1. Introduction
Let \(\mathcal{A\ }\) be the class of all analytic functions \(f\) in the open unit disk \(\Delta =\{z\in \mathbb{C} :\left\vert z\right\vert < 1\}\) and normalized by the conditions \(f(0)=0\) and \(f^{\prime }(0)=1.\) Also, by \(\wp\) we shall denote the subclass of all functions in \(\mathcal{A}\) which are univalent in \(\Delta.\) Let \(P\) denote the class of functions \(p(z)\) of the form \begin{equation*} p(z)=1+\sum\limits_{n=1}^{\infty }c_{_{n}}z^{n} \end{equation*} which are analytic in \(\Delta\) such that \begin{equation*} p(0)=1\text{and Re}\left\{ p(z)\right\} >0\ \ \ \left( z\in \Delta \right) . \end{equation*} If the functions \(f\) and \(g\) are analytic in \(\Delta ,\) then \(f\) is said to be subordinate to \(g,\) written \(f(z)\prec g(z),\) provided there is an analytic function \(w(z)\) defined on \(\Delta\) with \(w(0)=0\) and \(\left\vert w(z)\right\vert < 1\) so that \(f(z)=g(w(z)).\) Furthermore , if the function \(g(z)\) is univalent in \(\mathbb{\triangle },\) then we have the following equivalence (see for details, [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12]): \begin{equation*} f(z)\prec g(z)\Leftrightarrow f(0)=g(0)\ \textrm{and}\ f(\mathbb{\triangle })\subset g(\mathbb{\triangle }). \end{equation*} Some of the important and well-investigated subclasses of the univalent function class \(\wp\) include (for example) the class \(S(\alpha )\) of starlike functions of order \(\alpha\) in \(\Delta\) and the class \( C(\alpha )\) of convex functions of order \(\alpha\) in \(\Delta\). By definition, we have
\begin{equation} S(\alpha )=\left\{ f:f\in \wp \ \ \textrm{and}\ \ \textrm{Re}\frac{zf^{\prime }(z)}{ f(z)}>\alpha \ \ \ \ (z\in \Delta ,\ 0\leq \alpha < 1)\right\} \label{1.a} \end{equation}
(1)
and
\begin{equation} C(\alpha )=\left\{ f:f\in \wp \ \ \textrm{and}\ \ \textrm{Re}\left( 1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\right) >\alpha \ \ \ \ (z\in \Delta ,\ 0\leq \alpha < 1)\right\} . \label{1.b} \end{equation}
(2)
It readily follows from the definitions (1) and (2) that
\begin{equation} f(z)\in C(\alpha )\iff zf^{\prime }(z)\in S(\alpha ). \label{1.c} \end{equation}
(3)
It is well known that for each \(f\in \wp ,\) the koebe one-quarter theorem [
13] ensures the image of \(\Delta\) under \(f\) contains a disk of radius \(1/4.\) Thus every univalent function \(f\in \wp \) has an inverse \(f^{-1}\) which satisfies \begin{equation*} f^{-1}(f(z))=z\ (\left\vert z\right\vert < 1) \end{equation*} and \begin{equation*} f(f^{-1}(w))=w,\ \ \ (\left\vert w\right\vert < r_{0}(f),\ r_{0}(f)\geq 1/4). \end{equation*} In fact, the inverse function \(g=f^{-1}\) is defined by \begin{equation*} g(w)=f^{-1}(w)=w-a_{2}w^{2}+(2a_{2}^{2}-a_{3})w^{3}-(5a_{2}^{2}-5a_{2}a_{3}+a_{4})w^{4}+…. \end{equation*} A function \(f\in \mathcal{A}\) is said to bi-univalent in \(\Delta\) if both \(f\) and \(f^{-1}\) are univalent in \(\Delta.\) Let \(\sigma\) denote the class of bi-univalent functions defined in the unit disk \(\Delta\) and let \( \phi \in P\) and \(\phi (\Delta )\) is symmetric with respect to the the real axis, such a function has a Taylor series of the form:
\begin{equation} \phi (z)=1+B_{1}z+B_{2}z^{2}+B_{3}z^{3}+…\left( B_{1}>0\right) . \label{2.1} \end{equation}
(4)
In [
14], the authors introduced the class \(S(\phi)\) of the so-called Ma and Minda starlike functions and the class \(C(\phi )\) of Ma and Minda convex functions, unifying several previously studied classes related to those of starlike and convex functions. The class \(S(\phi)\) consists of all the functions \(f\in \mathcal{A}\) satisfying subordination \(\dfrac{zf^{\prime }(z)}{f(z)}\prec \phi (z),\) whereas \(C(\phi )\) is formed with functions \(f\in \mathcal{A}\) for which the subordination \(1+\) \(\dfrac{ zf^{\prime \prime }(z)}{f^{\prime }(z)}\prec \phi (z)\) holds. Lewin [
15] investigated the class \(\sigma\) and showed that \(\left\vert a_{2}\right\vert < 1.51\) for function \(f(z)=z+\sum\limits_{n=2}^{\infty }a_{_{n}}z^{n}\in \sigma\). Subsequently, Brannan and Clunie [
16] conjectured that \(\left\vert a_{2}\right\vert < \sqrt{2}.\) Netanyahu [
17], on the other hand, showed that max \(\left\vert a_{2}\right\vert =4/3\) if \(f(z)\in \sigma .\) Brannan and Taha [
18] and Taha[
19] introduced certain subclasses of bi-univalent functions, similar to the familiar subclasses of univalent functions consisting of strongly starlike and convex functions, they introduced bi-starlike functions and bi-convex functions and found non-sharp estimates on the first two Taylor-Maclaurin coefficients \(\left\vert a_{2}\right\vert\) and \(\left\vert a_{3}\right\vert .\) Recently, many authors investigated bounds for various subclasses of bi-univalent functions (see [
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33]). In [
34], Mitrinovic essentially investigated certain geometric properties of functions \(\psi\) of the form
\begin{equation} \psi (z)=\frac{z}{f(z)},\ \ \ f(z)=1+\sum\limits_{n=1}^{\infty }a_{_{n}}z^{n}. \label{1.1} \end{equation}
(5)
In [35], Reade et al. derived coefficient conditions that guarantee the univalence, starlikeness or convexity of rational functions of the form (5), these results have been improved and generalized in [36]. In this paper, estimates on the initial coefficients for bi-starlike of Ma-Minda type and bi-convex of Ma-Minda type of rational form (5) are obtained. Several related classes are also considered.
