Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions

Abstract

In this paper, we introduce and investigate two new subclasses $H_{\sigma }^{\mu }(\lambda ,\phi )$ and $M_{\sigma }^{\gamma }(\lambda ,\mu ,\phi )$ of Ma-Minda bi-univalent functions defined by using subordination in the open unit disk $D=\{z\in C:|z|<1\}$. For functions belonging to these new subclasses, we obtain estimates for the initial coefficients $|a_2|$ and $|a_3|$. The results presented in this paper would generalize those in related works of several earlier authors.

MSC:30C45, 30C80.

Dedication

Dedicated to Professor Hari M Srivastava

1 Introduction

Let C be a set of complex numbers and let $N=\{1,2,3,\dots \}=N_0\setminus \{0\}$ be a set of positive integers. Let A be a class of functions of the form

$$f(z)=z+\sum _{n=2}^{\mathrm{\infty }}{a}_n{z}^n,$$

(1.1)

which are analytic in the open unit disk $D=\{z\in C:|z|<1\}$. Also, let S denote a subclass of all functions in A which are univalent in D (for details, see [1, 2]).

Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk D. However, the famous Koebe one-quarter theorem [1] ensures that the image of the unit disk D under every function $f\in S$ contains a disk of radius 1/4. Thus, every univalent function $f\in S$ has an inverse $f^{-1}$ satisfying

$$f^{-1}(f(z))=z(z\in D)$$

and

$$f(f^{-1}(w))=w(|w|<r_0(f);r_0(f)\ge \frac{1}{4}),$$

where

$$f^{-1}(w)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\cdots .$$

(1.2)

A function $f\in A$ is said to be bi-univalent in D if both f and $f^{-1}$ are univalent in D. Let σ denote the class of bi-univalent functions defined in the unit disk D. In 1967, Lewin [3] first introduced the class σ of bi-univalent functions and showed that $|a_2|\le 1.51$ for every $f\in \sigma$. Subsequently, Branan and Clunie [4] conjectured that $|a_2|\le \sqrt{2}$ for $f\in \sigma$. Later, Netanyahu [5] proved that ${max}_{f\in \sigma }|a_2|=3/4$. The coefficient estimate problem for each of $|a_n|$ ($n\in N\setminus \{1,2\}$) is still an open problem.

Brannan and Taha [6] (see also [7]) introduced certain subclasses of a bi-univalent function class σ similar to the familiar subclasses $S^{\ast }(\alpha )$ and $K(\alpha )$ of starlike and convex functions of order α ($0<\alpha \le 1$), respectively (see [8]). Thus, following Brannan and Taha [6] (see also [7]), a function $f\in A$ is in the class $S_{\sigma }^{\ast }[\alpha ]$ of strongly bi-starlike functions of order α ($0<\alpha \le 1$) if both functions f and $f^{-1}$ are strongly starlike functions of order α. The classes $S_{\sigma }^{\ast }(\alpha )$ and $K_{\sigma }(\alpha )$ of bi-starlike functions of order α and bi-convex functions of order α, corresponding (respectively) to the function classes $S^{\ast }(\alpha )$ and $K(\alpha )$, were also introduced analogously. For each of the function classes $S_{\sigma }^{\ast }(\alpha )$ and $K_{\sigma }(\alpha )$, they found non-sharp estimates on the first two Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ (for details, see [6, 7]).

