscheweyh formula is indicated by the structure 
 formula \[\Phi^m\, \varphi(\xi)= \xi+ \sum^\infty_{n=2} C^m_{m+n-1} \,  \varphi_n  \xi^n.\]  While, the  S\`{a}l\`{a}gean operator admits the construction \[
 \Psi^m\, \varphi(\xi)= \xi+ \sum^\infty_{n=2}n^m \,  \varphi_n  \xi^n.\]
 Lupas operator is formulated by the structure 
 \[\lambda_\sigma^m\, \varphi(\xi)=\xi+ \sum^\infty_{n=2}  \, [\sigma n^m+(1-\sigma)\, 
 C^m_{m+n-1}] \varphi_n  \xi^n, \quad \xi \in \sqcup, \, \sigma \in [0,1]. \]
   Newly, Ibrahim and Darus \cite{r4} considered the next differential operator
\begin{equation*}
\begin{split}
&\Theta_\kappa ^0 \varphi(\xi)=\varphi(\xi)\\
&\Theta_\kappa ^1 \varphi(\xi)= \xi\, \varphi(\xi)'+\dfrac{\kappa}{2}\left(  \varphi(\xi)-\varphi(-\xi)-2\xi \right), \quad \kappa \in \mathbb{R}\\
&\vdots\\
& \Theta_\kappa^{m} \varphi(\xi)= \Theta_\kappa  (\Theta_\kappa^{m-1}\varphi(\xi))\\
& \,\,\,\,\, \quad \,\,\,\,\, \quad = \xi+ \sum^\infty_{n=2}[n+\dfrac{\kappa}{2} (1+(-1)^{n+1})]^m \,  \varphi_n  \xi^n.
\end{split}
\end{equation*}
When $\kappa=0,$we have $\Psi^m \, \varphi(\xi)$ In addition, it is a modified formula of the well-known Dunkl operator \cite{r5}, where $\kappa$ is known as the Dunkl order.  Proceeding, we define a generalized formula of 
$\lambda_\sigma^m $, as follows: 
\begin{equation}\label{e1} 
\begin{split}
 J_{\sigma,\kappa}^{m} \varphi(\xi)&= (1-\sigma) \Phi^m \varphi(\xi)+\sigma \Theta_\kappa^{m}\varphi(\xi)\\
& = \xi+ \sum^\infty_{n=2}[ (1-\sigma) C^m_{m+n-1}+ \sigma\big(n+\dfrac{\kappa}{2} (1+(-1)^{n+1})\big)^m] \,  \varphi_n  \xi^n.
\end{split}
\end{equation}
Clearly, the operator $ J_{\sigma,\kappa}^{m} \varphi(\xi)\in \Sigma.$

\begin{remark}\label{rem1} 

\noindent \begin{itemize}
\item $m=0 \Longrightarrow  J_{\sigma,\kappa}^{0} \varphi(\xi)=\varphi(\xi); $ 
\item $\kappa=0 \Longrightarrow  J_{\sigma,0}^{m} \varphi(\xi)=\lambda_\sigma^m\, \varphi(\xi); $ 
\item $\sigma=0 \Longrightarrow  J_{0,\kappa}^{m} \varphi(\xi)=\Phi^m\, \varphi(\xi); $ 
\item $ \sigma=1 \Longrightarrow  J_{1,\kappa}^{m} \varphi(\xi)=\Theta_\kappa^{m}\varphi(\xi); $
\item $\kappa=0, \sigma=1 \Longrightarrow  J_{1,0}^{m} \varphi(\xi)= \Psi^m\, \varphi(\xi). $ 
\end{itemize}
\end{remark}
\begin{definition}\label{def1} 
 Consider the following data $\epsilon \in [0,1),  \sigma \in [0,1], \kappa \geq 0,$ and $m \in \mathbb{N}.$  Then a function $\varphi \in \Sigma$ belongs to the set $\top_m(\sigma,\kappa,\epsilon)$  if and only if
  \[ \Re \Big( ( J_{\sigma,\kappa}^{m} \varphi(\xi))'  \Big)> \epsilon, \quad \xi \in \sqcup.\]
\end{definition}
 Observe that the set $\top_m(\sigma,\kappa,\epsilon)$ is an extension of the well known class of bounded turning functions (see \cite{r6}-\cite{r10}). Next results are requested to prove our results depending on the subordination concept  (see \cite{r11}). 

