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\begin{document}
\title [On Certain subclasses of bi-univalent functions]{Certain subclasses of bi-univalent functions \\associated with the  Hohlov operator}
\author{H. M. Srivastava, G. Murugusundaramoorthy and N. Magesh}
\maketitle
\begin{center}
Department of Mathematics and Statistics,\\
University of Victoria,\\
Victoria, British Columbia V8W 3R4, Canada\\ 
{\tt e-mail:} harimsri@math.uvic.ca
\end{center}
\begin{center}
School of Advanced Sciences, VIT University,\\ 
Vellore 632014, Tamilnadu, India\\ 
{\tt e-mail:} gmsmoorthy@yahoo.com
\end{center}
\begin{center}
Post-Graduate and Research Department of Mathematics,\\ 
Government Arts College for Men,\\
Krishnagiri 635001, Tamilnadu, India\\ 
{\tt e-mail:} nmagi\_2000@yahoo.co.in
\end{center}
\begin{abstract}
In this paper, we introduce and investigate two new subclasses of the function
class $\Sigma$ of bi-univalent functions defined in the open unit disk, which are associated with the Hohlov operator, that is, a familiar special case of the widely- (and extensively-) investigated Dziok-Srivastava linear operator. Furthermore, we find
estimates on the Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.\ 
\\
{\bf 2010 Mathematics Subject Classification.} Primary 30C45. \
\\
{\bf Key Words and Phrases.} Analytic functions; Univalent functions;  
Bi-univalent functions; Bi-starlike
and bi-convex functions; Hohlov operator, Gaussian hypergeometric
function; Dziok-Srivastava operator.
\end{abstract}

\maketitle

\section{Introduction, Definitions and Preliminaries} 
Let $\mathcal{A}$ denote the class of
functions of the form:
\begin{equation}\label{Int-e1}
f(z)=z+\sum\limits_{n=2}^{\infty}a_nz^n,
\end{equation}
which are analytic in the open unit disk 
$$\mathbb{U}=\{z: z \in
\mathbb{C}\quad {\rm and}\quad |z|<1\}.$$ 
Further, by $\mathcal{S}$
we shall denote the class of all functions in $\mathcal{A}$ which
are univalent in $\mathbb{U}.$

\par The convolution or Hadamard product of two functions $f,h\in \mathcal{A}$
is denoted by $f\ast h$ and is defined as
\begin{equation}
(f\ast h)(z)=z+\sum\limits_{n=2}^{\infty }a_{n}b_{n}z^{n},
\tag{1.2}
\end{equation}
where $f(z)$ is given by (\ref{Int-e1}) and
$h(z)=z+\sum\limits_{n=2}^{\infty }b_{n}z^{n}.$ In terms of the Hadamard product (or convolution), the Dziok-Srivastava linear operator involving the generalized hypergeometric function, was introduced and studied systematically
by Dziok and Srivastava \cite{dziok1,dziok2} and (subsequently) by many other authors. Here, in our present investigation, we recall a familiar convolution operator $\mathcal{I}_{a,b,c}$ due to Hohlov \cite{HO1,HO2}, which indeed is a very specialized case of the widely- (and extensively-) investigated Dziok-Srivastava operator.
\par For the complex parameters $a,~b~\text{and}~c$ with $c\neq
0,-1,-2,-3,\cdots$, the Gaussian
hypergeometric function $_{2}F_{1}(a,b,c;z)$ is defined as%
\begin{align} \label{Gauss-equ}
{}_{2}F_{1}(a,b,c;z)&=\overset{\infty }{\underset{n=0}{\sum }}\frac{%
(a)_{n}(b)_{n}}{(c)_{n}}\; \frac{z^{n}}{n!} \nonumber \\
& = 1+\overset{\infty }{\underset{n=2}{\sum }}\frac{%
(a)_{n-1}(b)_{n-1}}{(c)_{n-1}}\;\frac{z^{n-1}}{(n-1)!}\qquad (z \in
\mathbb{U}),
\end{align}
where $(\alpha )_{n}$ is the Pochhammer symbol (or the shifted
factorial) defined as follows:

