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\fancyhead[CE]{W.U. HAQ AND U.R. FASEEH AND F. M. SAKAR} 
\fancyhead[CO]{STUDY ON A NEW FAMILY OF CLOSED-TO-CONVEX FUNCTIONS...}



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{\noindent Journal of Mathematical Extension \\
 }\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
Original Research Paper\\
\vspace{9mm}
‎
\begin{center}

{\Large \bf  Study on a New Family of Close-to-Convex Functions in a Vertical Strip Involving Subordination and Convolution \\}
%{\bf Do You Have a Subtitle? \\ If so, Write It Here} 


\let\thefootnote\relax\footnote{\scriptsize Received: March 2025; Accepted: ?? 2025 }

{\bf  F. M. Sakar$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small  Dicle University} \vspace{2mm}

{\bf Wasim Ul Haq}\vspace*{-2mm}\\
\vspace{2mm} {\small  Abbottabad University of Science and
	Technology} \vspace{2mm}

{\bf Faseeh U Rahman}\vspace*{-2mm}\\
\vspace{2mm} {\small  Abbottabad University of Science and
	Technology} \vspace{2mm}


\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} 	In this present paper,we consider a subclass of close-to-convex functions associated with a vertical strip domain. We obtain several results concerned with integral coefficient inequalities,convolutions, and representations  for functions belonging to this class $ \mathbf{MC_c}$.
Furthermore, we consider  the Fekete-Szego, inclusion relations and  radius problems involving certain classes of strongly close-to-convex functions, parabolic close-to-convex functions, and other types of close-to-convex functions.
 
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 30C45, 30C80.
Holomorphic functions;  Univalent function; Close-to-convex function; Fekete-Szego.
\end{quotation}}
\section{Introduction}
Let $\mathbb{A}$ represent the class of the function $\xi(z)$ having the series form
\begin{equation}\label{g1}
\xi (\textit{z})~~=~~\textit{z}+\sum_{\ell=2}^{\infty}\mathfrak{a}_{\ell} \textit{z}^{\ell}
\end{equation}
which is holomorphic(analytic) and~univalent(one-one) in the~~open~~unit~~disc $ D = \{\textit{z}~\in~$$\mathbb{C} $ $~~and~~ |\textit{z}|~<~1 \}$.\\
Let $\xi$ be given in \eqref{g1}~ and~ $g \in~~\mathbb{A}$ and~given~as

\begin{equation*}
g(z) = z+ \sum_{\ell=2}^{\infty} b_{\ell} z^{\ell}
\end{equation*}
the convolution of which is illustrated by\\
$$( \xi\ast g ) (z)= z+ \sum_{ \ell=2}^{\infty} a_{\ell} b_{\ell} z^{\ell}.    $$\\


A~function~~$ \xi~\in~\mathbb{A} $  fulfil the~condition
\[    R\Bigg(\frac{z\xi '(\textit{z})}{\xi(\textit{z})}\Bigg)> \varphi ~~~~~~~~~~~ (\textit{z}\in D) \\ \]
is~said~to~be~~starlike~~of~order $ \varphi$ $  (0\leq \varphi< 1). $
$   S^* ( \varphi)   $ represents~~the~class~~of all such~functions. It is commonly understood that  $ S^* (\varphi) \subset  S^* (0) =  S^* \subset S $. \\

%  A funtion $ \xi~\in~\mathbb{A} $ is said to be starlike of order of
% $ \varphi$ $  (0\leq \varphi< 1) $ if it is satisfies the condition\\
%        \[    R\Bigg(\frac{z\xi '(\textit{z})}{\xi(\textit{z})}\Bigg)> \varphi ~~~~~~ (\textit{z}\in D).  \\ \]

%The class of all such function is represented  by $   S^* ( \varphi)   $ call it starlike functions of order $\varphi $. We also represented by $ S$ the class of all function in the normalized holomorphic function class $\mathbb{A}$ which are univalent(one-one) in D. We use that notations $S^*=S^*(0) ~~and ~~S=S(0)$.   It is well known that $ S^* (\varphi) \subset  S^* (0) =  S^* \subset S $ \\
Consider that 0 $\leq \varphi , \vartheta < 1 $. If there exists a function g $\in S^* ( \varphi) $ such that the inequality

