Coefficient Bounds for Certain Classes of Analytic Functions Associated with Faber Polynomial

Abstract

In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of $n-th$$(n\ge 3)$ Taylor–Maclaurin coefficients $\left|a_n\right|$ is obtained. Furthermore, the bounds value of the first two coefficients of such functions is established.

1. Introduction

Faber polynomials, which were introduced by Faber in 1903 [1], play an important role in the theory of functions of a complex variable and different areas of mathematics and there is a rich literature [2,3,4,5,6,7] describing their properties and their applications. Given a function $h\left(z\right)$ of the form

$$h\left(z\right)=z+b_0+b_1{z}^{-1}+b_2{z}^{-2}+\dots ,$$

consider the expansion

$$\frac{\varsigma h^{{}^{\prime }}\left(\zeta \right)}{h\left(\zeta \right)-w}=\sum _{n=0}^{\infty }{\Psi }_n\left(w\right){\zeta }^{-n},$$

valid for all $\zeta$ in some neighborhood of $\infty$. The function ${\Psi }_n\left(w\right)=w^n+{\displaystyle \sum _{k=1}^n}{a}_{nk}{w}^{n-k}$ is a polynomial of degree n, called the n-th Faber polynomial with respect to the function $h\left(z\right)$. In particular,

$$\begin{array}{cc}\hfill {\Psi }_0\left(w\right)& =1,{\Psi }_1\left(w\right)=w-b_0,\hfill \\ \hfill {\Psi }_2\left(w\right)& =w^2-2b_{0}w+(b_0^2-2b_1),\hfill \\ \hfill {\Psi }_3\left(w\right)& =w^3-3b_0{w}^2+(3b_0^2-3b_1)w+(b_0^3+3b_1{b}_0-3b_2).\hfill \end{array}$$

Let ${\Psi }_n\left(0\right)=F_n(b_0,b_1,\dots ,b_n),n\ge 0,$ see ([8], p. 118). Let A denote the class of all functions of the form:

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

(1)

which are analytic in the open unit disc $U=\{z:z\in \mathbb{C}$ and $\left|z\right|<1\}$ and let S be the class of all functions in A which are univalent in U. By using the Faber polynomial expansion of functions of the form (1), Airault and Bouali [9], p. 184 showed that

$$\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}=1-\sum _{j=2}^{\infty }{F}_{j-1}(a_2,a_3,\cdots ,a_j)z^{j-1},$$

(2)

where $F_{j-1}(a_2,a_3,\cdots ,a_j)$ is the Faber polynomial given by:

$$F_{j-1}(a_2,a_3,\cdots ,a_j)=\sum _{i_1+2i_2+\dots +(j-1)i_{j-1}=j-1}A(i_1,i_2,\dots ,i_{j-1})(a_2^{i_1},a_3^{i_2},\cdots ,a_j^{i_{j-1}})$$

and

$$A(i_1,i_2,\dots ,i_{j-1}):={(-1)}^{(j-1)+2i_1+\dots +j{i}_{j-1}}\frac{(i_1+i_2+\dots +i_{j-1}-1)!(j-1)}{\left(i_1\right)!\left(i_2\right)!\dots \left(i_{j-1}\right)!}.$$

The first few terms of the Faber polynomials $F_{j-1},j\ge 2,$ are given by (e.g., see [10], p. 52)

$$\begin{array}{cc}\hfill F_1& =-a_2,\hfill \\ \hfill F_2& =a_2^2-2a_3,\hfill \\ \hfill F_3& =-a_2^3+3a_2{a}_3-3a_4,\hfill \\ \hfill F_4& =a_2^4-4a_2^2{a}_3+4a_2{a}_4+2a_3^2-4a_5\hfill \\ \hfill F_5& =-a_2^5+5a_2^3{a}_3+5a_2^2{a}_4-5a_2(a_3^2-a_5)+5a_3{a}_4-5a_6.\hfill \end{array}$$

The Koebe one-quarter theorem [8], p. 31 ensures the range of every function of the class S contains the disc $\{w:\left|w\right|<\frac{1}{4}\}$. Thus every univalent function $f\in S$ has an inverse $f^{-1},$ which is defined by

$$f^{-1}\left(f\left(z\right)\right)=z(z\in U)$$

and

$$f\left(f^{-1}\left(\omega \right)\right)=\omega (\left|\omega \right|<\frac{1}{4}).$$

The inverse map $g:=f^{-1}$ of the function f$\in A$ has Taylor expansion given by (see [9], p. 185)

$$\begin{array}{cc}\hfill g\left(\omega \right)& =f^{-1}\left(\omega \right)=w+\sum _{n=2}^{\infty }\frac{1}{n}{K}_{n-1}^{-n}(a_2,a_3,\cdots ,a_n){\omega }^n\hfill \\ \hfill & =w-a_2{\omega }^2+(2a_2^2-a_3){\omega }^3-(5a_2^2-5a_2{a}_3+a_4){\omega }^4+\cdots ,\hfill \end{array}$$

where the coefficients $K_n^p(a_2,a_3,\cdots ,a_n)$ are given by

$$\begin{array}{cc}\hfill K_1^p& =p{a}_2,K_2^p=\frac{p(p-1)}{2}{a}_2^2+p{a}_3,\hfill \\ \hfill K_3^p& =p(p-1)a_2{a}_3+p{a}_4+\frac{p(p-1)(p-2)}{3!}{a}_2^3,\hfill \\ \hfill K_4^p& =p(p-1)a_2{a}_4+p{a}_5+\frac{p(p-1)}{2}{a}_3^2+\frac{p(p-1)(p-2)}{2}{a}_2^2{a}_3+\frac{p!}{(p-4)!4!}{a}_2^4,\hfill \\ \hfill & ⋮\hfill \\ \hfill K_n^p& =\frac{p!}{(p-n)!n!}{a}_2^n+\frac{p!}{(p-n+1)!(n-2)!}{a}_2^{n-2}{a}_3+\frac{p!}{(p-n+2)!(n-3)!}{a}_2^{n-3}{a}_4\hfill \\ \hfill & +\frac{p!}{(p-n+3)!(n-4)!}{a}_2^{n-4}\left[a_5+\frac{p-n+3}{2}{a}_3^2\right]\hfill \\ \hfill & +\frac{p!}{(p-n+4)!(n-5)!}{a}_2^{n-4}\left[a_6+(p-n+3)a_3{a}_4\right]+\sum _{j\ge 6}^{\infty }{a}_2^{n-j}{V}_j\hfill \end{array}$$

(3)

and $V_j$ is homogeneous polynomial of degree j in the variables $a_3,\cdots ,a_n,$ see ([11], p. 349 and [9], p. 183 and p. 205).

Lemma 1.

(Schwarz lemma [8], p. 3) Let $\omega \left(z\right)$ be analytic in the unit disc $U,$ with $\omega \left(0\right)=0$ and $\left|\omega \left(z\right)\right|<1$ in $U.$ Then $\left|\omega \left(z\right)\right|<\left|z\right|$ and $\left|{\omega }^{{}^{\prime }}\left(0\right)\right|<1$ in U.