\begin{equation}\label{Poch}
(\alpha )_{n}=\frac{\Gamma(\alpha+n)}{\Gamma(\alpha)}=\left\{
\begin{array}{ll}
1 &\qquad (n=0)
\\
\alpha (\alpha +1)(\alpha +2)\cdots(\alpha +n-1) & \qquad
(n=1,2,3,\cdots).
\end{array}
\right.
\end{equation}
By using the Gaussian hypergeometric function given by (\ref{Gauss-equ}),
Hohlov \cite{HO1,HO2} introduced the familiar convolution operator $\mathcal{I}_{a,b,c}$ as follows:
\begin{align}\label{Hohlov}
\mathcal{I}_{a,b;c}f(z)&=z~{}_{2}F_{1}(a,b,c;z)\ast f(z),\nonumber\\
&= z+\overset{\infty }{\underset{n=2}{\sum
}}\varphi_{n}a_{n}z^{n}\qquad (z \in \mathbb{U}),
\end{align}
where
\begin{equation}\label{varphi-n}
\varphi_{n}=\frac{(a)_{n-1}(b)_{n-1}}{(c)_{n-1}(n-1)!}.
\end{equation}
Hohlov \cite{HO1,HO2} discussed some interesting geometrical properties exhibited by the operator $\mathcal{I}_{a,b;c}$. The three-parameter family of operators
$\mathcal{I}_{a,b;c}$ contains, as its special cases, most of the
known linear integral or differential operators. In particular, if
$b=1$ in (\ref{Hohlov}), then $\mathcal{I}_{a,b;c}$ reduces to the
Carlson-Shaffer operator.
Similarly, it is easily seen that the Hohlov operator $\mathcal{I}_{a,b;c}$ is
also a generalization of the Ruscheweyh derivative operator as well as the Bernardi-Libera-Livingston operator.

\par Some of the important and well-investigated subclasses of the
univalent function class $\mathcal{S}$ include (for example) the
class $\mathcal{S}^*(\alpha)$ of starlike functions of order
$\alpha$ in $\mathbb{U}$ and the class $\mathcal{K}(\alpha)$ of
convex functions of order $\alpha$ in $\mathbb{U}.$ It is well known that every function $f\in \mathcal{S}$ has
an inverse $f^{-1},$ defined by
\[
f^{-1}(f(z))=z \qquad (z \in \mathbb{U})
\]
and
\[
f(f^{-1}(w))=w \qquad \left(|w| < r_0(f);\,\, r_0(f) \geqq \frac{1}{4}\right),
\]
where
\begin{equation}\label{Int-f-inver}
f^{-1}(w) = w - a_2w^2 + (2a_2^2-a_3)w^3 -
(5a_2^3-5a_2a_3+a_4)w^4+\cdots .
\end{equation}
\par A function $f \in \mathcal{A}$ is said to be bi-univalent in
$\mathbb{U}$ if both $f(z)$ and $f^{-1}(z)$ are univalent in
$\mathbb{U}.$ Let $\Sigma$ denote the class of bi-univalent
functions in $\mathbb{U}$ given by (\ref{Int-e1}).
\par In 1967, Lewin \cite{Lewin} investigated the bi-univalent
function class $\Sigma$ and showed that $|a_2|<1.51.$ On the other
hand, Brannan and Clunie \cite{Bran-1979} (see also
\cite{Branna1970,Bran1985,Taha1981}) and Netanyahu \cite{Netany} made
an attempt to introduce various subclasses of the bi-univalent
function class $\Sigma$ and obtained non-sharp coefficient
estimates on the first two coefficients $|a_2|$ and $|a_3|$ of
(\ref{Int-e1}). But the coefficient problem for each of the
following Taylor-Maclaurin coefficients: 
$$|a_n|\qquad (n\in\mathbb{N}\setminus\{1,2\};\;\;\mathbb{N}:=\{1,2,3,\cdots\}$$ is still an open problem. Following Brannan and Taha
\cite{Bran1985}, many researchers (see\cite{BAF-MKA,haya,Li-Wang,HMS-AKM-PG,Xu-HMS-AML,Xu-HMS-AMC}) have recently introduced and
investigated several interesting subclasses of the bi-univalent
function class $\Sigma$ and they have found non-sharp estimates on the
first two Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|.$