In order to derive our main results, we require the following lemma.
Lemma 1.1. (see 37) If \(p(z)\in P\), then
\begin{equation} \left\vert c_{n}\right\vert \leq 2\ \ \ \ \left( n\in \mathbb{N} =\left\{ 1,2,…\right\} \right) . \label{1.2} \end{equation}
(6)
2. Coefficients estimates
A function \(\psi (z)\in \mathcal{A}\) with Re \(\left( \psi ^{\prime }(z)\right) >0\) is known to be univalent. This motivates the following class of functions.
Definition 2.1. A function \(\psi \in \sigma\) given by (5) is said to be in the class \(\mathcal{H}_{\sigma }(\phi )\) if the following conditions are satisfied: \begin{equation*} \psi ^{\prime }(z)\prec \phi (z)\left( z\in \Delta \right) \ \text{and} \ g^{\prime }(w)\prec \phi (w)\left( w\in \Delta \right) ,\ \ \ \end{equation*} where \(g(w):=\psi ^{-1}(w).\)
If we set \begin{equation*} \phi (z)=\left( \frac{1+z}{1-z}\right) ^{\gamma }=1+2\gamma z+2\gamma ^{2}z^{2}+…\left( 0< \gamma \leq 1,\ z\in \Delta \right) \end{equation*} in Definition 2.1 of the bi-univalent function class \( \mathcal{H}_{\sigma }(\phi )\) we obtain a new class \(\mathcal{H}_{\sigma }(\gamma )\) given by Definition 2.2 below.
Definition 2.2. For \(0< \gamma \leq 1,\) a function \(\psi \in \sigma\) given by (5) is said to be in the class \(\mathcal{H}_{\sigma }(\gamma )\) if the following conditions are satisfied: \[\psi^{\prime }(z)\prec \left( \frac{1+z}{1-z}\right)^{\gamma }\left( z\in \Delta \right)\] and \[g^{\prime }(w)\prec \left( \frac{1+w}{1-w} \right) ^{\gamma }\left( w\in \Delta \right) , \] where \(g(w):=\psi ^{-1}(w).\)
If we set \begin{equation*} \phi (z)=\frac{1+(1-2\nu )z}{1-z}=1+2(1-\nu )z+2(1-\nu )z^{2}+…\left( 0< \nu \leq 1,\ z\in \Delta \right) \end{equation*} in Definition 2.1 of the bi-univalent function class \( \mathcal{H}_{\sigma }(\phi )\) we obtain, a new class \(\mathcal{H}_{\sigma }(\nu )\) given by Definition 2.3 below.
Definition 2.3. For \(0< \nu \leq 1,\) a function \(\psi \in \sigma\) given by (5) is said to be in the class \(\mathcal{H}_{\sigma }(\nu )\) if the following conditions hold true: \begin{equation*} \psi ^{\prime }(z)\prec \frac{1+(1-2\nu )z}{1-z}\left( z\in \Delta \right) \ \text{and} \ g^{\prime }(w)\prec \frac{1+(1-2\nu )w}{1-w}\left( w\in \Delta \right) , \end{equation*} where \(g(w):=\psi ^{-1}(w).\)
Theorem 2.4. Let \(\psi (z)\in \mathcal{H}_{\sigma }(\phi )\) be of the form (5). Then
\begin{equation} \left\vert a_{1}\right\vert \leq \frac{B_{1}\sqrt{B_{1}}}{\sqrt{\left\vert 3B_{1}^{2}-4B_{2}+4B_{1}\right\vert }}\ \ \ \text{and }\left\vert a_{2}\right\vert \leq \frac{1}{3}B_{1}. \label{2.2} \end{equation}
(7)
Proof. Let \(\psi (z)\in \mathcal{H}_{\sigma }(\phi )\) and \(g=\psi ^{-1}.\) Then there exist two functions \(u\) and \(v,\) analytic in \(\Delta,\) with \noindent \(u(0)=v(0)=0,\ \ \left\vert u(z)\right\vert < 1\) and \( \left\vert v(w)\right\vert < 1,\ z,w\in \Delta ,\) such that
\begin{equation} \psi ^{\prime }(z)=\phi (u(z)) \ \text{and} \ g^{\prime }(w)=\phi (v(w)). \label{2.3} \end{equation}
(8)
Next, define the functions \(p_{1}\) and \(p_{2}\) by \begin{equation*} p_{1}(z)=\frac{1+u(z)}{1-u(z)}=1+c_{1}z+c_{2}z^{2}+…\ \text{and}\ p_{2}(w)=\frac{1+v(w)}{1-v(w)}=1+b_{1}w+b_{2}^{2}w^{2}+…, \end{equation*} or, equivalently,
\begin{equation} u(z)=\frac{p_{1}(z)-1}{p_{1}(z)+1}=\frac{1}{2}\left[ c_{1}z+\left( c_{2}- \frac{c_{1}^{2}}{2}\right) z^{2}+…\right] , \label{2.4} \end{equation}
(9)
and
\begin{equation} v(w)=\frac{p_{2}(w)-1}{p_{2}(w)+1}=\frac{1}{2}\left[ b_{1}w+\left( b_{2}- \frac{b_{1}^{2}}{2}\right) w^{2}+…\right] . \label{2.5} \end{equation}
(7)
Then \(p_{1}\) and \(p_{2}\) analytic in \(\Delta\) with \( p_{1}(0)=1=p_{2}(0).\) Since \(u,v:\Delta \longrightarrow \Delta ,\) the functions \(p_{1}\) and \(p_{2}\) have a positive real part in \(\Delta,\) and \( \left\vert b_{i}\right\vert \leq 2\) and \(\left\vert c_{i}\right\vert \leq 2.\) Clearly, upon substituting from (9) and (10) into (8), if we make use of (4), we find that