An analytic function f is subordinate to an analytic function g, written $f\prec g$, if there is an analytic function w with $|w(z)|\le |z|$ such that $f=(g(w))$. If g is univalent, then $f\prec g$ if and only if $f(0)=g(0)$ and $f(D)\subseteq g(D)$. Ma and Minda [9] unified various subclasses of starlike and convex functions for which either of the quantities $z{f}^{\prime }(z)/f(z)$ or $1+z{f}^{″}(z)/f^{\prime }(z)$ is subordinate to a more general superordinate function. For this purpose, they considered an analytic function φ with positive real part in the unit disk D, $\phi (0)=1$, ${\phi }^{\prime }(0)>0$, and φ maps D onto a region starlike with respect to 1 and symmetric with respect to the real axis. The classes $S^{\ast }(\phi )$ and $K(\phi )$ of Ma-Minda starlike and Ma-Minda convex functions are respectively characterized by $z{f}^{\prime }(z)/f(z)\prec \phi (z)$ or $1+z{f}^{″}(z)/f^{\prime }(z)\prec \phi (z)$. A function f is bi-starlike of Ma-Minda type or bi-convex of Ma-Minda type if both f and $f^{-1}$ are respectively Ma-Minda starlike or convex. These classes are denoted respectively by $S_{\sigma }^{\ast }(\phi )$ and $K_{\sigma }(\phi )$. Recently, Srivastava et al. [10], Frasin and Aouf [11] and Caglar et al. [12] introduced and investigated various subclasses of bi-univalent functions and found estimates on the coefficients $|a_2|$ and $|a_3|$ for functions in these classes. Very recently, Ali et al. [13], Kumar et al. [14], Srivastava et al. [15] and Xu et al.[16] unified and extended some related results in [7, 10–12, 17] by generalizing their classes using subordination.

Motivated by Ali et al. [13] and Kumar et al. [14], we investigate the estimates for the initial coefficients $|a_2|$ and $|a_3|$ of bi-univalent functions of Ma-Minda type belonging to the classes $H_{\sigma }^{\mu }(\lambda ,\phi )$ and $M_{\sigma }^{\gamma }(\lambda ,\mu ,\phi )$ defined in Section 2. Our results generalize several well-known results in [10–14] and these are also pointed out.

2 Coefficient estimates

Throughout this paper, we assume that φ is an analytic univalent function with positive real part in D, $\phi (D)$ is symmetric with respect to the real axis and starlike with respect to $\phi (0)=1$, and ${\phi }^{\prime }(0)>0$. Such a function has series expansion of the form

$$\phi (z)=1+B_{1}z+B_2{z}_2+B_3{z}_3+\cdots (B_1>0).$$

(2.1)

With this assumption on φ, we now introduce the following subclasses of Ma-Minda bi-univalent functions.

Definition 2.1 A function $f\in \sigma$ given by (1.1) is said to be in the class $H_{\sigma }^{\mu }(\lambda ,\phi )$ if it satisfies

$$(1-\lambda ){\left(\frac{f(z)}{z}\right)}^{\mu }+\lambda f^{\prime }(z){\left(\frac{f(z)}{z}\right)}^{\mu -1}\prec \phi (z)(\lambda \ge 1,\mu \ge 1,z\in D)$$

(2.2)

and

$$(1-\lambda ){\left(\frac{g(w)}{w}\right)}^{\mu }+\lambda g^{\prime }(w){\left(\frac{g(w)}{z}\right)}^{\mu -1}\prec \phi (w)(\lambda \ge 1,\mu \ge 1,w\in D),$$

(2.3)

where the function g is given by

$$g(w)=f^{-1}(w)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\cdots .$$

(2.4)

We note that, for suitable choices λ, μ and φ, the class $H_{\sigma }^{\mu }(\lambda ,\phi )$ reduces to the following known classes.

• (1) $H_{\sigma }^{\mu }(\lambda ,{(\frac{1+z}{1-z})}^{\alpha })=H_{\sigma }^{\mu }(\lambda ,\alpha )$ ($\lambda \ge 1$, $0<\alpha \le 1$, $\mu \ge 0$) (see Caglar et al. [[12], Definition 2.1]);

• (2) $H_{\sigma }^{\mu }(\lambda ,\frac{1+(1-2\beta )z}{1-z})=H_{\sigma }^{\mu }(\lambda ,\beta )$ ($\lambda \ge 1$, $0\le \beta <1$, $\mu \ge 0$) (see Caglar et al. [[12], Definition 3.1]);

