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\begin{document}

\title{Coefficient Bounds for New Subclasses of Analytic and m-Fold Symmetric Bi-Univalent Functions}
\author{Abbas Kareem Wanas$^{1}$ and P$\acute{a}$ll-Szab$\acute{o}$ $\acute{A}$gnes Orsolya$^{2}$\\
%EndAName
$^{1}$Department of Mathematics\\
College of Science\\
University of Al-Qadisiyah, Iraq\\
abbas.kareem.w@qu.edu.iq\\
$^{2}$Department of Mathematics\\
Faculty of Mathematics and Computer Science\\
Babes-Bolyai University\\
Cluj-Napoca, Romania\\
pallszaboagnes@math.ubbcluj.ro}
\date{}
\maketitle

\begin{abstract}
In the present paper, we introduce and study two new subclasses of analytic and $m$-fold symmetric bi-univalent functions defined in the open unit disk $U$. Furthermore, for functions in each of the subclasses introduced here, we obtain upper bounds for the initial coefficients $\left| a_{m+1}\right|$ and $\left| a_{2m+1}\right|$. Also, we indicate certain special cases for our results.
\end{abstract}

\noindent \textbf{Keywords:} Analytic function, Univalent function, m-Fold symmetric bi-univalent function, Coefficient bound.

\noindent \textbf{2010 Mathematical Subject Classification:} 30C45, 30C50.

\section{Introduction}

Denote by $\mathcal{A}$ the class of functions $f$ of the form:
\begin{equation}\label{eq.1.1}
f(z)=z+\sum_{k=2}^{\infty}a_{k}z^{k},
\end{equation}
which are analytic in the open unit disk $U=\left\lbrace z\in \mathbb{C}:\left| z\right| <1 \right\rbrace$. Let $S$ be the subclass of $\mathcal{A}$ consisting in functions of the form (\ref{eq.1.1}) which are also univalent in $U$. The Koebe one-quarter theorem (see \cite{4}) states that the image of $U$ under every function $f\in S$ contains a disk of radius $\frac{1}{4}$. Therefore, every function $f\in S$ has an inverse $f^{-1}$ which satisfies $f^{-1}(f(z))=z$, $(z\in U)$ and $f(f^{-1}(w))=w$, $(\left| w\right|<r_{0}(f), r_{0}(f)\geq \frac{1}{4})$, where
\begin{equation}\label{eq.1.2}
g(w)=f^{-1}(w)=w-a_{2}w^{2}+\left( 2a_{2}^{2}-a_{3}\right)w^{3}-\left( 5a_{2}^{3}-5a_{2} a_{3}+a_{4}\right)w^{4}+ \cdots .
\end{equation}

A function $f\in \mathcal{A}$ is said to be bi-univalent in $U$ if both $f$ and $f^{-1}$ are univalent in $U$. We denote by $\Sigma$ the class of bi-univalent functions in $U$ given by (\ref{eq.1.1}). For a brief history and interesting examples in the class $\Sigma$ see \cite{14}, (see also \cite{6,7,10,11,12}).

For each function $f\in S$, the function $h(z)=\left(f(z^{m})\right)^{\frac{1}{m}}$, $(z\in U, m\in \mathbb{N})$ is univalent and maps the unit disk $U$ into a region with $m$-fold symmetry. A function is said to be $m$-fold symmetric (see \cite{8}) if it has the following normalized form:
\begin{equation}\label{eq.1.3}
f(z)=z+\sum_{k=1}^{\infty}a_{mk+1}z^{mk+1}, \; (z\in U, m\in \mathbb{N}).
\end{equation}

Let $S_{m}$ stands for the class of $m$-fold symmetric univalent functions in $U$, which are normalized by the series expansion (\ref{eq.1.3}). In fact, the functions in the class $S$ are one-fold symmetric.