\begin{equation} \psi ^{\prime }(z)=\phi (\frac{p_{1}(z)-1}{p_{1}(z)+1})=1+\frac{1}{2} B_{1}c_{1}z+\left[ \frac{1}{2}B_{1}\left( c_{2}-\frac{c_{1}^{2}}{2}\right) + \frac{1}{4}B_{2}c_{1}^{2}\right] z^{2}+…, \label{2.6} \end{equation}
(11)
and
\begin{equation} g^{\prime }(w)=\phi (\frac{p_{2}(w)-1}{p_{2}(w)+1})=1+\frac{1}{2} B_{1}b_{1}w+\left[ \frac{1}{2}B_{1}\left( b_{2}-\frac{b_{1}^{2}}{2}\right) + \frac{1}{4}B_{2}b_{1}^{2}\right] w^{2}+…\ .. \label{2.61} \end{equation}
(12)
Since \(\psi \in \sigma \) has the Maclaurin’s series given by
\begin{equation} \psi (z)=z-a_{1}z^{2}+(a_{1}^{2}-a_{2})z^{3}+…, \label{2.9} \end{equation}
(13)
a computation shows that its inverse \(g=\psi ^{-1}\) has the expansion
\begin{equation} g(w)=\psi ^{-1}(w)=w+a_{1}w^{2}+(a_{1}^{2}+a_{2})w^{3}+…\ . \label{2.91} \end{equation}
(14)
Using (13) and (14) in (11) and (12) respectively, we get
\begin{equation} -2a_{1}=\frac{1}{2}B_{1}c_{1} \label{2.10} \end{equation}
(15)
\begin{equation} 3(a_{1}^{2}-a_{2})=\frac{1}{2}B_{1}(c_{2}-\frac{c_{1}^{2}}{2})+\frac{1}{4} B_{2}c_{1}^{2}, \label{2.11} \end{equation}
(16)
\begin{equation} 2a_{1}=\frac{1}{2}B_{1}b_{1} \label{2.12} \end{equation}
(17)
and
\begin{equation} 3(a_{1}^{2}+a_{2})=\frac{1}{2}B_{1}(b_{2}-\frac{b_{1}^{2}}{2})+\frac{1}{4} B_{2}b_{1}^{2}. \label{2.13} \end{equation}
(18)
From (15) and (17), we have
\begin{equation} c_{1}=-b_{1}. \label{2.14} \end{equation}
(19)
Adding (16) and (18) and then using (15) and (19), we get \begin{equation*} a_{1}^{2}=\frac{B_{1}^{3}(c_{2}+b_{2})}{4(3B_{1}^{2}-4B_{2}+4B_{1})}, \end{equation*} and now, by applying Lemma 1.1 for the coefficients \(b_{2}\) and \(c_{2},\) the last equation gives the bound of \(\left\vert a_{1}\right\vert\) from (7). By subtracting (18) from (16), further computations using (19) lead to \begin{equation*} a_{2}=\frac{1}{12}B_{1}(b_{2}-c_{2}). \end{equation*} The bound of \(\left\vert a_{2}\right\vert ,\) as asserted in (7), is now a consequence of Lemma 1.1, and this completes our proof. Using the parameter setting of Definition 2.2 in Theorem 2.4, we get the following corollary.
Corollary 2.5. For \(0< \gamma \leq 1,\) let the function \(\psi \in \mathcal{H} _{\sigma }(\gamma )\) be of the form (5). Then \begin{equation*} \left\vert a_{1}\right\vert \leq \frac{\sqrt{2}\gamma }{\sqrt{\gamma +2}}\ \ \ \text{and}\ \left\vert a_{2}\right\vert \leq \frac{2}{3}\gamma . \end{equation*}
Using the parameter setting of Definition 2.3 in Theorem 2.1, we get the following corollary.
Corollary 2.6. For \(0< \nu \leq 1,\) let the function \(\psi \in \mathcal{H} _{\sigma }(\nu )\) be given by (5). Then \begin{equation*} \left\vert a_{1}\right\vert \leq \sqrt{\frac{2}{3}\left( 1-\nu \right) }\ \ \ \text{and }\left\vert a_{2}\right\vert \leq \frac{2}{3}\left( 1-\nu \right) . \end{equation*}
Definition 2.7. A function \(\psi \in \sigma\) is given by (5) is said to be in the class \(S_{\sigma }(\alpha ,\phi )\) if the following subordinations hold: \begin{equation*} \frac{z\psi ^{\prime }(z)}{\psi (z)}+\frac{\alpha z^{2}\psi ^{\prime \prime }(z)}{\psi (z)}\prec \phi (z)\left( z\in \Delta \right) \ \ \ \text{and } \frac{wg^{\prime }(w)}{g(w)}+\frac{\alpha w^{2}g^{\prime \prime }(w)}{g(w)} \prec \phi (w)\left( w\in \Delta \right) , \end{equation*} where \(g(w):=\psi ^{-1}(w).\)
If we set \begin{equation*} \phi (z)=\left( \frac{1+z}{1-z}\right) ^{\gamma }=1+2\gamma z+2 \gamma ^{2}z^{2}+…\left(0< \gamma \leq 1,\ z\in \Delta \right) \end{equation*} in Definition 2.4 of the bi-univalent function class \( S_{\sigma }(\alpha ,\phi ),\) we obtain a new class \(S_{\sigma }(\alpha ,\gamma )\) given by Definition 2.5 below.