• (3) $H_{\sigma }^1(\lambda ,\phi )=H_{\sigma }(\lambda ,\phi )$ ($\lambda \ge 1$) (see Kumar et al. [[14], Definition 1.1]);

• (4) $H_{\sigma }^{\mu }(1,\phi )=H_{\sigma }^{\mu }(\phi )$ ($\mu \ge 0$) (see Kumar et al. [[14], Definition 2.1]);

• (5) $H_{\sigma }^1(1,\phi )=H_{\sigma }(\phi )$ (see Ali et al. [[13], p.345]);

• (6) $H_{\sigma }^1(\lambda ,{(\frac{1+z}{1-z})}^{\alpha })=H_{\sigma }(\lambda ,\alpha )$ ($\lambda \ge 1$, $0<\alpha \le 1$) (see Frasin and Aouf [[11], Definition 2.1]);

• (7) $H_{\sigma }^1(\lambda ,\frac{1+(1-2\beta )z}{1-z})=H_{\sigma }(\lambda ,\beta )$ ($\lambda \ge 1$, $0\le \beta <1$) (see Frasin and Aouf [[11], Definition 3.1]);

• (8) $H_{\sigma }^1(1,{(\frac{1+z}{1-z})}^{\alpha })=H_{\sigma }(\alpha )$ ($0<\alpha \le 1$) (see Srivastava et al. [[10], Definition 1]);

• (9) $H_{\sigma }^1(1,\frac{1+(1-2\beta )z}{1-z})=H_{\sigma }(\beta )$ ($0\le \beta <1$) (see Srivastava et al. [[10], Definition 2]).

For functions in the class $H_{\sigma }^{\mu }(\lambda ,\phi )$, the following estimates are obtained.

Theorem 2.1 Let the function f given by (1.1) be in the class $H_{\sigma }^{\mu }(\lambda ,\phi )$, $\lambda \ge 1$ and $\mu \ge 0$. Then

$$|a_2|\le min\{\frac{B_1}{\lambda +\mu },\sqrt{\frac{2(B_1+|B_2-B_1|)}{(1+\mu )(2\lambda +\mu )}}\}$$

(2.5)

and

$$|a_3|\le \{\begin{array}{cc}min\{\frac{B_1}{2\lambda +\mu }+\frac{B_1^2}{{(\lambda +\mu )}^2},\frac{2(B_1+|B_2-B_1|)}{(1+\mu )(2\lambda +\mu )}\},\hfill & 0\le \mu <1,\hfill \\ \frac{B_1}{2\lambda +\mu }+\frac{2|B_2-B_1|}{(1+\mu )(2\lambda +\mu )},\hfill & \mu \ge 1.\hfill \end{array}$$

(2.6)

Proof Since $f\in H_{\sigma }^{\mu }(\lambda ,\phi )$, there exist two analytic functions $u,v:D\to D$, with $u(0)=v(0)=0$, such that

$$(1-\lambda ){\left(\frac{f(z)}{z}\right)}^{\mu }+\lambda f^{\prime }(z){\left(\frac{f(z)}{z}\right)}^{\mu -1}=\phi (u(z))$$

(2.7)

and

$$(1-\lambda ){\left(\frac{g(w)}{w}\right)}^{\mu }+\lambda g^{\prime }(w){\left(\frac{g(w)}{z}\right)}^{\mu -1}=\phi (v(w)).$$

(2.8)

Define the functions p and q by

$$\begin{array}{r}p(z)=\frac{1+u(z)}{1-u(z)}=1+p_{1}z+p_2{z}^2+\cdots \text{and}\\ q(z)=\frac{1+v(z)}{1-v(z)}=1+q_{1}z+q_2{z}^2+\cdots ,\end{array}$$

(2.9)

or, equivalently,

$$u(z)=\frac{p(z)-1}{p(z)+1}=\frac{1}{2}(p_{1}z+(p_2-\frac{p_1^2}{2})z^2+\cdots )$$

