tailieunhanh - Báo cáo hóa học: "Research Article Sufficient Conditions for Univalence of an Integral Operator Defined by Al-Oboudi Differential 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 Sufficient Conditions for Univalence of an Integral Operator Defined by Al-Oboudi Differential Operator | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 957042 5 pages doi 2008 957042 Research Article Sufficient Conditions for Univalence of an Integral Operator Defined by Al-Oboudi Differential Operator Serap Bulut Civil Aviation College Kocaeli University Arslanbey Campus 41285 Izmit-Kocaeli Turkey Correspondence should be addressed to Serap Bulut Received 10 June 2008 Accepted 21 July 2008 Recommended by Narendra Kumar Govil We investigate the univalence of an integral operator defined by Al-Oboudi differential operator. Copyright 2008 Serap Bulut. 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 A denote the class of all functions of the form f z Ệ akzk which are analytic in the open unit disk U z e C z 1 and S f e A f is univalent in U . For f e A Al-Oboudi 1 introduced the following operator D f z f z D1f z 1 - Ô f z Ôzf z Df z Ỗ 0 Dnf z D6Dn-1f z Ỵ n e N 1 2 3 . . If f is given by then from and we see that Dnf z z 1 k - 1 ỗ nakzk n e No N u 0 k 2 with Dnf 0 0. When Ỗ 1 we get Salagean s differential operator 2 . By using the Al-Oboudi differential operator we introduce the following integral operator. 2 Journal of Inequalities and Applications Definition . Let n m e No and ai e C 1 i m. We define the integral operator I 1 fm Am A I Dfl a . Dfh a-dt z e U where fi eA and Dn is the Al-Oboudi differential operator. Remark . i For n 0 m 1 a1 1 a2 a3 am 0 and D0f1 z D f z f z e A we have Alexander integral operator If z fdt 0 t which was introduced in 3 . ii For n 0 m 1 a1 a e 0 1 a2 a3 am 0 and Dof1 z D f z f z e S we have the integral operator Iaf z Jz ft di that was studied in 4 . iii For n 0 m e N0 ai e C Dũfi z ffiz e S 1 i m we have the integral .