Definition 2.8. For \(0\leq \alpha \leq 1\) and \(0< \gamma \leq 1,\) a function \( \psi \in \sigma \) given by (5) is said to be in the class \( S_{\sigma }(\alpha ,\gamma )\) if the following subordinations hold: \begin{equation*} \frac{z\psi ^{\prime }(z)}{\psi (z)}+\frac{\alpha z^{2}\psi ^{\prime \prime }(z)}{\psi (z)}\prec \left( \frac{1+z}{1-z}\right) ^{\gamma }\left( z\in \Delta \right) , \end{equation*} and \begin{equation*} \frac{wg^{\prime }(w)}{g(w)}+\frac{\alpha w^{2}g^{\prime \prime }(w)}{g(w)} \prec \left( \frac{1+w}{1-w}\right) ^{\gamma }\left( w\in \Delta \right) , \end{equation*} where \(g(w):=\psi ^{-1}(w).\)
If we set \begin{equation*} \phi (z)=\frac{1+(1-2\nu )z}{1-z}=1+2(1-\nu )z+2(1-\nu )z^{2}+…\left( 0< \nu \leq 1,\ z\in \Delta \right) \end{equation*} in Definition 2.4 of the bi-univalent function class \( S_{\sigma }(\alpha ,\phi )\) we obtain a new class \(S_{\sigma }(\alpha ,\nu )\) given by Definition 2.6 below.
Definition 2.9 For \(0\leq \alpha \leq 1\) and \(0< \nu \leq 1,\) a function \( \psi \in \sigma \) given by (5) is said to be in the class \( S_{\sigma }(\alpha ,\nu )\) if the following subordinations hold: \begin{equation*} \frac{z\psi ^{\prime }(z)}{\psi (z)}+\frac{\alpha z^{2}\psi ^{\prime \prime }(z)}{\psi (z)}\prec \frac{1+(1-2\nu )z}{1-z}\left( z\in \Delta \right) \end{equation*} and \begin{equation*} \frac{wg^{\prime }(w)}{g(w)}+\frac{\alpha w^{2}g^{\prime \prime }(w)}{g(w)} \prec \frac{1+(1-2\nu )w}{1-w}\left( w\in \Delta \right) , \end{equation*} where \(g(w)=\psi ^{-1}(w).\)
Note that \(S(\phi )=S_{\sigma }(0,\phi ).\) For functions in the class \(S_{\sigma }(\alpha ,\phi ),\) the following coefficient estimates are obtained,
Theorem 2.10 Let \(\psi (z)\in S_{\sigma }(\alpha ,\phi )\) be of the form (5). Then
\begin{equation} \left\vert a_{1}\right\vert \leq \frac{B_{1}\sqrt{B_{1}}}{\sqrt{\left\vert B_{1}^{2}(1+4\alpha )+(B_{1}-B_{2})(1+2\alpha )^{2}\right\vert }},\ \ \ \label{2.15} \end{equation}
(20)
and
\begin{equation} \left\vert a_{2}\right\vert \leq \frac{B_{1}}{1+3\alpha }. \label{2.16} \end{equation}
(21)
Proof. Let \(\psi \in S_{\sigma }(\alpha ,\phi ),\) there are two Schwarz functions \(u\) and \(v\) defined by (9) and (10) respectively, such that
\begin{equation} \frac{z\psi ^{\prime }(z)}{\psi (z)}+\frac{\alpha z^{2}\psi ^{\prime \prime }(z)}{\psi (z)}=\phi (u(z))\ \ \ \text{and }\frac{wg^{\prime }(w)}{g(w)}+ \frac{\alpha w^{2}g^{\prime \prime }(w)}{g(w)}=\phi (v(w)),\ \ \ \left( g=\psi ^{-1}\right) . \label{2.17} \end{equation}
(22)
Since \begin{equation*} \frac{z\psi ^{\prime }(z)}{\psi (z)}+\frac{\alpha z^{2}\psi ^{\prime \prime }(z)}{\psi (z)}=1-\left( 1+2\alpha \right) a_{1}z+\left[ \left( 1+4\alpha \right) a_{1}^{2}-2\left( 1+3\alpha \right) a_{2}\right] z^{2}+… \end{equation*} and \begin{equation*} \frac{wg^{\prime }(w)}{g(w)}+\frac{\alpha w^{2}g^{\prime \prime }(w)}{g(w)} =1+\left( 1+2\alpha \right) a_{1}w+\left[ \left( 1+4\alpha \right) a_{1}^{2}+2\left( 1+3\alpha \right) a_{2}\right] w^{2}+…, \end{equation*} then (11), (12) and (22) yields
\begin{equation} -(1+2\alpha )a_{1}=\frac{1}{2}B_{1}c_{1} \label{2.18} \end{equation}
(23)
\begin{equation} (1+4\alpha )a_{1}^{2}-2(1+3\alpha )a_{2}=\frac{1}{2}B_{1}(c_{2}-\frac{c_{1}^{2}}{2})+\frac{1}{4}B_{2}c_{1}^{2}, \label{2.19} \end{equation}
(24)
\begin{equation} (1+2\alpha )a_{1}=\frac{1}{2}B_{1}b_{1} \label{2.20} \end{equation}
(25)
and
\begin{equation} (1+4\alpha )a_{1}^{2}+2(1+3\alpha )a_{2}=\frac{1}{2}B_{1}(b_{2}-\frac{b_{1}^{2}}{2})+\frac{1}{4}B_{2}b_{1}^{2}. \label{2.21} \end{equation}
(26)
From (23) and (25), we get
\begin{equation} c_{1}=-b_{1}, \label{2.22} \end{equation}
(27)
and after some further calculations using (24)-(27) we find \begin{equation*} a_{1}^{2}=\frac{B_{1}^{3}(c_{2}+b_{2})}{4\left[ B_{1}^{2}(1+4\alpha )+(B_{1}-B_{2})(1+2\alpha )^{2}\right] }, \end{equation*} and \begin{equation*} a_{2}=\frac{B_{1}(b_{2}-c_{2})}{4(1+3\alpha )}. \end{equation*} Applying Lemma 1.1, the estimates in (20) and (21) follow. For \(\alpha =0,\) Theorem 2.2 readily yields the following coefficient estimates for Ma-Minda bi-starlike functions.