(2.10)

and

$$v(z)=\frac{q(z)-1}{q(z)+1}=\frac{1}{2}(q_{1}z+(q_2-\frac{q_1^2}{2})z^2+\cdots ).$$

(2.11)

It is clear that p and q are analytic in D and $p(0)=q(0)=1$. Since $u,v:D\to D$, the functions p and q have positive real part in D, and hence $|p_i|\le 2$ and $|q_i|\le 2$ ($i=1,2,\dots$). By virtue of (2.7), (2.8), (2.10) and (2.11), we have

$$(1-\lambda ){\left(\frac{f(z)}{z}\right)}^{\mu }+\lambda f^{\prime }(z){\left(\frac{f(z)}{z}\right)}^{\mu -1}=\phi \left(\frac{p(z)-1}{p(z)+1}\right)$$

(2.12)

and

$$(1-\lambda ){\left(\frac{g(w)}{w}\right)}^{\mu }+\lambda g^{\prime }(w){\left(\frac{g(w)}{z}\right)}^{\mu -1}=\phi \left(\frac{q(w)-1}{q(w)+1}\right).$$

(2.13)

Using (2.10), (2.11), together with (2.1), we easily obtain

$$\phi \left(\frac{p(z)-1}{p(z)+1}\right)=1+\frac{1}{2}{B}_1{p}_{1}z+(\frac{1}{2}{B}_1(p_2-\frac{1}{2}{p}_1^2)+\frac{1}{4}{B}_2{p}_1^2)z^2+\cdots$$

(2.14)

and

$$\phi \left(\frac{q(w)-1}{q(w)+1}\right)=1+\frac{1}{2}{B}_1{q}_{1}w+(\frac{1}{2}{B}_1(q_2-\frac{1}{2}{q}_1^2)+\frac{1}{4}{B}_2{q}_1^2)w^2+\cdots .$$

(2.15)

Since $f\in \sigma$ has the Maclaurin series given by (1.1), a computation shows that its inverse $g=f^{-1}$ has the expansion given by (1.2). Also, since

$$\begin{array}{r}{f}^{\prime }(z)=1+2a_{2}z+3a_3{z}^2+\cdots \text{and}\\ g^{\prime }(w)=1-2a_{2}w+3(2a_2-a_3)w^2-\cdots ,\end{array}$$

it follows from (2.12)-(2.15) that

$$(\lambda +\mu )a_2=\frac{1}{2}{B}_1{p}_1,$$

(2.16)

$$(2\lambda +\mu )a_3+\frac{(\mu -1)(2\lambda +\mu )}{2}{a}_2^2=\frac{1}{2}{B}_1(p_2-\frac{1}{2}{p}_1^2)+\frac{1}{4}{B}_2{p}_1^2,$$

(2.17)

$$-(\lambda +\mu )a_2=\frac{1}{2}{B}_1{q}_1$$

(2.18)

and

$$-(2\lambda +\mu )a_3+\frac{(3+\mu )(2\lambda +\mu )}{2}{a}_2^2=\frac{1}{2}{B}_1(q_2-\frac{1}{2}{q}_1^2)+\frac{1}{4}{B}_2{q}_1^2.$$

(2.19)

From (2.16) and (2.18), we get

$$p_1=-q_1$$

(2.20)

and

$$8{(\lambda +\mu )}^2{a}_2^2=B_1^2(p_1^2+q_1^2).$$

(2.21)

Also, from (2.17) and (2.19), we obtain

$$(1+\mu )(2\lambda +\mu )a_2^2=\frac{1}{2}{B}_1(p_2+q_2)+\frac{1}{4}(B_2-B_1)(p_1^2+q_1^2),$$

or

$$a_2^2=\frac{2B_1(p_2+q_2)+(B_2-B_1)(p_1^2+q_1^2)}{4(1+\mu )(2\lambda +\mu )}.$$

(2.22)

Since $|p_i|\le 2$ and $|q_i|\le 2$ ($i=1,2$), it follows from (2.21) and (2.22) that

$$|a_2|\le \frac{B_1}{\lambda +\mu }$$

(2.23)

and

$$|a_2|\le \sqrt{\frac{2(B_1+|B_2-B_1|)}{(1+\mu )(2\lambda +\mu )}},$$

(2.24)

which yields the desired estimate on $|a_2|$ as asserted in (2.5).