In \cite{15} Srivastava et al. defined $m$-fold symmetric bi-univalent functions analogues to the concept of $m$-fold symmetric univalent functions. They gave some important results, such as each function $f\in \Sigma$ generates an $m$-fold symmetric bi-univalent function for each $m\in \mathbb{N}$. Furthermore, for the normalized form of $f$ given by (\ref{eq.1.3}), they obtained the series expansion for $f^{-1}$ as follows:
\begin{align}\label{eq.1.4}
& g(w)=w-a_{m+1}w^{m+1}+\left[(m+1)a_{m+1}^{2}-a_{2m+1}\right]w^{2m+1} \notag \\
& -\left[\frac{1}{2}(m+1)(3m+2)a_{m+1}^{3}-(3m+2)a_{m+1}a_{2m+1}+a_{3m+1}\right] w^{3m+1}+\cdots ,
\end{align}
where $f^{-1}=g$. We denote by $\Sigma_{m}$ the class of $m$-fold symmetric bi-univalent functions in $U$. It is easily seen that for $m=1$, the formula (\ref{eq.1.4}) coincides with the formula (\ref{eq.1.2}) of the class $\Sigma$. Some examples of $m$-fold symmetric bi-univalent functions are given as follows:
\begin{equation*}
\left( \frac{z^{m}}{1-z^{m}}\right)^{\frac{1}{m}}, \; \left[\frac{1}{2} \log \left( \frac{1+z^{m}}{1-z^{m}}\right) \right]^{\frac{1}{m}} \; and \; \left[- \log \left( 1-z^{m}\right) \right]^{\frac{1}{m}}
\end{equation*}
with the corresponding inverse functions
\begin{equation*}
\left( \frac{w^{m}}{1+w^{m}}\right)^{\frac{1}{m}}, \; \left( \frac{e^{2w^{m}}-1}{e^{2w^{m}}+1}\right)^{\frac{1}{m}} \; and \; \left( \frac{e^{w^{m}}-1}{e^{w^{m}}}\right)^{\frac{1}{m}},
\end{equation*}
respectively.

Recently, many authors investigated bounds for various subclasses of $m$-fold bi-univalent functions (see \cite{1,2,5,13,15,16,17}).

The purpose of the present investigation is to introduce the new subclasses $\mathcal{AS}_{\Sigma_{m}}(\gamma,\lambda;\alpha)$ and $\mathcal{AS}_{\Sigma_{m}}^{*}(\gamma,\lambda;\beta)$ of $\Sigma_{m}$ and find estimates on the coefficients $\left| a_{m+1}\right|$ and $\left| a_{2m+1}\right|$ for functions in each of these new subclasses.

We will require the following lemma in proving our main results.

\begin{lemma}\label{lem1.1} \cite{3}
If $h\in \mathcal{P}$, then $\left| c_{k}\right|\leq 2$ for each $k\in \mathbb{N}$, where $\mathcal{P}$ is the family of all functions $h$ analytic in $U$ for which
\begin{equation*}
Re \left( h(z)\right)>0, \quad(z\in U),
\end{equation*}
where
\begin{equation*}
h(z)=1+c_{1}z+c_{2}z^{2}+\cdots , \quad (z\in U).
\end{equation*}
\end{lemma}

\section{Coefficient bounds for the function class $\mathcal{AS}_{\Sigma_{m}}(\gamma,\lambda;\alpha)$}

\begin{definition}\label{def2.1}
A function $f\in \Sigma_{m}$ given by (\ref{eq.1.3}) is said to be in the class $\mathcal{AS}_{\Sigma_{m}}(\gamma,\lambda;\alpha)$ if it satisfies the following conditions:
\begin{equation}\label{eq.2.1}
\left| arg \left(\left[(1-\lambda) \frac{z f^{\prime}(z)}{f(z)}+\lambda\left(1+\frac{z f^{\prime \prime}(z)}{f^{\prime}(z)}\right)\right]^{\gamma}\right) \right|<\frac{\alpha \pi}{2}
\end{equation}
and
\begin{equation}\label{eq.2.2}
\left| arg \left(\left[(1-\lambda) \frac{w g^{\prime}(w)}{g(w)}+\lambda\left(1+\frac{w g^{\prime \prime}(w)}{g^{\prime}(w)}\right)\right]^{\gamma}\right) \right|<\frac{\alpha \pi}{2},
\end{equation}
\begin{equation*}
\left( z,w\in U, 0<\alpha \leq 1, \: 0\leq\gamma \leq 1, \: 0\leq\lambda \leq 1, \: m\in \mathbb{N}\right),
\end{equation*}
where the function $g=f^{-1}$ is given by (\ref{eq.1.4}).
\end{definition}