\begin{lemma}\label{lem1} 
Suppose that $\hbar$ is convex function such that 
$\hbar(0) = \flat,$ and there is a complex number with a positive real part $\mu.$  If $\flat\in \mathfrak{H}[\flat, n],$ where  \[\mathfrak{H}[\flat,n] = \{ \flat \in \mathfrak{H}: \flat(\xi)= \flat+ \flat_{n}\xi^n+\flat_{n+1}\xi^{n+1}+...  \}\] (the space of  holomorphic functions) and
\[\flat(\xi)+\dfrac{1}{\mu} \xi \flat'(\xi) \prec \hbar(\xi), \quad \xi \in \sqcup,\]
then 
\[\flat(\xi)  \prec \iota(\xi) \prec \hbar(\xi),\] with
 \[ \iota(\xi)= \dfrac{\mu}{n\, \xi^{\mu/n}}\int_0^\xi  \hbar(\tau) \tau^{\dfrac{\mu}{(n-1)}}d \tau, \quad \xi \in \sqcup.\]
\end{lemma}

\begin{lemma}\label{lem2}
Suppose that the convex function  $\flat(\xi)$ satisfies the functional $\hbar(\xi)=\flat(\xi)+n \mu (\xi \, \flat'(\xi))$ for $\mu>0$ and $n$ is a positive integer. If $\flat \in \mathfrak{H}[\hbar(0), n],$ and $\flat(\xi)+\mu \xi \flat'(\xi) \prec \hbar(\xi), \, \xi \in \sqcup$ 
then $\flat(\xi)  \prec \hbar(\xi),$ and this outcome is sharp.
\end{lemma}

 \begin{lemma}\label{lem3}
\begin{itemize}
\noindent 
\item[(i)] If $\lambda>0, \gamma>0$, $\beta=\beta(\gamma,\lambda,n)$ and ${\flat}\in \mathfrak{H}[1,n]$ then
\[ {\flat}(\xi)+\lambda \xi{\flat}'(\xi) \prec \left[\dfrac{1+\xi}{1-\xi}\right]^\beta \Rightarrow {\flat}(\xi)\prec \left[\dfrac{1+\xi}{1-\xi}\right]^\gamma.\]

\item[(ii)] If $\epsilon \in [0,1)$, $\lambda=\lambda(\epsilon,n)$ and ${\flat}\in \mathfrak{H}[1,n]$ then $ \Re\Big({\flat}^2(\xi)+2{\flat}(\xi).\xi{\flat}'(\xi) \Big)>\epsilon \Rightarrow \Re({\flat}(\xi))> \lambda.$\end{itemize}
\end{lemma}

\section{Results} In this section, we investigate  some geometric conducts of the operator \eqref{e1}. 
 
\begin{theorem}\label{thm1} 
The set $\top_m(\sigma,\kappa,\epsilon)$  is convex. 
\end{theorem}

\begin{proof} 
Suppose that $\varphi_i, i=1,2$ are two functions belonging to $\top_m(\sigma,\kappa,\epsilon)$ satisfying $\varphi_1(\xi)= \xi+\sum^\infty_{n=2} \varphi_n \xi^n$ and $\varphi_2(\xi)= \xi+\sum^\infty_{n=2} \phi_n \xi^n.$ It is sufficient to prove that the function 
 $$\Pi(x\i)= \wp_1\varphi_1(\xi)+\wp_2\varphi_2(\xi), \quad \xi \in \sqcup$$ is in $\top_m(\sigma,\kappa,\epsilon),$ where $\wp_1>0, \wp_2>0$ and $\wp_1+\wp_2=1.$
The formula of $\Pi(z)$  yields  
\[\Pi(\xi)=\xi+\sum^\infty_{n=2} (\wp_1\varphi_n+\wp_2\phi_n) \xi^n.  \] 
Thus, under the operator \eqref{e1}, we get
\[ J_{\sigma,\kappa}^{m}  \Pi(\xi)=\xi+\sum^\infty_{n=2} (\wp_1\varphi_n+\wp_2\phi_n) [ (1-\sigma) C^m_{m+n-1}+ \sigma\big(n+\dfrac{\kappa}{2} (1+(-1)^{n+1})\big)^m] \xi^n.  \]  By making a differentiation, we obtain 
 \begin{equation*}
\begin{split}
\Re\{(J_{\alpha,\kappa}^{m}  \Pi(\xi))' \}=& 1+\wp_1\Re\{\sum^\infty_{n=2}  n[ (1-\sigma) C^m_{m+n-1}+ \sigma\big(n+\dfrac{\kappa}{2} (1+(-1)^{n+1})\big)^m] \varphi_n \xi^{n-1} \}\\
&+\wp_2\Re\{\sum^\infty_{n=2}  n[ (1-\sigma) C^m_{m+n-1}+\sigma
\big(n+\dfrac{\kappa}{2} (1+(-1)^{n+1})\big)^m] \phi_n \, \xi^{n-1} \}\\
&=\epsilon.
\end{split}
\end{equation*}
\end{proof}