Next, in order to find the bound on $|a_3|$, by subtracting (2.19) from (2.17), we get

$$2(2\lambda +\mu )(a_3-a_2^2)=\frac{1}{2}{B}_1(p_2-q_2)+\frac{1}{4}(B_2-B_1)(p_1^2-q_1^2).$$

(2.25)

Using (2.20) and (2.21) in (2.25), we have

$$a_3=\frac{1}{4(2\lambda +\mu )}{B}_1(p_2-q_2)+\frac{1}{4{(\lambda +\mu )}^2}{B}_1^2{p}_1^2,$$

which evidently yields

$$|a_3|\le \frac{B_1}{2\lambda +\mu }+\frac{B_1^2}{{(\lambda +\mu )}^2}.$$

(2.26)

On the other hand, by using (2.20) and (2.22) in (2.25), we obtain

$$a_3=\frac{B_1[(\mu +3)p_2+(1-\mu )q_2]+(B_2-B_1)(p_1^2+q_1^2)}{4(1+\mu )(2\lambda +\mu )},$$

(2.27)

and applying $|p_i|\le 2$ and $|q_i|\le 2$ ($i=1,2$) for (2.27), we get

$$|a_3|\le \frac{B_1}{2(2\lambda +\mu )}[\frac{\mu +3}{1+\mu }+\frac{|1-\mu |}{1+\mu }]+\frac{2|B_2-B_1|}{(1+\mu )(2\lambda +\mu )}.$$

(2.28)

Now, we consider the bounds on $|a_3|$ according to μ.

Case 1. If $0\le \mu <1$, then from (2.28)

$$|a_3|\le \frac{2(B_1+|B_2-B_1|)}{(1+\mu )(2\lambda +\mu )}.$$

(2.29)

Case 2. If $\mu \ge 1$, then from (2.28)

$$|a_3|\le \frac{B_1}{2\lambda +\mu }+\frac{2|B_2-B_1|}{(1+\mu )(2\lambda +\mu )}.$$

(2.30)

Thus, from (2.26), (2.29) and (2.30), we obtain the desired estimate on $|a_3|$ given in (2.6). This completes the proof of Theorem 2.1. □

Putting $\mu =1$ and $\lambda =\mu =1$ in Theorem 2.1, we respectively get the following Corollaries 2.1 and 2.2.

Corollary 2.1 If $f\in H_{\sigma }(\lambda ,\phi )$ ($\lambda \ge 1$), then

$$|a_2|\le min\{\frac{B_1}{\lambda +1},\sqrt{\frac{B_1+|B_2-B_1|}{2\lambda +1}}\}$$

and

$$|a_3|\le \{\begin{array}{cc}min\{\frac{B_1}{2\lambda +1}+\frac{B_1^2}{{(\lambda +1)}^2},\frac{B_1+|B_2-B_1|}{2\lambda +1}\},\hfill & 0\le \mu <1,\hfill \\ \frac{B_1+|B_2-B_1|}{2\lambda +1},\hfill & \mu \ge 1.\hfill \end{array}$$

Corollary 2.2 If $f\in H_{\sigma }(\phi )$, then

$$|a_2|\le min\{\frac{B_1}{2},\sqrt{\frac{B_1+|B_2-B_1|}{3}}\}$$

and

$$|a_3|\le \{\begin{array}{cc}min\{\frac{B_1}{3}+\frac{B_1^2}{4},\frac{B_1+|B_2-B_1|}{3}\},\hfill & 0\le \mu <1,\hfill \\ \frac{B_1+|B_2-B_1|}{3},\hfill & \mu \ge 1.\hfill \end{array}$$

Remark 2.1 The estimates of the coefficients $|a_2|$ and $|a_3|$ of Corollaries 2.1 and 2.2 are the improvement of the estimates obtained in [[14], Theorem 2.1] and [[13], Theorem 2.1], respectively.

