A study of Pescar's univalence criteria for space of analytic functions

Faisal Imran, Darus, Darus Maslina

Description

An attempt has been made to give a criteria to a family of functions defined in the space of analytic functions to be univalent. Such criteria extended earlier univalence criteria of Pescar's-type of analytic functions. 2000 MSC : 30C45. An attempt has been made to give a criteria to a family of functions defined in the space of analytic functions to be univalent. Such criteria extended earlier univalence criteria of Pescar's-type of analytic functions. 2000 MSC : 30C45.

Sujets

• Analytic function
• Univalent function
• Integral transform

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 11
• Langue: English

Extrait

Faisal and DarusJournal of Inequalities and Applications2011,2011:109 http://www.journalofinequalitiesandapplications.com/content/2011/1/109

R E S E A R C HOpen Access A study of Pescar’s univalence criteria for space of analytic functions * Imran Faisal and Maslina Darus

* Correspondence: maslina@ukm. my School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia

Abstract An attempt has been made to give a criteria to a family of functions defined in the space of analytic functions to be univalent. Such criteria extended earlier univalence criteria of Pescar’stype of analytic functions. 2000 MSC: 30C45. Keywords:analytic functions, univalent functions, integral operator

1. Introduction and preliminaries  ∞ LetAdenote the class of analytic functions of the form f(z) =z+a k z in the =2 open unit disk U =z:z<1 normalized byf(0) =f’(0)  1 = 0. We denote bySthe subclass ofAconsisting of functions which are univalent in. The results in this communication are motivated by Pescar [1]. In [1], a new criteria for an analytic function to be univalent is introduced which is true only for two fixed natural numbers. Then, Breaz and Breaz [2] introduced a new integral operator using productnmultiply analytic functions and gave another univalence criteria for such analytic integral operators. Using such integral operator, we extend the criteria given by Pescar in 2005 and prove that it is true for any two consecutive natural numbers. First, we recall the main results of Pescar introduced in 1996 and later 2005 as follow: Lemma 1.1. [1,3] Letabe a complex number withRea> 0 such thatcÎℂ, c≤1,c=− . IffÎAsatisfies the condition  zf(z) 2α2α  c | z | + (1 − | z | )   ≤1,∀ z ∈ U αf(z)  z α α−1 then the function(F α (z)) =αt f(t)dis analytic and univalent in. 2 z f(z) Lemma 1.2. [1] Let the functionfÎAsatisfies −1 ≤1,∀z∈ . Also, let 2  f z 3 3−2α ∈ R (α∈[1, ] andcÎℂ. If | c | ≤ (c=−1 and |g(z)|≤1, then the function  z α− α G a (z) defined by (G α (z)) =α[f(t)] is in the univalent function classS.  z f(z) Lemma 1.3. [4] IffÎAsatisfies the condition −1 ≤1,∀z∈ , then the 2  f z functionfis univalent in.

© 2011 Faisal and Darus; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.