On some new subclass of bi-univalent functions associated with the Opoola differential operator

Let \(\mathcal{J}\) denote the subclass of \(\mathcal{G}\) which is of the form

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 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:

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 and 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

and

Proof. It follows from (3) and (4) that

and 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 From (11) and (13), we get and 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 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:

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 and 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

and

Proof. From (18) and (19), where \(q(z),t(z)\in \mathcal{P}\),

and 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 From (26) and (28), we have and From (27) and (29), we have 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.