Remark 2.2 If we set

$$\phi (z)=\frac{1+(1-2\beta )z}{1-z}=1+2(1-\beta )z+2(1-\beta )z^2+\cdots (0\le \beta <1)$$

in Corollaries 2.1 and 2.2, the results obtained improve the results in [[11], Theorem 3.2, inequalities (3.3) and (3.4)] and [[10], Theorem 2, inequality (3.3)], respectively.

Definition 2.2 Let $\gamma \in C^{\ast }=C\setminus \{0\}$, $\lambda \ge 0$ and $\mu \ge 0$. A function $f\in \sigma$ given by (1.1) is said to be in the class $M_{\sigma }^{\gamma }(\lambda ,\mu ,\phi )$, if the following subordinations hold:

$$1+\frac{1}{\gamma }(\frac{z{f}^{\prime }(z)+(2\lambda \mu +\lambda -\mu )z^2{f}^{″}(z)+\lambda \mu z^3{f}^{‴}(z)}{(1-\lambda +\mu )f(z)+(\lambda -\mu )z{f}^{\prime }(z)+\lambda \mu z^2{f}^{″}(z)}-1)\prec \phi (z)$$

and

$$1+\frac{1}{\gamma }(\frac{w{g}^{\prime }(w)+(2\lambda \mu +\lambda -\mu )w^2{g}^{″}(w)+\lambda \mu w^3{g}^{‴}(w)}{(1-\lambda +\mu )g(w)+(\lambda -\mu )w{g}^{\prime }(w)+\lambda \mu w^2{g}^{″}(w)}-1)\prec \phi (w),$$

where the function g is defined by (2.4).

We note that, by choosing appropriate values for λ, μ, γ and φ, the class $M_{\sigma }^{\gamma }(\lambda ,\mu ,\phi )$ reduces to several earlier known classes.

• (1) $M_{\sigma }^{\gamma }(\lambda ,0,\phi )=N_{\sigma ,\gamma }^{\lambda }(\phi )$ ($\lambda \ge 0$, $\gamma \in C^{\ast }$) (see Kumar et al. [[14], Definition 2.2]);

• (2) $M_{\sigma }^1(0,0,\frac{1+(1-2\beta )z}{1-z})=S_{\sigma }^{\ast }(\beta )$ ($0\le \beta <1$) (see Brannan and Taha [[6], Definition 3.1]);

• (3) $M_{\sigma }^1(1,0,\frac{1+(1-2\beta )z}{1-z})=K_{\sigma }(\beta )$ ($0\le \beta <1$) (see Brannan and Taha [[6], Definition 4.1]);

• (4) $M_{\sigma }^1(0,0,{(\frac{1+z}{1-z})}^{\alpha })=S_{\sigma }^{\ast }(\alpha )$ ($0<\alpha \le 1$) (see Taha [7]).

For functions in the class $M_{\sigma }^{\gamma }(\lambda ,\mu ,\phi )$, the following estimates are derived.