In particular, for one-fold symmetric bi-univalent functions, we denote the class $\mathcal{AS}_{\Sigma_{1}}(\gamma,\lambda;\alpha)=\mathcal{AS}_{\Sigma}(\gamma,\lambda;\alpha)$.

\begin{remark}
It should be remarked that the classes $\mathcal{AS}_{\Sigma_{m}}(\gamma,\lambda;\alpha)$ and $\mathcal{AS}_{\Sigma}(\gamma,\lambda;\alpha)$ are a generalization of well-known classes consider earlier. These classes are: \\ \smallskip
(1) For $\lambda=0$ and $\gamma=1$, the class $\mathcal{AS}_{\Sigma_{m}}(\gamma,\lambda;\alpha)$ reduce to the class $S_{\Sigma_{m}}^{\alpha}$ which was considered by Altinkaya and Yal\c{c}in \cite{1};\\
(2) For $\gamma=1$, the class $\mathcal{AS}_{\Sigma}(\gamma,\lambda;\alpha)$ reduce to the class $M_{\Sigma}(\alpha,\lambda)$ which was introduced by Liu and Wang \cite{9};\\
(3) For $\lambda=0$ and $\gamma=1$, the class $\mathcal{AS}_{\Sigma}(\gamma,\lambda;\alpha)$ reduce to the class $S_{\Sigma}^{*}(\alpha)$ which was given by Brannan and Taha \cite{3}.
\end{remark}

\begin{theorem}\label{thm2.1}
Let $f\in \mathcal{AS}_{\Sigma_{m}}(\gamma,\lambda;\alpha)$ $\left(0<\alpha \leq 1, \: 0\leq\gamma \leq 1, \: 0\leq\lambda \leq 1, \: m\in \mathbb{N}\right)$ be given by (\ref{eq.1.3}). Then
\begin{equation}\label{eq.2.3}
\left|a_{m+1}\right|\leq \frac{2\alpha}{m\sqrt{2\alpha\gamma(1+\lambda m)+\gamma(\gamma-\alpha)\left(1+\lambda m\right)^{2}}}
\end{equation}
and
\begin{equation}\label{eq.2.4}
\left| a_{2m+1}\right|\leq \frac{2\alpha^{2}(m+1)}{m^{2}\gamma^{2}\left(1+\lambda m\right)^{2}}+\frac{\alpha}{m\gamma(1+2\lambda m)}.
\end{equation}
\end{theorem}