\[ \Re\bigg(\frac{z\xi'(\textit{z})}{g(\textit{z})} \bigg)> \varphi ~~~~~ (\textit{z}\in D)  \\ \]
is satisfied, then the function $\xi \in$ $\mathbb{A }$ is~called~\textsl{\textsl{\textsl{close-to-convex}}} of order $ \varphi$ and type $\vartheta$.\\
% We represented the class which include of all function $  \xi\in$ $\mathbb{A}$ that is \textsl{close-to-convex} of order $\varphi$ and type $ \eta $ by $C_{c}$ ($\varphi$ ,$ \eta$).
We used $C_{c}$( $\varphi$ ,$ \vartheta$) to represent the class that consists of any function that is \textsl{\textsl{\textsl{close-to-convex}}} of order $\varphi$ and type $ \vartheta $.
This class is investigated by Libera \cite{libera1964some}. \\
More specifically, when $ \vartheta $ = 0 we have $C_{c}$($\varphi$,0) = $C_{c}$($\varphi$) of \textsl{\textsl{\textsl{close-to-convex}}} function of order $\varphi$, also we get $C_{c}$(0,0) = $C_{c}$ class of \textsl{\textsl{\textsl{close-to-convex}}} function~investigated~by~Kaplan \cite{kaplan1952close}.  it is well-known that $ S^* \subset$$C_{c}$ $\subset$ S.\\
The $\xi$ and h are two holomorphic functions  in D. We say that function $\xi$ in D is subordinate to function~~h~~and~~write~~as \\ \[ \xi(z) \prec h(z) ~~~~   (z\in D)  \]
\\ if~~there~~exist~~its~~a~~Schwarz funtion~$ v(z)$ such that \\ \[ \xi(z) =h\big( v(z )\big) ~~   ~~  (z\in D). \]
\\ ~~~ It~~is~known~~that~~if~~h is univalent(one-one)~in~D then\\
\[  \xi(z) \prec h(z) \Longleftrightarrow  \xi(0) ~~=~~ h(0\label{key}) ~~ and ~~ \xi(D) \subset  h(D). \]
\\
A function $\xi$ $\in$  $\mathbb{A }$  is said to be strongly  ~close-to-convex of order $(\gamma)$\cite{cho2003multiplier}\\ (0$\leq$ $\gamma$ $<$ 1)	if there is a function $g(z)\in$ $S^*$ such that
\begin{align*}
\bigg| arg\bigg(\frac{z\xi'(z)}{g(z)}\bigg) \bigg| \leq \frac{\pi}{2} \gamma ~~~~(z\in D).
\end{align*}
We denote it by $SC_{c}(\gamma)$. While a function $\xi$ $\in$ $\mathbb{A }$ is said to be parabolic close-to-convex function (\cite{Kumar1994}) if there is a function g $\in$ S* such that
\begin{equation*}
\bigg|\frac{z\xi'(z)}{\xi(z)} -1\bigg| \leq R\bigg(\frac{z\xi'(z)}{g(z)} \bigg)~~~~(z\in D).
\end{equation*}
We denote it by~$ PC_{c}$.\\
Srivastava et al. \cite{srivastava2019upper} investigated the class of close-to-convex functions associated with the lemniscate of Bernoulli denoted by $CL_{c}$. A function $\xi$ $\in$ $\mathbb{A }$ is said to be the class $CL_{c}$ if there exists a function g $\in$ S* such that
\begin{equation*}
\bigg(\frac{z\xi'(z)}{g(z)} \bigg) \prec \sqrt{1+z}  ~~~(z\in D).
\end{equation*}	\\
Recently Karger et al. \cite{kargar2017radius} developed the holomorphic function $ U_{\sigma}$ and vertical strip $\Phi_{\sigma}$ defined as follows:
\begin{equation}\label{F1}
U_{\sigma}(\textit{z})=\frac{1}{ 2 \iota \sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma)  }\textit{z}}{1+e^{(-\iota \sigma)}\textit{z}}\bigg)  ~~~~(\textit{z}\in D)
\end{equation}
and
\begin{equation}\label{2.2}
\Phi_{\sigma}  = \bigg\{v \in C :  \frac{ \sigma-\pi}{ 2 \sin ( \sigma )}< \Re(v)< \frac{ \sigma}{ 2 \sin ( \sigma )} \bigg \},
\end{equation}
where $(\frac{\pi}{2}\leq\sigma\pi)$.They also introduced the following class $M(\sigma)$:
\\ $M(\sigma)=\biggl\{\xi\in\mathbb{A}:[1 + \frac{\sigma-\pi}{2\sin(\sigma)}<\Re \bigg(\frac{z\xi'(\textit{z})}{\xi(\textit{z})} \bigg) < 1+\frac{\sigma}{2\sin(\sigma)}]\biggl\}$.
\\
It is clear that the function $ U_{\sigma}$ given in \eqref{F1} is convex and univalent (one-to-one) in $D$. Furthermore, $ U_{\sigma}$ maps $D$ onto $ \Phi_{\sigma}$ when $(\frac{\pi}{2}\leq\sigma< \pi)$. In other choices of $\sigma$, the image could be a triangle, a half strip or a trapezium (see article \cite{dorff2001convolutions}).\\
Note that the following form may be
used to write the function $ U_{\sigma}$:
%In other cases, the image can be, for example, a half strip, a quadrilateral, or a triange (see \cite{dorff2001convolutions}  ). \\
%Take note that the following form can be used to write the function $ U_{a}$.
%Note that the function $ U_{a}$ can be written in the form  \\
\[ U_{\sigma} (\textit{z})=\textit{z}+ \sum_{\ell= 2}^{\infty} B_{\ell} (\sigma) \textit{z}^{\ell}    ~~~~~~~~~~( \frac{\pi}{2} \leq  \sigma < \pi  ;\textit{z}\in D ),\]
where,
\begin{equation} \label{faseeh1}
B_{\ell}  (\sigma) = (-1)^{(\ell-1)}   \frac{\sin( \ell \sigma)}{\ell \sin(\sigma)}  ~~~~~~(\frac{\pi}{2} \leq  \sigma < \pi; \ell \in N ).
\end{equation}
In the last few years, significant results have been reported on the class of normalized holomorphic functions$\xi $ $ \in $ $\mathbb{A}$that map $D$ onto a vertical strip. For related study see \cite{sakar1, kuroki2012notes, kwon2014some, li2017coefficient, Olatunji, ronning1993uniformly, sun2017integral, wang2015harmonic}.\\
\textbf{Motivation}\\
Motivated by above works, we will define the following family of close-to-convex functions associated with the vertical strip $\Phi_{\sigma}$ due to its interesting geometry.
\begin{definition}
	A function $\xi\in\mathbb{A}$ is said to belong to the class $\mathbf{MC}(\sigma)(\pi/ 2 \leq \sigma < \pi)$ if it there exists a function $g \in M(\sigma)$ such that:
	