\begin{theorem}\label{thm2} 
Define the following functions: 
 $\varphi \in \top_m(\sigma,\kappa,\epsilon),$  $\phi$ be convex and 
\[F(\xi)= \dfrac{2+c}{\xi^{1+c}}  \int_0^\xi t^c\, \varphi(t) dt, \quad \xi \in \sqcup.   \]
Then 
\[ \Big(J_{\alpha,\kappa}^{m}  \varphi(\xi) \Big)' \prec \phi(\xi)+ \dfrac{(\xi\, \phi'(\xi))}{2+c}, \quad c >0, \] yields 
\[\Big(J_{\sigma,\kappa}^{m}  F(\xi) \Big)' \prec \phi(\xi),\]and this outcome is sharp.
\end{theorem}

\begin{proof}
By the assumptions, we have 
\[\Big(J_{\sigma,\kappa}^{m}  F(\xi) \Big)'+ \dfrac{\Big(J_{\sigma,\kappa}^{m}  F(\xi) \Big)''}{2+c}=\Big(J_{\sigma,\kappa}^{m}  \varphi(\xi) \Big)'.\]
Consequently, we get 
\[ \Big(J_{\sigma,\kappa}^{m}  F(\xi) \Big)'+ \dfrac{\Big(J_{\sigma,\kappa}^{m}  F(\xi) \Big)''}{2+c}\prec\phi(\xi)+ \dfrac{(\xi\phi'(\xi))}{2+c}. \]
Assuming 
\[\flat(\xi):= \Big(J_{\sigma,\kappa}^{m}  F(\xi) \Big)',\]
one can find 
\[\flat(\xi)+ \dfrac{(\xi\flat'(\xi))}{2+c}\prec  \phi(\xi)+ \dfrac{(\xi\phi'(\xi))}{2+c}.\]
In virtue of Lemma \ref{lem1}, we have 
\[\Big(J_{\sigma,\kappa}^{m}  F(\xi) \Big)' \prec \phi(\xi),\]and $\phi$ is the best dominant.
\end{proof}

\begin{theorem}\label{thm3} 
Assume the convex function  $\phi$ achieving $\phi(0)=1$ and for $\varphi \in \Sigma$ 
\[ \Big(J_{\sigma,\kappa}^{m}  \varphi(\xi) \Big)' \prec \phi(\xi)+ \xi\, \phi'(\xi), \quad \xi \in \sqcup, \]then 
\[\dfrac{J_{\sigma,\kappa}^{m} \varphi(\xi)}{\xi}  \prec \phi(\xi),\]and this outcome is sharp.
\end{theorem}

\begin{proof} Formulate the next functional 
\begin{equation}\label{e2.1} 
\flat(z):=\dfrac{J_{\sigma,\kappa}^{m}  \varphi(\xi)}{\xi} \in \mathfrak{H}[1,1]
\end{equation}
Consequently, we get 
\[J_{\sigma,\kappa}^{m}  \varphi(\xi)=\xi\, \flat(\xi) \Longrightarrow 
 \Big(J_{\sigma,\kappa}^{m}  \varphi(\xi) \Big)'= \flat(\xi)+\, \xi\flat'(\xi). \]
Therefore, we obtain the inequality  
\[ \flat(\xi)+ \xi\flat'(\xi) \prec  \phi(\xi)+\xi\phi'(\xi).\]
According to Lemma \ref{lem2}, we attain 
\[\dfrac{J_{\sigma,\kappa}^{m}  \varphi(\xi)}{\xi}  \prec \phi(\xi),\]and $\phi$ is the best dominant.
\end{proof}
%_______________________________________
 \begin{theorem}\label{thm4} 
For $\varphi \in \Sigma$ if the inequality 
 \[ (J_{\sigma,\kappa}^{m} \varphi(\xi))' \prec \left(\dfrac{1+\xi}{1-\xi}\right)^\beta,  \quad \xi \in \sqcup,\ \beta>0,\] achieves 
 then $$\Re \Big(\frac{ J_{\sigma,\kappa}^{m}\varphi(\xi)}{\xi}\Big)> \epsilon$$ for some $\epsilon \in [0,1).$
 \end{theorem}
 
