TAILIEUCHUNG - Báo cáo hóa học: " Generalized conditions for starlikeness and convexity of certain analytic functions"

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 .

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