If f and g are analytic functions in U, we say that f is subordinate to g, written $f\left(z\right)\prec g\left(z\right)$ if there exists a Schwarz function $\omega \left(z\right)$ such that $f\left(z\right)=g\left(\omega \right(z\left)\right).$ Let $\varphi$ be an analytic function with positive real part in U, satisfying $\varphi \left(0\right)=1,{\varphi }^{{}^{\prime }}\left(0\right)>0,$ and $\varphi \left(U\right)$ is symmetric with respect to the real axis. Such a function has a Taylor series of the form

$$\varphi \left(z\right)=1+B_{1}z+B_2{z}^2+B_3{z}^3+\cdots (B_1>0).$$

(4)

Using this $\varphi$, Ma and Minda [12] considered the classes

$$S\left(\varphi \right)=\left\{f\in A:\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}\prec \varphi \left(z\right),z\in U\right\}$$

and

$$K\left(\varphi \right)=\left\{f\in A:z{f}^{{}^{\prime }}\left(z\right)\in S\left(\varphi \right),z\in U\right\}$$

Several well-known classes can be obtained by specializing of the function $\varphi ,$ for instance

• By taking $\varphi \left(z\right)=\frac{1+Az}{1+Bz},-1\le B<A\le 1,$ we obtain the classes $S[A,B]$ and $K[A,B]$ of the well-known Janowski starlike and convex functions.
• If we set $\varphi \left(z\right)=\frac{1+(1-2\alpha )z}{1-z},$ we obtain the classes $S^{\ast }\left(\alpha \right)$ and $K\left(\alpha \right)$ of starlike and convex functions of order $\alpha (0\le \alpha <1)$.
• The class $S_L^{\ast }:=S\left(\sqrt{1+z}\right)$ was considered by Sokol and Stankieicz [13], consisting of functions f such that $\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}$ lies in the region bounded by the right half of the Bernoulli lemniscate given by $\left|w^2-1\right|<1$.
• Taking $\varphi \left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\delta }(0<\delta \le 1)$ yields the classes of strongly starlike and convex functions.
• The function class $S_{\nabla }^{\ast }:=S(z+\sqrt{1+z^2})$ was considered by Raina and Sokol [14], consisting of normalized starlike functions f satisfying the inequality

  $$\left|{\left\{\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}\right\}}^2-1\right|<2\left|\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}\right|.$$
• Kanas et al. [15] considered the family of analytic functions $S\left(\frac{1}{{(1-z)}^s}\right)$ and $K\left(\frac{1}{{(1-z)}^s}\right)$ with the property that $\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}$ and $1+\frac{z{f}^{{}^{{}^{″}}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)}$ lie in adomain bounded by the righ branch of a hyperbola $\rho =\rho \left(s\right)={(2cos\frac{\phi }{s})}^{-1}(0<s\le 1,\left|\phi \right|<\frac{\pi s}{2}).$
• The function class $S_e^{\ast }:=S\left(e^z\right)$ was introduced and studied by Mendiratta et al. [16]. The exponential function $\varphi \left(z\right)=e^z$ has positive real part in $U,$ maps U onto a domain $\varphi \left(U\right):=\{w\in C:|logw|<1\}$ is symmetric with respect to the real axis and starlike with respect to 1 and ${\varphi }^{{}^{\prime }}\left(0\right)>0.$
• The classes $S\left(\frac{2}{1+e^{-z}}\right)$ and $K\left(\frac{2}{1+e^{-z}}\right)$ were introduced and studied by Goel and Kumar [17]. The modified sigmoid function $\varphi \left(z\right)=\frac{2}{1+e^{-z}}$ maps U onto a domain $S_G:=\{w\in C:|log(w/(2-w))|<1\},$ which is symmetric about the real axis. Moreover, G is a convex and hence starlike function with respect to $G\left(0\right)=1.$

An interesting families of the domains that are bounded by a conic sections were introduced and studied by Shams et al. [18], they introduced the class $SD(\alpha ,\beta )$ of $\beta$- uniformly starlike functions of order $\alpha (0\le \alpha <1)$ in U consisting of functions $f\in A$ which satisfy the following inequality

$$\Re \left\{\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}-\alpha \right\}>\beta \left|\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}-1\right|(\beta \ge 0;0\le \alpha <1;z\in U).$$

(5)

and class $KD(\alpha ,\beta )$ of $\beta$-uniformly convex of order $\alpha (0\le \alpha <1),$ defined by

$$f\in KD(\alpha ,\beta )\iff z{f}^{{}^{\prime }}\left(z\right)\in SD(\alpha ,\beta ).$$

Since $Rew>\alpha |w-1|+\gamma$ if and only if $Re\{w(1+\alpha e^{i\theta })-\alpha e^{i\theta }\}>\gamma$(see [19]), then the condition (5) is equivalent to

$$\Re \left\{(1+\beta e^{i\theta })\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}-\beta e^{i\theta }\right\}>\alpha .$$

Motivated by the classes $SD(\alpha ,\beta )$ and $KD(\alpha ,\beta )$ we now introduce and investigate the following subclasses of A, and obtain some interesting results.

Definition 1.

A function $f\left(z\right)\in A$ is said to be in the class $M(\lambda ,\beta ,\gamma ,\varphi )$if it satisfies

$$(1+\beta e^{i\gamma })\frac{z{f}^{\prime }\left(z\right)+\lambda z^2{f}^{{}^{{}^{″}}}\left(z\right)}{(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)}-\beta e^{i\gamma }\prec \varphi \left(z\right)\left(z\in U\right),$$

where $\beta \ge 0,0\le \lambda \le 1$ and $-\pi \le \gamma <\pi .$

We note that:

• The class $M(0,0,\gamma ,\varphi )=S\left(\varphi \right)$ and the class $M(1,0,\gamma ,\varphi )=K\left(\varphi \right).$
• The class $M(0,\beta ,\gamma ,\frac{1+(1-2\alpha )z}{1-z})$=$SD(\alpha ,\beta )$ and the class $M(1,\beta ,\gamma ,\frac{1+(1-2\alpha )z}{1-z})$$=K$$D(\alpha ,\beta ).$
• The class $M(\lambda ,0,\gamma ,\frac{1+(1-2\alpha )z}{1-z})$ was introduced and studied by Aouf et al. [20].

Definition 2.

A function $f\left(z\right)\in A$ is said to be in the class $S(\lambda ,\beta ,\gamma ,\varphi )$if it satisfies

$$(1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda (1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)})\right]-\beta e^{i\gamma }\prec \varphi \left(z\right)\left(z\in U\right),$$

where $\beta \ge 0,0\le \lambda \le 1$ and $-\pi \le \gamma <\pi .$

We note that:

• The class $S(0,0,\gamma ,\varphi )=S\left(\varphi \right)$ and $S(1,0,\gamma ,\varphi )=K\left(\varphi \right).$
• $M(1,\beta ,\gamma ,\frac{1+(1-2\alpha )z}{1-z})$=$SD(\alpha ,\beta )$ and $S(0,\beta ,\gamma ,\frac{1+(1-2\alpha )z}{1-z})=K$$D(\alpha ,\beta ).$

A single-valued function f analytic in a domain $D\subset C$ is said to be univalent there if it never take the same value twice; that is, if $f\left(z_1\right)\ne$$f\left(z_2\right)$ for all points $z_1$ and $z_2$ in D with $z_1\ne z_2$ (see [8], p. 26). A function $f\in A$ is said to be bi-univalent in $U$ if f and its inverse map $f^{-1}$ are univalent in U. Let $\sigma$ denote the class of bi-univalent functions in U given by (1). The class of analytic bi-univalent functions was first introduced and studied by Lewin [21] and showed that $|a_2|<1.51$. Recently, many authors found non-sharp estimates on the first two Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for various subclasses of bi-univalent functions, see for example, ([22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]). For other related topics see also, ([44,45,46,47]).

Definition 3.

A function $f\in \sigma$ given by (1) is said to be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$ if both f and its inverse map $g=f^{-1}$ are in $M(\lambda ,\beta ,\gamma ,\varphi ).$

We note that:

• The class $M_{\sigma }(0,1,\gamma ,\varphi )={\Re }_{\sigma }\left(\varphi \right)$ was introduced and studied by Darwish et al. [48].
• The class $M_{\sigma }(0,0,\gamma ,\frac{1+Az}{1+Bz})=S[A,B]$ was introduced and studied by Hamidi and Jahangiri [49].

Definition 4.

A function $f\in \sigma$ given by (1) is said to be in the class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$ if both f and its inverse map $g=f^{-1}$ are in $S(\lambda ,\beta ,\gamma ,\varphi ).$

We note that:

• The class $S_{\sigma }(0,1,\gamma ,\varphi )={\Re }_{\sigma }\left(\varphi \right)$.
• The class $S_{\sigma }(0,0,\gamma ,\frac{1+Az}{1+Bz})=S[A,B]$.
• The class $S_{\sigma }(\lambda ,0,\gamma ,\varphi )=M_q^{\sigma }(\lambda ,\varphi )$ was introduced and studied by Goyal and Kumar [50], see also Zireh et al. [51].

In this paper, we use the Faber polynomial expansions to obtain bounds for the general coefficients $|a_n|$ of bi-univalent functions in $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$ and $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$ as well as we provide estimates for the initial coefficients of these functions.