\begin{proof}
It follows from conditions (\ref{eq.2.1}) and (\ref{eq.2.2}) that
\begin{equation}\label{eq.2.5}
\left[(1-\lambda) \frac{z f^{\prime}(z)}{f(z)}+\lambda\left(1+\frac{z f^{\prime \prime}(z)}{f^{\prime}(z)}\right)\right]^{\gamma}=\left[ p(z)\right]^{\alpha}
\end{equation}
and
\begin{equation}\label{eq.2.6}
\left[(1-\lambda) \frac{w g^{\prime}(w)}{g(w)}+\lambda\left(1+\frac{w g^{\prime \prime}(w)}{g^{\prime}(w)}\right)\right]^{\gamma}=\left[ q(w)\right]^{\alpha},
\end{equation}
where $g=f^{-1}$ and $p$, $q$ in $\mathcal{P}$ have the following series representations:
\begin{equation}\label{eq.2.7}
p(z)=1+p_{m}z^{m}+p_{2m}z^{2m}+p_{3m}z^{3m}+\cdots
\end{equation}
and
\begin{equation}\label{eq.2.8}
q(w)=1+q_{m}w^{m}+q_{2m}w^{2m}+q_{3m}w^{3m}+\cdots .
\end{equation}
Comparing the corresponding coefficients of (\ref{eq.2.5}) and (\ref{eq.2.6}) yields
\begin{equation}\label{eq.2.9}
m\gamma(1+\lambda m)a_{m+1}=\alpha p_{m},
\end{equation}
\begin{align}
& m\left[2\gamma(1+2\lambda m)a_{2m+1}-\gamma\left(\lambda m^{2}+2\lambda m+1\right)a_{m+1}^{2}\right]+\frac{m^{2}}{2}\gamma(1+\lambda m)(\gamma-1)(1+\lambda m)a_{m+1}^{2} \notag \\
& =\alpha p_{2m}+\frac{\alpha(\alpha-1)}{2} p_{m}^{2},\label{eq.2.10}
\end{align}
\begin{equation}\label{eq.2.11}
-m\gamma(1+\lambda m)a_{m+1}=\alpha q_{m}
\end{equation}
and
\begin{align}
& m\left[\gamma\left(3\lambda m^{2}+2(\lambda+1)m+1\right) a_{m+1}^{2}-2\gamma(1+2\lambda m)a_{2m+1}\right] \notag \\
&+\frac{m^{2}}{2}\gamma(1+\lambda m)(\gamma-1)(1+\lambda m)a_{m+1}^{2}=\alpha q_{2m}+\frac{\alpha(\alpha-1)}{2} q_{m}^{2}.\label{eq.2.12}
\end{align}
Making use of (\ref{eq.2.9}) and (\ref{eq.2.11}), we obtain
\begin{equation}\label{eq.2.13}
p_{m}=-q_{m}
\end{equation}
and
\begin{equation}\label{eq.2.14}
2m^{2}\gamma^{2}\left(1+\lambda m\right)^{2}a_{m+1}^{2}=\alpha^{2}(p_{m}^{2}+q_{m}^{2}).
\end{equation}
Also, from (\ref{eq.2.10}), (\ref{eq.2.12}) and (\ref{eq.2.14}), we find that
\begin{align*}
&m^{2}\left[2\gamma\left(1+\lambda m\right) +\gamma(\gamma-1)\left(1+\lambda m\right)^{2}\right]a_{m+1}^{2} \\
& =\alpha(p_{2m}+q_{2m})+\frac{\alpha(\alpha-1)}{2}\left(p_{m}^{2}+q_{m}^{2}\right) \\
&=\alpha(p_{2m}+q_{2m})+\frac{m^{2}\gamma^{2}(\alpha-1)\left(1+\lambda m\right)^{2}}{\alpha}a_{m+1}^{2}.
\end{align*}
Therefore, we have
\begin{equation}\label{eq.2.15}
a_{m+1}^{2}=\frac{\alpha^{2}(p_{2m}+q_{2m})}{m^{2}\left[2\alpha \gamma(1+\lambda m)+\gamma(\gamma-\alpha)\left(1+\lambda m\right)^{2}\right]}.
\end{equation}
Now, taking the absolute value of (\ref{eq.2.15}) and applying Lemma \ref{lem1.1} for the coefficients $p_{2m}$ and $q_{2m}$, we deduce that
\begin{equation*}
\left|a_{m+1}\right|\leq \frac{2\alpha}{m\sqrt{2\alpha\gamma(1+\lambda m)+\gamma(\gamma-\alpha)\left(1+\lambda m\right)^{2}}}.
\end{equation*}
This gives the desired estimate for $\left|a_{m+1}\right|$ as asserted in (\ref{eq.2.3}). \\
In order to find the bound on $\left|a_{2m+1}\right|$, by subtracting (\ref{eq.2.12}) from (\ref{eq.2.10}), we get
\begin{equation}\label{eq.2.16}
2m\gamma(1+2\lambda m)\left[2a_{2m+1}-(m+1)a_{m+1}^{2}\right]=\alpha \left(p_{2m}-q_{2m}\right)+\frac{\alpha(\alpha-1)}{2} \left(p_{m}^{2}-q_{m}^{2}\right).
\end{equation}
It follows from (\ref{eq.2.13}), (\ref{eq.2.14}) and (\ref{eq.2.16}) that
\begin{equation}\label{eq.2.17}
a_{2m+1}=\frac{\alpha^{2}(m+1)\left(p_{m}^{2}+q_{m}^{2}\right)}{4m^{2}\gamma^{2}\left(1+\lambda m\right)^{2}}+\frac{\alpha\left(p_{2m}-q_{2m}\right)}{4m\gamma(1+2\lambda m)}.
\end{equation}
Taking the absolute value of (\ref{eq.2.17}) and applying Lemma \ref{lem1.1} once again for the coefficients $p_{m}$, $p_{2m}$, $q_{m}$ and $q_{2m}$, we obtain
\begin{equation*}
\left| a_{2m+1}\right|\leq \frac{2\alpha^{2}(m+1)}{m^{2}\gamma^{2}\left(1+\lambda m\right)^{2}}+\frac{\alpha}{m\gamma(1+2\lambda m)},
\end{equation*}
which completes the proof of Theorem \ref{thm2.1}.
\end{proof}

