Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Generalized conditions for starlikeness and convexity of certain analytic functions | Uyanik and Owa Journal of Inequalities and Applications 2011 2011 87 http content 2011 1 87 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Generalized conditions for starlikeness and convexity of certain analytic functions Neslihan Uyanik 1 and Shigeyoshi Owa2 Correspondence owa@math. department of Mathematics Kinki University Higashi-Osaka Osaka 577-8502 Japan Full list of author information is available at the end of the article Springer Abstract For analytic functions f z in the open unit disk U with f 0 0 and f 0 1 Nunokawa et al. Turk J Math 34 333-337 2010 have shown some conditions for starlikeness and convexity of f z . The object of the present paper is to derive some generalized conditions for starlikeness and convexity of functions f z with examples. 2010 Mathematics Subject Classification Primary 30C45. Keywords Analytic function starlike function convex function 1 Introduction Let A denote the class of functions f z of the form X f z z anZn n 2 which are analytic in the open unit disk U z e C z 1 . Let 5 be the subclass of A consisting of functions f z which are univalent in U. A function f z e s is said to be starlike with respect to the origin in U if f U is the starlike domain. We denote by s the class of all starlike functions f z with respect to the origin in U. Furthermore if a function f z e s satisfies zf z e s then f z is said to be convex in U. We also denote by K the class of all convex functions in U. Note that K c s c s c A. To discuss the univalency of f z e A Nunokawa 1 has given Lemma If f z e Asatisfies If z I 1 z e U then f z e s. Also Mocanu 2 has shown that Lemma If f z e Asatisfies lf z - 11 Ự5 z e U then f z e s . In view of Lemmas and Nunokawa et al. 3 have proved the following results. Lemma If f z e Asatisfies 2 If z I . z e U Then f z e s . 2011 Uyanik and Owa licensee Springer. This is an Open .
