Coefficients of a Comprehensive Subclass of Meromorphic Bi-Univalent Functions Associated with the Faber Polynomial Expansion

Abstract

In this paper, we introduce a new comprehensive subclass ${\Sigma }_B(\lambda ,\mu ,\beta )$ of meromorphic bi-univalent functions in the open unit disk $\mathbb{U}$. We also find the upper bounds for the initial Taylor-Maclaurin coefficients $|b_0|$, $|b_1|$ and $|b_2|$ for functions in this comprehensive subclass. Moreover, we obtain estimates for the general coefficients $|b_n|(n\geqq 1)$ for functions in the subclass ${\Sigma }_B(\lambda ,\mu ,\beta )$ by making use of the Faber polynomial expansion method. The results presented in this paper would generalize and improve several recent works on the subject.

1. Introduction

Let $\mathcal{A}$ denote the class of functions f of the form:

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

(1)

which are analytic in the open unit disk

$$\mathbb{U}=\{z:z\in \mathbb{C}\mathrm{and}|z|<1\}.$$

We also let $\mathcal{S}$ be the class of functions $f\in \mathcal{A}$ which are univalent in $\mathbb{U}$.

It is well known that every function $f\in \mathcal{S}$ has an inverse $f^{-1}$, which is defined by

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

and

$$f\left(f^{-1}\left(w\right)\right)=w\left(\left|w\right|<r_0\left(f\right);r_0\left(f\right)\geqq \frac{1}{4}\right).$$

If f and $f^{-1}$ are univalent in $\mathbb{U}$, then f is said to be bi-univalent in $\mathbb{U}$. We denote by ${\sigma }_{\mathcal{B}}$ the class of bi-univalent functions in $\mathbb{U}$. For a brief history and interesting examples of functions in the class ${\sigma }_{\mathcal{B}}$, see the pioneering work [1]. In fact, this widely-cited work by Srivastava et al. [1] actually revived the study of analytic and bi-univalent functions in recent years, and it has also led to a flood of papers on the subject by (for example) Srivastava et al. [2,3,4,5,6,7,8,9,10,11,12,13,14] and by others [15,16].

In this paper, let $\Sigma$ be the family of meromorphic univalent functions f of the following form:

$$\begin{array}{c}\hfill f\left(z\right)=z+b_0+\sum _{n=1}^{\infty }\frac{b_n}{z^n},\end{array}$$

(2)

which are defined on the domain

$$\Delta =\{z:z\in \mathbb{C}\mathrm{and}1<|z|<\infty \}.$$

Since a function $f\in \Sigma$ is univalent, it has an inverse $f^{-1}$ that satisfies the following relationship:

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

and

$$f\left(f^{-1}\left(w\right)\right)=w\left(M<\left|w\right|<\infty ;M>0\right).$$

Furthermore, the inverse function $f^{-1}$ has a series expansion of the form [17]:

$$g\left(w\right)=f^{-1}\left(w\right)=w+\sum _{n=0}^{\infty }\frac{B_n}{w^n}\left(M<\left|w\right|<\infty \right).$$

A function $f\in \Sigma$ is said to be meromorphic bi-univalent if both f and $f^{-1}$ are meromorphic univalent in $\Delta$. The family of all meromorphic bi-univalent functions in $\Delta$ of the form (2) is denoted by ${\Sigma }_{\mathcal{M}}$. A simple calculation shows that (see also [18,19])

$$g\left(w\right)=f^{-1}\left(w\right)=w-b_0-\frac{b_1}{w}-\frac{b_2+b_0{b}_1}{w^2}-\cdots .$$

(3)

Moreover, the coefficients of $g=f^{-1}$ can be given in terms of the Faber polynomial [20] (see also [21,22,23]) as follows:

$$g\left(w\right)=f^{-1}\left(w\right)=w-b_0-\sum _{n=1}^{\infty }\frac{1}{n}{K}_{n+1}^n\frac{1}{w^n}\left(w\in \Delta \right),$$

(4)

where

$$\begin{array}{cc}\hfill K_{n+1}^n& =n{b}_0^{n-1}{b}_1+n(n-1)b_0^{n-2}{b}_2+\frac{1}{2}n(n-1)(n-2)b_0^{n-23}(b_3+b_1^2)\hfill \\ \hfill & +\frac{n(n-1)(n-2)(n-3)}{3!}{b}_0^{n-4}(b_4+3b_1{b}_2)+\sum _{j\geqq 5}{b}_0^{n-j}{V}_j\hfill \end{array}$$

and $V_j$ (with $5\leqq j\leqq n$) is a homogeneous polynomial of degree j in the variables $b_1,b_2,\cdots ,b_n$.