For one-fold symmetric bi-univalent functions, Theorem \ref{thm2.1} reduce to the following corollary:

\begin{corollary} \label{cor2.1}
Let $f\in \mathcal{AS}_{\Sigma}(\gamma,\lambda;\alpha)$ $\left(0<\alpha \leq 1, \: 0\leq\gamma \leq 1, \: 0\leq\lambda \leq 1\right)$ be given by (\ref{eq.1.1}). Then
\begin{equation*}
\left|a_{2}\right|\leq \frac{2\alpha}{\sqrt{2 \alpha \gamma(1+\lambda) +\gamma(\gamma-\alpha)\left(1+\lambda \right)^{2}}}
\end{equation*}
and
\begin{equation*}
\left| a_{3}\right|\leq \frac{4\alpha^{2}}{\gamma^{2}\left(1+\lambda\right)^{2}} +\frac{\alpha}{\gamma(1+2\lambda)}.
\end{equation*}
\end{corollary}

\section{Coefficient bounds for the function class $\mathcal{AS}_{\Sigma_{m}}^{*}(\gamma,\lambda;\beta)$}
\noindent
\begin{definition}\label{def3.1}
A function $f\in \Sigma_{m}$ given by (\ref{eq.1.3}) is said to be in the class $\mathcal{AS}_{\Sigma_{m}}^{*}(\gamma,\lambda;\beta)$ if it satisfies the following conditions:
\begin{equation}\label{eq.3.1}
Re \left\lbrace \left[(1-\lambda) \frac{z f^{\prime}(z)}{f(z)}+\lambda\left(1+\frac{z f^{\prime \prime}(z)}{f^{\prime}(z)}\right)\right]^{\gamma} \right\rbrace >\beta
\end{equation}
and
\begin{equation}\label{eq.3.2}
Re \left\lbrace \left[(1-\lambda)\frac{w g^{\prime}(w)}{g(w)}+\lambda\left(1+\frac{w g^{\prime \prime}(w)}{g^{\prime}(w)}\right)\right]^{\gamma} \right\rbrace >\beta,
\end{equation}
\begin{equation*}
\left( z,w\in U, 0\leq\beta<1, \: 0\leq\gamma \leq 1, \: 0\leq\lambda \leq 1, \: m\in \mathbb{N}\right),
\end{equation*}
where the function $g=f^{-1}$ is given by (\ref{eq.1.4}).
\end{definition}

In particular, for one-fold symmetric bi-univalent functions, we denote the class $\mathcal{AS}_{\Sigma_{1}}^{*}(\gamma,\lambda;\beta)=\mathcal{AS}_{\Sigma}^{*}(\gamma,\lambda;\beta)$.