2. Coefficient Estimates for the Class $M_{\sigma }(p,\lambda ,\tau ,\varphi )$

Theorem 1.

Let the function $f\in \sigma$ given by (1) be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ).$ If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{B_1}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Proof. 

If we set $F\left(z\right):=(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)=z+{\displaystyle \sum _{n=2}^{\infty }}[1+\lambda (n-1)]a_n{z}^n:=z+{\displaystyle \sum _{n=2}^{\infty }}{\delta }_n{z}^n,$ then

$$f\in M(\lambda ,\beta ,\gamma ,\varphi )\iff (1+\beta e^{i\gamma })\frac{z{F}^{{}^{\prime }}\left(z\right)}{F\left(z\right)}-\beta e^{i\gamma }\prec \varphi \left(z\right).$$

Since, both functions f and its inverse map $g=f^{-1}$ are in $M(\lambda ,\beta ,\gamma ,\varphi ),$ by the definition of subordination, there are analytic functions $u,v:U\to U$ with $u\left(0\right)=v\left(0\right)=0,\left|u\right(z\left)\right|<1$ and $\left|v\right(z\left)\right|<1,$ such that

$$(1+\beta e^{i\gamma })\frac{z{F}^{{}^{\prime }}\left(z\right)}{F\left(z\right)}-\beta e^{i\gamma }=\varphi \left(u\left(z\right)\right)(z\in U)$$

(6)

and

$$(1+\beta e^{i\gamma })\frac{w{G}^{{}^{\prime }}\left(w\right)}{G\left(w\right)}-\beta e^{i\gamma }=\varphi \left(v\left(w\right)\right)(z\in U),$$

(7)

where $G\left(z\right):=(1-\lambda )g\left(z\right)+\lambda z{g}^{{}^{\prime }}\left(z\right)=z+{\displaystyle \sum _{n=2}^{\infty }}[1+\lambda (n-1)]d_n{z}^n:=z+{\displaystyle \sum _{n=2}^{\infty }}{\zeta }_n{z}^n$ and $d_n=\frac{1}{n}{K}_{n-1}^{-n}(a_2,a_3,\cdots ,a_n).$ Define the functions $u\left(z\right)$ and $v\left(z\right)$ by

$$u\left(z\right)=\sum _{n=1}^{\infty }{b}_n{z}^n,v\left(z\right)=\sum _{n=1}^{\infty }{c}_n{z}^n(z\in U).$$

(8)

It is well known that (see Duren [8], p. 265)

$$\left|b_n\right|\le 1,\left|c_n\right|\le 1n=2,3,\dots .$$

(9)

By a simple calculation, we have

$$\begin{array}{cc}\hfill \varphi \left(u\right(z\left)\right)& =1-B_1\sum _{n=1}^{\infty }{K}_n^{-1}(b_1,b_2,\cdots ,b_n,B_1,B_1,B_2,B_3,\cdots ,B_n)z^n\hfill \\ \hfill & =1+B_1{b}_{1}z+(B_1{b}_2+B_2{b}_1^2)z^2+\cdots (z\in U),\hfill \end{array}$$

(10)

and

$$\begin{array}{cc}\hfill \varphi \left(v\right(\omega \left)\right)& =1-B_1\sum _{n=1}^{\infty }{K}_n^{-1}(c_1,c_2,\cdots ,c_n,B_1,B_2,B_3,\cdots ,B_n)w^n\hfill \\ \hfill & =1+B_1{c}_1\omega +(B_1{c}_2+B_2{c}_1^2){\omega }^2+\cdots (\omega \in U),\hfill \end{array}$$

(11)

In general (see [52], p. 649), the coefficients $K_n^p(k_1,k_2,\cdots ,k_n,B_1,B_2,B_3,\cdots ,B_n)$ are given by

$$\begin{array}{cc}\hfill & K_n^p(k_1,k_2,\cdots ,k_n,B_1,B_2,B_3,\cdots ,B_n)\hfill \\ \hfill & =\frac{p!}{(p-n)!n!}{k}_1^n\frac{{(-1)}^{n+1}{B}_n}{B_1}+\frac{p!}{(p-n+1)!(n-2)!}{k}_1^{n-2}{k}_2\frac{{(-1)}^n{B}_{n-1}}{B_1}\hfill \\ \hfill & +\frac{p!}{(p-n+2)!(n-3)!}{k}_1^{n-3}{k}_3\frac{{(-1)}^{n-1}{B}_{n-2}}{B_1}\hfill \\ \hfill & +\frac{p!}{(p-n+3)!(n-4)!}{k}_1^{n-4}[k_4\frac{{(-1)}^{n-2}{B}_{n-3}}{B_1}+\frac{p-n+3}{2}{k}_2^2{k}_3\frac{{(-1)}^{n-1}{B}_{n-2}}{B_1}]\hfill \\ \hfill & +\sum _{j\ge 5}^{\infty }{k}_1^{n-j}{X}_j,\hfill \end{array}$$

where $X_j$ is a homogeneous polynomial of degree j in the variables $k_2,\cdots ,k_n.$

Using the Faber polynomial expansion (2) yield the following identities

$$(1+\beta e^{i\gamma })\frac{z{F}^{{}^{\prime }}\left(z\right)}{F\left(z\right)}-\beta e^{i\gamma }=(1+\beta e^{i\gamma })[1-\sum _{j=2}^{\infty }{F}_{j-1}({\delta }_2,{\delta }_3,\cdots ,{\delta }_j)z^{j-1}]-\beta e^{i\gamma },$$

(12)

and

$$(1+\beta e^{i\gamma })\frac{w{G}^{{}^{\prime }}\left(w\right)}{G\left(w\right)}-\beta e^{i\gamma }=(1+\beta e^{i\gamma })[1-\sum _{j=2}^{\infty }{F}_{j-1}({\zeta }_2,{\zeta }_3,\cdots ,{\zeta }_j)w^{j-1}]-\beta e^{i\gamma }.$$

(13)

Comparing the corresponding coefficients of (10) and (12) yields

$$(1+\beta e^{i\gamma })F_{n-1}({\delta }_2,{\delta }_3,\cdots ,{\delta }_n)=B_1{K}_{n-1}^{-1}(b_1,b_2,\cdots ,b_{n-1},B_1,B_2,B_3,\cdots ,B_{n-1})$$

(14)

and similarly, from (11) and (13), we have

$$(1+\beta e^{i\gamma })F_{n-1}({\zeta }_2,{\zeta }_3,\cdots ,{\zeta }_n)=B_1{K}_{n-1}^{-1}(c_1,c_2,\cdots ,c_{n-1},B_1,B_2,B_3,\cdots ,B_{n-1}).$$

(15)

Since $a_k=0$ for $2\le k\le n-1,$ by substituting ${\delta }_n=[1+\lambda (n-1)]a_n,{\zeta }_n=[1+\lambda (n-1)]d_n$ and $d_n=$$-a_n$ in (14) and (15), we have

$$(1+\beta e^{i\gamma })(n-1)[1+\lambda (n-1)]a_n=B_1{b}_{n-1}$$

and

$$-(1+\beta e^{i\gamma })(n-1)[1+\lambda (n-1)]a_n=B_1{c}_{n-1}.$$

By using (9), we conclude that

$$\left|a_n\right|\le \frac{B_1}{\left|1+\beta e^{i\gamma }\right|(n-1)[1+\lambda (n-1)]},$$

this completes the proof. □

To prove our next theorem, we shall need the following lemma.

Lemma 2.

Ref. [52] Let the function $\Phi \left(z\right)={\displaystyle \sum _{n=1}^{\infty }}{\Phi }_n{z}^n$ be a Schwarz function with $\left|\Phi \left(z\right)\right|<1$, $z\in U$. Then for $-\infty <\rho <\infty .$

$$\left|{\Phi }_2+\rho {\Phi }_1^2\right|\le \left\{\begin{array}{c}1-(1-\rho )\left|{\Phi }_1^2\right|\rho >0\\ \\ 1-(1+\rho )\left|{\Phi }_1^2\right|\rho \le 0\end{array}\right.$$

Theorem 2.