Estimates on the coefficients of meromorphic univalent functions were widely investigated in the literature. For example, Schiffer [24] obtained the estimate $|b_2|\leqq 2/3$ for meromorphic univalent functions $f\in \Sigma$ with $b_0=0$ and Duren [25] proved that

$$|b_n|\leqq \frac{2}{n+1}\left(f\in \Sigma ;b_k=0;1\leqq k<\frac{n}{2}\right).$$

Many researchers introduced and studied subclasses of meromorphic bi-univalent functions (see, for instance, Janani et al. [26], Orhan et al. [27] and others [28,29,30]).

Recently, Srivastava et al. [31] introduced a new class ${\Sigma }_{B*}(\lambda ,\beta )$ of meromorphic bi-univalent functions and obtained the estimates on the initial Taylor–Maclaurin coefficients $|b_0|$ and $|b_1|$ for functions in this class.

Definition 1

(see [31]). A function $f\in {\Sigma }_{\mathcal{M}},$ given by (2), is said to be in the class ${\Sigma }_{B*}(\lambda ,\beta )$ $(\lambda \geqq 1;0\leqq \beta <1),$ if the following conditions are satisfied:

$$\Re \left(\frac{z{\left(f^{\prime }\left(z\right)\right)}^{\lambda }}{f\left(z\right)}\right)>\beta$$

and

$$\Re \left(\frac{w{\left(g^{\prime }\left(w\right)\right)}^{\lambda }}{g\left(w\right)}\right)>\beta ,$$

where the function g, given by $\left(3\right)$ is the inverse of f and $z,w\in \Delta$.

Theorem 1

(see [31]). Let the function $f\in {\Sigma }_{\mathcal{M}},$ given by $\left(2\right),$ be in the class ${\Sigma }_{B*}(\lambda ,\beta )$. Then,

$$|b_0|\leqq 2(1-\beta )and|b_1|\leqq \frac{2(1-\beta )\sqrt{4{\beta }^2-8\beta +5}}{1+\lambda }.$$

In this paper, we introduce a new comprehensive subclass ${\Sigma }_B(\lambda ,\mu ,\beta )$ of the meromorphic bi-univalent function class ${\Sigma }_{\mathcal{M}}$. We also obtain estimates for the initial Taylor–Maclaurin coefficients $b_0$, $b_1$ and $b_2$ for functions in this subclass. Furthermore, we find estimates for the general coefficients $b_n(n\geqq 1)$ for functions in this comprehensive subclass ${\Sigma }_B(\lambda ,\mu ,\beta )$ by using the Faber polynomials [20]. Our results for the meromorphic bi-univalent function subclass ${\Sigma }_B(\lambda ,\mu ,\beta )$ would generalize and improve some recent works by Srivastava et al. [31], Hamidi et al. [32] and Jahangiri et al. [33] (see also the recent works [34,35]).

2. Preliminary Results

For finding the coefficients of functions belonging to the function class ${\Sigma }_B(\lambda ,\mu ,\beta )$, we need the following lemmas and remarks.

Lemma 1

(see [21,22]). Let f be the function given by

$$f\left(z\right)=z+b_0+\frac{b_1}{z}+\frac{b_2}{z^2}+\cdots$$

be a meromorphic univalent function defined on the domain Δ. Then, for any $\rho \in \mathbb{R},$ there are polynomials $K_n^{\rho }$ such that

$${\left(\frac{f\left(z\right)}{z}\right)}^{\rho }=1+\sum _{n=1}^{\infty }\frac{K_n^{\rho }(b_0,b_1,\cdots ,b_{n-1})}{z^n},$$

where

$$K_n^{\rho }(b_0,b_1,\cdots ,b_{n-1})=\rho b_{n-1}+\frac{\rho (\rho -1)}{2}{D}_n^2+\frac{\rho !}{(\rho -3)!3!}{D}_n^3+\cdots +\frac{\rho !}{(\rho -n)!n!}{D}_n^n$$

and

$$D_n^k(x_1,x_2,\cdots ,x_{n-k+1})=\sum \frac{k!{\left(x_1\right)}^{{\mu }_1}\cdots {\left(x_{n-k+1}\right)}^{{\mu }_{n-k+1}}}{{\mu }_1!\cdots {\mu }_{n-k+1}!},$$

in which the sum is taken over all non-negative integers ${\mu }_1,\cdots ,{\mu }_{n-k+1}$ such that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mu }_1+{\mu }_2+\cdots +{\mu }_{n-k+1}=k\hfill \\ \\ {\mu }_1+2{\mu }_2+\cdots +(n-k+1){\mu }_{n-k+1}=n.\hfill \end{array}\right.\end{array}$$