	\[ 1 + \frac{\sigma-\pi}{2\sin(\sigma)}<\Re \bigg(\frac{z\xi'(\textit{z})}{g(\textit{z})} \bigg) < 1+\frac{\sigma}{2\sin(\sigma)}~~~~ (\textit{z}\in D )\].
	
\end{definition}
\begin{rem}
	From the article \cite{kargar2017radius} we have
	$[ 1-\frac{\pi}{4} \leq 1+\frac{\sigma-\pi}{2 \sin( \sigma )} <\frac{1}{2}~ and~ 1+\frac{ \sigma}{2\sin (\sigma)}\geq 1+\frac{\pi}{4} (\pi/2\leq\sigma < \pi )]$
	It is clear that
	$[C_{c} \supset \mathbf{MC_c} (\sigma)  (\frac{\pi}{2} \leq \sigma < \pi ) and  C_{c}( \frac{4-\pi}{4} , \frac{4+\pi}{4}  ) \supset \mathbf{MC_c}( \pi / 2 ),]$
	where the class $ C_{c} (\varphi,\eta), 0 \leq \varphi < 1 < \eta $,
	was recently introduced by S.Bulut \cite{Bulut18}.
\end{rem}

We aim to prove certain coefficient bounds Fekete-Szego type results, inclusion relations and radius problems involving functions in the class $MC_{c}(\sigma)$.
\section{Main Results}