Let the function $f\in \sigma$ given by (1) be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ then

$$\left|a_2\right|\le \left\{\begin{array}{c}\frac{B_1\sqrt{B_1}}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+2\lambda -{\lambda }^2)B_1^2+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(B_1+B_2)}}(B_2\le 0,B_1+B_2\ge 0)\\ \frac{B_1\sqrt{B_1}}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+2\lambda -{\lambda }^2)B_1^2+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(B_1-B_2)}}(B_2>0,B_1-B_2\ge 0)\end{array}\right.$$

(16)

and

$$\left|a_3-a_2^2\right|\le \left\{\begin{array}{c}\frac{B_1}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}(B_1\ge \left|B_2\right|)\\ \frac{\left|B_2\right|}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}(B_1<\left|B_2\right|).\end{array}\right.$$

(17)

Proof. 

Replacing $n=2$ and 3 in (14) and (15), respectively, we find that

$$(1+\beta e^{i\gamma })(1+\lambda )a_2=B_1{b}_1,$$

(18)

$$(1+\beta e^{i\gamma })[2(1+2\lambda )a_3-{(1+\lambda )}^2{a}_2^2]=[B_1{b}_2+B_2{b}_1^2],$$

(19)

$$(-1+\beta e^{i\gamma })(1+\lambda )a_2=B_1{c}_1,$$

(20)

$$(1+\beta e^{i\gamma })\{-2(1+2\lambda )a_3+[4(1+2\lambda )-{(1+\lambda )}^2]a_2^2\}=[B_1{c}_2+B_2{c}_1^2].$$

(21)

It follows from (18) and (20) that

$$b_1=-c_1.$$

(22)

Adding (19) to (21) implies

$$2(1+\beta e^{i\gamma })[2(1+2\lambda )-{(1+\lambda )}^2]a_2^2=B_1(b_2+c_2)+B_2\left(b_1^2+c_1^2\right).$$

(23)

Taking the absolute values of both sides of the above equation, we get

$${\left|a_2\right|}^2\le \frac{B_1}{2\left|1+\beta e^{i\gamma }\right|(1+2\lambda -{\lambda }^2)}\left(\left|b_2+\frac{B_2}{B_1}{b}_1^2\right|+\left|c_2+\frac{B_2}{B_1}{c}_1^2\right|\right).$$

(24)

Case 1. Let $B_2\le 0.$ Applying Lemma 2 with $\rho =\frac{B_2}{B_1}\le 0$ and using (22) we obtain

$${\left|a_2\right|}^2\le \frac{B_1}{\left|1+\beta e^{i\gamma }\right|(1+2\lambda -{\lambda }^2)}\left(1-[\frac{B_1+B_2}{B_1}]{\left|b_1\right|}^2\right).$$

If $B_1+B_2\ge 0$, then (18) yields

$$\left|a_2\right|\le \frac{B_1\sqrt{B_1}}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+2\lambda -{\lambda }^2)B_1^2+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(B_1+B_2)}}.$$

(25)

Case 2. Let $B_2>0.$ Applying Lemma 2 with $\rho =\frac{B_2}{B_1}>0$ and using (22), we obtain

$$\left|a_2^2\right|\le \frac{B_1}{\left|1+\beta e^{i\gamma }\right|[2(1+2\lambda )-{(1+\lambda )}^2]}\left(1-[\frac{B_1-B_2}{B_1}]{\left|b_1\right|}^2\right).$$

If $B_1-B_2\ge 0,$ then (18) gives

$$\left|a_2\right|\le \frac{B_1\sqrt{B_1}}{\sqrt{\left|1+\beta e^{i\gamma }\right|[1+2\lambda -{\lambda }^2]B_1^2+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(B_1-B_2)}}.$$

(26)

From (25) and (26) we obtain the desired estimate of $\left|a_2\right|$ given by (16). Next, from (19) and (21), we have

$$\left|a_3-a_2^2\right|\le \frac{B_1}{4(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}\left(\left|b_2+\frac{B_2}{B_1}{b}_1^2\right|+\left|c_2+\frac{B_2}{B_1}{c}_1^2\right|\right).$$

(27)

Let $B_2\le 0$. Applying Lemma 2 for $\rho =\frac{B_2}{B_1}\le 0,$ we get

$$\left|a_3-a_2^2\right|\le \frac{B_1}{4(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}\left(\left[1-\frac{B_1+B_2}{B_1}{\left|b_1\right|}^2\right]+\left[1-\frac{B_1+B_2}{B_1}{\left|c_1\right|}^2\right]\right).$$

(28)

If $B_1+B_2\ge 0,$ then (28) gives

$$\left|a_3-a_2^2\right|\le \frac{B_1}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If $B_1+B_2<0$, then (9) and (28) lead to

$$\left|a_3-a_2^2\right|\le \frac{B_1}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}\left[1-\frac{B_1+B_2}{B_1}\right]=-\frac{B_2}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

Let $B_2>0$. Applying Lemma 2 for $\rho =\frac{B_2}{B_1}>0,$ (27) gives

$$\left|a_3-a_2^2\right|\le \frac{B_1}{4(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}\left(\left[1-\frac{B_1-B_2}{B_1}{\left|b_1\right|}^2\right]+\left[1-\frac{B_1-B_2}{B_1}{\left|c_1\right|}^2\right]\right).$$

(29)

If $B_1-B_2\ge 0,$ then (29) gives

$$\left|a_3-a_2^2\right|\le \frac{B_1}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If $B_1-B_2<0$, then from (9) and (29) we have

$$\left|a_3-a_2^2\right|\le \frac{B_1}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}\left[1-\frac{B_1-B_2}{B_1}\right]=\frac{B_2}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

Which is the second part of assertion (17). This completes the proof of Theorem 2. □

Remark 1.

If we take $\beta =0$ in Theorem 2 we obtain that the bounds on $\left|a_3-a_2^2\right|$ given by Deniz et al. [52] when $\gamma =1$.

If we set

$$\varphi \left(z\right)=\frac{1+Az}{1+Bz}=1+(A-B)z-B(A-B)z^2+B^2(A-B)z^3\cdots$$

in Definition 3 of the bi-univalent function class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $M_{\sigma }(\lambda ,\beta ,\gamma ,A,B)$ given by Definition 5 below.

Definition 5.

A function $f\in \sigma$ given by (1) is said to be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,A,B),$$-1\le B<A\le 1,$ if the following conditions are satisfied:

$$(1+\beta e^{i\gamma })\frac{z{f}^{{}^{\prime }}\left(z\right)+\lambda z^2{f}^{{}^{{}^{″}}}\left(z\right)}{(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)}-\beta e^{i\gamma }\prec \frac{1+Az}{1+Bz}(z\in U)$$

and

$$(1+\beta e^{i\gamma })\frac{z{g}^{{}^{\prime }}\left(\omega \right)+\lambda z^2{g}^{{}^{{}^{″}}}\left(\omega \right)}{(1-\lambda )g\left(\omega \right)+\lambda z{g}^{{}^{\prime }}\left(\omega \right)}-\beta e^{i\gamma }\prec \frac{1+A\omega }{1+B\omega }(\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 5 in Theorems 1 and 2, respectively, we get the following corollaries.

Corollary 1.

Let the function $f\in M_{\sigma }(\lambda ,\beta ,\gamma ,A,B)$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{(A-B)}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 2.

If the function $f\in \sigma$ given by (1) be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,A,B)$, then

$$\left|a_2\right|\le \left\{\begin{array}{c}\frac{(A-B)}{\sqrt{\left|1+\beta e^{i\gamma }\right|[1+2\lambda -{\lambda }^2](A-B)+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(1-B)}}(B\ge 0)\\ \frac{(A-B)}{\sqrt{\left|1+\beta e^{i\gamma }\right|[1+2\lambda -{\lambda }^2](A-B)+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(1+B)}}(-1\le B<0)\end{array}\right.$$

and

$$\left|a_3-a_2^2\right|\le \frac{A-B}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If we set

$$\varphi \left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\delta }=1+2\delta z+2{\delta }^2{z}^2+\cdots (0<\delta \le 1,z\in U)$$

in Definition 3 of the bi-univalent function class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $M_{\sigma }(\lambda ,\beta ,\gamma ,\delta )$ given by Definition 6 below.