\par Making use of the Hohlov operator $\mathcal{I}_{a,b;c},$
we introduce the following two new subclasses of the function class
$\Sigma.$

\begin{definition}
A function $f(z)$ given by (\ref{Int-e1}) is said to be in the
class $\mathcal{S}^{a,b;c}_{\Sigma}(\alpha, \lambda)$ if the
following conditions are satisfied:
\begin{equation}\label{Defi-1-e1}
f \in \Sigma \,\, \text{and} \,\,\left |\arg \left
(\frac{z(\mathcal{I}_{a,b;c}f(z))'}{(1-\lambda)z + \lambda
\mathcal{I}_{a,b;c}f(z)} \right )\right | < \frac{\alpha
\pi}{2}\qquad (0 < \alpha \leqq 1;\,\, 0 \leqq \lambda \leqq 1;
\,\, z \in \mathbb{U})
\end{equation}
and
\begin{equation}\label{Defi-1-e2}
\left |\arg \left (
\frac{w(\mathcal{I}_{a,b;c}g(w))'}{(1-\lambda)w + \lambda
\mathcal{I}_{a,b;c}g(w)} \right )\right | < \frac{\alpha
\pi}{2}\qquad (0 < \alpha \leqq 1;\,\, 0 \leqq \lambda \leqq 1;
\,\, w \in \mathbb{U}),
\end{equation}
where the function $g$ is given by
\begin{equation}\label{g-e}
g(w) = w - a_2w^2 + (2a_2^2-a_3)w^3 -
(5a_2^3-5a_2a_3+a_4)w^4+\cdots,
\end{equation}
that is, the extension of $f^{-1}$ to $\mathbb{U}.$
\end{definition}
We note that for $\lambda = 0,$ $a=c$ and $b=1,$ the class
$\mathcal{S}^{a,b;c}_{\Sigma}(\alpha, \lambda)$ reduces to the
class $\mathcal{H}_{\Sigma}^{\alpha}$ introduced and studied by
Srivastava et al. \cite{HMS-AKM-PG}. Putting $\lambda =1,$ $a=c$
and $b=1,$ the class $\mathcal{S}^{a,b;c}_{\Sigma}(\alpha,
\lambda)$ reduces to the class of strongly bi-starlike functions
of order $\alpha (0<\alpha \leqq 1)$ and denoted by
$\mathcal{S}^*_{\Sigma}(\alpha).$

\begin{definition}
A function $f(z)$ given by (\ref{Int-e1}) is said to be in the
class $\mathcal{M}^{a,b;c}_{\Sigma}(\beta, \lambda)$ if the
following conditions are satisfied:
\begin{equation}\label{Defi-2-e1}
f \in \Sigma\quad \text{and} \quad  \Re \left (
\frac{z(\mathcal{I}_{a,b;c}f(z))'}{(1-\lambda)z + \lambda
\mathcal{I}_{a,b;c}f(z)} \right ) > \beta \qquad (0 \leqq \beta <
1;\,\, 0 \leqq \lambda \leqq 1; \,\, z \in \mathbb{U})
\end{equation}
and
\begin{equation}\label{Defi-2-e2}
\Re \left ( \frac{w(\mathcal{I}_{a,b;c}g(w))'}{(1-\lambda)w +
\lambda \mathcal{I}_{a,b;c}g(w)} \right )>\beta\qquad (0 \leqq \beta
< 1;\,\, 0 \leqq \lambda \leqq 1; \,\, w \in \mathbb{U}),
\end{equation}
where the function $g$ is given by (\ref{g-e}).
\end{definition}
It is interesting to note that, for $\lambda = 0,$ $a=c$ and $b=1,$
the class $\mathcal{M}^{a,b;c}_{\Sigma}(\beta, \lambda)$ reduces
to the class $\mathcal{H}_{\Sigma}^{\beta}$ introduced and studied
by Srivastava et al. \cite{HMS-AKM-PG}. Putting $\lambda =1,$
$a=c$ and $b=1,$ the class $\mathcal{M}^{a,b;c}_{\Sigma}(\beta,
\lambda)$ reduces to the class of bi-starlike functions of order
$\beta (0<\beta \leqq 1)$ and denoted by
$\mathcal{S}_{\Sigma}(\beta).$

\par The object of the present paper is to find estimates on the
coefficients $|a_2|$ and $|a_3|$ for functions in the above-defined 
subclasses $\mathcal{S}^{a,b;c}_{\Sigma}(\alpha,\lambda)$ and $\mathcal{M}^{a,b;c}_{\Sigma}(\alpha,\lambda)$ of the function class $\Sigma$ by employing the techniques used earlier by Srivastava et al. \cite{HMS-AKM-PG}. 