Corollary 2.11. Let \(\psi\) given by (7) be in the class \(S(\phi ).\) Then \begin{equation*} \left\vert a_{1}\right\vert \leq \frac{B_{1}\sqrt{B_{1}}}{\sqrt{\left\vert B_{1}^{2}+B_{1}-B_{2}\right\vert }},\ \ \ and\ \ \ \left\vert a_{2}\right\vert \leq B_{1}. \end{equation*}
Using the parameter setting of Definition 2.8 in Theorem 2.10, we get the following corollary.
Corollary 2.12. For \(0\leq \alpha \leq 1\) and \(0< \gamma \leq 1,\) let the function \(\psi \in S_{\sigma }(\alpha ,\gamma )\) be of the form (5). Then \begin{equation*} \left\vert a_{1}\right\vert \leq \frac{2\gamma }{\sqrt{\left( 1+2\alpha \right) ^{2}+\gamma \left[ 1+4\alpha -4\alpha ^{2}\right] }}\ \ \ \text{and} \ ~\ \left\vert a_{2}\right\vert \leq \frac{2\gamma }{1+3\alpha }. \end{equation*}
Using the parameter setting of Definition 2.9 in Theorem 2.10 we get the following corollary.
Corollary 2.13. For \(0\leq \alpha \leq 1\) and \(0< \nu \leq 1,\) let the function \(\psi \in S_{\sigma }(\alpha ,\nu )\) be of the form (5). Then \begin{equation*} \left\vert a_{1}\right\vert \leq \sqrt{\frac{2\left( 1-\nu \right) }{ 1+4\alpha }}\ \ \ \text{and }\left\vert a_{2}\right\vert \leq \frac{2\left( 1-\nu \right) }{1+3\alpha }. \end{equation*}
Definition 2.14. A function \(\psi \in \sigma\) given by (5) belongs to the class \(M_{\sigma }(\alpha ,\phi )\) \(\left( 0\leq \alpha \leq 1\right),\) if the following subordinations hold: \begin{equation*} (1-\alpha )\frac{z\psi ^{\prime }(z)}{\psi (z)}+\alpha (1+\frac{z\psi ^{\prime \prime }(z)}{\psi ^{\prime }(z)})\prec \phi (z)\left( z\in \Delta \right) , \end{equation*} and \begin{equation*} (1-\alpha )\frac{wg^{\prime }(w)}{g(w)}+\alpha (1+\frac{wg^{\prime \prime }(w)}{g^{\prime }(w)})\prec \phi (w),\left( w\in \Delta \right) , \end{equation*} where \(g(w):=\psi ^{-1}(w).\)
If we set \begin{equation*} \phi (z)=\left( \frac{1+z}{1-z}\right) ^{\gamma }=1+2\gamma z+2\gamma ^{2}z^{2}+…\left( 0< \gamma \leq 1,\ z\in \Delta \right) \end{equation*} in Definition 2.14 of the bi-univalent function class \( M_{\sigma }(\alpha ,\phi ),\) we obtain a new class \(M_{\sigma }(\alpha ,\gamma )\) given by Definition 2.15 below.
Definition 2.15. For \( 0\leq \alpha \leq 1\) and \(0< \gamma \leq 1,\) a function \( \psi \in \sigma \) given by (5) is said to be in the class \( M_{\sigma }(\alpha ,\gamma )\) if the following subordinations hold: \begin{equation*} (1-\alpha )\frac{z\psi ^{\prime }(z)}{\psi (z)}+\alpha (1+\frac{z\psi ^{\prime \prime }(z)}{\psi ^{\prime }(z)})\prec \left( \frac{1+z}{1-z} \right) ^{\gamma }\left( z\in \Delta \right) , \end{equation*} and \begin{equation*} (1-\alpha )\frac{wg^{\prime }(w)}{g(w)}+\alpha (1+\frac{wg^{\prime \prime }(w)}{g^{\prime }(w)})\prec \left( \frac{1+w}{1-w}\right) ^{\gamma }\left( w\in \Delta \right) , \end{equation*} \(g(w):=\psi ^{-1}(w).\)
Corollary 2.16. If we set \begin{equation*} \phi (z)=\frac{1+(1-2\nu )z}{1-z}=1+2(1-\nu )z+2(1-\nu )z^{2}+…\left( 0< \nu \leq 1,\ z\in \Delta \right) \end{equation*} in Definition 2.14 of the bi-univalent function class \( M_{\sigma }(\alpha ,\phi )\) we obtain a new class \(M_{\sigma }(\alpha ,\nu)\) given by Definition 2.17 below.