Definition 6.

Let $0<\delta \le 1.$ A function $f\in \sigma$ given by (1) is said to be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,\delta ),$ if the following conditions are satisfied:

$$\left|arg\left((1+\beta e^{i\gamma })\frac{z{f}^{{}^{\prime }}\left(z\right)+\lambda z^2{f}^{{}^{{}^{″}}}\left(z\right)}{(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)}-\beta e^{i\gamma }\right)\right|\le \frac{\pi }{2}\delta (z\in U)$$

and

$$\left|arg\left((1+\beta e^{i\gamma })\frac{z{g}^{{}^{\prime }}\left(\omega \right)+\lambda z^2{g}^{{}^{{}^{″}}}\left(\omega \right)}{(1-\lambda )g\left(\omega \right)+\lambda z{g}^{{}^{\prime }}\left(\omega \right)}-\beta e^{i\gamma }\right)\right|\le \frac{\pi }{2}\delta (\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 6 in Theorems 1 and 2, respectively, we get the following corollaries.

Corollary 3.

Let the function $f\in$$M_{\sigma }(\lambda ,\beta ,\gamma ,\delta )$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{2\delta }{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 4.

Let $0<\delta \le 1.$ If the function $f\in \sigma$ given by (1) be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,\delta )$, then

$$\left|a_2\right|\le \frac{2\delta }{\sqrt{\left|1+\beta e^{i\gamma }\right|[1+2\lambda -{\lambda }^2]2\delta +{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(1-\delta )}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{\delta }{(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If we set

$$\varphi \left(z\right)=\frac{1+(1-2\upsilon )z}{1-z}=1+2(1-\upsilon )z+2(1-\upsilon )z^2+\cdots (0\le \upsilon <1,z\in U)$$

in Definition 3 of the bi-univalent function class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $M_{\sigma }^{\upsilon }(\lambda ,\beta ,\gamma )$ given by Definition 7 below.

Definition 7.

Let $0\le \upsilon <1.$ A function $f\in \sigma$ given by (1) is said to be in the class $M_{\sigma }^{\upsilon }(\lambda ,\beta ,\gamma )$ if the following conditions hold true:

$$\Re \left((1+\beta e^{i\gamma })\frac{z{f}^{{}^{\prime }}\left(z\right)+\lambda z^2{f}^{{}^{{}^{″}}}\left(z\right)}{(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)}-\beta e^{i\gamma }\right)>\upsilon (z\in U)$$

and

$$\Re \left((1+\beta e^{i\gamma })\frac{z{g}^{{}^{\prime }}\left(\omega \right)+\lambda z^2{g}^{{}^{{}^{″}}}\left(\omega \right)}{(1-\lambda )g\left(\omega \right)+\lambda z{g}^{{}^{\prime }}\left(\omega \right)}-\beta e^{i\gamma }\right)>\upsilon (\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 7 in Theorems 1 and 2, respectively, we get the following corollaries.

Corollary 5.

Let the function $f\in$$M_{\sigma }^{\upsilon }(\lambda ,\beta ,\gamma )$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{2(1-\upsilon )}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 6.

Let the function $f\in$$M_{\sigma }^{\upsilon }(\lambda ,\beta ,\gamma )$ be given by (1). Then

$$\left|a_2\right|\le \sqrt{\frac{2(1-\upsilon )}{(1+2\lambda -{\lambda }^2)\left|1+\beta e^{i\gamma }\right|}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{(1-\upsilon )}{(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If we set

$$\varphi \left(z\right)=\sqrt{1+z}=1+\frac{1}{2}z-\frac{1}{8}{z}^2+\cdots (z\in U)$$

in Definition 3 of the bi-univalent function class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $M_{L\sigma }(\lambda ,\beta ,\gamma )$ given by Definition 8 below.

Definition 8.

A function $f\in \sigma$ given by (1) is said to be in the class $M_{L\sigma }(\lambda ,\beta ,\gamma ),$ if the following conditions are satisfied:

$$\left|{\left((1+\beta e^{i\gamma })\frac{z{f}^{{}^{\prime }}\left(z\right)+\lambda z^2{f}^{{}^{{}^{″}}}\left(z\right)}{(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)}-\beta e^{i\gamma }\right)}^2-1\right|<1(z\in U)$$

and

$$\left|{\left((1+\beta e^{i\gamma })\frac{z{g}^{{}^{\prime }}\left(\omega \right)+\lambda z^2{g}^{{}^{{}^{″}}}\left(\omega \right)}{(1-\lambda )g\left(\omega \right)+\lambda z{g}^{{}^{\prime }}\left(\omega \right)}-\beta e^{i\gamma }\right)}^2-1\right|<1(\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 8 in Theorems 1 and 2, respectively, we get the following corollaries.

Corollary 7.

Let the function $f\in$$M_{L\sigma }(\lambda ,\beta ,\gamma )$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{1}{2(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 8.

If the function $f\in \sigma$ given by (1) be in the class $M_{L\sigma }(\lambda ,\beta ,\gamma )$, then

$$\left|a_2\right|\le \frac{1}{\sqrt{2\left|1+\beta e^{i\gamma }\right|[1+2\lambda -{\lambda }^2]+3{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{1}{4(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If we set

$$\varphi \left(z\right)=z+\sqrt{1+z^2}=1+z+\frac{1}{2}{z}^2-\frac{1}{8}{z}^4\cdots (z\in U),$$

in Definition 3 of the bi-univalent function class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $M_{\sigma }^{\Delta }(\lambda ,\beta ,\gamma )$ given by Definition 9 below.

Definition 9.

A function $f\in \sigma$ given by (1) is said to be in the class $M_{\sigma }^{\Delta }(\lambda ,\beta ,\gamma )$ if the following conditions are satisfied:

$$\begin{array}{cc}\hfill & \left|{\left((1+\beta e^{i\gamma })\frac{z{f}^{{}^{\prime }}\left(z\right)+\lambda z^2{f}^{{}^{{}^{″}}}\left(z\right)}{(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)}-\beta e^{i\gamma }\right)}^2-1\right|\hfill \\ \hfill & <2\left|(1+\beta e^{i\gamma })\frac{z{f}^{{}^{\prime }}\left(z\right)+\lambda z^2{f}^{{}^{{}^{″}}}\left(z\right)}{(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)}-\beta e^{i\gamma }\right|(z\in U)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \left|{\left((1+\beta e^{i\gamma })\frac{z{g}^{{}^{\prime }}\left(\omega \right)+\lambda z^2{g}^{{}^{{}^{″}}}\left(\omega \right)}{(1-\lambda )g\left(\omega \right)+\lambda z{g}^{{}^{\prime }}\left(\omega \right)}-\beta e^{i\gamma }\right)}^2-1\right|\hfill \\ \hfill & <2\left|(1+\beta e^{i\gamma })\frac{z{g}^{{}^{\prime }}\left(\omega \right)+\lambda z^2{g}^{{}^{{}^{″}}}\left(\omega \right)}{(1-\lambda )g\left(\omega \right)+\lambda z{g}^{{}^{\prime }}\left(\omega \right)}-\beta e^{i\gamma }\right|(\omega \in U)\hfill \end{array}$$

where $g=f^{-1}.$

Using the parameter setting of Definition 9 in Theorems 1 and 2, respectively, we get the following corollaries.

Corollary 9.

Let the function $f\in$$M_{\sigma }^{\Delta }(\lambda ,\beta ,\gamma )$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{1}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 10.

If the function $f\in \sigma$ given by (1) be in the class $M_{\sigma }^{\Delta }(\lambda ,\beta ,\gamma )$, then

$$\left|a_2\right|\le \sqrt{\frac{2}{2\left|1+\beta e^{i\gamma }\right|[1+2\lambda -{\lambda }^2]+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{1}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If we set

$$\begin{array}{cc}\hfill \varphi \left(z\right)& =\frac{1}{{(1-z)}^s}=1+sz+\frac{s(s+1)}{2}{z}^2+\frac{s(s+1)(s+2)}{6}{z}^3\cdots \hfill \\ \hfill & =1+{\displaystyle \sum _{n=1}^{\infty }}\frac{s(s+1)\dots (s+n-1)}{n!}{z}^n(z\in U),\hfill \end{array}$$

in Definition 3 of the bi-univalent function class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $M_{\sigma }(\lambda ,\beta ,\gamma ,s)$ given by Definition 10 below.

