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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 be the family of functions of the formthat are univalent in the punctured unit disk .
For the real constants and (; ), let consist of functions so thatwhere is a Schwarz function; that is, is analytic in the open unit disk and . Note that (Duren ) and the functions in are meromorphic starlike in the punctured unit disk (e.g., see Clunie  and Karunakaran ). It has been proved by Libera and Livingston  and Karunakaran  that for .
The coefficients of , the inverse map of , are given by the Faber polynomial expansion (e.g., see Airault and Bouali  or Airault and Ren [6, page 349]) where ,and is a homogeneous polynomial of degree in the variables .
In 1923, Löwner  proved that the inverse of the Koebe function 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 ). The interest on the bounds for the coefficients of subclasses of biunivalent functions picked up by the publications [10–14] 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.  also declared that finding the bounds for , , is an open problem. In this paper, we use the Faber polynomial expansions of the functions and in to obtain bounds for their general coefficients and provide estimates for the early coefficients of these types of functions.
Lemma 1. Let be so that for . If , then
Consequently, we have the following lemma, in which we will provide a short proof for the sake of completeness.
Lemma 2. Consider the Schwarz function , where for . If , then
Proof. Writewhere is so that for . Comparing the corresponding coefficients of powers of in shows that and .
By substituting for and in (5), we obtain or Now, (6) follows upon substitution of in the above inequality.
In the following theorem, we will observe the unpredictability of the early coefficients of the functions and its inverse map in , providing an estimate for the general coefficients of such functions.
Theorem 3. For and , let the function and its inverse map be in . Then,
Proof. Consider the function given by (1). Therefore (see [5, 6])where is a Faber polynomial of degree . We note that , , , , and . In general (Bouali [16, page 52]),where Similarly, for the inverse map , we have where is a Faber polynomial of degree given by where is a homogeneous polynomial of degree in the variables and Since both and its inverse map are in , the Faber polynomial expansion yields (also see Duren [1, pages 118-119])where and are two Schwarz functions; that is, for and for .
In general (see Airault  or Airault and Bouali ), the coefficients are given by where is a homogeneous polynomial of degree in the variables .
Comparing the corresponding coefficients of (11) and (17) impliesSimilarly, comparing the corresponding coefficients of (14) and (18) gives Substituting , , and in (16), (20), and (21), respectively, yieldsTaking the absolute values of either equation in (22), we obtain . Obviously, from (22), we note that . Solving the equations in (23) for and then adding them givesNow, in light of (22), we conclude thatOnce again, solving for and taking the square root of both sides, we obtainNow, the first part of Theorem 3 follows since for it is easy to see thatAdding the equations in (23) and using the fact that , we obtain Dividing by 4 and taking the absolute values of both sides yieldOn the other hand, from the second equations in (22) and (23), we obtain Taking the absolute values of both sides and applying Lemma 2, it follows thatThis concludes the second part of Theorem 3 since for we haveSubstituting (22) in (23), we obtainFollowing a simple algebraic manipulation, we obtain the coefficient bodyFinally, for , , (20) yields Solving for and taking the absolute values of both sides, we obtain
Remark 5. In ([3, Theorem 1]), the bound was declared to be sharp for the coefficients of the function . The coefficient estimates and given by Theorem 3 show that the coefficient bound is not sharp for the meromorphic bistarlike functions, that is, if both and its inverse map are in . Finding sharp coefficient bound for meromorphic bistarlike functions remains an open problem.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2015 Jay M. Jahangiri and Samaneh G. Hamidi. 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.