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Báo cáo hóa học: " Research Article A Convexity Property for an Integral Operator on the Class SP β"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Convexity Property for an Integral Operator on the Class SP β | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 143869 4 pages doi 10.1155 2008 143869 Research Article A Convexity Property for an Integral Operator on the Class Sp b Daniel Breaz Department of Mathematics 1 Decembrie 1918 University Alba lulia 510009 Romania Correspondence should be addressed to Daniel Breaz dbreaz@uab.ro Received 30 October 2007 Accepted 30 December 2007 Recommended by Narendra Kumar K. Govil We consider an integral operator F z for analytic functions fi z in the open unit disk U. The object of this paper is to prove the convexity properties for the integral operator F z on the class Sp b . Copyright 2008 Daniel Breaz. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let U z C z 1 be the unit disc of the complex plane and denote by H U the class of the holomorphic functions in U. Let A f H U f z z a2z2 a3z3 z U be the class of analytic functions in U and S f A f is univalent in U . Denote with K the class of convex functions in U defined by K if A Rei Zc 1 0 z . f z 1.1 A function f S is the convex function of order a 0 a 1 and denote this class by K 0 if f verifies the inequality Ref z0z 1 a z u . I f z J 1.2 Consider the class Sp b which was introduced by Ronning 1 and which is defined by f Sv B zf zZ -1 Reizf zZ - bL 1.3 f Sp p f z 1 Re f z PỊ U.3 where b is a real number with the property -1 b 1. 2 Journal of Inequalities and Applications For fi z A and a. 0 i 1 . n we define the integral operator Fn z given by p r f1 t a1.fA Fn z - 0 t .Ự t dt 1.4 This integral operator was first defined by B. Breaz and N. Breaz 2 . It is easy to see that FnZ A. 2. Main results Theorem 2.1. Let ai 0 for i 1 . n let p. be real numbers with the property -1 p. 1 and let fi SpiPị for i 1 . n . If 0 1 - Pi 1 2.1 i 1 then the function Fn .