Let \(\mathcal{J}\) denote the subclass of \(\mathcal{G}\) which is of the form
\begin{equation}
\label{main}
\Im(z)=z+\sum_{k=2}^{\infty}l_kz^k
\end{equation}
(1)
consisting of functionas which are holomorphic and univalent in the unit disk \(\bigtriangledown\).
Let \(\Im^{-1}\) be inverse of the function \(\Im(z)\), then we have
\[\Im^{-1}(\Im(z))=z\]
and
\[\Im^{-1}(\Im(b))=b,\quad |b| < r_0(\Im);r_0(\Im)\ge\frac{1}{4}\]
where
\begin{equation}
\Im^{-1}(\Im(b))=b-l_2b^2+(2l_2^2-l_3)b^3-(5l_2^3-5l_2l_3+l_4)b^4+\cdots .
\end{equation}
(2)
A function \(\Im(z)\in \mathcal{G}\) denoted by \(\mathcal{H}\) is said to be bi-univalent in \(\bigtriangledown\) if both \(\Im(z)\) and \(\Im^{-1}(z)\) are univalent in \(\bigtriangleup\) [
1]. Subclasses of \(\mathcal{H}\), such as class of bi-convex and starlike functions and bi-strongly convex and starlike function similar to the well known subclasses \(\mathcal{L}^*(\vartheta)\) and \(\mathcal{K}(\vartheta)\) of starlike and convex functions of order \(\vartheta(0< \vartheta< 1)\) respectively [
2].
Recently, numerous researchers [1,3,4] obtained the coefficient \(|l_2|\) and \(|l_3|\) of bi-univalent functions for the several subclasses of functions in the class \(\mathcal{H}\). Motivated by the work of Darus and Singh [5], we introduce the subclasses \(\mathcal{M}_\mathcal{H}^{\mu,\beta}(m,\psi,k,\tau)\) and \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(m,\xi,k,\tau)\) of the function class \(\mathcal{H}\), which are associated with the Opoola differential operator and to obtain estimates on the coefficients \(|l_2|\) and \(|l_3|\) for functions in these new subclasses of the function class \(\mathcal{H}\) applying the techniques used earlier by Darus and Singh [5], Frasin and Aouf [4] and Srivastava et al., [1].
Lemma 1. [6]
Suppose \(u(z)\in \mathcal{P}\) and \(z\in \bigtriangledown\), then \(|w_k|\leq2\) for each \(k\), where \(\mathcal{P}\) is the family of all function \(u\) analytic in \(\bigtriangledown\) for which \(\Re(u(z))>0\),
\[u(z)=1+w_1z+w_2z^2+\cdots .\]
Definition 1.
A function \(\Im(z)\in \mathcal{G}\) is in the class \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(m,\psi,\tau)\) if the following condition are fulfilled:
\begin{equation}
\label{eq1}
\left|\arg\left[\frac{(1-\sigma)D^{m,\mu}_{\tau,\beta}\Im(z)+\sigma D^{m+1,\mu}_{\tau,\beta}\Im(z)}{z}\right]\right|< \frac{\psi\pi}{2},
\end{equation}
(3)
\begin{equation}
\label{eq2}
\left|\arg\left[\frac{(1-\sigma)D^{m,\mu}_{\tau,\beta}h(b)+\sigma D^{m+1,\mu}_{\tau,\beta}h(b)}{b}\right]\right|< \frac{\psi\pi}{2}
\end{equation}
(4)
where \(0< \psi\leq1\), \(\sigma\ge 1\), \(\tau\ge0\), \(z\in \bigtriangleup\), \(b\in \bigtriangleup\), \(0\leq\mu\leq\beta\), \(m\in \mathcal{N}_0\) and
\begin{equation}
h(b)=b-l_2b^2+(2l_2^2-l_3)b^3-(5l_2^3-5l_2l_3+l_4)b^4+\cdots
\end{equation}
(5)
and
\begin{equation}
\label{third}
D^{m,\mu}_{\tau,\beta}\Im(z)=z+\sum_{k=2}^{\infty} (1+(k+\mu-\beta-1)\tau)^m l_kz^k
\end{equation}
(6)
where \(\quad0\leq\mu\leq\beta, \tau\ge0\) and \( m\in \mathbb{N}_0=\{0,1,2,3\cdots\}\)
is the generalized Al-oboudi derivative defined by Opoola [
7].
Remark 1.
- 1. \(\mathcal{M}_{\mathcal{H},1}^{\mu,\beta}(0,\psi,\tau)\)=\(\mathcal{M}_{\mathcal{H}}(\psi)\) which Srivastava et al., [1] presented and studied.