Theorem 2.2 Let $\gamma \in C^{\ast }$, $\lambda \ge 0$ and $\mu \ge 0$. If $f\in M_{\sigma }^{\gamma }(\lambda ,\mu ,\phi )$, then

$$|a_2|\le \frac{|\gamma |B_1\sqrt{B_1}}{\sqrt{|[2(6\lambda \mu +2\lambda -2\mu +1)-{(2\lambda \mu +\lambda -\mu +1)}^2]B_1^2\gamma +2{(2\lambda \mu +\lambda -\mu +1)}^2(B_1-B_2)|}}$$

(2.31)

and

$$|a_3|\le \frac{|\gamma |(B_1+|B_2-B_1|)}{|2(6\lambda \mu +2\lambda -2\mu +1)-{(2\lambda \mu +\lambda -\mu +1)}^2|}.$$

(2.32)

Proof If $f\in M_{\sigma }^{\gamma }(\lambda ,\mu ,\phi )$, then there are analytic functions $u,v:D\to D$, with $u(0)=v(0)=0$, satisfying

$$1+\frac{1}{\gamma }(\frac{z{f}^{\prime }(z)+(2\lambda \mu +\lambda -\mu )z^2{f}^{″}(z)+\lambda \mu z^3{f}^{‴}(z)}{(1-\lambda +\mu )f(z)+(\lambda -\mu )z{f}^{\prime }(z)+\lambda \mu z^2{f}^{″}(z)}-1)=\phi (u(z))$$

(2.33)

and

$$1+\frac{1}{\gamma }(\frac{w{g}^{\prime }(w)+(2\lambda \mu +\lambda -\mu )w^2{g}^{″}(w)+\lambda \mu w^3{g}^{‴}(w)}{(1-\lambda +\mu )g(w)+(\lambda -\mu )w{g}^{\prime }(w)+\lambda \mu w^2{g}^{″}(w)}-1)=\phi (v(w)).$$

(2.34)

Let p and q be defined as in (2.8), then it is clear from (2.33), (2.34), (2.9) and (2.10) that

$$\begin{array}{r}1+\frac{1}{\gamma }(\frac{z{f}^{\prime }(z)+(2\lambda \mu +\lambda -\mu )z^2{f}^{″}(z)+\lambda \mu z^3{f}^{‴}(z)}{(1-\lambda +\mu )f(z)+(\lambda -\mu )z{f}^{\prime }(z)+\lambda \mu z^2{f}^{″}(z)}-1)\\ =\phi \left(\frac{p(z)-1}{p(z)+1}\right)\end{array}$$

(2.35)

and

$$\begin{array}{r}1+\frac{1}{\gamma }(\frac{w{g}^{\prime }(w)+(2\lambda \mu +\lambda -\mu )w^2{g}^{″}(w)+\lambda \mu w^3{g}^{‴}(w)}{(1-\lambda +\mu )g(w)+(\lambda -\mu )w{g}^{\prime }(w)+\lambda \mu w^2{g}^{″}(w)}-1)\\ =\phi \left(\frac{q(w)-1}{q(w)+1}\right).\end{array}$$

(2.36)

It follows from (2.35), (2.36), (2.14) and (2.15) that

$$\begin{array}{r}(2\lambda \mu +\lambda -\mu +1)a_2=\frac{1}{2}{B}_1{p}_1\gamma ,\end{array}$$

(2.37)

$$\begin{array}{r}-{(2\lambda \mu +\lambda -\mu +1)}^2{a}_2^2+2(6\lambda \mu +2\lambda -2\mu +1)a_3\\ =\gamma [\frac{1}{2}{B}_1(p_2-\frac{1}{2}{p}_1^2)+\frac{1}{4}{B}_2{p}_1^2],\end{array}$$

(2.38)

$$\begin{array}{r}-(2\lambda \mu +\lambda -\mu +1)a_2=\frac{1}{2}{B}_1{q}_1\gamma \end{array}$$

(2.39)

and

$$\begin{array}{r}[4(6\lambda \mu +2\lambda -2\mu +1)-{(2\lambda \mu +\lambda -\mu +1)}^2]a_2^2-2(6\lambda \mu +2\lambda -2\mu +1)a_3\\ =\gamma [\frac{1}{2}{B}_1(q_2-\frac{1}{2}{q}_1^2)+\frac{1}{4}{B}_2{q}_1^2].\end{array}$$

(2.40)