\begin{remark}
It should be remarked that the classes $\mathcal{AS}_{\Sigma_{m}}^{*}(\gamma,\lambda;\beta)$ and $\mathcal{AS}_{\Sigma}^{*}(\gamma,\lambda;\beta)$ are a generalization of well-known classes consider earlier. These classes are: \\ \smallskip
(1) For $\lambda=0$ and $\gamma=1$, the class $\mathcal{AS}_{\Sigma_{m}}^{*}(\gamma,\lambda;\beta)$ reduce to the class $S_{\Sigma_{m}}^{\beta}$ which was considered by Altinkaya and Yal\c{c}in \cite{1};\\
(2) For $\gamma=1$, the class $\mathcal{AS}_{\Sigma}^{*}(\gamma,\lambda;\beta)$ reduce to the class $B_{\Sigma}(\beta,\tau)$ which was introduced by Liu and Wang \cite{9};\\
(3) For $\lambda=0$ and $\gamma=1$, the class $\mathcal{AS}_{\Sigma}^{*}(\gamma,\lambda;\beta)$ reduce to the class $S_{\Sigma}^{*}(\beta)$ which was given by Brannan and Taha \cite{3}.
\end{remark}

\begin{theorem}\label{thm3.1}
Let $f\in \mathcal{AS}_{\Sigma_{m}}^{*}(\gamma,\lambda;\beta)$ $\left(0\leq\beta<1, \: 0\leq\gamma \leq 1, \: 0\leq\lambda \leq 1, \: m\in \mathbb{N}\right)$ be given by (\ref{eq.1.3}). Then
\begin{equation}\label{eq.3.3}
\left|a_{m+1}\right|\leq \frac{2}{m}\sqrt{\frac{1-\beta}{2\gamma(1+\lambda m)+\gamma(\gamma-1)\left(1+\lambda m\right)^{2}}}
\end{equation}
and
\begin{equation}\label{eq.3.4}
\left| a_{2m+1}\right|\leq \frac{2(m+1)\left(1-\beta\right)^{2}}{m^{2}\gamma^{2}\left(1+\lambda m\right)^{2}}+\frac{1-\beta}{m\gamma(1+2\lambda m)}.
\end{equation}
\end{theorem}