- 2. \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(0,\psi,\tau)\)=\(\mathcal{M}_{\mathcal{H},\sigma}(\psi)\) which Frasin and Aouf [4] presented and studied.
- 3. \(\mathcal{M}_{\mathcal{H},\sigma}^{1,1}(m,\psi,1)\)=\(\mathcal{M}_{\mathcal{H},\sigma}(m,\psi)\) which Porwal and Darus [8] presented and studied.
- 4. \(\mathcal{M}_{\mathcal{H},\sigma}^{1,1}(m,\psi,\tau)\)=\(\mathcal{M}_{\mathcal{H},\sigma}(m,\psi,\tau)\) which Darus and Singh [5] presented and studied.
2. Coefficient Bounds For The Function Class \(\mathcal{M}_\mathcal{H}^{\mu,\beta}(m,\psi,k,\tau)\)
Theorem 1.
Let \(\Im(z)\in \mathcal{G}\) be in the class \(\mathcal{M}_\mathcal{H}^{\mu,\beta}(m,\psi,k,\tau)\), \(0< \psi\leq1\), \(\sigma\ge 1\), \(\tau\ge0\), \(z\in \bigtriangleup\), \(b\in \bigtriangleup\), \(0\leq\mu\leq\beta\), \(m\in \mathcal{N}_0\) , then
\begin{equation}
\label{new}
|l_2|\leq\frac{2\psi}{\sqrt{{2\psi[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]-}{\psi(\psi-1)[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2}}}
\end{equation}
(7)
and
\begin{eqnarray}
|l_3|&\leq&
\frac{2\psi}{[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}\notag\\&&+\frac{4\psi^2}{[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}.
\end{eqnarray}
(8)
Proof.
It follows from (3) and (4) that
\begin{equation}
\label{mb}
\frac{(1-\sigma)D^{m,\mu}_{\tau,\beta}\Im(z)+\sigma D^{m+1,\mu}_{\tau,\beta}\Im(z)}{z}=(q(z))^\psi,
\end{equation}
(9)
and
\begin{equation}
\label{mb1}
\frac{(1-\sigma)D^{m,\mu}_{\tau,\beta}h(b)+\sigma D^{m+1,\mu}_{\tau,\beta}h(b)}{b}=(t(b))^\psi,
\end{equation}
(10)
where \(q(z)=1+q_1z+q_2z^2+q_3z^3+\cdots\)
and \(t(b)=1+t_1b+t_2b^2+t_3b^3\cdots\)
are in \(\mathcal{P}\).
Equating the coefficient in (9) and (10), we have
\begin{equation}
\label{mb2}
[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]l_2=\psi q_1,
\end{equation}
(11)
\begin{equation}
\label{mb3}
[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]l_3=\psi q_2+\frac{\psi(\psi-1)}{2}q_1^2,
\end{equation}
(12)
\begin{equation}
\label{mb4}
-[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]l_2=\psi t_1,
\end{equation}
(13)
\begin{equation}
\label{mb5}
[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}](2l_2^2-l_3)=\psi t_2+\frac{\psi(\psi-1)}{2}t_1^2.
\end{equation}
(14)
From (11) and (13), we get
\begin{equation}
\label{mb6}
q_1=-t_1,
\end{equation}
(15)
and
\begin{equation}
\label{mb7}
2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2l_2^2=\psi^2 (q_1^2+t_1^2).
\end{equation}
(16)
From (12),(14) and (16), we get
\begin{eqnarray*}
&&2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2l_2^2\\&&-[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2l_3=\psi t_2+\frac{\psi(\psi-1)}{2}t_1^2
\end{eqnarray*}
implies
\begin{eqnarray*}
&&2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2l_2^2\\&&=[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2l_3+\psi t_2+\frac{\psi(\psi-1)}{2}t_1^2.
\end{eqnarray*}
Then from (12), we have
\begin{multline*}
2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2l_2^2=\psi q_2+\frac{\psi(\psi-1)}{2}q_1^2+\psi t_2+\frac{\psi(\psi-1)}{2}t_1^2,
\end{multline*}
implies
\begin{multline*}
2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2l_2^2=\psi (q_2+t_2)+\frac{\psi(\psi-1)}{2}(q_1^2+t_1^2).
\end{multline*}
Then from (16), we get
\begin{eqnarray*}
&&2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2l_2^2\\&&=\psi (q_2+t_2)+\frac{\psi(\psi-1)}{2}\frac{2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2}{\psi^2}l_2^2,
\end{eqnarray*}
implies
\begin{align}\label{mb8} l_2^2=\frac{\psi^2(q_2+t_2)}{ {2\psi[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]-}{\psi(\psi-1)[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2}}.