The first three terms of $K_n^{\rho }$ are given by

$$K_1^{\rho }\left(b_0\right)=\rho b_0,$$

$$K_2^{\rho }(b_0,b_1)=\rho b_1+\frac{\rho (\rho -1)}{2}{b}_0^2$$

and

$$K_3^{\rho }(b_0,b_1,b_2)=\rho b_2+\rho (\rho -1)b_0{b}_1+\frac{\rho (\rho -1)(\rho -2)}{3!}{b}_0^3.$$

Remark 1.

In the special case when

$$b_0=b_1=\cdots =b_{n-1}=0,$$

it is easily seen that

$$K_i^{\rho }(b_0,\cdots ,b_{i-1})=0(1\leqq i\leqq n)$$

and

$$K_{n+1}^{\rho }(b_0,b_1,\cdots ,b_n)=\rho b_n.$$

Lemma 2

(see [21,22]). Let f be the function given by

$$f\left(z\right)=z+b_0+\frac{b_1}{z}+\frac{b_2}{z^2}+\cdots$$

be a meromorphic univalent function defined on the domain Δ. Then, the Faber polynomials $F_n$ of $f\left(z\right)$ are given by

$$\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}=1+\sum _{n=1}^{\infty }\frac{F_n(b_0,b_1,\cdots ,b_{n-1})}{z^n},$$

(5)

where $F_n(b_0,b_1,\cdots ,b_{n-1})$ is a homogeneous polynomial of degree n.

Remark 2

(see [36]). For any integer $n\geqq 1,$ the polynomials $F_n(b_0,b_1,\cdots ,b_{n-1})$ are given by

$$F_n(b_0,b_1,\cdots ,b_{n-1})=\sum _{i_1+2i_2+\cdots +n{i}_n=n}{A}_{(i_1,i_2,\cdots ,i_n)}{b}_0^{i_1}{b}_1^{i_2}\cdots b_{n-1}^{i_n},$$

where

$$A_{(i_1,i_2,\cdots ,i_n)}:={(-1)}^{n+2i_1+3i_2+\cdots +(n+1)i_n}\frac{(i_1+i_2+\cdots +i_n-1)!n}{i_1!i_2!\cdots i_n!}.$$

The first three terms of $F_n$ are given by

$$F_1\left(b_0\right)=-b_0,$$

$$F_2(b_0,b_1)=b_0^2-2b_1$$

and

$$F_3(b_0,b_1,b_2)=-b_0^3+3b_0{b}_1-3b_2.$$

Remark 3.

In the special case when $b_0=b_1=\cdots =b_{n-1}=0,$ it is readily observed that

$$F_i(b_0,\cdots ,b_{i-1})=0(1\leqq i\leqq n)$$

and

$$F_{n+1}(b_0,b_1,\cdots ,b_n)={(-1)}^{2n+3}(n+1)b_n=-(n+1)b_n.$$

Lemma 3.

Let f be the function given by

$$f\left(z\right)=z+b_0+\frac{b_1}{z}+\frac{b_2}{z^2}+\cdots$$

be a meromorphic univalent function defined on the domain Δ. Then, for $\lambda \geqq 1$ and $\mu \geqq 0,$

$${\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)}^{\lambda }{\left(\frac{f\left(z\right)}{z}\right)}^{\mu }=1+\sum _{n=1}^{\infty }\frac{L_n(b_0,b_1,\cdots ,b_{n-1})}{z^n},$$

where

$$L_n(b_0,b_1,\cdots ,b_{n-1})=\sum _{i=0}^n{K}_{n-i}^{\lambda }(F_1,\cdots ,F_{n-i})K_i^{\mu }(b_0,\cdots ,b_{i-1})\left(K_0^{\lambda }=K_0^{\mu }=1\right)$$

and $F_n=F_n(b_0,b_1,\cdots ,b_{n-1})$ is given by $\left(5\right)$.

Proof. 

