Faber polynomial coefficients for meromorphic bi-subordinate functions of complex order

1.   Introduction

Let $\Sigma$ be the class of meromorphic univalent functions in the domain $\Delta = \{z\in \mathbb{ \mathbb{C} }:$ $1 < \left\vert z\right\vert < \infty \}$ of the form

Since $f\in \Sigma$ is univalent, it has an inverse $f^{-1},$ that satisfy

and

A simple calculation shows that the function $g: = f^{-1}$ is given by

Analogous to the bi-univalent analytic functions, a function $f\in \Sigma$ is said to be meromorphic bi-univalent if $f^{-1}\in \Sigma$. We denote the family of all meromorphic bi-univalent functions by $\mathcal{M}_{\Sigma }$. Estimates on the coefficients of meromorphic univalent functions were widely investigated in theliterature, for example; Schiffer ^[18] obtained the estimate$|b_{2}|\leq 2/3$ for meromorphic univalent functions $f\in \Sigma$ with $b_{0} = 0$ and Duren ^[6] proved that $|b_{n}|\leq 2/(n+1)$ for $f\in \Sigma$ with $b_{k} = 0$ for $1\leq k\leq n/2$. For the coefficient of the inverse of meromorphic univalent functions Springer ^[20] proved that

and conjectured that

In 1977, Kubota ^[13] has proved that the Springer's conjecture is true for $n = 3, 4, 5$ and subsequently Schober ^[19] obtained a sharp bounds for the coefficients $B_{2n-1}$, $1\leq n\leq 7.$ Recently, Kapoor and Mishra ^[12] found the coefficient estimates for a class consisting of inverses of meromorphic starlike univalent functions of order $\alpha$ in $\Delta$.

For a brief history and interesting examples of functions which are in (or are not in) the class $\mathcal{M}_{\Sigma }$, including various properties of such functions we refer the reader to the work of Hamidi et al. ^[9,10] and references therein. Bounds for the first few coefficients of various subclasses of bi-univalent functions were obtained by a variety of authors including ^{[4,8,11,15,16,22]}. Not much was known about the bounds of the general coefficients $b_{n}; n\geq 1$ of subclasses of $\mathcal{M}_{\Sigma }$ up until the publication of the article ^[9,10] by Hamidi, Halim and Jahangiri and followed by a number of related publications (see ^[5,14,21]). In this paper, we apply the Faber polynomial expansions to certain subclass of bi-univalent functions and obtain bounds for their $n-th; (n\geq 1)$ coefficients subject to a given gap series condition.

2.   Coefficient estimates

An analytic function $f$ is subordinate to an analytic function $g$, written by $f\prec g$, provided that there is an analytic function $w$ defined on $\mathbb{D = }\{z\in \mathbb{ \mathbb{C} }:$ $\left\vert z\right\vert < 1\}$ with $w(0) = 0$ and $|w(z)| < 1$ satisfying $f(z) = g(w(z))$.

In the sequel, it is assumed that $\varphi$ is an analytic function with positive real part in the unit disk $\mathbb{D}$, satisfying $\varphi (0) = 1,$ $\varphi ^{\prime }(0) > 0,$ and $\varphi (\mathbb{D})$ is symmetric with respect to the real axis. Such a function is known to be typically real with the series expansion $\varphi (z) = 1+B_{1}z+B_{2}z^{2}+B_{3}z^{3}+...$ where $B_{1}$, $B_{2}$ are real and $B_{1} > 0.$ We define the following comprehensive class of meromorphic functions:

Definition 2.1. For $0\leq \lambda < 1$ and $\gamma \in \mathbb{C} \backslash \left\{ 0\right\}$, a function $f\in \Sigma$ given by (1.1) is said to be in the class $\mathcal{M}_{\Sigma }(\lambda, \gamma; \varphi)$ if the following conditions are satisfied:

and

where $z, w\in$ $\Delta$ and the function $g$ is given by (1.2).

A function $f\in \mathcal{M}_{\Sigma }(\lambda, \gamma; \varphi)$ is said the be generalized meromorphic bi-subordinate of complex order $\gamma$ and type $\lambda.$

By suitably specializing the parameters $\lambda,$ $\gamma$ and the function $\varphi$, we state new subclass of meromorphic bi-univalent functions as illustrated in the following examples.

(1) $\mathcal{M}_{\Sigma }\left(0, 1;\varphi)\right) = \mathcal{M}_{\Sigma }(\varphi)$ is class of meromorphic Ma-Minda bi-starlike functions,

(2) $\mathcal{M}_{\Sigma }\left(0, 1;(1+Az)\diagup (1+Bz)\right) = \mathcal{M} _{\Sigma }[A, B]$ $\left(-1\leq B < A\leq 1\right)$ is class of meromorphic Janowski bi-starlike functions,

(3) $\mathcal{M}_{\Sigma }(0, (1-\beta)e^{-i\delta }\cos \delta; (1+z)\diagup (1-z)) = \Sigma ^{\ast }[\delta, \beta]$ $\left(\left\vert \delta \right\vert < \pi \diagup 2, \text{ }0\leq \beta < 1\right)$ is class of meromorphic bi- $\delta$-spirallike functions of order $\beta,$

