Research Article | Open Access
Faber Polynomial Coefficient Estimates for Meromorphic Bi-Starlike Functions
We consider meromorphic starlike univalent functions that are also bi-starlike and find Faber polynomial coefficient estimates for these types of functions. A function is said to be bi-starlike if both the function and its inverse are starlike univalent.
In general (also see Bouali [3, page 52]) where
The coefficients of , the inverse map of are given by where and with is a homogeneous polynomial of degree in the variables . (Also see [1, page 349].)
Similarly where is a Faber polynomial of degree and where for is a homogeneous polynomial of degree in the variables .
The Faber polynomials introduced by Faber  play an important role in various areas of mathematical sciences, especially in geometric function theory (e.g., see Gong  and Schiffer ). The recent interest in the calculus of the Faber polynomials, especially when it involves the function , the inverse map of (see [2, page 186]) beautifully fits the case for the meromorphic bi-univalent functions.
The function is said to be meromorphic bi-univalent in if both and its inverse are meromorphic univalent in . By the same token, the function is said to be meromorphic bi-starlike of order in if both and its inverse map are meromorphic starlike of order in , that is,
Estimates on the coefficients of meromorphic univalent functions were widely investigated in the literature. For example, Schiffer  obtained the estimate for meromorphic univalent functions with and Duren ( or [8, Theorem 4.9, page 139]) proved that if for then . He then proved that this bound also holds for meromorphic starlike univalent functions of order zero (Duren [8, Theorem 4.8, page 137]). So far, the latest known results are given by the following two articles. Kapoor and Mishra  found sharp bounds for the coefficients of starlike univalent functions of order ; in and for its inverse functions they obtained the bound when . More recently, Srivastava et al.  found sharp bounds for the coefficients of starlike univalent functions of order , , having -fold gaps in their series representation in and also for their inverse functions. The above two articles settled the coefficient bounds for starlike functions and their inverses but they have not considered the bi-starlike case. The problem arises when the bi-univalency condition is imposed on the meromorphic functions . The bi-univalency requirement makes the task of finding bounds for the coefficients of and its inverse map more involved. In this paper, for the first time, we use the Faber polynomial expansions to study the coefficients of meromorphic bi-starlike functions. As a result, we are able to prove.
Theorem 1. Let be meromorphic bi-starlike of order in . If for being odd or if for being even, then
Proof. Suppose that the function is a meromorphic bi-starlike function of order in . Then both and its inverse are starlike of order in . Therefore, by definition, there exist two functions and with positive real parts in of the form so that Note that, according to the Caratheodory lemma (see Duren [8, page 41]), and for . On the other hand, comparing the corresponding coefficients of the functions and , we obtain Now, from and , upon noting that there are just two choices of and or and , we obtain Since for the second system of equation we can write Therefore, for odd , we obtain the system of equations Hence Applying the Caratheodory Lemma yields Similarly, for even with , we obtain Hence which upon applying the Caratheodory Lemma, we obtain Relaxing the coefficient restrictions imposed on Theorem 1, we can prove the following.
Theorem 2. Let be meromorphic bi-starlike of order in . Then , .
Proof. Comparing the corresponding coefficients of
Similarly, comparing the corresponding coefficients of we obtain
Adding and , we obtain which, upon applying the Caratheodory Lemma, yields
On the other hand, subtracting from , we obtain which upon, applying the Caratheodory Lemma, yields
Remark 3. For the estimates of the first two coefficients of certain subclasses of analytic and bi-univalent functions, also see recent publications by Srivastava et al.  and Frasin and Aouf .
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Copyright © 2013 Samaneh G. Hamidi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.