By using Lemmas 1 and 2, we have

$$\begin{array}{cc}\hfill {\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)}^{\lambda }{\left(\frac{f\left(z\right)}{z}\right)}^{\mu }& ={\left(1+\sum _{m=1}^{\infty }\frac{F_m(b_0,b_1,\cdots ,b_{m-1})}{z^m}\right)}^{\lambda }\hfill \\ \hfill & ·\left(1+\sum _{m=1}^{\infty }\frac{K_m^{\mu }(b_0,b_1,\cdots ,b_{m-1})}{z^m}\right).\hfill \end{array}$$

In addition, by applying Lemma 1 once again, we obtain

$$\begin{array}{cc}\hfill {\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)}^{\lambda }{\left(\frac{f\left(z\right)}{z}\right)}^{\mu }& =\left(1+\sum _{m=1}^{\infty }\frac{K_m^{\lambda }(F_1,\cdots ,F_m)}{z^m}\right)\hfill \\ \hfill & ·\left(1+\sum _{m=1}^{\infty }\frac{K_m^{\mu }(b_0,\cdots ,b_{m-1})}{z^m}\right)\hfill \\ \hfill & =1+\sum _{n=1}^{\infty }\sum _{i=0}^n{K}_{n-i}^{\lambda }(F_1,\cdots ,F_{n-i})K_i^{\mu }(b_0,\cdots ,b_{i-1})\frac{1}{z^n}\hfill \end{array}$$

$$\left(K_0^{\lambda }=K_0^{\mu }=1\right).$$

Our demonstration of Lemma 3 is thus completed. □

The first three terms of $L_n$ are given by

$$L_1\left(b_0\right)=(\mu -\lambda )b_0,$$

$$L_2(b_0,b_1)=\frac{\lambda (1+\lambda -2\mu )+\mu (\mu -1)}{2}{b}_0^2+(\mu -2\lambda )b_1$$

and

$$\begin{array}{cc}\hfill L_3(b_0,b_1,b_2)& =\left(\frac{\lambda (2-\mu )(\mu -\lambda )}{2}+\frac{\mu (\mu -1)(\mu -2)-\lambda (\lambda -1)(\lambda -2)}{6}\right)b_0^3\hfill \\ \hfill & +\left[\lambda (2\lambda +1)+\mu (\mu -3\lambda -1)\right]b_0{b}_1+(\mu -3\lambda )b_2.\hfill \end{array}$$

Remark 4.

In the special case when $b_0=b_1=\cdots =b_{n-1}=0,$ we easily find that

$$L_i(b_0,\cdots ,b_{i-1})=0(1\leqq i\leqq n)$$

and

$$L_{n+1}(b_0,b_1,\cdots ,b_n)=\left(\mu -(n+1)\lambda \right)b_n.$$

Lemma 4

(see [37]). If the function $p\in \mathcal{P},$ then $|c_k|\leqq 2$ for each $k,$ where $\mathcal{P}$ is the family of all functions $p,$ which are analytic in the domain Δ given by

$$\Delta =\{z:z\in \mathbb{C}and1<|z|<\infty \}$$

for which

$$\Re \left(p\left(z\right)\right)>0(z\in \Delta ),$$

where

$$p\left(z\right)=1+\frac{c_1}{z}+\frac{c_2}{z^2}+\frac{c_3}{z^3}+\cdots .$$

3. The Comprehensive Class ${\mathbf{\Sigma }}_{\mathit{B}}\mathbf{(}\mathit{\lambda }\mathbf{,}\mathit{\mu }\mathbf{,}\mathit{\beta }\mathbf{)}$

In this section, we introduce and investigate the comprehensive class ${\Sigma }_B(\lambda ,\mu ,\beta )$ of meromorphic bi-univalent functions defined on the domain $\Delta$.

Definition 2.

A function $f\in {\Sigma }_{\mathcal{M}},$ given by $\left(2\right),$ is said to be in the class

$${\Sigma }_B(\lambda ,\mu ,\beta )(\lambda \geqq 1;\mu \geqq 0;0\leqq \beta <1)$$

of meromorphic bi-univalent functions of order β and type $\mu ,$ if the following conditions are satisfied:

$$\Re \left({\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)}^{\lambda }{\left(\frac{f\left(z\right)}{z}\right)}^{\mu }\right)>\beta$$

and

$$\Re \left({\left(\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}\right)}^{\lambda }{\left(\frac{g\left(w\right)}{w}\right)}^{\mu }\right)>\beta ,$$

where the function g given by $\left(4\right)$, is the inverse of f and $z,w\in \Delta$.

Remark 5.

