Lobachevskii Journal of Mathematics

, Volume 29, Issue 4, pp 221–229 | Cite as

Coefficient inequalities for a new class of univalent functions

Article

Abstract

In the present article, a new class Σ α , 0 ⩽ α < 1, of analytic and univalent functions f: U\( \bar C \), where U is an open unit disk, satisfying the standard normalization f(0) = f′(0) − 1 = 0 is considered. Assume that f ∈ Σ α takes the form
$$ f(z) = z + \sum\limits_{n = 2}^\infty {A_{n,\alpha } (f)z^{n + \alpha } } , 0 \leqslant \alpha < 1, $$
such that A 0,0 = 0 and A 1,0 = 1. Also, we define the family Co(p), where p ∈ (0, 1), of functions f: U\( \bar C \) that satisfy the following conditions:
  1. (i)

    f ∈ Σ α is meromorphic in U and has a simple pole at the point p.

     
  2. (ii)

    f(0) = f′(0) − 1 = 0.

     
  3. (iii)

    f maps U conformally onto a set whose complement with respect to \( \bar C \) is convex.

     
We call such functions concave univalent functions. We prove some coefficient estimates for functions in this class when f has the expansion
$$ f(z) = \sum\limits_{n = 0}^\infty {a_{n,\alpha } (z - p)^{n + \alpha } } , \left| {z - p} \right| < 1 - p, p \in (0,1), 0 \leqslant \alpha < 1. $$
The second part of the article concerns some properties of a generalized Sălăgean operator for functions in Σ α . Moreover, a result on subordination for the functions f ∈ Σ α is given.

Key words and phrases

meromorphic univalent functions concave functions convex set Sălăgean operator 

2000 Mathematics Subject Classification

30C45 

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Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Faculty of Science and TechnologyUniversiti Kebangsaan MalaysiaBangiMalaysia

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