Definition 10.

Let $0<s\le 1.$ A function $f\in \sigma$ given by (1) is said to be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,s),$ if the following conditions are satisfied:

$$(1+\beta e^{i\gamma })\frac{z{f}^{{}^{\prime }}\left(z\right)+\lambda z^2{f}^{{}^{{}^{″}}}\left(z\right)}{(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)}-\beta e^{i\gamma }\prec \frac{1}{{(1-z)}^s}(z\in U)$$

and

$$(1+\beta e^{i\gamma })\frac{z{g}^{{}^{\prime }}\left(\omega \right)+\lambda z^2{g}^{{}^{{}^{″}}}\left(\omega \right)}{(1-\lambda )g\left(\omega \right)+\lambda z{g}^{{}^{\prime }}\left(\omega \right)}-\beta e^{i\gamma }\prec \frac{1}{{(1-\omega )}^s}(\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 10 in Theorems 1 and 2, respectively, we get the following corollaries.

Corollary 11.

Let the function $f\in$$M_{\sigma }(\lambda ,\beta ,\gamma ,s)$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{s}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 12.

If the function $f\in \sigma$ given by (1) be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,s)$, then

$$\left|a_2\right|\le \frac{\sqrt{2}s}{\sqrt{\left|1+\beta e^{i\gamma }\right|[1+2\lambda -{\lambda }^2]2s+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(1-s)}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{s}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If we set

$$\varphi \left(z\right)=e^z=1+z+\frac{1}{2}{z}^2+\frac{1}{6}{z}^3+\cdots (z\in U),$$

in Definition 3 of the bi-univalent function class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $M_{\sigma e}(\lambda ,\beta ,\gamma )$ given by Definition 11 below.

Definition 11.

A function $f\in \sigma$ given by (1) is said to be in the class $M_{\sigma e}(\lambda ,\beta ,\gamma )$ if the following conditions are satisfied:

$$\left|log\left((1+\beta e^{i\gamma })\frac{z{f}^{{}^{\prime }}\left(z\right)+\lambda z^2{f}^{{}^{{}^{″}}}\left(z\right)}{(1-\lambda )f\left(z\right)+\lambda z{f}^{{}^{\prime }}\left(z\right)}-\beta e^{i\gamma }\right)\right|<1(z\in U)$$

and

$$\left|log\left((1+\beta e^{i\gamma })\frac{z{g}^{{}^{\prime }}\left(\omega \right)+\lambda z^2{g}^{{}^{{}^{″}}}\left(\omega \right)}{(1-\lambda )g\left(\omega \right)+\lambda z{g}^{{}^{\prime }}\left(\omega \right)}-\beta e^{i\gamma }\right)\right|<1(\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 11 in Theorems 1 and 2, respectively, we get the following corollaries.

Corollary 13.

Let the function $f\in$$M_{\sigma }(\lambda ,\beta ,\gamma ,s),$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{1}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 14.

If the function $f\in \sigma$ given by (1) be in the class $M_{\sigma }(\lambda ,\beta ,\gamma ,s)$, then

$$\left|a_2\right|\le \sqrt{\frac{2}{2\left|1+\beta e^{i\gamma }\right|[1+2\lambda -{\lambda }^2]+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{1}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

3. Coefficient Estimates for the Class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$

Theorem 3.

Let the function $f\in \sigma$ given by (1) be in the class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$. If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{B_1}{\left|1+\beta e^{i\gamma }\right|(n-1)[1+(n-1)\lambda ]},n\ge 3.$$

Proof. 

Since, both functions f and its inverse map $g=f^{-1}$ are in $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ by the definition of subordination, there are analytic functions $u,v:U\to U$ given by (8) such that

$$(1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda \frac{{\left(z{f}^{{}^{\prime }}\left(z\right)\right)}^{{}^{\prime }}}{f^{{}^{\prime }}\left(z\right)}\right]-\beta e^{i\gamma }=\varphi \left(u\left(z\right)\right)$$

and

$$(1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda \frac{{\left(w{g}^{{}^{\prime }}\left(w\right)\right)}^{{}^{\prime }}}{g^{{}^{\prime }}\left(w\right)}\right]-\beta e^{i\gamma }=\varphi \left(v\left(\omega \right)\right).$$

Now, from (2), we get that

$$\begin{array}{cc}\hfill & (1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda \frac{{\left(z{f}^{{}^{\prime }}\left(z\right)\right)}^{{}^{\prime }}}{f^{{}^{\prime }}\left(z\right)}\right]-\beta e^{i\gamma }\hfill \\ \hfill & =1-(1+\beta e^{i\gamma })\sum _{j=2}^{\infty }[(1-\lambda )F_{j-1}(a_2,a_3,\cdots ,a_j)\hfill \\ \hfill & +\lambda F_{j-1}(2a_2,3a_3,\cdots ,j{a}_j)]z^{j-1},\hfill \end{array}$$

(30)

and

$$\begin{array}{cc}\hfill & (1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda \frac{{\left(w{g}^{{}^{\prime }}\left(w\right)\right)}^{{}^{\prime }}}{g^{{}^{\prime }}\left(w\right)}\right]-\beta e^{i\gamma }\hfill \\ \hfill & =1-(1+\beta e^{i\gamma })\sum _{j=2}^{\infty }[(1-\lambda )F_{j-1}(d_2,d_3,\cdots ,d_j)\hfill \\ \hfill & +\lambda F_{j-1}(2d_2,3d_3,\cdots ,j{d}_j)]w^{j-1},\hfill \end{array}$$

(31)

where $d_n=\frac{1}{n}{K}_{n-1}^{-n}(a_2,a_3,\cdots ,a_n).$ Now, upon comparing the corresponding coefficients in (10) and (30), we get

$$\begin{array}{cc}\hfill & (1+\beta e^{i\gamma })[(1-\lambda )F_{n-1}(a_2,a_3,\cdots ,a_n)+\lambda F_{n-1}(2a_2,3a_3,\cdots ,n{a}_n)]\hfill \\ \hfill & =B_1{K}_{n-1}^{-1}(b_1,b_2,\cdots ,b_{n-1},B_1,B_1,B_2,B_3,\cdots ,B_{n-1})\hfill \end{array}$$

(32)

and similarly, from (11) and (31), we have

$$\begin{array}{cc}\hfill & (1+\beta e^{i\gamma })[(1-\lambda )F_{n-1}(d_2,d_3,\cdots ,d_n)+\lambda F_{n-1}(2d_2,3d_3,\cdots ,n{d}_n)]\hfill \\ \hfill & =B_1{K}_{n-1}^{-1}(c_1,c_2,\cdots ,c_{n-1},B_1,B_2,B_3,\cdots ,B_{n-1}).\hfill \end{array}$$

(33)

Since $a_k=0$ for $2\le k\le n-1,$ by using $d_n=-a_n$ and $F_{n-1}(a_2,a_3,\cdots ,a_n)=-(n-1)a_n$, we have

$$(1+\beta e^{i\gamma })(n-1)[1+(n-1)\lambda ]a_n=B_1{b}_{n-1}$$

(34)

and

$$-(1+\beta e^{i\gamma })(n-1)[1+(n-1)\lambda ]a_n=B_1{c}_{n-1}.$$

(35)

By using (9), we conclude that

$$\left|a_n\right|\le \frac{B_1}{\left|1+\beta e^{i\gamma }\right|(n-1)[1+(n-1)\lambda ]}.$$

□

Remark 2.

If we take $\beta =0$ in Theorem 3, then we have the results which were given by Zireh et al. [51] when $\phi \left(z\right)=1.$

Theorem 4.

