\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{lipsum} \usepackage{afterpage} \usepackage[labelfont=bf]{caption} \usepackage[nottoc,notlof,notlot]{tocbibind} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{lipsum} \usepackage{fancyhdr} \usepackage{amsmath,amsthm,amscd} \usepackage{amssymb,amsfonts} \usepackage{indentfirst} \usepackage{amsmath, amssymb} \usepackage{hyperref} \usepackage{mathrsfs} \usepackage{xcolor} \usepackage{epsfig} \usepackage{float} \usepackage{setspace} \usepackage{etoolbox} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} %\newtheorem{theorem}{Theorem} %\newtheorem{lemma}{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{assumption}{Assumption}[section] \usepackage{times} \newtheorem{prop}[thm]{Proposition} \newtheorem{rem}[thm]{Remark} \newtheorem{con}[thm]{Conclusion} \newtheorem{exm}[thm]{Example} \renewcommand{\theequation}{\thesection.\arabic{equation}} \numberwithin{equation}{section} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \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...} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \theoremstyle{definition} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\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. 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Acta Mathematica Sinica}, Acta Mathematica Sinica, \textbf{31}, (2015), 1970$ \textendash $1976. \end{thebibliography} \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}