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 On Bounded Boundary and Bounded Radius Rotations | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 813687 12 pages doi 2009 813687 Research Article On Bounded Boundary and Bounded Radius Rotations K. I. Noor W. Ul-Haq M. Arif and S. Mustafa Department of Mathematics COMSATS Institute of Information Technology 44000 Islamabad Pakistan Correspondence should be addressed to M. Arif marifmaths@ Received 6 January 2009 Revised 6 March 2009 Accepted 19 March 2009 Recommended by Narendra Kumar Govil We establish a relation between the functions of bounded boundary and bounded radius rotations by using three different techniques. A well-known result is observed as a special case from our main result. An interesting application of our work is also being investigated. Copyright 2009 K. I. Noor et al. 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 be the class of functions f of the form TO f z z 2 anZn n 2 which are analytic in the unit disc E z z 1 . We say that f e A is subordinate to g e A written as f g if there exists a Schwarz function w z which by definition is analytic in E with w 0 0 and w z 1 z e E such that f z g w z . In particular when g is univalent then the above subordination is equivalent to f 0 g 0 and f E c g E . For any two analytic functions TO f z ữnzn n 0 TO g z bnzn z e E n 0 the convolution Hadamard product of f and g is defined by TO f g z ữnbnzn z e E . n 0 2 Journal of Inequalities and Applications We denote by S à C a 0 a 1 the classes of starlike and convex functions of order a respectively defined by S a zf z f z a z e E C a f e A zf z e S a z e E . For a 0 we have the well-known classes of starlike and convex univalent functions denoted by S and C respectively. Let Pk a be the class of functions p z analytic in the unit disc E .