 \begin{proof} 
For the function  $\flat(\xi)$ in \eqref{e2.1}, we have
\[(J_{\sigma,\kappa}^{m} \varphi(\xi))'=\xi\flat'(\xi)+ \flat(\xi)\prec \left(\dfrac{1+\xi}{1-\xi}\right)^\beta.\]
According to  Lemma \ref{lem3}.i, there occurs a constant $\gamma>0$ with $ \beta=\beta(\gamma)$ with
\[ \frac{J_{\sigma,\kappa}^{m} \varphi(\xi)}{\xi}\prec \left(\dfrac{1+\xi}{1-\xi}\right)^\gamma. \]
This yields  $\Re (J_{\sigma,\kappa}^{m} \varphi(\xi)/\xi)>\epsilon,$ for some $\epsilon \in [0,1).$
\end{proof}

%_________________

 \begin{theorem}\label{thm5} 
Assume that  $\varphi \in \Sigma$ achieves the inequality 
 \[\Re\Big ((J_{\sigma,\kappa}^{m} \varphi(\xi))' \frac{J_{\sigma,\kappa}^{m} \varphi(\xi)}{\xi} \Big )> \dfrac{\sigma}{2},  \quad \xi \in \sqcup, \, \sigma\in [0,1).\] Then $J_{\sigma,\kappa}^{m}\varphi(\xi) \in \top_m(\sigma,\kappa,\epsilon) $ for some $\epsilon \in [0,1).$ In addition, it is univalent of bounded turning in $\sqcup$. 
 \end{theorem}
 
 \begin{proof} 
Assume the function $\flat(\xi)$ as in  \eqref{e2.1}.  A Calculation implies that 
 \begin{equation}\label{e4}
\Re\Big (\flat^2(\xi)+2\flat(\xi) . \xi \flat'(\xi) \Big )=2\Re\Big ((J_{\sigma,\kappa}^{m} \varphi(\xi))' \frac{J_{\sigma,\kappa}^{m}\varphi(\xi)}{\xi} \Big )\ >\ \sigma.\\
\end{equation}
  Lemma \ref{lem3}.ii, implies that there occurs a constant $\lambda(\sigma)$ satisfying $\Re(\flat(\xi))>\lambda(\sigma).$ Thus, we obtain 
$\Re (\flat(\xi)\ )> \epsilon$ for some $\epsilon \in [0,1).$  It yields from \eqref{e4} that $\Re\Big (J_{\sigma,\kappa}^{m} \varphi(\xi))'  \Big )>\epsilon$ and by Noshiro-Warschawski and Kaplan Theorems (see \cite{dur}),  we have that $J_{\sigma,\kappa}^{m} \varphi(\xi)$ is univalent and of bounded turning in $\sqcup$.
\end{proof}

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\bibitem{r2}  Ruscheweyh, St., New criteria for univalent functions, Proc. Amet. Math. Soc., \textbf{49}(1975), 109--115.

\bibitem{r3} Salagean, G. St. , \textit{Subclasses of univalent functions}, Lecture Notes in Math., Springer Verlag,
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%______________
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\bibitem{r7}  Ibrahim, R.W., Darus M.,  \textit{Extremal bounds for functions of bounded turning,}  Int. Math. Forum, \textbf{6}(2011), 1623--1630.

\bibitem{r8}  Krishna, D.,  et al., \textit{Third Hankel determinant for bounded turning functions of order alpha}, Journal of the Nigerian Mathematical Society, \textbf{34.2} (2015), 121--127.

\bibitem{r9} Ibrahim, R.W., \textit{Geometric properties of the differential shift plus complex Volterra operator, } Asian-European Journal of Mathematics, \textbf{11.01} (2018) 1850013.

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\bibitem{r11}  Miller, S.S.,  Mocanu, P. T.  \textit{Differential subordinations: theory and applications}. CRC Press, 2000.


\bibitem{dur} Duren, P. , \emph{Univalent Functions}, Grundlehren der mathematischen Wissenschaften; Springer-Verlag New York Inc. 1983.
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\end{thebibliography}

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