There are several choices of the parameters λ and μ which would provide interesting subclasses of meromorphic bi-univalent functions. For example, we have the following special cases:

• By putting $\lambda =1$ and $0\leqq \mu <1,$ the class ${\Sigma }_B(\lambda ,\mu ,\beta )$ reduces to the subclass $B(\beta ,\mu )$ of meromorphic bi-Bazilevič functions of order β and type $\mu ,$ which was considered by Jahangiri et al. [33].
• By putting $\lambda =1$ and $\mu =0,$ the class ${\Sigma }_B(\lambda ,\mu ,\beta )$ reduces to the subclass ${\Sigma }_B^{*}\left(\beta \right)$ of meromorphic bi-starlike functions of order $\beta ,$ which was considered by Hamidi et al. [32].
• By putting $\mu =\lambda -1,$ the class ${\Sigma }_B(\lambda ,\mu ,\beta )$ reduces to the class ${\Sigma }_{B^{*}}(\lambda ,\beta )$ in Definition 1.

Theorem 2.

Let $f\in {\Sigma }_B(\lambda ,\mu ,\beta )$. If $b_0=b_1=\cdots =b_{n-1}=0,$ then

$$|b_n|\leqq \frac{2(1-\beta )}{\left|\right(n+1)\lambda -\mu |}(n\geqq 1).$$

Proof. 

By using Lemma 3 for the meromorphic bi-univalent function f given by

$$f\left(z\right)=z+b_0+\sum _{n=1}^{\infty }\frac{b_n}{z^n},$$

we have

$${\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)}^{\lambda }{\left(\frac{f\left(z\right)}{z}\right)}^{\mu }=1+\sum _{n=0}^{\infty }\frac{L_{n+1}(b_0,b_1,\cdots ,b_n)}{z^{n+1}}.$$

(6)

Similarly, for its inverse map g given by

$$g\left(w\right)=f^{-1}\left(w\right)=w+B_0+\sum _{n=1}^{\infty }\frac{B_n}{w^n},$$

we find that

$${\left(\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}\right)}^{\lambda }{\left(\frac{g\left(w\right)}{w}\right)}^{\mu }=1+\sum _{n=0}^{\infty }\frac{L_{n+1}(B_0,B_1,\cdots ,B_n)}{w^{n+1}}.$$

(7)

Furthermore, since $f\in {\Sigma }_B(\lambda ,\mu ,\beta )$, by using Definition 2, there exist two positive real-part functions

$$c\left(z\right)=1+\sum _{n=1}^{\infty }{c}_n{z}^{-n}$$

and

$$d\left(w\right)=1+\sum _{n=1}^{\infty }{d}_n{w}^{-n}$$

for which

$$\Re \left(c\left(z\right)\right)>0\mathrm{and}\Re \left(d\left(w\right)\right)>0(z,w\in \Delta ),$$

such that

$${\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)}^{\lambda }{\left(\frac{f\left(z\right)}{z}\right)}^{\mu }=1+(1-\beta )\sum _{n=0}^{\infty }{K}_{n+1}^1(c_1,c_2,\cdots ,c_{n+1})\frac{1}{z^{n+1}}$$

(8)

and

$${\left(\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}\right)}^{\lambda }{\left(\frac{g\left(w\right)}{w}\right)}^{\mu }=1+(1-\beta )\sum _{n=0}^{\infty }{K}_{n+1}^1(d_1,d_2,\cdots ,d_{n+1})\frac{1}{w^{n+1}}.$$

(9)

Upon equating the corresponding coefficients in $\left(6\right)$ and $\left(8\right)$, we get

$$L_{n+1}(b_0,b_1,\cdots ,b_n)=(1-\beta )K_{n+1}^1(c_1,c_2,\cdots ,c_{n+1}).$$

(10)

Similarly, from $\left(7\right)$ and $\left(9\right)$, we obtain

$$\begin{array}{c}\hfill L_{n+1}(B_0,B_1,\cdots ,B_n)=(1-\beta )K_{n+1}^1(d_1,d_2,\cdots ,d_{n+1}).\end{array}$$

(11)

Now, since $b_i=0(0\leqq i\leqq n-1)$, we have

$$B_i=0(0\leqq i\leqq n-1)\mathrm{and}{B}_n=-b_n.$$

Hence, by using Remark 4, Equations (10) and (11) can be rewritten as follows:

$$\left(\mu -(n+1)\lambda \right)b_n=(1-\beta )c_{n+1}$$

(12)

and

$$-\left(\mu -(n+1)\lambda \right)b_n=(1-\beta )d_{n+1},$$

(13)

respectively. Thus, from (12) and (13), we find that

$$2(\mu -(n+1)\lambda )b_n=(1-\beta )(c_{n+1}-d_{n+1}).$$

Finally, by applying Lemma 4, we get

$$|b_n|=\frac{(1-\beta )|c_{n+1}-d_{n+1}|}{2\left|\right(n+1)\lambda -\mu |}\leqq \frac{2(1-\beta )}{\left|\right(n+1)\lambda -\mu |},$$

which completes the proof of Theorem 2 □

Theorem 3.