\par In order to derive our main results, we shall need the following lemma.
\begin{lemma}{\rm (see \cite{Pom})}\label{lem-pom}
If $h\in \mathcal{P},$ then $|c_k|\leqq 2$ for each $k,$ where $\mathcal{P}$ is the family of all functions $h,$ analytic in $\mathbb{U},$ for which
$$\Re\{h(z)\}>0\qquad (z \in \mathbb{U}),$$ where 
$$h(z)=1+c_1z+c_2z^2+\cdots \qquad (z \in
\mathbb{U}).$$
\end{lemma}

\section{Coefficient Bounds for the Function Class $\mathcal{S}^{a,b;c}_{\Sigma}(\alpha,\lambda)$}

\par We begin by finding the estimates on the coefficients
$|a_2|$ and $|a_3|$ for functions in the class
$\mathcal{S}^{a,b;c}_{\Sigma}(\alpha, \lambda).$
\begin{theorem}\label{Bi-th1}
Let the function $f(z)$ given by $(\ref{Int-e1})$ be in the following class$:$
$$\mathcal{S}^{a,b;c}_{\Sigma}(\alpha, \lambda)\qquad(0 < \alpha \leqq
1;\;\;0 \leqq \lambda \leqq 1).$$ Then
\begin{equation}\label{bi-th1-b-a2}
|a_2|\leqq\frac{2\alpha}{\sqrt{[2\alpha(\lambda^2-2\lambda)+(1-\alpha)(2-\lambda)^2]\varphi_2^2+2\alpha(3-\lambda)\varphi_3}}
\end{equation}
and
\begin{equation}\label{bi-th1-b-a3}
|a_3| \leqq \frac{2\alpha}{(3-\lambda)\varphi_3}.
\end{equation}
\end{theorem}