(4) $\mathcal{M}_{\Sigma }(0, 1;(1+(1-2\beta)z)\diagup (1-z)) = \Sigma ^{\ast }(\beta)$ $\left(0\leq \beta < 1\right)$ is class of meromorphic bi-starlike functions of order $\beta,$

(5) $\mathcal{M}_{\Sigma }(0, 1;(1+z)\diagup (1-z)) = \Sigma ^{\ast }$ is class of meromorphic bi-starlike functions,

(6) $\mathcal{M}_{\Sigma }\left(0, 1;\left(\frac{1+z}{1-z}\right) ^{\beta }\right) = \Sigma _{\beta }^{\ast }$ is class of meromorphic strongly bi-starlike functions of order $\beta,$

(7) $\mathcal{M}_{\Sigma }(0, \gamma; (1+z)\diagup (1-z)) = \mathcal{S}^{\ast }[\gamma]$ is class of meromorphic bi-starlike functions of complex order.

In the following theorem, we use the Faber polynomials introduced by Faber ^[7] to obtain a bound for the general coefficients $\left\vert b_{n}\right\vert$ of the bi-univalent functions in $\mathcal{M}_{\Sigma }(\lambda, \gamma; \varphi)$ subject to a gap series condition.

Theorem 2.1. Let $f\in \Sigma$ given by (1.1) in the class $\mathcal{M} _{\Sigma }(\lambda, \gamma; \varphi).$ If, $b_{m}\mathrm{ = }0\mathrm{, \; 1\leq }m\leq n-1$ for $n$ being odd or if $b_{m}\mathrm{ = }0 \mathrm{, \; 0\leq }m\leq n-1$ for $n$ being even, then

Proof. If we write $\Lambda (f(z)) = \lambda zf^{\prime }(z)+(1-\lambda)f(z)$, then

Also, for the function $f\left(z\right) = z+b_{0}+\sum_{n = 1}^{\infty }\frac{ b_{n}}{z^{n}}$ we have $\Lambda (f(z)) = z+\sum\nolimits_{n = 0}^{\infty }a_{n}z^{n}$ where $a_{n} = (1-\lambda (n+1))b_{n}$.

Now, an application of Faber polynomial expansion to the power series $\mathcal{M}_{\Sigma }(\lambda, \gamma; \varphi)$ (e.g., see ^[2] or [3,equation (1.6)]) yields

where $F_{n+1}\left(a_{0}, a_{1}, a_{2}, \cdots, a_{n}\right)$ is a Faber polynomial of degree $n+1,$ i.e.,

and

The first few terms of $F_{n+1}\left(a_{0}, a_{1}, a_{2}, \cdots, a_{n}\right)$ are

By the same token, the coefficients of the inverse map $\mathit{g\mathrm{ = } f^{-1}}$may be expressed by

where

and $V_{j}$ for $5\leq j\leq n$ is a homogeneous polynomial in the variables $b_{1}, b_{2}, ..., b_{n}$.

The first few terms of ${K_{n+1}^{n}}$ are

Obviously,

where $A_{n} = (1-\lambda (n+1))\beta _{n}$. Since, the function $f$ in the class $\mathcal{M}_{\Sigma }(\lambda, \gamma; \varphi)$, by the definition of subordination, there exist two Schwarz functions $u(z)\mathrm{ = }\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+...+\frac{c_{n}}{z^{n}}+...,$ $\left\vert u(z)\right\vert < 1,$ $z\in \Delta$ and $v(w)\mathrm{ = } \frac{d_{1}}{w}+\frac{d_{2}}{w^{2}}+...+\frac{d_{n}}{w^{n}}+...,$ $\left\vert v(w)\right\vert < 1,$ $w\in \Delta$, so that

and

In general (e.g., see ^[1] and [2,equation (1.6)]), the coefficients ${K_{n}^{p}}$ $: = K_{n}^{p}\left(k_{1}, k_{2}, ..., k_{n}, B_{1}, B_{2}, ..., B_{n}\right)$ are given by

where $X_{j}$ is a homogeneous polynomial of degree $j$ in the variables $k_{2}, k_{3}, ..., k_{n}.$

For the coefficients of the Schwarz functions $u\left(z\right)$ and $v\left(w\right),$ we have $\left\vert c_{n}\right\vert \leq 1$and $\left\vert d_{n}\right\vert \leq 1$ (e.g., see ^[17]).