\end{align}
(17)
Applying Lemma 1 for (17), we get
\begin{equation*} |l_2|\leq\frac{2\psi}{\sqrt{ {2\psi[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]-}{\psi(\psi-1)[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2}}}
\end{equation*}
which gives the desired estimate on \(|l_2|\) in (7).
Hence in order to find the bound on \(|l_3|\),
\begin{eqnarray*}
&&[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]l_3-[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}\\&&+\sigma(1+\tau(2+\mu-\beta))^{m+1}](2l_2^2-l_3)=\psi q_2+\frac{\psi(\psi-1)}{2}q_1^2-[\psi t_2+\frac{\psi(\psi-1)}{2}t_1^2],
\end{eqnarray*}
implies
\begin{eqnarray*}
&&2[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]l_3\\&&=[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]2l_2^2+\psi (q_2-t_2)+\frac{\psi(\psi-1)}{2}(q_1^2-t_1^2).
\end{eqnarray*}
Since \((q_1)^2=(-t_1)^2\Longrightarrow q_1^2=t_1^2\), then we have
\begin{eqnarray*}
&&2[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]l_3\\&&=\psi (q_2-t_2)+[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]2l_2^2
\end{eqnarray*}
\begin{eqnarray*}
l_3&=& \frac{\psi(q_2-t_2)}{2[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}\\&&+\frac{[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}{2[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}2l_2^2.
\end{eqnarray*}
From (16), we have
\begin{eqnarray*}
l_3&=& \frac{\psi(q_2-t_2)}{2[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}\\&&+\frac{\psi^2 (q_1^2+t_1^2)}{2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2}.
\end{eqnarray*}
Applying Lemma 1 for coefficient \(q_1,q_2,t_1\) and \(t_2\), we have
\begin{eqnarray*}
|l_3|&\leq& \frac{2\psi}{[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}\\&&+\frac{4\psi^2 }{[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2}.
\end{eqnarray*}
3. Coefficient bounds for the function class \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(m,\xi,k,\tau)\)
Definition 2.
A function \(\Im(z)\in \mathcal{G}\) is said to be in the class \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(m,\xi,k,\tau)\) if the following condition are fulfilled:
\begin{equation}
\label{eq3}
\Re\left[\frac{(1-\sigma)D^{m,\mu}_{\tau,\beta}\Im(z)+\sigma D^{m+1,\mu}_{\tau,\beta}\Im(z)}{z}\right]>\xi,
\end{equation}
(18)
\begin{equation}
\label{eq4}
\Re\left[\frac{(1-\sigma)D^{m,\mu}_{\tau,\beta}h(b)+\sigma D^{m+1,\mu}_{\tau,\beta}h(b)}{b}\right]>\xi,
\end{equation}
(19)
where \(\Im(z)\in \mathcal{H}\), \(0\leq\xi< 1\), \(\sigma\ge 1\), \(\tau\ge0\), \(z\in \bigtriangleup\), \(b\in \bigtriangleup\), \(0\leq\mu\leq\beta\), \(m\in \mathcal{N}_0,\) and
\begin{equation}
h(b)=b-l_2b^2+(2l_2^2-l_3)b^3-(5l_2^3-5l_2l_3+l_4)b^4+\cdots,
\end{equation}
(20)
and
\begin{equation}
D^{m,\mu}_{\tau,\beta}\Im(z)=z+\sum_{k=2}^{\infty} (1+(k+\mu-\beta-1)\tau)^m l_kz^k,
\end{equation}
(21)
where \(\quad0\leq\mu\leq\beta, \tau\ge0\) and \( m\in \mathbb{N}_0=\{0,1,2,3\cdots\}\)
is the generalized Al-oboudi derivative defined by Opoola [
7].
Remark 2.
- 1. \(\mathcal{M}_{\mathcal{H},1}^{\mu,\beta}(0,\xi,\tau)\)=\(\mathcal{M}_{\mathcal{H}}(\xi)\) which Srivastava et al., [1] presented and studied.
- 2. \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(0,\xi,\tau)\)=\(\mathcal{M}_{\mathcal{H},\sigma}(\xi)\) which Frasin and Aouf [4] presented and studied.
- 3. \(\mathcal{M}_{\mathcal{H},\sigma}^{1,1}(m,\xi,1)\)=\(\mathcal{M}_{\mathcal{H},\sigma}(m,\xi)\) which Porwal and Darus [8] presented and studied.
- 4. \(\mathcal{M}_{\mathcal{H},\sigma}^{1,1}(m,\xi,\tau)\)=\(\mathcal{M}_{\mathcal{H},\sigma}(m,\xi,\tau)\) which Darus and Singh [5] presented and studied.
Theorem 2.