Firstly, we need the following  lemmas to prove the main results.
\begin{lemma} \label{lamma 4no.1}
	Let $\xi \in\mathbf{MC_c} (\sigma)~~~~ ( \frac{\pi}{2} \leq  \sigma < \pi )$. Then
	\begin{equation}\label{ch43}
	\bigg(\frac{z\xi'(\textit{z})}{g(\textit{z})} -1 \bigg) \prec U_{\sigma} = \frac{1}{ 2 \iota \sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma) }z}{1+e^{(-\iota \sigma)}\textit{z}}\bigg) ~~~~(\textit{z}\in D).
	\end{equation}	
\end{lemma}
\begin{proof}
	From \eqref{2.2}, we see that
	\begin{equation*}
	v(z)= \bigg\{\frac{z\xi'(z)}{g(z)}\bigg\}- 1
	\end{equation*}
	lies in a strip $\Phi_{a}$ and it is known that $ U_{\sigma}(D)$ =$\Phi_{a}$. Because $ U_{\sigma}(z)$  is univalent then by the subordination principle we get \eqref{ch43}.
\end{proof}
\begin{lemma}\label{L2}\cite{miller2000differential} \\
	Let h be convex in D and let G : $D\longrightarrow $ $\verb"C"$, with $ R(G(\textit{z}))$ $> 0. $ If G is holomorphic in D, then
	\begin{align}
	q(\textit{z})+G(\textit{z}) zq'(\textit{z}) ~~\prec~~ h(\textit{z})  \\ \nonumber \Rightarrow ~~~~~~~~q(\textit{z})\prec h(\textit{z}).
	\end{align}
\end{lemma}
\begin{lemma} \label{L3}\cite{rogosinski1945coefficients} \\
	Suppose~that~u($\textit{z}$) is holomorphic, univalen and maps D onto~a~convex~domain. Assume that ~u(z)~given~by
	\[   u(\textit{z}) = \sum_{\ell=1}^{\infty} C_{\ell} \textit{z}^{\ell}\]
	is holomorphic and univalent in D. Also assume that
	\[ s(\textit{z})= \sum_{\ell=1}^{\infty}A_{\ell} \textit{z}^{\ell}  \]
	is holomorphic in D and that the subordination relation
	\[ s(\textit{z})~~~\prec~~u(\textit{z})~~~~~~~~~~~~~~  (\textit{z}\in D)\]
	is satisfied. Then,
	\[|A_{\ell}| ~\leq~ |C_{1}|~~~~~~~~~~~~ (\textit{z}\in N).\]\\
\end{lemma}
\begin{lemma} \label{chena6} \cite{alsoboh2019fekete}\\
	If~~ p(z) = 1+ ${c}_{1}\textit{z}$+${c}_{2}$ $\textit{z}^{2}$+${c}_{3}\textit{z}^{3}$+${c}_{4} \textit{z}^{4}+$... is a function with a positive real part in D and $\varPsi$ $\in$ $\mathbb{R}$(real number) then
	\begin{equation*}
	|c_2 - \varPsi c_2^2 | \leq 2 max \{1; |2\varPsi -1| \}.
	\end{equation*}
\end{lemma}


\begin{theorem}
	A function $\xi $ $ \in$  $ \mathbf{MC_c}$ ($ \sigma  $)~~( $\frac{\pi}{2} \leq  \sigma < \pi )$ if and only if
	\begin{equation}\label{11}
	\xi(\textit{z})  =\int_{0}^{z} \bigg[ \frac{g(\textit{x})}{x} + \frac{g(\textit{x})}{ 2 \iota ~x~\sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma) }}{1+e^{(-\iota \sigma)}\textit{x}}\bigg)\bigg] dx ~~~~(\textit{z}\in D).
	\end{equation}
\end{theorem}
\begin{proof}
	For $\xi\in\mathbf{MC_c}(\sigma)$, we know from lemma \eqref{lamma 4no.1} that relation \eqref{ch43} holds. It follows that
	\begin{equation}\label{ch45}
	\frac{z\xi'(z)}{g(z)} - 1 = \frac{1}{ 2 \iota \sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma) })v(z)}{1+e^{(-\iota \sigma)} v(z) }\bigg) ~~~~(z\in D),
	\end{equation}
	where the Schwarz function $v(z)$ is analytic in D with $v(0) = 0$ and $\big| v(z)\big|<1$ ($z\in$ D ). We next see from \eqref{ch45}  that
	\[ \frac{z\xi'(z)}{g(z)} - \frac{1}{z}= \frac{1}{ 2 \iota z\sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma) } v(z)}{1+e^{(-\iota \sigma)} v(z) }\bigg) ~~~~(z\in D)\]
	%which, upon integration , yields.
	\begin{equation}\label{??}
	\xi'(z)- \frac{g(z)}{z} = \frac{g(z)}{ 2 \iota~ z~ \sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma) })z}{1+e^{(-\iota \sigma)} z }\bigg) ~~~~(z\in D)
	\end{equation}
	
	\begin{equation}\label{??}
	\xi'(z) = \frac{g(z)}{z} + \frac{g(z)}{ 2 \iota ~z~\sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma) })z}{1+e^{(-\iota \sigma)} z }\bigg) ~~~~(z\in D)
	\end{equation}
	Integrating both sides from $0$ to $z$, we get the desired result.
\end{proof}