Let the function $f\in \mathcal{M},$ given by $\left(2\right),$ be in the class

$${\Sigma }_B(\lambda ,\mu ,\beta )(\lambda \geqq 1;\mu \geqq 0;0\leqq \beta <1).$$

Then,

$$|b_0|\leqq min\left\{\frac{2(1-\beta )}{|\mu -\lambda |},2\sqrt{\frac{1-\beta }{\left|\lambda \right(1+\lambda -2\mu )+\mu (\mu -1\left)\right|}}\right\},$$

$$|b_1|\leqq \frac{2(1-\beta )}{|\mu -2\lambda |}$$

and

$$\begin{array}{cc}\hfill |b_2|& \leqq \frac{2\left\{\right|\lambda (2\lambda +4)+\mu (\mu -3\lambda -2)|+|\lambda (2\lambda +1)+\mu (\mu -3\lambda -1)\left|\right\}(1-\beta )}{\left|\right(\mu -3\lambda \left)\right[\lambda (4\lambda +5)+\mu (2\mu -6\lambda -3)\left]\right|}\hfill \\ \hfill & +\frac{{8\left|T(\mu ,\lambda )\right|(1-\beta )}^3}{{|(\mu -3\lambda )(\mu -\lambda )}^3|},\hfill \end{array}$$

where

$$T(\mu ,\lambda )=\frac{\lambda (2-\mu )(\mu -\lambda )}{2}+\frac{\mu (\mu -1)(\mu -2)-\lambda (\lambda -1)(\lambda -2)}{6}.$$

Proof. 

By putting $n=0,1,2$ in $\left(10\right)$, we get

$$\begin{array}{c}\hfill (\mu -\lambda )b_0=(1-\beta )c_1,\end{array}$$

(14)

$$\begin{array}{c}\hfill \frac{\lambda (1+\lambda -2\mu )+\mu (\mu -1)}{2}{b}_0^2+(\mu -2\lambda )b_1=(1-\beta )c_2\end{array}$$

(15)

and

$$T(\mu ,\lambda )b_0^3+[\lambda (2\lambda +1)+\mu (\mu -3\lambda -1)]b_0{b}_1+(\mu -3\lambda )b_2=(1-\beta )c_3.$$

(16)

Similarly, by putting $n=0,1,2$ in $\left(11\right)$, we have

$$\begin{array}{c}\hfill -(\mu -\lambda )b_0=(1-\beta )d_1,\end{array}$$

(17)

$$\begin{array}{c}\hfill \frac{\lambda (1+\lambda -2\mu )+\mu (\mu -1)}{2}{b}_0^2-(\mu -2\lambda )b_1=(1-\beta )d_2\end{array}$$

(18)

and

$$-T(\mu ,\lambda )b_0^3+(\lambda (2\lambda +4)+\mu (\mu -3\lambda -2))b_0{b}_1-(\mu -3\lambda )b_2=(1-\beta )d_3.$$

(19)

Clearly, from $\left(14\right)$ and $\left(17\right)$, we get

$$c_1=-d_1$$

(20)

and

$$b_0=\frac{(1-\beta )c_1}{\mu -\lambda }.$$

(21)

Adding $\left(15\right)$ and $\left(18\right)$, we obtain

$$b_0^2=\frac{(1-\beta )(c_2+d_2)}{\lambda (1+\lambda -2\mu )+\mu (\mu -1)}.$$

(22)

In view of the Equations (21) and (22), by applying Lemma 4, we get

$$|b_0|\leqq \frac{2(1-\beta )}{|\mu -\lambda |}\mathrm{and}{\left|b_0\right|}^2\leqq \frac{4(1-\beta )}{\left|\lambda \right(1+\lambda -2\mu )+\mu (\mu -1\left)\right|},$$

respectively. Thus, we get the desired estimate on the coefficient $|b_0|$.