If the function $f\in \sigma$ given by (1) be in the class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$, then

$$\left|a_2\right|\le \left\{\begin{array}{c}\frac{B_1\sqrt{B_1}}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+\lambda )B_1^2+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(B_1+B_2)}}(B_2\le 0,B_1+B_2\ge 0)\\ \frac{B_1\sqrt{B_1}}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+\lambda )B_1^2+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(B_1-B_2)}}(B_2>0,B_1-B_2\ge 0)\end{array}\right.,$$

(36)

and

$$\left|a_3-a_2^2\right|\le \left\{\begin{array}{c}\frac{B_1}{2\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}(B_1\ge \left|B_2\right|)\\ \frac{\left|B_2\right|}{2\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}(B_1<\left|B_2\right|).\end{array}\right.$$

(37)

Proof. 

Letting $n=2$ and 3 in (32) and (33), respectively, we find that

$$(1+\beta e^{i\gamma })(1+\lambda )a_2=B_1{b}_1,$$

(38)

$$(1+\beta e^{i\gamma })\left[2(1+2\lambda )a_3-(1+3\lambda )a_2^2\right]=B_1{b}_2+B_2{b}_1^2,$$

(39)

$$-(1+\beta e^{i\gamma })(1+\lambda )a_2=B_1{c}_1,$$

(40)

$$(1+\beta e^{i\gamma })\{-2(1+2\lambda )a_3+[4(1+2\lambda )-(1+3\lambda )]a_2^2\}=B_1{c}_2+B_2{c}_1^2.$$

(41)

Equations (38) and (40) lead to

$$b_1=-c_1.$$

(42)

Adding (39) and (41) yields

$$2(1+\beta e^{i\gamma })(1+\lambda )a_2^2=B_1(b_2+c_2)+B_2\left(b_1^2+c_1^2\right)$$

(43)

or

$$\left|a_2^2\right|\le \frac{B_1}{2\left|1+\beta e^{i\gamma }\right|(1+\lambda )}\left(\left|b_2+\frac{B_2}{B_1}{b}_1^2\right|+\left|c_2+\frac{B_2}{B_1}{c}_1^2\right|\right).$$

First, let $B_2\le 0$. Applying Lemma 2 with $\rho =\frac{B_2}{B_1}\le 0$ and using (42), we get

$$\left|a_2^2\right|\le \frac{B_1}{\left|1+\beta e^{i\gamma }\right|(1+\lambda )}\left(1-[\frac{B_1+B_2}{B_1}]\left|b_1^2\right|\right)$$

(44)

If $B_1+B_2\ge 0$, then (38) yields

$$\left|a_2\right|\le \frac{B_1\sqrt{B_1}}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+\lambda )B_1^2+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(B_1+B_2)}}$$

(45)

Similarly, for $B_2>0(\rho =\frac{B_2}{B_1}>0,B_1-B_2\ge 0)$, we have

$$\left|a_2\right|\le \frac{B_1\sqrt{B_1}}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+\lambda )B_1^2+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(B_1-B_2)}}$$

(46)

From (45) and (46) we obtain the desired estimate of $\left|a_2\right|$ given by (36).

Next, in order to find the bound on $\left|a_3-a_2^2\right|$, by subtracting (41) from (39), we have

$$\left|a_3-a_2^2\right|\le \frac{B_1}{4\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}\left(\left|b_2+\frac{B_2}{B_1}{b}_1^2\right|+\left|c_2+\frac{B_2}{B_1}{c}_1^2\right|\right).$$

(47)

Let $B_2\le 0$. Applying Lemma 2 with $\rho =\frac{B_2}{B_1}\le 0,$ we get

$$\left|a_3-a_2^2\right|\le \frac{B_1}{4\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}\left(\left[1-\frac{B_1+B_2}{B_1}{\left|b_1\right|}^2\right]+\left[1-\frac{B_1+B_2}{B_1}{\left|c_1\right|}^2\right]\right).$$

(48)

If $B_1+B_2\ge 0,$ then (48) gives $\left|a_3-a_2^2\right|\le \frac{B_1}{2\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}.$

If $B_1+B_2<0$, then (9) and (48) give

$$\left|a_3-a_2^2\right|\le \frac{B_1}{2\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}\left[1-\frac{B_1+B_2}{B_1}\right]=-\frac{B_2}{2\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}.$$

Let $B_2>0$. Applying Lemma 2 with $\rho =\frac{B_2}{B_1}>0$, (47) gives

$$\left|a_3-a_2^2\right|\le \frac{B_1}{4\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}\left(\left[1-\frac{B_1-B_2}{B_1}{\left|b_1\right|}^2\right]+\left[1-\frac{B_1-B_2}{B_1}{\left|c_1\right|}^2\right]\right).$$

(49)

If $B_1-B_2\ge 0,$ then (49) gives

$$\left|a_3-a_2^2\right|\le \frac{B_1}{2\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}.$$

If $B_1-B_2<0$, then from (9) and (49) we get

$$\left|a_3-a_2^2\right|\le \frac{B_1}{2\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}\left[1-\frac{B_1-B_2}{B_1}\right]=\frac{B_2}{2\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}.$$

This completes the proof of Theorem 3. □

Remark 3.

If we set $\beta =0$ in Theorem 4, then we obtain the results of Goyal and Kumar [50] when $\phi \left(z\right)=1.$

If we set $\varphi \left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\delta }(0<\delta \le 1,z\in U)$ in Definition 4 of the bi-univalent function class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$, we obtain a new class $S_{\sigma }^{\delta }(\lambda ,\beta ,\gamma )$ given by Definition 12 below.

Definition 12.

Let $0<\delta \le 1.$ A function $f\in \sigma$ given by (1) is said to be in the class $S_{\sigma }^{\delta }(\lambda ,\beta ,\gamma )$ if the following subordinations hold:

$$\left|arg\left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda (1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)})\right]-\beta e^{i\gamma }\right)\right|\le \frac{\pi }{2}\delta (z\in U)$$

and

$$\left|arg\left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda (1+\frac{w{g}^{{}^{″}}\left(w\right)}{g^{{}^{\prime }}\left(w\right)})\right]-\beta e^{i\gamma }\right)\right|\le \frac{\pi }{2}\delta (\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 12 in Theorems 3 and 4, respectively, we get the following corollaries.

Corollary 15.

Let the function $f\in S_{\sigma }^{\delta }(\lambda ,\beta ,\gamma )$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{2\delta }{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 16.

Let $0<\gamma \le 1.$ If the function $f\in \sigma$ given by (1) be in the class $S_{\sigma }^{\delta }(\lambda ,\beta ,\gamma ),$ then

$$\left|a_2\right|\le \frac{2\delta }{\sqrt{2\left|1+\beta e^{i\gamma }\right|(1+\lambda )\delta +{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(1-\delta )}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{\delta }{\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}.$$

If we set $\varphi \left(z\right)=\frac{1+(1-2\upsilon )z}{1-z}(0\le \upsilon <1,z\in U)$ in Definition 4 of the bi-univalent function class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$, we obtain a new class $S_{\sigma }(\lambda ,\beta ,\gamma ,\upsilon )$ given by Definition 13 below.

Definition 13.

Let $0\le \upsilon <1.$ A function $f\in \sigma$ given by (1) is said to be in the class $S_{\sigma }(\lambda ,\beta ,\gamma ,\upsilon )$, if the following conditions are satisfied:

$$\Re \left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda (1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)})\right]-\beta e^{i\gamma }\right)>\upsilon (z\in U)$$

and

$$\Re \left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda (1+\frac{w{g}^{{}^{″}}\left(w\right)}{g^{{}^{\prime }}\left(w\right)})\right]-\beta e^{i\gamma }\right)>\upsilon (\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 13 in Theorems 3 and 4, respectively, we get the following corollaries.

Corollary 17.

Let the function $f\in S_{\sigma }(\lambda ,\beta ,\gamma ,\upsilon )$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{2(1-\upsilon )}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 18.

Let $0\le \upsilon <1.$ If the function $f\in$$S_{\sigma }(\lambda ,\beta ,\gamma ,\upsilon )$be of the form (1), then

$$\left|a_2\right|\le \sqrt{\frac{2(1-\upsilon )}{\left|1+\beta e^{i\gamma }\right|(1+\lambda )}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{(1-\upsilon )}{\left|1+\beta e^{i\gamma }\right|(1+2\lambda )}.$$

If we set $\varphi \left(z\right)=\frac{1+Az}{1+Bz}$ in Definition 4 of the bi-univalent function class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$, we obtain a new class $S_{\sigma }(\lambda ,\beta ,\gamma ,A,B)$ given by Definition 14 below.