\begin{proof}
It follows from (\ref{Defi-1-e1}) and (\ref{Defi-1-e2}) that
\begin{equation}\label{bi-th1-pr-e1}
\frac{z\big(\mathcal{I}_{a,b;c}f(z)\big)'}{(1-\lambda)z + \lambda
\mathcal{I}_{a,b;c}f(z)} = [p(z)]^{\alpha}
\end{equation}
and
\begin{equation}\label{bi-th1-pr-e2}
\frac{w\big(\mathcal{I}_{a,b;c}g(w)\big)'}{(1-\lambda)w + \lambda
\mathcal{I}_{a,b;c}g(w)} = [q(w)]^{\alpha},
\end{equation}
where $p(z)$ and $q(w)$ in $\wp$ and have the following forms:
\begin{equation}\label{Exp-p(z)}
p(z)=1+p_1z+p_2z^2+\cdots
\end{equation}
and
\begin{equation}\label{Exp-q(w)}
q(z)=1+q_1w+q_2w^2+\cdots ,
\end{equation}
respectively. Now, equating the coefficients in (\ref{bi-th1-pr-e1}) and
(\ref{bi-th1-pr-e2}), we get
\begin{equation}\label{th1-ceof-p1}
(2-\lambda)\varphi_2 a_2 = \alpha p_1,
\end{equation}
\begin{equation}\label{th1-ceof-p2}
(\lambda^2-2\lambda)\varphi_2^2 a_2^2+(3-\lambda)\varphi_3 a_3 =
\frac{1}{2}\left [\alpha(\alpha-1)p_1^2 + 2\alpha p_2\right ],
\end{equation}
\begin{equation}\label{th1-ceof-q1}
-(2-\lambda)\varphi_2 a_2 = \alpha q_1
\end{equation}
and
\begin{equation}\label{th1-ceof-q2}
(\lambda^2-2\lambda)\varphi_2^2 a_2^2 +(3-\lambda)\varphi_3
(2a_2^2-a_3) = \frac{1}{2}\left [\alpha(\alpha-1)q_1^2 + 2\alpha
q_2\right ].
\end{equation}
From (\ref{th1-ceof-p1}) and (\ref{th1-ceof-q1}), we find that
\begin{equation}\label{th1-pr-p1=q1-a2}
a_2=\frac{\alpha p_1}{(2-\lambda)\varphi_2}=\frac{-\alpha
q_1}{(2-\lambda)\varphi_2},
\end{equation}
which implies
\begin{equation}\label{th1-pr-p1=q1}
p_1=-q_1.
\end{equation}
Adding (\ref{th1-ceof-p2}) and (\ref{th1-ceof-q2}), we obtain
\begin{equation}\label{th1-a2-star}
[2(\lambda^2-2\lambda)\varphi_2^2+2(3-\lambda)\varphi_3]a_2^2 =
\frac{\alpha(\alpha-1)}{2}(p_1^2+q_1^2)+\alpha(p_2+q_2).
\end{equation}
Substituting the value of $a_2$ from (\ref{th1-pr-p1=q1-a2}) and
(\ref{th1-pr-p1=q1}) into (\ref{th1-a2-star}), we get
\begin{equation}\label{th1-p1-square}
p_1^2=\frac{(2-\lambda)^2\varphi_2^2(p_2+q_2)}{[2\alpha(\lambda^2-2\lambda)+(1-\alpha)(2-\lambda)^2]\varphi_2^2+2\alpha(3-\lambda)\varphi_3}.
\end{equation}
Applying Lemma \ref{lem-pom} for the coefficients $p_2$ and $q_2,$
we immediately have
\begin{equation}\label{th1-|p1|}
|p_1|\leqq\frac{2(2-\lambda)\varphi_2}{\sqrt{[2\alpha(\lambda^2-2\lambda)+(1-\alpha)(2-\lambda)^2]\varphi_2^2+2\alpha(3-\lambda)\varphi_3}}.
\end{equation}
Thus, (\ref{th1-pr-p1=q1-a2}) gives
\[
|a_2|\leqq\frac{2\alpha}{\sqrt{[2\alpha(\lambda^2-2\lambda)+(1-\alpha)(2-\lambda)^2]\varphi_2^2+2\alpha(3-\lambda)\varphi_3}}.