Definition 2.17. For \(0\leq \alpha \leq 1\) and \(0< \nu \leq 1,\) a function \( \psi \in \sigma \) given by (5) is said to be in the class \( M_{\sigma }(\alpha ,\nu )\) if the following subordinations hold: \begin{equation*} (1-\alpha )\frac{z\psi ^{\prime }(z)}{\psi (z)}+\alpha (1+\frac{z\psi ^{\prime \prime }(z)}{\psi ^{\prime }(z)})\prec \frac{1+(1-2\nu )z}{1-z} \left( z\in \Delta \right) , \end{equation*}
and \begin{equation*} (1-\alpha )\frac{w\psi ^{\prime }(w)}{\psi (w)}+\alpha (1+\frac{w\psi ^{\prime \prime }(w)}{\psi ^{\prime }(w)})\prec \frac{1+(1-2\nu )w}{1-w} \left( w\in \Delta \right) , \end{equation*} where \(g(w):=\psi ^{-1}(w).\) A function in the class \(M_{\sigma }(\alpha ,\phi )\) is called bi-Mocanu-convex function of Ma-Minda type. This class unifies the classes \( S(\alpha )\) and \(C(\alpha ).\) For functions in the class \(M_{\sigma }(\alpha ,\phi ),\) the following coefficients estimates hold.
Theorem 2.18 Let \(\psi (z)\in M_{\sigma }(\alpha ,\phi )\) be of the form (5). Then
\begin{equation} \left\vert a_{1}\right\vert \leq \frac{B_{1}\sqrt{B_{1}}}{\sqrt{(1+\alpha )\left\vert B_{1}^{2}+(1+\alpha )(B_{1}-B_{2})\right\vert }}, \label{2.23} \end{equation}
(28)
and
\begin{equation} \left\vert a_{2}\right\vert \leq \frac{B_{1}}{2(1+2\alpha )}. \label{2.24} \end{equation}
(29)
Proof. If \(\psi \in M_{\sigma }(\alpha ,\phi ),\) then there exist are two Schwarz functions \(u\) and \(v\) defined by (9) and (10) respectively, such that
\begin{equation} (1-\alpha )\frac{z\psi ^{\prime }(z)}{\psi (z)}+\alpha (1+\frac{z\psi ^{\prime \prime }(z)}{\psi ^{\prime }(z)})=\phi (u(z)), \label{2.25} \end{equation}
(30)
and
\begin{equation} (1-\alpha )\frac{wg^{\prime }(w)}{g(w)}+\alpha (1+\frac{wg^{\prime \prime }(w)}{g^{\prime }(w)})=\phi (v(w)). \label{2.26} \end{equation}
(31)
Since \begin{equation*} (1-\alpha )\frac{z\psi ^{\prime }(z)}{\psi (z)}+\alpha (1+\frac{z\psi ^{\prime \prime }(z)}{\psi ^{\prime }(z)})=1-\left( 1+\alpha \right) a_{1}z+ \left[ \left( 1+\alpha \right) a_{1}^{2}-2\left( 1+2\alpha \right) a_{2} \right] z^{2}+… \end{equation*} and \begin{equation*} (1-\alpha )\frac{wg^{\prime }(w)}{g(w)}+\alpha (1+\frac{wg^{\prime \prime }(w)}{g^{\prime }(w)})=1+\left( 1+\alpha \right) a_{1}w+\left[ \left( 1+\alpha \right) a_{1}^{2}+2\left( 1+2\alpha \right) a_{2}\right] w^{2}+…, \end{equation*} from (11), (12), (30) and (31), it follows that
\begin{equation} -(1+\alpha )a_{1}=\frac{1}{2}B_{1}c_{1}, \label{2.27} \end{equation}
(32)
\begin{equation} (1+\alpha )a_{1}^{2}-2(1+2\alpha )a_{2}=\frac{1}{2}B_{1}(c_{2}-\frac{ c_{1}^{2}}{2})+\frac{1}{4}B_{2}c_{1}^{2}, \label{2.28} \end{equation}
(33)
\begin{equation} (1+\alpha )a_{1}=\frac{1}{2}B_{1}b_{1}, \label{2.29} \end{equation}
(34)
and
\begin{equation} (1+\alpha )a_{1}^{2}+2(1+2\alpha )a_{2}=\frac{1}{2}B_{1}(b_{2}-\frac{ b_{1}^{2}}{2})+\frac{1}{4}B_{2}b_{1}^{2}, \label{2.30} \end{equation}
(35)
Equations (32) and (34) yields
\begin{equation} c_{1}=-b_{1}, \label{2.31} \end{equation}
(36)
and after some further calculations using (33)-(35) we find \begin{equation*} a_{1}^{2}=\frac{B_{1}^{3}(c_{2}+b_{2})}{4(1+\alpha )\left[ B_{1}^{2}+(1+\alpha )(B_{1}-B_{2})\right] }, \end{equation*} and \begin{equation*} a_{2}=\frac{B_{1}\left( b_{2}-c_{2}\right) }{8(1+2\alpha )}, \end{equation*} Applying Lemma 1.1, the estimates in (28) and (29) follow. For \(\alpha =0,\) Theorem 2.18 gives the coefficient estimates for Ma-Minda bi-starlike functions, while for \(\alpha =1,\) it gives the following estimates for Ma-Minda bi-convex functions.