Next, in order to find the bound on the coefficient $|b_1|$, we subtract $\left(18\right)$ from $\left(15\right)$. We thus obtain

$$b_1=\frac{(1-\beta )(c_2-d_2)}{2(\mu -2\lambda )}.$$

(23)

Applying Lemma 4 once again, we get

$$|b_1|\leqq \frac{2(1-\beta )}{|\mu -2\lambda |}.$$

Finally, in order to determine the bound on $|b_2|$, we consider the sum of the Equations $\left(16\right)$ and $\left(19\right)$ with $c_1=-d_1$. This yields

$$b_0{b}_1=\frac{(1-\beta )(c_3+d_3)}{\lambda (4\lambda +5)+\mu (2\mu -6\lambda -3)}.$$

(24)

Subtracting $\left(19\right)$ from $\left(16\right)$ with $c_1=-d_1$, we obtain

$$2(\mu -3\lambda )b_2+(\mu -3\lambda )b_0{b}_1+2T(\mu ,\lambda )b_0^3=(1-\beta )(c_3-d_3).$$

(25)

In addition, by using $\left(21\right)$ and $\left(24\right)$ in $\left(25\right)$, we get

$$b_2=\frac{(1-\beta )(c_3-d_3)}{2(\mu -3\lambda )}-\frac{(1-\beta )(c_3+d_3)}{2\left[\lambda \right(4\lambda +5)+\mu (2\mu -6\lambda -3\left)\right]}-\frac{T(\mu ,\lambda ){(1-\beta )}^3{c}_1^3}{(\mu -3\lambda ){(\mu -\lambda )}^3}.$$

Hence,

$$\begin{array}{cc}\hfill b_2& =\frac{\left\{[\lambda (2\lambda +4)+\mu (\mu -3\lambda -2)]c_3-[\lambda (2\lambda +1)+\mu (\mu -3\lambda -1)]d_3\right\}(1-\beta )}{(\mu -3\lambda )\left[\lambda \right(4\lambda +5)+\mu (2\mu -6\lambda -3\left)\right]}\hfill \\ \hfill & -\frac{T(\mu ,\lambda ){(1-\beta )}^3{c}_1^3}{(\mu -3\lambda ){(\mu -\lambda )}^3}.\hfill \end{array}$$

Thus, by applying Lemma 4 once again, we get

$$\begin{array}{cc}\hfill |b_2|& \leqq \frac{2\left\{\right|\lambda (2\lambda +4)+\mu (\mu -3\lambda -2)|+|\lambda (2\lambda +1)+\mu (\mu -3\lambda -1)\left|\right\}(1-\beta )}{\left|\right(\mu -3\lambda \left)\right[\lambda (4\lambda +5)+\mu (2\mu -6\lambda -3)\left]\right|}\hfill \\ \hfill & +\frac{{8\left|T(\mu ,\lambda )\right|(1-\beta )}^3}{{|(\mu -3\lambda )(\mu -\lambda )}^3|}.\hfill \end{array}$$

This completes the proof of Theorem 3. □

4. A Set of Corollaries and Consequences

By setting $\lambda =1$ and $0\leqq \mu <1$ in Theorem 2, we have the following result.

Corollary 1.

Let the function $f\in \mathcal{M},$ given by $\left(2\right),$ be in the subclass $B(\beta ,\mu )$ of meromorphic bi-Bazilevič functions of order β and type μ. If

$$b_0=b_1=\cdots =b_{n-1}=0,$$

then

$$|b_n|\leqq \frac{2(1-\beta )}{n+1-\mu }(n\geqq 1).$$

Remark 6.

The estimate of $|b_n|,$ given in Corollary 1, is the same as the corresponding estimate given by Hamidi et al. [38] Corollary 3.3.

By setting $\mu =0$ in Corollary 1, we have the following result.

Corollary 2.

Let the function $f\in \mathcal{M},$ given by $\left(2\right),$ be in the subclass ${\Sigma }_B^{*}\left(\beta \right)$ of meromorphic bi-starlike functions of order β. If

$$b_0=b_1=\cdots =b_{n-1}=0,$$

then

$$|b_n|\leqq \frac{2(1-\beta )}{n+1}(n\geqq 1).$$

Remark 7.

The estimate of $|b_n|,$ given in Corollary 2, is the same as the corresponding estimate given by Hamidi et al. [38] Corollary 3.4.

By setting $\mu =\lambda -1$ in Theorem 2, we have the following result.

Corollary 3.

Let the function $f\in \mathcal{M},$ given by $\left(2\right),$ be in the subclass ${\Sigma }_{B*}(\lambda ,\beta )$. If

$$b_0=b_1=\cdots =b_{n-1}=0,$$

then

$$|b_n|\leqq \frac{2(1-\beta )}{n\lambda +1}(n\geqq 1).$$

Remark 8.

