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

: In this paper, we introduce a new comprehensive subclass Σ B ( λ , µ , β ) of meromorphic bi-univalent functions in the open unit disk 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 (cid:61) 1 ) for functions in the subclass Σ B ( λ , µ , β ) 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.


Introduction
Let A denote the class of functions f of the form: which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1}.
We also let S be the class of functions f ∈ A which are univalent in U.
It is well known that every function f ∈ S has an inverse f −1 , which is defined by and f f −1 (w) = w |w| < r 0 ( f ); r 0 ( f ) 1 4 .
If f and f −1 are univalent in U, then f is said to be bi-univalent in U. We denote by σ B the class of bi-univalent functions in U. For a brief history and interesting examples of functions in the class σ 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 Σ be the family of meromorphic univalent functions f of the following form: which are defined on the domain ∆ = {z : z ∈ C and 1 < |z| < ∞}.
Since a function f ∈ Σ is univalent, it has an inverse f −1 that satisfies the following relationship: Furthermore, the inverse function f −1 has a series expansion of the form [17]: .
A function f ∈ Σ is said to be meromorphic bi-univalent if both f and f −1 are meromorphic univalent in ∆. The family of all meromorphic bi-univalent functions in ∆ of the form (2) is denoted by Σ M . A simple calculation shows that (see also [18,19]) Moreover, the coefficients of g = f −1 can be given in terms of the Faber polynomial [20] (see also [21][22][23]) as follows: where K n n+1 = nb n−1 Estimates on the coefficients of meromorphic univalent functions were widely investigated in the literature. For example, Schiffer [24] obtained the estimate |b 2 | 2/3 for meromorphic univalent functions f ∈ Σ with b 0 = 0 and Duren [25] proved that 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 Σ B * (λ, β) of meromorphic biunivalent functions and obtained the estimates on the initial Taylor-Maclaurin coefficients |b 0 | and |b 1 | for functions in this class.
Theorem 1 (see [31]). Let the function f ∈ Σ M , given by (2), be in the class Σ B * (λ, β). Then, In this paper, we introduce a new comprehensive subclass Σ B (λ, µ, β) of the meromorphic bi-univalent function class Σ 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 1) for functions in this comprehensive subclass Σ B (λ, µ, β) by using the Faber polynomials [20]. Our results for the meromorphic bi-univalent function subclass Σ B (λ, µ, β) 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]).

Preliminary Results
For finding the coefficients of functions belonging to the function class Σ B (λ, µ, β), we need the following lemmas and remarks.
Lemma 1 (see [21,22]). Let f be the function given by be a meromorphic univalent function defined on the domain ∆. Then, for any ρ ∈ R, there are polynomials K ρ n such that in which the sum is taken over all non-negative integers µ 1 , · · · , µ n−k+1 such that The first three terms of K ρ n are given by

Remark 1.
In the special case when Lemma 2 (see [21,22]). Let f be the function given by be a meromorphic univalent function defined on the domain ∆. Then, the Faber polynomials F n of f (z) are given by where F n (b 0 , b 1 , · · · , b n−1 ) is a homogeneous polynomial of degree n.
Remark 2 (see [36]). For any integer n 1, the polynomials F n (b 0 , b 1 , · · · , b n−1 ) are given by The first three terms of F n are given by Remark 3. In the special case when b 0 = b 1 = · · · = b n−1 = 0, it is readily observed that Lemma 3. Let f be the function given by be a meromorphic univalent function defined on the domain ∆. Then, for λ 1 and µ 0, Proof. By using Lemmas 1 and 2, we have In addition, by applying Lemma 1 once again, we obtain Our demonstration of Lemma 3 is thus completed.
The first three terms of L n are given by

Remark 4.
In the special case when b 0 = b 1 = · · · = b n−1 = 0, we easily find that Lemma 4 (see [37]). If the function p ∈ P, then |c k | 2 for each k, where P is the family of all functions p, which are analytic in the domain ∆ given by

The Comprehensive Class
In this section, we introduce and investigate the comprehensive class Σ B (λ, µ, β) of meromorphic bi-univalent functions defined on the domain ∆.

Definition 2.
A function f ∈ Σ M , given by (2), is said to be in the class of meromorphic bi-univalent functions of order β and type µ, if the following conditions are satisfied: where the function g given by (4), is the inverse of f and z, w ∈ ∆.

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 λ = 1 and 0 µ < 1, the class Σ B (λ, µ, β) reduces to the subclass B(β, µ) of meromorphic bi-Bazilevič functions of order β and type µ, which was considered by Jahangiri et al. [33].
Proof. By using Lemma 3 for the meromorphic bi-univalent function f given by Similarly, for its inverse map g given by Furthermore, since f ∈ Σ B (λ, µ, β), by using Definition 2, there exist two positive real-part functions such that and Upon equating the corresponding coefficients in (6) and (8), we get Similarly, from (7) and (9), we obtain Now, since b i = 0 (0 i n − 1), we have Hence, by using Remark 4, Equations (10) and (11) can be rewritten as follows: and respectively. Thus, from (12) and (13), we find that Finally, by applying Lemma 4, we get which completes the proof of Theorem 2 Theorem 3. Let the function f ∈ M, given by (2), be in the class Then, Proof. By putting n = 0, 1, 2 in (10), we get and Similarly, by putting n = 0, 1, 2 in (11), we have and Clearly, from (14) and (17), we get Adding (15) and (18), we obtain In view of the Equations (21) and (22), by applying Lemma 4, we get 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 (18) from (15). We thus obtain Applying Lemma 4 once again, we get Finally, in order to determine the bound on |b 2 |, we consider the sum of the Equations (16) and (19) with c 1 = −d 1 . This yields In addition, by using (21) and (24) in (25), we get Hence, Thus, by applying Lemma 4 once again, we get This completes the proof of Theorem 3.

A Set of Corollaries and Consequences
By setting λ = 1 and 0 µ < 1 in Theorem 2, we have the following result.

Corollary 1.
Let the function f ∈ M, given by (2), be in the subclass B(β, µ) of meromorphic bi-Bazilevič functions of order β and type µ. By setting µ = 0 in Corollary 1, we have the following result.

Corollary 2.
Let the function f ∈ M, given by (2), be in the subclass Σ * B (β) of meromorphic bi-starlike functions of order β. If

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 µ = λ − 1 in Theorem 2, we have the following result.

Remark 8.
Corollary 3 is a generalization of a result presented in Theorem 1, which was proved by Srivastava et al. [31].
By setting λ = 1 and 0 µ < 1 in Theorem 3, we have the following result.
By setting µ = 0 in Corollary 4, we have the following result.

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 Σ * B (β). 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 µ = λ − 1 in Theorem 3, we have the following result.

Corollary 6.
Let the function f ∈ M, given by (2), be in the subclass Σ B * (λ, β). Then, 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 Σ B * (λ, β).

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 < |q| < p 1) would obviously be inconsequential because the additional parameter p is redundant.
Author Contributions: All three authors contributed equally to this investigation. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.