Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A study of Pescar’s univalence criteria for space of analytic functions | Faisal and Darus Journal of Inequalities and Applications 2011 2011 109 http content 2011 1 109 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open 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 Let A denote the class of analytic functions of the form f z z 2 akZk in the open unit disk U z z 1 normalized byf 0 f 0 -1 0. We denote by S the subclass of A consisting of functions which are univalent in U. 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 product u-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 3 Let a be a complex number with Re a 0 such that c e c c 1 c 1. If f e A satisfies the condition c z 2a 1 z 2a zyz af z 1 Vz e U then the function Fa z a a f0 ta 1 f t dt is analytic and univalent in U. Lemma . 1 Let the function f e A satisfies z2f z f 2 z 1 Vz e U. Also let a e R a e 1 3 and c e c. If c 3 2a a c 1 and g z 1 then the