\begin{theorem} \label{4.1.2}
	A function  $\xi \in\mathbf{MC_c}$ ($ \sigma$)($\frac{\pi}{2} \leq  \sigma < \pi )$, if and only if
	\begin{equation}\label{ch4,13}
	\bigg\{ \xi(\textit{z}) \ast \frac{z}{(1-z)^2  } \bigg\}-\bigg\{ 1+ \frac{1}{ 2 \iota \sin (\sigma) } \log \bigg( \frac{1+e^{\iota(\phi- \sigma) })}{1+e^{\iota(\phi - \sigma)} }\bigg)\bigg\} \ast g(\textit{z}) \neq 0 ~~~~(\textit{z}\in D),
	\end{equation}
	where "$\ast $" denote the ~ ``Hadamard ~  product'', $( 0 \leq \phi < 2 \pi ) $ and~~ $ \phi - \sigma = \pi $.
	
\end{theorem}

\begin{proof}
	Let $\xi\in\mathbf{MC_c}$ ($\sigma$). Then by lemma \eqref{lamma 4no.1} we observe that \eqref{ch43} holds.
	
	\begin{equation}\label{ch46}
	\implies~	\frac{z\xi'(z)}{g(z)} \neq 1+ \frac{1}{ 2 \iota \sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma) }e^{(\phi) })z}{1+e^{(-\iota \sigma)} e^{( \phi) }}\bigg),
	\end{equation}
	where  $( 0 \leq \phi < 2 \pi )$ and $(\phi -\sigma \neq \pi )( z\in D)$.
	The condition \eqref{ch46} can now be written as follows: \begin{equation}\label{ch4,15}
	\implies~	\frac{z\xi'(z)}{g(z)} - \bigg[ 1- \frac{1}{ 2 \iota \sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma) }e^{(\phi) })}{1+e^{(-\iota \sigma)} e^{( \phi) }}\bigg) \bigg]g(z)\neq 0 
	\end{equation}
	where  $( 0 \leq \phi < 2 \pi )$ and $(\phi -\sigma \neq \pi )( z\in D)$. Note that,
	\begin{equation}\label{ch4,16}
	g(z)= g(z)*  \bigg(\frac{z}{1-z}\bigg)~~~~and~~~~ z\xi'(z)=\xi(z)* \bigg(\frac{z}{(1-z)^2}\bigg).
	\end{equation}
	Thus by \eqref{ch4,15} and \eqref{ch4,16}, we obtain assertion \eqref{ch4,13}  of theorem \ref{4.1.2}.
	
\end{proof}
\begin{theorem}
	Let $\xi\in A$ satisfy the subordination
	\begin{equation}\label{??}
	\frac{(z\xi'(z))'}{g'(z)} \prec 1+ U_{\sigma}(z) ~~~~({z}\in D).
	\end{equation}
	for $g\in M(\sigma)$. Then
	\begin{equation}\label{ch4,18}
	\frac{z\xi'(z)}{g(z)} \prec 1+ U_{\sigma}(z) ~~~~({z}\in D).
	\end{equation}
	That is, $\xi\in\mathbf{MC_c} $ ($\sigma$).
	
\end{theorem}
\begin{proof}
	Suppose that the function $p(z)$ such that
	\begin{equation}\label{s10}
	p(z) = \frac{z\xi'(z)}{g(z)} ~~({z}\in D).
	\end{equation}
	Then
	\begin{equation*}
	\frac{\big(z\xi'(z)\big)'}{g'(z)}= p(z) + zp'(z)\frac{g(z)}{zg'(z)}.
	\end{equation*}
	Let $ G(z) =\frac{g(z)}{zg'(z)}$. Then
	\begin{equation}\label{??}
	p(z) + zp'(z)G(z) =\frac{\big(z\xi'(z)\big)'}{g'(z)} \prec U_{\sigma}(z) ~~~~({z}\in D).
	\end{equation}
	Note that
	\begin{equation}\label{??}
	p(0) = 0 = U_{\sigma}(0)~~~ and ~~~ \Re(1+U_{\sigma}(z) ) > 0 ~~~~(\frac{\pi}{2} \leq \sigma < \pi ; z\in U).
	\end{equation}
	Moreover, by Lemma \ref{L2}
	\begin{equation*}
	p(z) \prec U_{\sigma}(z), ~~~ \Longrightarrow  ~~~ \frac{z\xi'(z)}{g(z)} ~~\prec~~ 1+U_{\sigma}(z).
	\end{equation*}
\end{proof}