\]
This gives the bound on $|a_2|$ as asserted in
(\ref{bi-th1-b-a2}).
\par Next, in order to find the bound on
$|a_3|$, by subtracting (\ref{th1-ceof-q2}) from
(\ref{th1-ceof-p2}), we get
\begin{equation}\label{th1-a3-cal-e1}
2(3-\lambda)\varphi_3a_3 - 2(3-\lambda)\varphi_3a_2^2 = \alpha
(p_2-q_2)+\frac{\alpha(\alpha-1)}{2}(p_1^2-q_1^2).
\end{equation}
It follows from (\ref{th1-pr-p1=q1-a2}), (\ref{th1-pr-p1=q1}) and
(\ref{th1-a3-cal-e1}) that
\begin{align*}
2(3-\lambda)\varphi_3a_3
&=\left[\frac{2(3-\lambda)\alpha^2\varphi_3}{[2\alpha(\lambda^2-2\lambda)+(1-\alpha)(2-\lambda)^2]\varphi_2^2+2\alpha(3-\lambda)\varphi_3}+\alpha\right]p_2
\nonumber\\ 
&\qquad +\left[\frac{2(3-\lambda)\alpha^2\varphi_3}{[2\alpha(\lambda^2-2\lambda)+(1-\alpha)(2-\lambda)^2]\varphi_2^2+2\alpha(3-\lambda)\varphi_3}-\alpha\right]q_2.
\end{align*}
Applying Lemma \ref{lem-pom} once again for the coefficients $p_2$
and $q_2,$ we readily get
\[
|a_3| \leqq \frac{2\alpha}{(3-\lambda)\varphi_3}.
\]
This completes the proof of Theorem \ref{Bi-th1}.
\end{proof}
Putting $\lambda =0$ in Theorem \ref{Bi-th1}, we have the
following corollary.
\begin{corollary}\label{Bi-cor1}
Let the function $f(z)$ given by $(\ref{Int-e1})$ be in the class
$\mathcal{S}^{a,b;c}_{\Sigma}(\alpha)\;\;(0 < \alpha \leqq 1).$
Then
\begin{equation}\label{bi-cor1-b-a2}
|a_2| \leqq \alpha
\sqrt{\frac{2}{2(1-\alpha)\varphi^2_2+3\alpha\varphi_3}}
\end{equation}
and
\begin{equation}\label{bi-cor1-b-a3}
|a_3| \leqq \frac{2\alpha}{3\varphi_3}.
\end{equation}
\end{corollary}
Taking $a=c$ and $b=1,$ in Corollary \ref{Bi-cor1}, we get the
following corollary.
\begin{corollary}\label{Bi-cor1a}
Let the function $f(z)$ given by $(\ref{Int-e1})$ be in the class
$\mathcal{H}^{\alpha}_{\Sigma}\;\;(0 < \alpha \leqq 1).$ Then
\begin{equation}\label{bi-cor1a-b-a2}
|a_2| \leqq \alpha \sqrt{\frac{2}{2+\alpha}}
\end{equation}
and
\begin{equation}\label{bi-cor1a-b-a3}
|a_3| \leqq \frac{2\alpha}{3}.
\end{equation}
\end{corollary}
\begin{remark}
The bound on $|a_3|$ in Corollary \ref{Bi-cor1a} provides an
improvement over the result of Srivastava et al. \cite{HMS-AKM-PG}.
\end{remark}
When $\lambda =1$ in Theorem \ref{Bi-th1}, we have the following
corollary.
\begin{corollary}\label{Bi-cor2}
Let the function $f(z)$ given by $(\ref{Int-e1})$ be in the class
$\mathcal{S}_{\Sigma}^{a,b,c}(\alpha,1)\;\;(0 < \alpha \leqq 1).$
Then
\[
|a_2| \leqq
\frac{2\alpha}{\sqrt{(1-3\alpha)\varphi^2_2+4\alpha\varphi_3}}\qquad
{\it and} \qquad |a_3| \leqq \frac{\alpha}{\varphi_3}.
\]
\end{corollary}