Equations (2.37) and (2.39) yield

$$p_1=-q_1$$

(2.41)

and

$$8{(2\lambda \mu +\lambda -\mu +1)}^2{a}_2^2=B_1^2{\gamma }^2(p_1^2+q_1^2).$$

(2.42)

From (2.38), (2.40), (2.41) and (2.42), it follows that

$$a_2^2=\frac{{\gamma }^2{B}_1^3(p_2+q_2)}{4[(2(6\lambda \mu +2\lambda -2\mu +1)-{(2\lambda \mu +\lambda -\mu +1)}^2)B_1^2\gamma +{(2\lambda \mu +\lambda -\mu +1)}^2(B_1-B_2)]}$$

which yields the desired estimate on $|a_2|$ as described in (2.31).

Similarly, it can be obtained from (2.38), (2.40) and (2.41) that

$$\begin{array}{rcl}{a}_3& =& \frac{\gamma B_1[p_2(4(6\lambda \mu +2\lambda -2\mu +1)-{(2\lambda \mu +\lambda -\mu +1)}^2)+q_2{(2\lambda \mu +\lambda -\mu +1)}^2]}{8[2(6\lambda \mu +2\lambda -2\mu +1)-{(2\lambda \mu +\lambda -\mu +1)}^2](6\lambda \mu +2\lambda -2\mu +1)}\\ +\frac{2\gamma (B_2-B_1)(6\lambda \mu +2\lambda -2\mu +1)p_1^2}{8[2(6\lambda \mu +2\lambda -2\mu +1)-{(2\lambda \mu +\lambda -\mu +1)}^2](6\lambda \mu +2\lambda -2\mu +1)}\end{array}$$

which easily leads to the desired estimate (2.32) on $|a_3|$. □

Taking $\mu =0$ in Theorem 2.2, we obtain the following corollary.

Corollary 2.3 [[14], Theorem 2.3]

If $f\in N_{\sigma ,\gamma }^{\lambda }(\phi )$, then

$$|a_2|\le \frac{|\gamma |B_1\sqrt{B_1}}{\sqrt{|(1+2\lambda -{\lambda }^2)B_1^2\gamma +{(1+\lambda )}^2(B_1-B_2)|}}\mathit{\text{and}}|a_3|\le \frac{|\gamma |(B_1+|B_2-B_1|)}{|1+2\lambda -{\lambda }^2|}.$$

Further, for $\gamma =1$, putting $\lambda =0$ and $\lambda =1$ in Corollary 2.3, respectively, we have the following Corollaries 2.4 and 2.5.

Corollary 2.4 [[13], Corollary 2.1]

If $f\in M_{\sigma }^1(0,0,\phi )=S{T}_{\sigma }(\phi )$, then

$$|a_2|\le \frac{B_1\sqrt{B_1}}{\sqrt{|B_1^2+B_1-B_2|}}\mathit{\text{and}}|a_3|\le B_1+|B_2-B_1|.$$

Corollary 2.5 [[13], Corollary 2.2]

If $f\in M_{\sigma }^1(1,0,\phi )=C{V}_{\sigma }(\phi )$, then

$$|a_2|\le \frac{B_1\sqrt{B_1}}{\sqrt{2|B_1^2+2B_1-2B_2|}}\mathit{\text{and}}|a_3|\le \frac{1}{2}(B_1+|B_2-B_1|).$$

Remark 2.3 If we set

$$\phi (z)={\left(\frac{1+z}{1-z}\right)}^{\alpha }=1+2\alpha z+2{\alpha }^2{z}^2+\cdots (0<\alpha \le 1)$$

and

$$\phi (z)=\frac{1+(1-2\beta )z}{1-z}=1+2(1-\beta )z+2(1-\beta )z^2+\cdots (0\le \beta <1)$$

in Corollaries 2.4 and 2.5, we obtain the results of Brannan and Taha [[6], Theorems 2.1, 3.1 and 4.1, respectively].