Definition 14.

A function $f\in \sigma$ given by (1) is said to be in the class $S_{\sigma }(\lambda ,\beta ,\gamma ,A,B),$$-1\le B<A\le 1,$ if the following conditions are satisfied:

$$(1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda (1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)})\right]-\beta e^{i\gamma }\prec \frac{1+Az}{1+Bz}(z\in U)$$

and

$$(1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda (1+\frac{w{g}^{{}^{″}}\left(w\right)}{g^{{}^{\prime }}\left(w\right)})\right]-\beta e^{i\gamma }\prec \frac{1+A\omega }{1+B\omega }(\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 14 in Theorems 3 and 4, respectively, we get the following corollaries.

Corollary 19.

Let the function $f\in S_{\sigma }(\lambda ,\beta ,\gamma ,A,B)$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{(A-B)}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 20.

If the function $f\in \sigma$ given by (1) be in the class $S_{\sigma }(\lambda ,\beta ,\gamma ,A,B)$, then

$$\left|a_2\right|\le \left\{\begin{array}{c}\frac{(A-B)}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+\lambda )(A-B)+{\left|1+\beta e^{i\gamma }\right|}^2(1-B)}}(B\ge 0)\\ \frac{(A-B)}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+\lambda )(A-B)+{\left|1+\beta e^{i\gamma }\right|}^2(1+B)}}(-1\le B<0)\end{array}\right.$$

and

$$\left|a_3-a_2^2\right|\le \frac{A-B}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

Remark 4.

If we put $\beta =\lambda =0$ in Corollaries 19 and 20, then we obtain the results of Hamidi and Jahangiri [49]

If we set $\varphi \left(z\right)=\sqrt{1+z}$ in Definition 4 of the bi-univalent function class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $S_{L\sigma }(\lambda ,\beta ,\gamma )$ given by Definition 15 below.

Definition 15.

A function $f\in \sigma$ given by (1) is said to be in the class $S_{L\sigma }(\lambda ,\beta ,\gamma )$ if the following conditions are satisfied:

$$\left|{\left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda (1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)})\right]-\beta e^{i\gamma }\right)}^2-1\right|<1(z\in U)$$

and

$$\left|{\left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda (1+\frac{w{g}^{{}^{″}}\left(w\right)}{g^{{}^{\prime }}\left(w\right)})\right]-\beta e^{i\gamma }\right)}^2-1\right|<1(\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 15 in Theorems 3 and 4, respectively, we get the following corollaries.

Corollary 21.

Let the function $f\in$$S_{L\sigma }(\lambda ,\beta ,\gamma ),$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{1}{2(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 22.

If the function $f\in \sigma$ given by (1) be in the class $S_{L\sigma }(\lambda ,\beta ,\gamma )$, then

$$\left|a_2\right|\le \frac{1}{\sqrt{2\left|1+\beta e^{i\gamma }\right|(1+\lambda )+3{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{1}{4(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If we set $\varphi \left(z\right)=z+\sqrt{1+z^2}$ in Definition 4 of the bi-univalent function class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $S_{\sigma }^{\Delta }(\lambda ,\beta ,\gamma )$ given by Definition 16 below.

Definition 16.

A function $f\in \sigma$ given by (1) is said to be in the class $S_{\sigma }^{\Delta }(\lambda ,\beta ,\gamma )$ if the following conditions are satisfied:

$$\begin{array}{cc}\hfill & \left|{\left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda (1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)})\right]-\beta e^{i\gamma }\right)}^2-1\right|\hfill \\ \hfill & <2\left|(1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda (1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)})\right]-\beta e^{i\gamma }\right|(z\in U)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \left|{\left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda (1+\frac{w{g}^{{}^{″}}\left(w\right)}{g^{{}^{\prime }}\left(w\right)})\right]-\beta e^{i\gamma }\right)}^2-1\right|\hfill \\ \hfill & <2\left|(1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda (1+\frac{w{g}^{{}^{″}}\left(w\right)}{g^{{}^{\prime }}\left(w\right)})\right]-\beta e^{i\gamma }\right|(\omega \in U)\hfill \end{array}$$

where $g=f^{-1}.$

Using the parameter setting of Definition 9 in Theorems 3 and 4, respectively, we get the following corollaries.

Corollary 23.

Let the function $f\in$$S_{\sigma }^{\Delta }(\lambda ,\beta ,\gamma ),$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{1}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 24.

If the function $f\in \sigma$ given by (1) be in the class $S_{\sigma }^{\Delta }(\lambda ,\beta ,\gamma )$, then

$$\left|a_2\right|\le \sqrt{\frac{2}{2\left|1+\beta e^{i\gamma }\right|(1+\lambda )+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{1}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If we set $\varphi \left(z\right)=\frac{1}{{(1-z)}^s}$ in Definition 4 of the bi-univalent function class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $S_{\sigma }(\lambda ,\beta ,\gamma ,s)$ given by Definition 17 below.

Definition 17.

Let $0<s\le 1.$ A function $f\in \sigma$ given by (1) is said to be in the class $S_{\sigma }(\lambda ,\beta ,\gamma ,s),$ if the following conditions are satisfied:

$$(1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda (1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)})\right]-\beta e^{i\gamma }\prec \frac{1}{{(1-z)}^s}(z\in U)$$

and

$$(1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda (1+\frac{w{g}^{{}^{″}}\left(w\right)}{g^{{}^{\prime }}\left(w\right)})\right]-\beta e^{i\gamma }\prec \frac{1}{{(1-\omega )}^s}(\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 17 in Theorems 3 and 4, respectively, we get the following corollaries.

Corollary 25.

Let the function $f\in$$S_{\sigma }(\lambda ,\beta ,\gamma ,s)$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{s}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 26.

If the function $f\in \sigma$ given by (1) be in the class $S_{\sigma }(\lambda ,\beta ,\gamma ,s)$, then

$$\left|a_2\right|\le \frac{\sqrt{2}s}{\sqrt{\left|1+\beta e^{i\gamma }\right|(1+\lambda )2s+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2(1-s)}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{s}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$

If we set $\varphi \left(z\right)=e^z$ in Definition 4 of the bi-univalent function class $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $S_{\sigma e}(\lambda ,\beta ,\gamma )$ given by Definition 18 below.

Definition 18.

A function $f\in \sigma$ given by (1) is said to be in the class $S_{\sigma e}(\lambda ,\beta ,\gamma ),$ if the following conditions are satisfied:

$$\left|log\left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}+\lambda (1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)})\right]-\beta e^{i\gamma }\right)\right|<1(z\in U)$$

and

$$\left|log\left((1+\beta e^{i\gamma })\left[(1-\lambda )\frac{w{g}^{{}^{\prime }}\left(w\right)}{g\left(w\right)}+\lambda (1+\frac{w{g}^{{}^{″}}\left(w\right)}{g^{{}^{\prime }}\left(w\right)})\right]-\beta e^{i\gamma }\right)\right|<1(\omega \in U),$$

where $g=f^{-1}.$

Using the parameter setting of Definition 18 in Theorems 3 and 4, respectively, we get the following corollaries.

Corollary 27.

Let the function $f\in$$S_{\sigma }(\lambda ,\beta ,\gamma ,s),$ be given by (1). If $a_k=0$ for $2\le k\le n-1,$ then

$$\left|a_n\right|\le \frac{1}{(n-1)[1+\lambda (n-1)]\left|1+\beta e^{i\gamma }\right|},n\ge 3.$$

Corollary 28.

If the function $f\in \sigma$ given by (1) be in the class $S_{\sigma }(\lambda ,\beta ,\gamma ,s)$, then

$$\left|a_2\right|\le \sqrt{\frac{2}{2\left|1+\beta e^{i\gamma }\right|(1+\lambda )+{\left|1+\beta e^{i\gamma }\right|}^2{(1+\lambda )}^2}}$$

and

$$\left|a_3-a_2^2\right|\le \frac{1}{2(1+2\lambda )\left|1+\beta e^{i\gamma }\right|}.$$