\section{Coefficient Bounds for the Function Class $\mathcal{M}^{a,b;c}_{\Sigma}(\beta,
\lambda)$}
\par In this section, we find the estimates on the coefficients $|a_2|$ and
$|a_3|$ for functions in the class
$\mathcal{M}^{a,b;c}_{\Sigma}(\beta, \lambda).$
\begin{theorem}\label{Bi-th2}
Let the function $f(z)$ given by $(\ref{Int-e1})$ be in the following class$:$
$$\mathcal{M}^{a,b;c}_{\Sigma}(\beta, \lambda)\qquad(0 \leqq \beta < 1;\;\; 0 \leqq \lambda \leqq 1).$$ Then
\begin{equation}\label{bi-th2-b-a2}
|a_2|\leqq\sqrt{\frac{2(1-\beta)}{(\lambda^2-2\lambda)\varphi_2^2+(3-\lambda)\varphi_3}}
\end{equation}
and
\begin{equation}\label{bi-th2-b-a3}
|a_3| \leqq \frac{2(1-\beta)}{(3-\lambda)\varphi_3}.
\end{equation}
\end{theorem}
\begin{proof}
It follows from (\ref{Defi-2-e1}) and (\ref{Defi-2-e2}) that there
exist $p,q \in \wp$ such that
\begin{equation}\label{bi-th2-pr-e1}
\frac{z\big(\mathcal{I}_{a,b;c}f(z)\big)'}{(1-\lambda)z + \lambda
\mathcal{I}_{a,b;c}f(z)} = \beta + (1-\beta)p(z)
\end{equation}
and
\begin{equation}\label{bi-th2-pr-e2}
\frac{w\big(\mathcal{I}_{a,b;c}g(w)\big)'}{(1-\lambda)w + \lambda
\mathcal{I}_{a,b;c}g(w)} = \beta + (1-\beta)q(w),
\end{equation}
where $p(z)$ and $q(w)$ have the forms (\ref{Exp-p(z)}) and
(\ref{Exp-q(w)}), respectively. Equating coefficients in
(\ref{bi-th2-pr-e1}) and (\ref{bi-th2-pr-e2}), we get
\begin{equation}\label{th2-ceof-p1}
(2-\lambda)\varphi_2a_2 = (1-\beta) p_1,
\end{equation}
\begin{equation}\label{th2-ceof-p2}
(\lambda^2-2\lambda)\varphi_2^2a_2^2+(3-\lambda)\varphi_3a_3 =
(1-\beta)p_2,
\end{equation}
\begin{equation}\label{th2-ceof-q1}
-(2-\lambda)\varphi_2a_2 = (1-\beta) q_1
\end{equation}
and
\begin{equation}\label{th2-ceof-q2}
(\lambda^2-2\lambda)\varphi_2^2a_2^2+(3-\lambda)\varphi_3(2a_2^2-a_3)
= (1-\beta)q_2.
\end{equation}
From (\ref{th2-ceof-p1}) and (\ref{th2-ceof-q1}), we get
\begin{equation}\label{th2-pr-p1=q1-a2}
a_2=\frac{(1-\beta)}{(2-\lambda)\varphi_2}p_1=\frac{-(1-\beta)}{(2-\lambda)\varphi_2}q_1
\end{equation}
which implies
\begin{equation}\label{th2-pr-p1=q1}
p_1=-q_1.
\end{equation}
From (\ref{th2-ceof-p2}) and (\ref{th2-ceof-q2}), we have
\begin{equation}\label{th2-a2-square}
[2(\lambda^2-2\lambda)^2\varphi_2^2+2(3-\lambda)\varphi_3]a_2^2 =
(1-\beta)(p_2+q_2).
\end{equation}
Also, by using (\ref{th2-pr-p1=q1-a2}) and (\ref{th2-a2-square}), we
obtain
\begin{equation}\label{th2-p-1-square}
p_1^2=\frac{(2-\lambda)^2\varphi_2^2(p_2+q_2)}{2[(\lambda^2-2\lambda)^2\varphi_2^2+(3-\lambda)\varphi_3](1-\beta)}.
\end{equation}
Applying Lemma \ref{lem-pom}, we get
\begin{equation}\label{th2-|p-1|}
|p_1|
\leqq(2-\lambda)\varphi_2\sqrt{\frac{2}{[(\lambda^2-2\lambda)^2\varphi_2^2+(3-\lambda)\varphi_3](1-\beta)}}.
\end{equation}
Again, by applying Lemma \ref{lem-pom} to (\ref{th2-pr-p1=q1-a2}) and
using (\ref{th2-|p-1|}), we immediately find that
\[
|a_2|\leqq\sqrt{\frac{2(1-\beta)}{(\lambda^2-2\lambda)\varphi_2^2+(3-\lambda)\varphi_3}}.
\]
This gives the bound on $|a_2|$ as asserted in
(\ref{bi-th2-b-a2}).
\par Next, in order to find the bound on $|a_3|$, by subtracting
(\ref{th2-ceof-q2}) from (\ref{th2-ceof-p2}), we get
\begin{equation}\label{th2-a3-cal-e1}
2(3-\lambda)\varphi_3a_3 - 2(3-\lambda)\varphi_3a_2^2 = (1-\beta)
(p_2-q_2).
\end{equation}
It follows from (\ref{th2-a2-square} and \ref{th2-a3-cal-e1} that
\begin{align*}
2(3-\lambda)\varphi_3a_3 &= \frac{2(3-\lambda)\varphi_3(1-\beta)
+(\lambda^2-2\lambda)\varphi_2^2(1-\beta)}{(\lambda^2-2\lambda)\varphi_2^2+(3-\lambda)\varphi_3}p_2\\
& \qquad - \frac{(\lambda^2-2\lambda)\varphi_2^2(1-\beta)}{(\lambda^2-2\lambda)\varphi_2^2+(3-\lambda)\varphi_3}q_2.
\end{align*}
Applying Lemma \ref{lem-pom} once again for the coefficients $p_2$
and $q_2,$ we readily get
\[
|a_3| \leqq \frac{2(1-\beta)}{(3-\lambda)\varphi_3}.
\]
This completes the proof of Theorem \ref{Bi-th2}.
\end{proof}