Corollary 2.19 Let \(\psi \) given by (5) be in the class \(C(\phi ).\) Then \begin{equation*} \left\vert a_{1}\right\vert \leq \frac{B_{1}\sqrt{B_{1}}}{2\left\vert B_{1}^{2}+2(B_{1}-B_{2})\right\vert },\ \ \ \text{and}\ \ \ \left\vert a_{2}\right\vert \leq \frac{B_{1}}{6}. \end{equation*}
Using the parameter setting of Definition 15 in Theorem 18 we get the following corollary.
Corollary 2.20. For \(0\leq \alpha \leq 1\) and \(0< \gamma \leq 1,\) let the function \(\psi \in M_{\sigma }(\alpha ,\gamma )\) be of the form (5). Then \begin{equation*} \left\vert a_{1}\right\vert \leq \frac{2\gamma }{\sqrt{\left( 1+\alpha \right) \left[ \left( 1+\alpha \right) +\gamma \left( 1-\alpha \right) \right] }}\ \ \ \text{and \ \ }\left\vert a_{2}\right\vert \leq \frac{\gamma }{1+2\alpha }. \end{equation*}
Using the parameter setting of Definition 17 in Theorem 18 we get the following corollary.
Corollary 2.21. For \(0\leq \alpha \leq 1\) and \(0< \nu \leq 1,\) let the function \(\psi \in M_{\sigma }(\alpha ,\nu )\) be of the form (5). Then \begin{equation*} \left\vert a_{1}\right\vert \leq \sqrt{\frac{2\left( 1-\nu \right) }{ 1+\alpha }}\ \ \ \text{and }\left\vert a_{2}\right\vert \leq \frac{\left( 1-\nu \right) }{1+2\alpha }. \end{equation*}
Definition 2.22. A function \(\psi \in \sigma \) given by (5) is said to be in the class \(\Im _{\alpha }(\alpha ,\phi )\left( 0\leq \alpha \leq 1\right) ,\) if the following subordinations hold: \begin{equation*} \left( \frac{z\psi ^{\prime }(z)}{\psi (z)}\right) ^{\alpha }\left( 1+\frac{ z\psi ^{\prime \prime }(z)}{\psi ^{\prime }(z)}\right) ^{1-\alpha }\prec \phi (z)\left( z\in \Delta \right) , \end{equation*} and \begin{equation*} \left( \frac{wg^{\prime }(w)}{g(w)}\right) ^{\alpha }\left( 1+\frac{ wg^{\prime \prime }(w}{g^{\prime }(w)}\right) ^{1-\alpha }\prec \phi (w)\left( w\in \Delta \right) , \end{equation*} \(g(w):=\psi ^{-1}(w).\) This class also reduces to classes of Ma-Minda bi-starlike and bi-convex functions. For functions in this class, the following coefficient estimates are obtained.
Theorem 2.23 Let \(\psi (z)\in \Im _{\alpha }(\alpha ,\phi )\) be of the form(5). Then
\begin{equation} \left\vert a_{1}\right\vert \leq \frac{2B_{1}\sqrt{B_{1}}}{\sqrt{\left\vert 2\left( \alpha ^{2}-3\alpha +4\right) B_{1}^{2}+4(\alpha -2)^{2}(B_{1}-B_{2})\right\vert }}, \label{2.32} \end{equation}
(37)
and
\begin{equation} \left\vert a_{2}\right\vert \leq \frac{B_{1}}{2\left\vert 3-2\alpha \right\vert }. \label{2.33} \end{equation}
(38)
Proof. Let \(\psi \in \Im _{\alpha }(\alpha ,\phi ),\) then there exist are two Schwarz functions \(u\) and \(v\) defined by (9) and (10) respectively, such that
\begin{equation} \left( \frac{z\psi ^{\prime }(z)}{\psi (z)}\right) ^{\alpha }\left( 1+\frac{ z\psi ^{\prime \prime }(z)}{\psi ^{\prime }(z)}\right) ^{1-\alpha }=\phi (u(z)) \label{2.34} \end{equation}
(39)
and
\begin{equation} \left( \frac{wg^{\prime }(w)}{g(w)}\right) ^{\alpha }\left( 1+\frac{ wg^{\prime \prime }(w}{g^{\prime }(w)}\right) ^{1-\alpha }=\phi (v(w)). \label{2.35} \end{equation}
(40)
Since \begin{equation*} \left( \frac{z\psi ^{\prime }(z)}{\psi (z)}\right) ^{\alpha }\left( 1+\frac{ z\psi ^{\prime \prime }(z)}{\psi ^{\prime }(z)}\right) ^{1-\alpha }=1-\left( 2-\alpha \right) a_{1}z \end{equation*} \begin{equation*} +\left[ \frac{\alpha ^{2}-3\alpha +4}{2}a_{1}^{2}-2\left( 3-2\alpha \right) a_{2}\right] z^{2}+…\ \ . \end{equation*} Also \begin{equation*} \left( \frac{wg^{\prime }(w)}{g(w)}\right) ^{\alpha }\left( 1+\frac{ wg^{\prime \prime }(w}{g^{\prime }(w)}\right) ^{1-\alpha }=1+\left( 2-\alpha \right) a_{1}w \end{equation*} \begin{equation*} +\left[ \frac{\alpha ^{2}-3\alpha +4}{2}a_{1}^{2}+2\left( 3-2\alpha \right) a_{2}\right] w^{2}+…, \end{equation*} from (11), (12), (39) and (40), it follows that
\begin{equation} -(2-\alpha )a_{1}=\frac{1}{2}B_{1}c_{1}, \label{2.36} \end{equation}
(41)
\begin{equation} \frac{\alpha ^{2}-3\alpha +4}{2}a_{1}^{2}-2\left( 3-2\alpha \right) a_{2}= \frac{1}{2}B_{1}(c_{2}-\frac{c_{1}^{2}}{2})+\frac{1}{4}B_{2}c_{1}^{2}, \label{2.37} \end{equation}
(42)
\begin{equation} (2-\alpha )a_{1}=\frac{1}{2}B_{1}b_{1} \label{2.38} \end{equation} and
\begin{equation} \frac{\alpha ^{2}-3\alpha +4}{2}a_{1}^{2}+2\left( 3-2\alpha \right) a_{2}= \frac{1}{2}B_{1}(b_{2}-\frac{b_{1}^{2}}{2})+\frac{1}{4}B_{2}b_{1}^{2}. \label{2.39} \end{equation}
(43)
Equations (41) and (43) obviously yield
\begin{equation} c_{1}=-b_{1}. \label{2.40} \end{equation}
(44)
Eqs. (42)-(44) and (45) lead to \begin{equation*} a_{1}^{2}=\frac{B_{1}^{3}(c_{2}+b_{2})}{2\left( \alpha ^{2}-3\alpha +4\right) B_{1}^{2}+4(\alpha -2)^{2}(B_{1}-B_{2})}. \end{equation*} By applying Lemma 1.1, we get the desired estimate of \( \left\vert a_{1}\right\vert \) as asserted in (37). Proceeding similarly as in the earlier proof, using (42)-(45), it follows that \begin{equation*} a_{2}=\frac{B_{1}(b_{2}-c_{2})}{8(3-2\alpha )}, \end{equation*} which, in view of Lemma 1.1, yields the estimate (38).
