Journal of Complex Analysis

Journal of Complex Analysis / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 263917 | 4 pages | https://doi.org/10.1155/2014/263917

Coefficients of Meromorphic Bi-Bazilevic Functions

Academic Editor: Yan Xu
Received19 Nov 2013
Accepted06 Feb 2014
Published09 Mar 2014

Abstract

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.

1. Introduction

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 [1], p. 349 or [24]).

For , , , and , let denote the class of bi-Bazilevic functions of order and type (see Bazilevic [5]) if and only if

Estimates on the coefficients of classes of meromorphic univalent functions were widely investigated in the literature. For example, Schiffer [4] obtained the estimate for meromorphic univalent functions with and Duren ([6] or [7]) proved that if for , then . Schober [8] 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 [9] 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. [10] (also see [11]) improved the coefficient estimate given by Kapoor and Mishra in [9]. 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 [9] 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 [12] 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 [3]) is given by where and . Here we note that the sum is taken over all nonnegative integers satisfying and . Evidently, , [12]. 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 [13] play an important role in various areas of mathematical sciences, especially in geometric function theory (Gong [14] Chapter III and Schiffer [4]). 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., [7]), 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.

References

  1. H. Airault and J. Ren, “An algebra of differential operators and generating functions on the set of univalent functions,” Bulletin des Sciences Mathematiques, vol. 126, no. 5, pp. 343–367, 2002. View at: Publisher Site | Google Scholar | MathSciNet
  2. H. Airault, “Symmetric sums associated to the factorization of Grunsky coefficients,” in Conference, Groups and Symmetries, Montreal, Canada, April 2007. View at: Google Scholar
  3. H. Airault and A. Bouali, “Differential calculus on the Faber polynomials,” Bulletin des Sciences Mathematiques, vol. 130, no. 3, pp. 179–222, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. M. Schiffer, “Faber polynomials in the theory of univalent functions,” Bulletin of the American Mathematical Society, vol. 54, pp. 503–517, 1948. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. I. E. Bazilevic, “On a case of integrability in quadratures of the Loewner-Kufarev equation,” Matematicheskii Sbornik, vol. 37, no. 79, pp. 471–476, 1955. View at: Google Scholar | MathSciNet
  6. P. L. Duren, “Coefficients of meromorphic schlicht functions,” Proceedings of the American Mathematical Society, vol. 28, pp. 169–172, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1983. View at: MathSciNet
  8. G. Schober, “Coefficients of inverses of meromorphic univalent functions,” Proceedings of the American Mathematical Society, vol. 67, no. 1, pp. 111–116, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. G. P. Kapoor and A. K. Mishra, “Coefficient estimates for inverses of starlike functions of positive order,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 922–934, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. S. G. Hamidi, S. A. Halim, and J. M. Jahangiri, “Coefficient estimates for a class of meromorphic bi-univalent functions,” Comptes Rendus Mathematique, vol. 351, no. 9-10, pp. 349–352, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. S. G. Hamidi, S. A. Halim, and J. M. Jahangiri, “Faber polynomial coefficient estimates for meromorphic bi-starlike functions,” International Journal of Mathematics and Mathematical Sciences, vol. 2013, Article ID 498159, 4 pages, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. H. Airault, “Remarks on Faber polynomials,” International Mathematical Forum, vol. 3, no. 9–12, pp. 449–456, 2008. View at: Google Scholar | MathSciNet
  13. G. Faber, “Über polynomische Entwickelungen,” Mathematische Annalen, vol. 57, no. 3, pp. 389–408, 1903. View at: Publisher Site | Google Scholar | MathSciNet
  14. S. Gong, The Bieberbach Conjecture, vol. 12 of AMS/IP Studies in Advanced Mathematics, American Mathematical Society, Providence, RI, USA, 1999. View at: MathSciNet
  15. P. G. Todorov, “On the Faber polynomials of the univalent functions of class Σ,” Journal of Mathematical Analysis and Applications, vol. 162, no. 1, pp. 268–276, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. M. Jahangiri, “On the coefficients of powers of a class of Bazilevic functions,” Indian Journal of Pure and Applied Mathematics, vol. 17, no. 9, pp. 1140–1144, 1986. View at: Google Scholar | MathSciNet

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.

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