tailieunhanh - Báo cáo hóa học: " Research Article A Univalence Preserving Integral Operator"

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 Univalence Preserving Integral Operator | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 263408 10 pages doi 2008 263408 Research Article A Univalence Preserving Integral Operator Georgia Irina Oros Department of Mathematics University ofOradea Universitatii street No. 1 410087 Oradea Romania Correspondence should be addressed to Georgia Irina Oros georgia_oros_ro@ Received 23 July 2008 Accepted 8 September 2008 Recommended by Vijay Gupta We define an integral operator denoted by I and we give sufficient conditions such that F I f1 f2 --- fm is univalent. Copyright 2008 Georgia Irina Oros. 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 and preliminaries Let U denote the unit disc of the complex plane U z e C z 1 . Let H U denote the space of holomorphic functions in U and let An f e HfU f z z an 1zn 1 z e U with A1 A. Let S f e A f is univalent in U . Definition Ruscheweyh 1 . For f e A n e N u 0 let Rn be the operator defined by Rn A A R f z f z n 1 Rn 1f z z Rnf z nRnf z z e U. 2 Journal of Inequalities and Applications Remark . If f e A TO f z z y ajZj z e U then TO Rnf z z z cnn j j z e U. In order to prove our main results we shall use the following lemmas. Lemma A see 2 . Let a be a complex number Re a 0 and f e A. If 1 - z 2Rea zf z Re a f z 1 Vz e U then the function Fa z a ta 1f f dt 0 1 a is in the class S. Lemma B see 3 . Let a be a complex number Re a 0 and let f z z a2z2 function in U. If 1 z 2Re a zf fz Re a f z 1 Vz e U then for any complex number f with Re f Re a the function Fp z f f tp-1f t dt 0 1 f be a regular is in the class S. 2. Main results By using the Ruscheweyh differential operator given by Definition we introduce the following integral operator. Georgia Irina Oros 3 Definition . .