Faber Polynomial Coefficients of Classes of Meromorphic Bistarlike Functions

Applying the Faber polynomial coefficient expansions to certain classes of meromorphic bistarlike functions, 
we demonstrate the unpredictability of their early coefficients and also obtain general coefficient estimates for 
such functions subject to a given gap series condition. Our results improve some of the coefficient bounds 
published earlier.

Let Σ be the family of functions  of the form that are univalent in the punctured unit disk D := { : 0 < || < 1}.

International Journal of Mathematics and Mathematical Sciences
In 1923, Löwner [7] proved that the inverse of the Koebe function () = /(1−) 2 provides the best upper bounds for the coefficients of the inverses of analytic univalent functions.Although the estimates for the coefficients of the inverses of analytic univalent functions have been obtained in a surprisingly straightforward way (e.g., see [8, page 104]), the case turns out to be a challenge when the biunivalency condition is imposed on these functions.A function is said to be biunivalent in a given domain if both the function and its inverse are univalent there.By the same token, a function is said to be bistarlike in a given domain if both the function and its inverse are starlike there.Finding bounds for the coefficients of classes of biunivalent functions dates back to 1967 (see Lewin [9]).The interest on the bounds for the coefficients of subclasses of biunivalent functions picked up by the publications [10][11][12][13][14] where the estimates for the first two coefficients of certain classes of biunivalent functions were provided.Not much is known about the higher coefficients of the subclasses biunivalent functions as Ali et al. [13] also declared that finding the bounds for |  |,  ≥ 4, is an open problem.In this paper, we use the Faber polynomial expansions of the functions  and ℎ =  −1 in Σ(, ) to obtain bounds for their general coefficients |  | and provide estimates for the early coefficients of these types of functions.
We will need the following well-known two lemmas, the first of which can be found in [15] (also see Duren [1]).
Consequently, we have the following lemma, in which we will provide a short proof for the sake of completeness.
Comparing the corresponding coefficients of ( 11) and ( 17) implies Similarly, comparing the corresponding coefficients of ( 14) and (18) gives Substituting  = 0,  = 1, and  = 2 in ( 16), (20), and (21), respectively, yields Taking the absolute values of either equation in (22), we obtain | 0 | ≤  − .Obviously, from (22), we note that  1 = − 1 .Solving the equations in (23) for  2 0 and then adding them gives International Journal of Mathematics and Mathematical Sciences Now, in light of (22), we conclude that Once again, solving for  2 0 and taking the square root of both sides, we obtain Now, the first part of Theorem 3 follows since for 2 −  > 1 it is easy to see that Adding the equations in (23) and using the fact that  1 = − 1 , we obtain Dividing by 4 and taking the absolute values of both sides yield On the other hand, from the second equations in ( 22) and (23), we obtain (31) This concludes the second part of Theorem 3 since for 0 <  < 1 − 2 we have Substituting ( 22) in (23), we obtain