\begin{theorem}
	Consider the function \[\xi(z) = \sum_{\ell=2}^{\infty} a_{\ell} z^{\ell} \in \mathbf{MC_c} ( \sigma  ).\] Then
	\[ |a_{\ell}| \leq 1~~~~~~ (\ell\in N).\]
\end{theorem}
\begin{proof}
	For given $\sigma$ $(\frac{\pi}{2} \leq \sigma < \pi),$ we define the functions $ q(z)$ and $p(z)$ by
	\begin{equation}\label{S11}
	q(z)= \frac{z\xi'(z)}{g(z)}  ~~~~~~~( z  \in D ),
	\end{equation}
	and
	\begin{equation} \label{ch4,23}
	p(z)= 1 + \frac{1}{ 2 \iota \sin (\sigma) } \log \bigg( \frac{1+e^{(\iota \sigma)  }\textit{z}}{1+e^{(-\iota \sigma)}\textit{z}}\bigg)    ~~~~~~~~~~(\textit{z}\in D).
	\end{equation}
	Then the subordination \eqref{ch43} can be written as follows;
	\begin{equation}
	q(z)\prec p(z)~~~~~~~~~~(\textit{z}\in D).
	\end{equation}
	Note that the function p(z) defined by \eqref{ch4,23} is convex in D and has the form
	\begin{equation}\label{ch4.21}
	p(z) = 1+\sum_{\ell=1}^{\infty} B_{\ell} (\sigma) z^{\ell}~~~~~~~~~(\textit{z}\in D),
	\end{equation}
	where $B_{\ell} (\sigma)$ is given by \eqref{faseeh1} and $q(z)$
	\begin{equation}\label{??}
	q(z) = 1 + \sum_{\ell=1}^{\infty} A_{\ell} z^{\ell}  ~~~~~~~~~(\textit{z}\in D),
	\end{equation}
	then by Lemma\ref{L3} we see that the subordination relation \eqref{ch4,15} implies that
	\begin{equation}\label{ch4,27}
	\big| A_{\ell}\big| \leq \big| B_{1}\big|  ~~~~~(\ell\in N ).
	\end{equation}
	Now \eqref{S11} implies that
	\begin{equation}\label{??}
	z\xi'(z) = g(z) q(z)  ~~~~~~~~~(z\in D).
	\end{equation}
	Then by equating the coefficient of $z^{\ell}$ both sides we get
	\begin{equation}\label{??}
	a_{\ell}=\frac{1}{(\ell-1)} \bigg[A_{\ell-1} + b_{2} A_{\ell-2}+ b_{3} A_{\ell-3}+...+b_{\ell-1} A_{1}   \bigg] ~~~~~~(\ell\in N\diagdown \{1\} ).
	\end{equation}
	A simple calculation  combined with inequality \eqref{ch4,27} yields $|a_{2} = |A_{1}| \leq 1 $ and
	\begin{equation}\label{??}
	\big| a_{\ell} \big| =\frac{1}{(\ell-1)} \bigg|A_{\ell-1} + b_{2} A_{\ell-2}+ b_{3} A_{\ell-3}+...+b_{\ell-1} A_{1}   \bigg|
	\end{equation}
	\begin{equation}\label{??}
	\big| a_{\ell} \big|  \leq  \frac{1}{(\ell-1)} \bigg[\big|A_{\ell-1}\big| +\big| b_{2}\big| \big|A_{\ell-2}\big|+ \big|b_{3}\big| \big|A_{\ell-3}\big|+...+\big|b_{\ell-1}\big| \big|A_{1}\big|   \bigg].
	\end{equation}
	Using \ref{ch4,27} and $|b_{k}| \leq 1$ (see \cite{Sun2019}), we have
	\begin{equation}\label{??}
	\big| a_{\ell} \big|\leq\big|B_{1}\big|= 1.
	\end{equation}
	Hence proved.
	