Corollary 3 is a generalization of a result presented in Theorem 1, which was proved by Srivastava et al. [31].

By setting $\lambda =1$ and $0\leqq \mu <1$ in Theorem 3, we have the following result.

Corollary 4.

Let the function $f\in \mathcal{M},$ given by $\left(2\right),$ be in the subclass $B(\beta ,\mu )$ of meromorphic bi-Bazilevič functions of order β and type μ. Then,

$$|b_0|\leqq \left\{\begin{array}{cc}\sqrt{\frac{4(1-\beta )}{(1-\mu )(2-\mu )}}\hfill & \left(0\leqq \beta \leqq \frac{1}{2-\mu }\right)\hfill \\ \\ \frac{2(1-\beta )}{1-\mu }\hfill & \left(\frac{1}{2-\mu }\leqq \beta <1\right),\hfill \end{array}\right.$$

$$|b_1|\leqq \frac{2(1-\beta )}{2-\mu }$$

and

$$|b_2|\leqq \frac{2(1-\beta )}{3-\mu }+\frac{4(2-\mu ){(1-\beta )}^3}{3{(1-\mu )}^2}.$$

Remark 9.

Corollary 4 also contains the estimate of the Taylor–Maclaurin coefficient $|b_2|$ of functions in the subclass $B(\beta ,\mu )$ (see [33]).

By setting $\mu =0$ in Corollary 4, we have the following result.

Corollary 5.

Let the function $f\in \mathcal{M},$ given by $\left(2\right),$ be in the subclass ${\Sigma }_B^{*}\left(\beta \right)$ of meromorphic bi-starlike functions of order β. Then,

$$|b_0|\leqq \left\{\begin{array}{cc}\sqrt{2(1-\beta )}\hfill & \left(0\leqq \beta \leqq \frac{1}{2}\right)\hfill \\ \\ 2(1-\beta )\hfill & \left(\frac{1}{2}\leqq \beta <1\right),\hfill \end{array}\right.$$

$$|b_1|\leqq 1-\beta$$

and

$$|b_2|\leqq \frac{2(1-\beta )}{3}+\frac{8{(1-\beta )}^3}{3}.$$

Remark 10.

Corollary 5 not only improves the estimate of the Taylor–Maclaurin coefficient $|b_0|,$ which was given by Hamidi et al. [32] Theorem 2, but it also provides an improvement of the known estimate of the Taylor–Maclaurin coefficient $|b_2|$ of functions in the subclass ${\Sigma }_B^{*}\left(\beta \right)$. Furthermore, the estimate of $|b_0|,$ presented in Corollary 5, is the same as the corresponding estimate given by Hamidi et al. [38] Corollary 3.5.

By setting $\mu =\lambda -1$ in Theorem 3, we have the following result.

Corollary 6.

Let the function $f\in \mathcal{M},$ given by $\left(2\right),$ be in the subclass ${\Sigma }_{B*}(\lambda ,\beta )$. Then,

$$|b_0|\leqq \left\{\begin{array}{cc}\sqrt{2(1-\beta )}\hfill & \left(0\leqq \beta \leqq \frac{1}{2}\right)\hfill \\ \\ 2(1-\beta )\hfill & \left(\frac{1}{2}\leqq \beta <1\right),\hfill \end{array}\right.$$

$$|b_1|\leqq \frac{2(1-\beta )}{\lambda +1}$$

and

$$|b_2|\leqq \frac{2(1-\beta )}{2\lambda +1}+\frac{8{(1-\beta )}^3}{2\lambda +1}.$$

Remark 11.

Corollary 6 improves the estimates of the Taylor–Maclaurin coefficients $|b_0|$ and $|b_1|$ in Theorem 1 of Srivastava et al. [31]. In fact, it also provides an improvement of the known estimate of the Taylor–Maclaurin coefficient $|b_2|$ of functions in the subclass ${\Sigma }_{B*}(\lambda ,\beta )$.

Remark 12.

In his recently-published survey-cum-expository review article, Srivastava [39] demonstrated how the theories of the basic (or q-) calculus and the fractional q-calculus have significantly encouraged and motivated further developments in Geometric Function Theory of Complex Analysis (see, for example, [8,40,41,42]). This direction of research is applicable also to the results which we have presented in this article. However, as pointed out by Srivastava [39] (p. 340), any further attempts to easily (and possibly trivially) translate the suggested q-results into the corresponding $(p,q)$-results (with $0<\left|q\right|<p\leqq 1$) would obviously be inconsequential because the additional parameter p is redundant.