Putting $\lambda =0$ in Theorem \ref{Bi-th2}, we have the
following corollary.
\begin{corollary}\label{Bi-cor3}
Let the function $f(z)$ given by $(\ref{Int-e1})$ be in the class
$\mathcal{M}^{a,b,;c}_{\Sigma}(\beta)\;\;(0 \leqq \beta < 1).$
Then
\begin{equation}\label{bi-cor3-b-a2}
|a_2| \leqq \sqrt{\frac{2(1-\beta)}{3\varphi_3}}
\end{equation}
and
\begin{equation}\label{bi-cor3-b-a3}
|a_3| \leqq \frac{2(1-\beta)}{3\varphi_3}.
\end{equation}
\end{corollary}

\begin{remark}
For $a=c$ and $b=1,$ the bound on $|a_3|$ in Corollary
\ref{Bi-cor3} is provides an improvement over the result of
Srivastava et al. \cite[Theorem2,1191]{HMS-AKM-PG}.
\end{remark}
When $\lambda =1$ in Theorem \ref{Bi-th2}, we  
get the following corollary for the well-known
class $\mathcal{M}^{a,b;c}_{\Sigma}(\beta,1)$ of bi-starlike
functions of order $\beta$.
\begin{corollary}\label{Bi-cor4}
Let the function $f(z)$ given by $(\ref{Int-e1})$ be in the class
$\mathcal{S}^{a,b;c}_{\Sigma}(\beta)\;\;(0 \leqq \beta < 1).$ Then
\[
|a_2| \leqq  \sqrt{\frac{2-2\beta}{2\varphi_3-\varphi^2_2}}\qquad
{\it and} \qquad |a_3| \leqq \frac{1-\beta}{\varphi_3}.
\]
\end{corollary}

\begin{remark} Various other interesting corollaries and consequences of our main results (which are asserted by Theorems \ref{Bi-th1} and \ref{Bi-th2} above) can be derived similarly. The details involved may be left as an exercise for the interested reader.
\end{remark}

\begin{thebibliography}{0}

\bibitem{Bran-1979}
D.A. Brannan and J.G. Clunie (Editors), {\it Aspects of Contemporary Complex Analysis}, Academic Press, London, 1980.

\bibitem{Branna1970}
D. A. Brannan, J. Clunie and W. E. Kirwan, Coefficient estimates for a class of star-like functions, {\it Canad. J. Math.} {\bf 22} (1970), 476--485.

\bibitem{Bran1985}
D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, {\it Studia Univ. Babe\c s-Bolyai Math.} {\bf 31} (2) (1986), 70--77.

\bibitem{dziok2}
J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, {\it Appl. Math. Comput.} {\bf 103} (1999), 1--13.

\bibitem{dziok1}
J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, {\it Intergral Transforms Spec. Funct.} {\bf 14} (2003), 7--18.

\bibitem{BAF-MKA}
B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, {\it Appl. Math. Lett.} {\bf 24} (2011), 1569--1573.

\bibitem{haya}
T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, {\it Pan Amer. Math. J.} {\bf 22} (4) (2012), 15--26.

\bibitem{HO1}
Yu. E. Hohlov, Convolution operators that preserve univalent functions, {\it Ukrain. Mat. Zh.} {\bf 37} (1985), 220--226.

\bibitem{HO2}
Yu. E. Hohlov, Hadamard convolutions, hypergeometric functions and linear operators in the class of univalent functions, {\it Dokl. Akad. Nauk Ukrain. SSR Ser. A} {\bf 1984} (7) (1984), 25--27.

\bibitem{Lewin}
M. Lewin, On a coefficient problem for bi-univalent functions, {\it Proc. Amer. Math. Soc.} {\bf 18} (1967), 63--68.

\bibitem{Li-Wang}
X.-F. Li and A.-P. Wang, Two new subclasses of bi-univalent functions, {\it Internat. Math. Forum} {\bf 7} (2012), 1495--1504.

\bibitem{Netany}
E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $z<1$, {\it Arch. Rational Mech. Anal.} {\bf 32} (1969), 100--112.

\bibitem{Pom}
C. Pommerenke, {\it Univalent Functions}, Vandenhoeck \&\ Ruprecht, G\"ottingen, 1975.

\bibitem{HMS-AKM-PG}
H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, {\it Appl. Math. Lett.} {\bf 23} (2010), 1188--1192.

\bibitem{Taha1981}
T. S. Taha, {\it Topics in Univalent Function Theory}, Ph.D. Thesis,
University of London, 1981.

\bibitem{Xu-HMS-AML}
Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, {\it Appl. Math. Lett.} {\bf 25} (2012), 990--994.

\bibitem{Xu-HMS-AMC}
Q.-H. Xu, H.-G. Xiao and H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, {\it Appl. Math. Comput.} {\bf 218} (2012), 11461--11465.

\end{thebibliography}
\end{document}