Definition 2.24. A function \(\psi \in \sigma\) given by (5) is said to be in the class \(\beta _{\alpha }(\lambda ,\phi ),\ \lambda \geq 0,\) if the following subordinations hold: \begin{equation*} \left( 1-\lambda \right) \frac{\psi (z)}{z}+\lambda \psi ^{\prime }(z)\prec \phi (z)\left( z\in \Delta \right) , \end{equation*} and \begin{equation*} \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)\prec \phi (w)\left( w\in \Delta \right) , \end{equation*} where \(g(w):=\psi ^{-1}(w).\)
Theorem 2.25. Let \(\psi (z)\in \beta _{\alpha }(\lambda ,\phi ),\ \lambda \geq 0\) be of the form (5). Then
\begin{equation} \left\vert a_{1}\right\vert \leq \frac{B_{1}\sqrt{B_{1}}}{\sqrt{\left\vert \left( 1+2\lambda \right) B_{1}^{2}+(1+\lambda )^{2}(B_{1}-B_{2})\right\vert }}, \label{2.41} \end{equation}
(46)
and
\begin{equation} \left\vert a_{2}\right\vert \leq \frac{B_{1}}{1+2\lambda }. \label{2.42} \end{equation}
(47)
Proof. Let \(\psi \in \beta _{\alpha }(\lambda ,\phi ),\) then there exist are two Schwarz functions \(u\) and \(v\) defined by (9) and (10) respectively, such that
\begin{equation} \left( 1-\lambda \right) \frac{\psi (z)}{z}+\lambda \psi ^{\prime }(z)=\phi (u(z)) \label{2.43} \end{equation}
(48)
and
\begin{equation} \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)=\phi (v(w)). \label{2.44} \end{equation}
(49)
Since \begin{equation*} \left( 1-\lambda \right) \frac{\psi (z)}{z}+\lambda \psi ^{\prime }(z)=1-\left( 1+\lambda \right) a_{1}z+\left[ \left( 1+2\lambda \right) \left( a_{1}^{2}-a_{2}\right) \right] z^{2}+…, \end{equation*} and \begin{equation*} \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)=1+\left( 1+\lambda \right) a_{1}w+\left[ \left( 1+2\lambda \right) \left( a_{1}^{2}+a_{2}\right) \right] w^{2}+…, \end{equation*} from (11), (12), (48) and (49), it follows that
\begin{equation} -(1+\lambda )a_{1}=\frac{1}{2}B_{1}c_{1}, \label{2.45} \end{equation}
(50)
\begin{equation} (1+2\lambda )(a_{1}^{2}-a_{2})=\frac{1}{2}B_{1}(c_{2}-\frac{c_{1}^{2}}{2})+ \frac{1}{4}B_{2}c_{1}^{2}, \label{2.46} \end{equation}
(51)
\begin{equation} (1+\lambda )a_{1}=\frac{1}{2}B_{1}b_{1} \label{2.47} \end{equation}
(52)
and
\begin{equation} (1+2\lambda )(a_{1}^{2}+a_{2})=\frac{1}{2}B_{1}(b_{2}-\frac{b_{1}^{2}}{2})+ \frac{1}{4}B_{2}b_{1}^{2}. \label{2.48} \end{equation}
(53)
Now (50) and (52) clearly yield
\begin{equation} c_{1}=-b_{1}. \label{2.49} \end{equation}
(54)
Equations (51), (53) and (54) lead to \begin{equation*} a_{1}^{2}=\frac{B_{1}^{3}(c_{2}+b_{2})}{4\left[ \left( 1+2\lambda \right) B_{1}^{2}+\left( 1+\lambda \right) ^{2}(B_{1}-B_{2})\right] }, \end{equation*} By applying Lemma 1.1, we get the desired estimate of \(\left\vert a_{1}\right\vert\) as asserted in (46). Proceeding similarly as in the earlier proof, using (51)-(54), it follows that \begin{equation*} a_{2}=\frac{B_{1}(b_{2}-c_{2})}{4(1+2\lambda )}, \end{equation*} which, in view of Lemma 1.1, yields the estimate (47).
Competing Interests
The authors declares that he has no competing interests.