Let \(\Im(z)\in \mathcal{G}\) be in the class \(\mathcal{M}_\mathcal{H}^{\mu,\beta}(m,\xi,k,\tau)\), \(0\leq\xi< 1\), \(\sigma\ge 1\), \(\tau\ge0\), \(z\in \bigtriangleup\), \(b\in \bigtriangleup\), \(0\leq\mu\leq\beta\), \(m\in \mathcal{N}_0\), then
\begin{equation}
\label{tb1}
|l_2|\leq \sqrt{\frac{2(1-\xi)}{[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}},
\end{equation}
(22)
and
\begin{eqnarray}\label{tb2}
|l_3|&\leq& \frac{4(1-\xi)^2}{[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]^2}\notag\\&&+\frac{2(1-\xi)}{[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}.
\end{eqnarray}
(23)
Proof.
From (18) and (19), where \(q(z),t(z)\in \mathcal{P}\),
\begin{equation}
\label{tb3}
\frac{(1-\sigma)D^{m,\mu}_{\tau,\beta}\Im(z)+\sigma D^{m+1,\mu}_{\tau,\beta}\Im(z)}{z}=\xi+(1-\xi)q(z),
\end{equation}
(24)
and
\begin{equation}
\label{tb4}
\frac{(1-\sigma)D^{m,\mu}_{\tau,\beta}h(b)+\sigma D^{m+1,\mu}_{\tau,\beta}h(b)}{b}=\xi+(1-\xi)t(b),
\end{equation}
(25)
where \(q(z)=1+q_1z+q_2z^2+q_3z^3+\cdots\)
and \(t(b)=1+t_1b+t_2b^2+t_3b^3\cdots\).
Now on equating the coefficient in (24) and (25), we have
\begin{equation}
\label{tb5}
[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]l_2=(1-\xi) q_1,
\end{equation}
(26)
\begin{equation}
\label{tb6}
[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]l_3=(1-\xi) q_2,
\end{equation}
(27)
\begin{equation}
\label{tb7}
-[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]l_2=(1-\xi) t_1,
\end{equation}
(28)
\begin{equation}
\label{tb8}
[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}](2l_2^2-l_3)=(1-\xi) t_2.
\end{equation}
(29)
From (26) and (28), we have
\begin{equation}
\label{tb88}
q_1=-t_1,
\end{equation}
(30)
and
\begin{equation}
\label{tb9}
2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2l_2^2=(1-\xi)^2 (q_1^2+t_1^2).
\end{equation}
(31)
From (27) and (29), we have
\begin{equation}
\label{mb10}
2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]l_2^2=(1-\xi) (q_2+t_2),
\end{equation}
(32)
or we have
\begin{equation*}\label{mb11}
l_2^2=\frac{(1-\xi) (q_2+t_2)}{2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]}
\end{equation*}
implies
\begin{equation*}
|l_2^2|\leq\frac{2(1-\xi) }{[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]}
\end{equation*}
which is the bound on \(|l_2|\) as given in (22). Hence in order to find the bound on \(|l_3|\), we subtract (27) and (29) and get
\begin{eqnarray*}
&&[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]l_3\\&&-[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}](2l_2^2-l_3)=(1-\xi) q_2-[(1-\xi) t_2],
\end{eqnarray*}
implies
\begin{eqnarray*}
&&2 [(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]l_3\\&&=[(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]2l_2^2+(1-\xi) (q_2-t_2),
\end{eqnarray*}
implies
\begin{equation*}
l_3=l_2^2+\frac{(1-\xi) (q_2-t_2)}{2 [(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}.
\end{equation*}
Then from (31), we have
\begin{eqnarray*}
l_3&=&\frac{(1-\xi)^2 (q_1^2+t_1^2)}{2[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2}\notag\\&&+\frac{(1-\xi) (q_2-t_2)}{2 [(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]}.
\end{eqnarray*}
Applying Lemma 1 for the coefficient \(q_1,q_2,t_1\) and \(t_2\), we get
\begin{eqnarray*}
|l_3|&\leq&\frac{4(1-\xi)^2 }{[(1-\sigma)(1+\tau(1+\mu-\beta))^{m}+\sigma(1+\tau(1+\mu-\beta))^{m+1}]^2}\notag\\&&+\frac{2(1-\xi) }{ [(1-\sigma)(1+\tau(2+\mu-\beta))^{m}+\sigma(1+\tau(2+\mu-\beta))^{m+1}]},
\end{eqnarray*}
which is the bond on \(|l_3|\) in (23).
4. Conclusion
In this present paper, two new subclasses of bi-univalent functions associated with Opoola differential operator \(D^{m,\mu}_{\tau,\beta}\) were introduced and worked on. Furthermore, the coefficient bounds for \(|l_2|\) and \(|l_3|\) of functions in these classes are obtained.
Conflict of Interests
The author declares no conflict of interest.
