Research Article | Open Access
Coefficients of Meromorphic Bi-Bazilevic Functions
A function is said to be bi-Bazilevic in a given domain if both the function and its inverse map are Bazilevic there. Applying the Faber polynomial expansions to the meromorphic Bazilevic functions, we obtain the general coefficient bounds for bi-Bazilevic functions. We also demonstrate the unpredictability of the behavior of early coefficients of bi-Bazilevic functions.
Let denote the family of meromorphic functions of the form which are univalent in . The coefficients of , the inverse map of the function , are given by the Faber polynomial expansion: where and with is a homogeneous polynomial of degree in the variables (see , p. 349 or [2–4]).
For , , , and , let denote the class of bi-Bazilevic functions of order and type (see Bazilevic ) if and only if
Estimates on the coefficients of classes of meromorphic univalent functions were widely investigated in the literature. For example, Schiffer  obtained the estimate for meromorphic univalent functions with and Duren ( or ) proved that if for , then . Schober  considered the case where and obtained the estimate for the odd coefficients of the inverse function subject to the restrictions if is even or that if is odd. Kapoor and Mishra  considered the inverse function , where , and obtained the bound if . This restriction imposed on is a very tight restriction since the class shrinks for large values of . More recently, Hamidi et al.  (also see ) improved the coefficient estimate given by Kapoor and Mishra in . The real difficulty arises when the bi-univalency condition is imposed on the meromorphic functions and its inverse . The unexpected and unusual behavior of the coefficients of meromorphic functions and their inverses prove the investigation of the coefficient bounds for bi-univalent functions to be very challenging. In this paper we extend the results of Kapoor and Mishra  and Hamidi et al. [10, 11] to a larger class of meromorphic bi-univalent functions, namely, . We conclude our paper with an examination of the unexpected behavior of the early coefficients of meromorphic bi-Bazilevic functions which is the best estimate yet appeared in the literature.
2. Main Results
Applying a result of Airault  or [1, 3] to meromorphic functions of the form (1), for real values of we can write where and is a homogeneous polynomial of degree in the variables . A simple calculation reveals that the first three terms of may be expressed as In general, for any real number , an expansion of (e.g., see [2, equation (4)] or ) is given by where and . Here we note that the sum is taken over all nonnegative integers satisfying and . Evidently, , . A similar Faber polynomial expansion formula holds for the coefficients of , the inverse map of (e.g., see [1, p. 349]).
The Faber polynomials introduced by Faber  play an important role in various areas of mathematical sciences, especially in geometric function theory (Gong  Chapter III and Schiffer ). The recent interest in the calculus of the Faber polynomials, especially when it involves , the inverse of (see [1, 3, 12, 15]), beautifully fits our case for the meromorphic bi-univalent functions. As a result, we are able to state and prove the following.
Theorem 1. For and let g and . If for being odd or if for being even, then
Proof. For and for , there exist positive real part functions and in so that
Note that, according to the Caratheodory lemma (e.g., ), and .
On the other hand, by the Faber polynomial expansion, we observe that Comparing the corresponding coefficients of (11) and (13) we obtain Similarly, from (12) and (14) we obtain For the case ( = odd), (15) and (16), respectively, upon using a simple algebraic manipulation and the fact that , reduce to Multiplying (18) by −1 and adding it to (17) we obtain For the other case ( = even) (15) and (16), respectively, reduce to
Solving either of (19), (20), or (21) for , taking the absolute values, and applying the Caratheodory lemma we obtain .
Relaxing the coefficient restrictions imposed on Theorem 1, we experience the unpredictable behavior of the coefficients of bi-univalent functions.
Theorem 2. Let , , , be bi-univalent in . Then(i)(ii)(iii)
Proof. (i) For (15) yields
Similarly, for (16) yields
From either of the relations (25) or (28) we obtain
On the other hand, adding (26) and (29) yields
Solve the above equation for , take the absolute values of both sides, and apply the Caratheodory lemma to obtain
Now the bounds given in Theorem 2 (i) for follow upon noting that
(ii) Multiply (29) by −1 and adding it to (26) we obtain Solve the above equation for , take the absolute values of both sides, and apply the Caratheodory lemma to obtain the bound .
(iii) From (29) we have
Substituting for and taking the absolute values of both sides we obtain
Using the fact if which is due to the first author [16, Lemma 1] and noting that if , we obtain
Now substituting back for we obtain
Remark 3. For the special case we obtain the class of meromorphic bi-starlike functions. Consequently, the bound given for by our Theorem 2 (i) is an improvement to that given in Hamidi et al. [11, Theorem 2.i.].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2014 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.