\end{proof}


\section{Fekete-Szego problem}
In this section , first give results on the Fekete-Szego problem involving the function class $ \mathbf{MC_c}$ ($ \sigma $).  The basic mathod of the proof in the following theorem is similar to that used.\\



\begin{theorem}
	Let 	\[(\xi(z)) = 1+  \mathfrak{a}_{2} \textit{z}^{2} +  \mathfrak{a}_{3} \textit{z}^{3}  +  \mathfrak{a}_{4} \textit{z}^{4}+... \] be in $ \mathbf{MC_c}$ ($ \sigma $). Then for  $\varLambda \in~ \mathbb{R} $,
	\begin{equation}
	|\mathfrak{a}_{3} -\varLambda \mathfrak{a}_{2}^2|= \frac{1}{3|1-\varpi|}Max \{    1; | 2 \varLambda_1 -1|\},
	\end{equation}
	where
	
	$\varpi$=$\frac{ \sigma - \pi  }{ 2 \sin ( \sigma )}$ ~~~,~~  \\ $\varLambda_1~$=$\frac{-3(1-\varpi)\Upsilon}{d_{1}^2}-\varLambda \frac{3}{4(1-\varpi)}$
	and\\$\Upsilon~= \bigg[ \frac{b_3}{3}  +\frac{b_2d_{1}}{3(1-\varpi)}- \varLambda \frac{b_2 ^2}{4}-\varLambda \frac{b_2 d_{1}}{2(1-\varpi)} \bigg]$.
\end{theorem}
\begin{proof}
	For $v\in \Phi_{\sigma}$, we have
	$\Re(v(z)) >\varpi$. That is
	\begin{equation*}\label{??}
	(\frac{v(z)-\varpi}{1-\varpi}) \in P,
	\end{equation*}
	where $P$ denotes the class of functions with positive real part. Setting 
	\begin{equation*}\label{??}
	(\frac{v(z)-\varpi}{1-\varpi}) =p(z)=1+\sum_{\ell=1}^{\infty} c_{\ell} z^{\ell}.
	\end{equation*}
	Then
	\begin{equation*}\label{??}
	v(z)=1+\sum_{\ell=1}^{\infty} (1-\varpi)c_{\ell} z^{\ell}.
	\end{equation*}
	