Note that for $a_{m}\mathrm{ = }0\mathrm{, \; 1\leq }m\leq n-1$, we have

Comparing the corresponding coefficients of (2.1) and (2.3) yields

Then, under the assumption $a_{m}\mathrm{ = }0\mathrm{, \; 1\leq }m\leq n-1,$ we get

Similarly, comparing the corresponding coefficients of (2.2) and (2.4) gives

Note that, for $A_{m}\mathrm{ = }0\mathrm{, }$ $\mathrm{1\leq }m\leq n-1$, we have

On the other hand, comparing the corresponding coefficients of the functions $f$ and $g = f^{-1}$, we obtain $\beta _{0} = -b_{0}$ and $\beta _{n} = -b_{n}$ for $b_{m}\mathrm{ = }0\mathrm{, \; 1\leq }m\leq n-1.$

Hence, when $n$ is odd, by using Eqs. (2.6), (2.8) and $\beta _{0} = -b_{0}$ and $\beta _{n} = -b_{n},$ we obtain following system

Subtracting two above equation, we have

Applying the $\left\vert c_{n}\right\vert \leq 1$and $\left\vert d_{n}\right\vert \leq 1$ yields

Similarly, when $n$ is even, by using Eqs. (2.6), (2.8) with $b_{m}\mathrm{ = }0\mathrm{, \; 0\leq }m\leq n-1,$ we obtain following system

Hence

Applying the $\left\vert c_{n}\right\vert \leq 1$and $\left\vert d_{n}\right\vert \leq 1$ yields

Corollary 2.2. Let $f\in \Sigma$ given by (1.1) in the class $\mathcal{M}_{\Sigma }(\varphi).$ If, $b_{m}\mathrm{ = }0\mathrm{, \; 1\leq }m\leq n-1$ for $n$ being odd or if $b_{m}\mathrm{ = }0 \mathrm{, \; 0\leq }m\leq n-1$ for $n$ being even, then

For functions in the class $\mathcal{M}_{\Sigma }(\lambda, \gamma; \varphi)$, the following initial coefficients estimation holds. To prove our next theorem, we shall need the following well-known lemma (see ^[17]).

Lemma 2.1. (^[17]) If $p\in \mathcal{P}$, the class of all functions with $\Re\left(p\left(z\right) \right) > 0$ $\left(z\in \mathbb{D} \right),$ then $\left\vert p_{n}\right\vert \leq 2$ $(n\in \mathbb{N} = \{1, 2, \cdots \}),$ where $p\left(z\right) \mathrm{ = 1}+\sum_{n\mathrm{ = } 1}^{\infty }p_{n}z^{n}.$

We know that $p(z)\in \mathcal{P}$ $\left(z\in \mathbb{D}\right) \mathcal{ \Leftrightarrow }p\left(\frac{1}{z}\right) \in \mathcal{P}$ $\left(z\in \Delta\right).$ Define the functions $p$ and $q$ in $\mathcal{P}$ given by

and

where $u(z)\mathrm{ = }\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+...+\frac{c_{n}}{ z^{n}}+...,$ $\left\vert u(z)\right\vert < 1,$ $z\in \Delta$ and $v(z)\mathrm{ = }\frac{d_{1}}{z}+\frac{d_{2}}{z^{2}}+...+\frac{d_{n}}{z^{n}} +...,$ $\left\vert v(z)\right\vert < 1,$ $z\in \Delta$ are Schwarz functions (e.g., see ^[17]). It follows that

and

Theorem 2.3. Let $\mathit{f}$ given by (1.1) be in the class $\mathcal{M}_{\Sigma }(\lambda, \gamma; \varphi)$. Then

and

Proof. Let $f\in$ $\mathcal{M}_{\Sigma }(\lambda, \gamma; \varphi)$. Then, there are analytic functions $u, v:\Delta \rightarrow \mathbb{ \mathbb{C} }$, with $u(\infty) = v(\infty) = 0$, satisfying

Since

and

then (2.3) and (2.4) yield

and

Now, considering (2.12) and (2.14), we get

Also, from (2.13) and (2.15), we find that

which, in view of the inequalities $\left\vert c_{n}\right\vert \leq 1$and $\left\vert d_{n}\right\vert \leq 1$ yield

Since $B_{1} > 0$, the last inequality gives the desired first estimate on $\left\vert b_{0}\right\vert$ given in the theorem. On the other hand, comparing the coefficients of (2.9) and (2.10) with (2.11), we have

and

From (2.17) and (2.19), we get $p_{1} = -q_{1}.$ Considering the sums of (2.18) and (2.20) with $p_{1} = -q_{1},$ we have

Applying Lemma 2.1 for the coefficients ${p}_{1}$, ${p_{2}}$ and ${q}_{2},$ we obtain

that gives the second estimate on $\left\vert b_{0}\right\vert$ given in the theorem.

Next, in order to find the bound on $\left\vert b_{1}\right\vert$, by further computations from (2.13), (2.15) and (2.16) lead to

Applying the inequalities $\left\vert c_{n}\right\vert \leq 1$and $\left\vert d_{n}\right\vert \leq 1,$ we readily get

which is the bound on $\left\vert b_{1}\right\vert$.

Corollary 2.4. Let $\mathit{f}$ given by (1.1) be in the class $\mathcal{M}_{\Sigma }(\varphi)$. Then,

and

Remark 2.5. Taking $\varphi (z) = (1+(1-2\beta)z)\diagup (1-z)$ in Corollary 2.2 and 2.4, we obtain results of ^[10].

Conflict of interest

The authors declare no conflict of interest.