\begin{proof}
It follows from conditions (\ref{eq.3.1}) and (\ref{eq.3.2}) that there exist $p,q \in \mathcal{P}$ such that
\begin{equation}\label{eq.3.5}
\left[(1-\lambda) \frac{z f^{\prime}(z)}{f(z)}+\lambda\left(1+\frac{z f^{\prime \prime}(z)}{f^{\prime}(z)}\right)\right]^{\gamma}=\beta+(1-\beta)p(z)
\end{equation}
and
\begin{equation}\label{eq.3.6}
\left[(1-\lambda) \frac{w g^{\prime}(w)}{g(w)}+\lambda\left(1+\frac{w g^{\prime \prime}(w)}{g^{\prime}(w)}\right)\right]^{\gamma}=\beta+(1-\beta)q(w),
\end{equation}
where $p(z)$ and $q(w)$ have the forms (\ref{eq.2.7}) and (\ref{eq.2.8}), respectively. Equating coefficients (\ref{eq.3.5}) and (\ref{eq.3.6}) yields
\begin{equation}\label{eq.3.7}
m\gamma(1+\lambda m)a_{m+1}=(1-\beta)p_{m},
\end{equation}
\begin{align}
& m\left[2\gamma(1+2\lambda m)a_{2m+1}-\gamma\left(\lambda m^{2}+2\lambda m+1\right)a_{m+1}^{2}\right] \notag \\
&+\frac{m^{2}\gamma}{2}(1+\lambda m)(\gamma-1)(1+\lambda m)a_{m+1}^{2}=(1-\beta)p_{2m}, \label{eq.3.8}
\end{align}
\begin{equation}\label{eq.3.9}
-m\gamma(1+\lambda m)a_{m+1}=(1-\beta)q_{m}
\end{equation}
and
\begin{align}
& m\left[\gamma\left(3\lambda m^{2}+2(\lambda+1)m+1\right) a_{m+1}^{2}-2\gamma(1+2\lambda m)a_{2m+1}\right] \notag \\
&+\frac{m^{2}\gamma}{2}(1+\lambda m)(\gamma-1)(1+\lambda m)a_{m+1}^{2}=(1-\beta)q_{2m}.\label{eq.3.10}
\end{align}
From (\ref{eq.3.7}) and (\ref{eq.3.9}), we get
\begin{equation}\label{eq.3.11}
p_{m}=-q_{m}
\end{equation}
and
\begin{equation}\label{eq.3.12}
2m^{2}\gamma^{2}\left(1+\lambda m\right)^{2}a_{m+1}^{2}=\left(1-\beta\right)^{2}(p_{m}^{2}+q_{m}^{2}).
\end{equation}
Adding (\ref{eq.3.8}) and (\ref{eq.3.10}), we obtain
\begin{equation}\label{eq.3.13}
m^{2}\left[2\gamma\left(1+\lambda m\right)+\gamma(\gamma-1)\left(1+\lambda m\right)^{2}\right]a_{m+1}^{2}=(1-\beta)(p_{2m}+q_{2m}).
\end{equation}
Therefore, we have
\begin{equation*}
a_{m+1}^{2}=\frac{(1-\beta)(p_{2m}+q_{2m})}{m^{2}\left[2\gamma\left(1+\lambda m\right)+\gamma(\gamma-1)\left(1+\lambda m\right)^{2}\right]}.
\end{equation*}
Applying Lemma \ref{lem1.1} for the coefficients $p_{2m}$ and $q_{2m}$, we obtain
\begin{equation*}
\left|a_{m+1}\right|\leq \frac{2}{m}\sqrt{\frac{1-\beta}{2\gamma(1+\lambda m)+\gamma(\gamma-1)\left(1+\lambda m\right)^{2}}}.
\end{equation*}
This gives the desired estimate for $\left|a_{m+1}\right|$ as asserted in (\ref{eq.3.3}). \\
In order to find the bound on $\left|a_{2m+1}\right|$, by subtracting (\ref{eq.3.10}) from (\ref{eq.3.8}), we get
\begin{equation*}
2m\gamma(1+2\lambda m) \left[2a_{2m+1}-(m+1)a_{m+1}^{2}\right]=(1-\beta)\left(p_{2m}-q_{2m}\right).
\end{equation*}
or equivalently
\begin{equation*}
a_{2m+1}=\frac{m+1}{2}a_{m+1}^{2}+\frac{(1-\beta)\left(p_{2m}-q_{2m}\right)}{4m\gamma(1+2\lambda m)}.
\end{equation*}
Upon substituting the value of $a_{m+1}^{2}$ from (\ref{eq.3.12}), it follows that
\begin{equation*}
a_{2m+1}=\frac{(m+1)\left(1-\beta\right)^{2}(p_{m}^{2}+q_{m}^{2})}{4m^{2}\gamma^{2}\left(1+\lambda m\right)^{2}}+\frac{\left(1-\beta\right)\left(p_{2m}-q_{2m}\right)}{4m\gamma(1+2\lambda m)}.
\end{equation*}
Applying Lemma \ref{lem1.1} once again for the coefficients $p_{m}$, $p_{2m}$, $q_{m}$ and $q_{2m}$, we obtain
\begin{equation*}
\left| a_{2m+1}\right|\leq \frac{2(m+1)\left(1-\beta\right)^{2}}{m^{2}\gamma^{2}\left(1+\lambda m\right)^{2}}+\frac{1-\beta}{m\gamma(1+2\lambda m)}.
\end{equation*}
which completes the proof of Theorem \ref{thm3.1}.
\end{proof}

For one-fold symmetric bi-univalent functions, Theorem \ref{thm3.1} reduce to the following corollary:

\begin{corollary} \label{cor3.1}
Let $f\in \mathcal{AS}_{\Sigma}^{*}(\gamma,\lambda;\beta)$ $\left(0\leq\beta<1, \: 0\leq\gamma \leq 1, \: 0\leq\lambda \leq 1\right)$ be given by (\ref{eq.1.1}). Then
\begin{equation*}
\left|a_{2}\right|\leq 2\sqrt{\frac{1-\beta}{2\gamma(1+\lambda)+\gamma(\gamma-1)\left(1+\lambda \right)^{2}}}
\end{equation*}
and
\begin{equation*}
\left| a_{3}\right|\leq \frac{4\left(1-\beta\right)^{2}}{\gamma^{2}\left(1+\lambda\right)^{2}}+\frac{1-\beta}{\gamma(1+2\lambda)}.
\end{equation*}
\end{corollary}