	Let
	\begin{equation}
	v(z)= \frac{z\xi'(z)}{g(z)}-1  ~~~~~~~( z \in D )
	\end{equation}
	where $g(z)=z+\sum_{\ell=2}^{\infty} b_{\ell} z^{\ell}~~~\in M(\sigma)$.
	Then
	\begin{equation*}
	(v(z)+1 )g(z)= z\xi'(z).
	\end{equation*}
	Now
	\begin{equation*}
	(v(z)+1 )g(z)=  \bigg(2+\sum_{\ell=1}^{\infty} (1-\varpi)c_{\ell} z^{\ell} \bigg)\bigg( z+\sum_{\ell=2}^{\infty} b_{\ell} z^{\ell}  \bigg).
	\end{equation*}
	In series form
	\begin{equation*}
	(v(z)+1 )g(z)= \bigg(2+(1-\varpi)c_{1}  z^{1} +(1-\varpi)c_{2}  z^{2} +(1-\varpi)c_{3}  z^{3} +...+(1-\varpi)c_{\ell}  z^{\ell}+... \bigg)\times
	\end{equation*}
	\begin{equation*}
	\bigg( z+ b_2 z^2+ b_3 z^3 + b_4 z^4 +... +b_\ell z^\ell+...\bigg)
	\end{equation*}
	\begin{align}\label{ch22}
	=2z+ 2b_2 z^2+ 2b_3 z^3 + 2b_4 z^4 +...\nonumber\\  +(1-\varpi)c_{1}  z^{2}+ (1-\varpi)b_{2}c_{1} z^3+ (1-\varpi)b_{3}c_{1} z^4 + (1-\varpi)b_{4}c_{1}z^5 +...\nonumber\\ +(1-\varpi)c_{2} z^3+ (1-\varpi)b_{2}c_{2} z^4 + (1-\varpi)b_{3}c_{2} z^3 z^5 + (1-\varpi)b_{4}c_{2} z^6 +...
	\end{align}
	Also
	\begin{equation} \label{ch21}
	z(\xi'(z)) = z+ 2 \mathfrak{a}_{2} \textit{z}^{2} + 3 \mathfrak{a}_{3} \textit{z}^{3}+... +\ell \mathfrak{a}_{\ell}{z}^{\ell}+...
	\end{equation}
	Comparing the coefficients  \eqref{ch22} and \eqref{ch21} of $z^2 and ~z^3$
	\begin{align}
	\mathfrak{a}_{2}  = \frac{1}{2}\bigg( 2b_2+(1-\varpi)c_{1} \bigg)  \nonumber\\
	\mathfrak{a}_{3} = \frac{1}{3}\bigg( 2b_3  +(1-\varpi)b_{2}c_{1}  + (1-\varpi)c_{2} \bigg).
	\end{align}
	Let $\varLambda \in~ \mathbb{R} $
	\begin{equation}
	\mathfrak{a}_{3} -\varLambda \mathfrak{a}_{2}^2= \bigg[ \frac{1}{3}\bigg( 2b_3  +(1-\varpi)b_{2}c_{1}  + (1-\varpi)c_{2} \bigg)-\varLambda \bigg\{  \frac{1}{2}\bigg( 2b_2+(1-\varpi)c_{1} \bigg) \bigg\}^2  \bigg].
	\end{equation}
	Then
	\begin{equation*}
	\mathfrak{a}_{3} -\varLambda \mathfrak{a}_{2}^2= \bigg[ \frac{2b_3}{3}  + \frac{(1-\varpi)b_2d_{1}}{3} + \frac{(1-\varpi)d_{2}}{3}- \varLambda b_2 ^{2}-\varLambda \frac{(1-\varpi)b_2 c_{1}}{2}-	\varLambda \frac{(1-\varpi)^2c_{1}^2}{4}   \bigg].
	\end{equation*}
	Let~~$\Upsilon~= \bigg[  \frac{2b_3}{3}  + \frac{(1-\varpi)b_2d_{1}}{3}- - \varLambda b_2 ^{2}-\varLambda \frac{(1-\varpi)b_2 c_{1}}{2} \bigg]$. Then
	\begin{equation*}
	\mathfrak{a}_{3} -\varLambda \mathfrak{a}_{2}^2=\bigg[	\Upsilon+\frac{(1-\varpi)d_{2}}{3}-	\varLambda \frac{(1-\varpi)^2c_{1}^2}{4} \bigg]
	\end{equation*}
	\begin{equation*}
	\mathfrak{a}_{3} -\varLambda \mathfrak{a}_{2}^2 =  \frac{(1-\varpi)}{3} \bigg[	3\frac{\Upsilon}{1-\varpi}+c_{2}-\varLambda \frac{3(1-\varpi)c_{1}^2}{4}     \bigg].
	\end{equation*}
	Let$\varLambda_1~=\frac{-3\Upsilon}{(1-\varpi)c_{1}^2}-\varLambda\frac{3(1-\varpi)}{4}$
	\begin{equation}\label{mar1}
	\mathfrak{a}_{3} -\varLambda \mathfrak{a}_{2}^2 = \frac{1}{3(1-\varpi)} \bigg[	c_{2} -\varLambda_1 c_{1}^2    \bigg].
	\end{equation}
	Give modulus on both side \eqref{mar1} and applying the Lemma \eqref{chena6}, we get
	\begin{equation}
	|\mathfrak{a}_{3} -\varLambda \mathfrak{a}_{2}^2| \leq \frac{|1-\varpi|}{3}Max \{1; | 2 \varLambda_1-1|\}.
	\end{equation}
\end{proof}


\section{Conclusion}
The problem of studying different classes of analytic functions is well-known specially with some nice geometrical description e.g starlikeness and convexity. The recent study of starlikeness and convexity with the geometry of vertical strip domain is given, see \cite{kargar2017radius,kuroki2012notes,kwon2014some,li2017coefficient}. Motivated by these works, in the present article we aim to study another geometric property known as close-to-convexity associated with domain of vertical strip.
In concluding our present work one may investigate other geometrical description of analytic functions in a similar fashion. The results involving convolution techniques are also point of further interest and investigation in this direction.





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\newpage

{\small
	
	\noindent{\bf Fethiye M\"{u}ge Sakar}
	
	\noindent Professor of Mathematics
	
	\noindent Department of Management
	
	\noindent Faculty of Economics and Administrative Sciences
	
	\noindent Dicle University
	
	\noindent 21280 Diyarbak{\i}r, Turkey
	
	\noindent E-mail: mugesakar@hotmail.com}\\




{\small
\noindent{\bf Wasim Ul Haq }

\noindent Professor of Mathematics

\noindent Department of Mathematicsn

\noindent Abbottabad University of Science and
Technology

\noindent Abbottabad 22010, Pakistan

\noindent E-mail: dr.wasim@aust.edu.pk}\\

{\small
	
	\noindent{\bf Faseeh U Rahman}
	
	\noindent Associated Professor of Mathematics
	
	\noindent Department of Mathematicsn
	
	\noindent Abbottabad University of Science and
	Technology
	
	\noindent Abbottabad 22010, Pakistan
	
	\noindent E-mail: fasseeh6264@gmail.com}\\


\end{document}