\begin{thebibliography}{20}

\bibitem{1} Altinkaya, \c{S}., Yal\c{c}in, S., \emph{Coefficient bounds for certain subclasses of m-fold symmetric bi-univalent functions}, Journal of Mathematics, Art. ID \textbf{241683}(2015), 1-5.
\bibitem{2} Altinkaya, \c{S}., Yal\c{c}in, S., \emph{On some subclasses of m-fold symmetric bi-univalent functions}, Commun. Fac. Sci. Univ. Ank. Series A1, \textbf{67}(2018), no. 1, 29-36.
\bibitem{3} Brannan, D.A., Taha, T.S., \emph{On Some classes of bi-univalent functions}, Studia Univ. Babes-Bolyai Math., \textbf{31}(1986), no. 2, 70-77.
\bibitem{4} Duren, P.L., \emph{Univalent Functions}, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
\bibitem{5} Eker, S.S., \emph{Coefficient bounds for subclasses of m-fold symmetric bi-univalent functions}, Turk. J. Math., \textbf{40}(2016), 641-646.
\bibitem{6} Frasin, B.A., Aouf, M.K., \emph{New subclasses of bi-univalent functions}, Appl. Math. Lett., \textbf{24}(2011), 1569–1573.
\bibitem{7} Goyal, S.P., Goswami, P., \emph{Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives}, J. Egyptian Math. Soc., \textbf{20}(2012), 179-182.
\bibitem{8} Koepf, W., \emph{Coefficients of symmetric functions of bounded boundary rotations}, Proc. Amer. Math. Soc., \textbf{105}(1989), 324-329.
\bibitem{9} Li, X.–F., Wang, A.–P., \emph{Tow new subclasses of bi-univalent functions}, Int. Math. Forum, \textbf{7}(2012), 1495-1504.
\bibitem{10} Srivastava, H.M., Bansal, D., \emph{Coefficient estimates for a subclass of analytic and bi-univalent functions}, J. Egyptian Math. Soc., \textbf{23}(2015), 242-246.
\bibitem{11} Srivastava, H.M., Bulut, S., Caglar, M., Yagmur, N., \emph{Coefficient estimates for a general subclass of analytic and bi-univalent functions}, Filomat, \textbf{27}(2013), no. 5, 831-842.
\bibitem{12} Srivastava, H.M., Eker, S.S., Ali, R.M., \emph{Coefficient bounds for a certain class of analytic and bi-univalent functions}, Filomat, \textbf{29}(2015), 1839-1845.
\bibitem{13} Srivastava, H.M., Gaboury, S., Ghanim, F., \emph{Initial coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions}, Acta Mathematica Scientia, \textbf{36B}(2016), no. 3, 863-871.
\bibitem{14} Srivastava, H.M., Mishra, A.K., Gochhayat, P., \emph{Certain subclasses of analytic and bi-univalent functions}, Appl. Math. Lett., \textbf{23}(2010), 1188-1192.
\bibitem{15} Srivastava, H.M., Sivasubramanian, S., Sivakumar, R., \emph{Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions}, Tbilisi Math. J., \textbf{7}(2014), no. 2, 1-10.
\bibitem{16} Srivastava, H.M., Wanas, A.K., \emph{Initial Maclaurin Coefficient Bounds for New Subclasses of Analytic and m-Fold Symmetric Bi-Univalent Functions Defined by a Linear Combination}, Kyungpook Mathematical Journal, \textbf{59}(2019), no. 3, 493-503.
\bibitem{17} Tang, H., Srivastava, H.M., Sivasubramanian, S., Gurusamy, P., \emph{The Fekete-Szeg$\ddot{o}$ functional problems for some subclasses of m-fold symmetric bi-univalent functions}, J. Math. Inequal., \textbf{10}(2016), 1063-1092.

\end{thebibliography}
\end{document}