tailieunhanh - Báo cáo hóa học: " Research Article Integral Means Inequalities for Fractional Derivatives of a Unified Subclass of Prestarlike Functions with Negative Coefficients"

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 Integral Means Inequalities for Fractional Derivatives of a Unified Subclass of Prestarlike Functions with Negative Coefficients | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 97135 9 pages doi 2007 97135 Research Article Integral Means Inequalities for Fractional Derivatives of a Unified Subclass of Prestarlike Functions with Negative Coefficients H. Ozlem Guney and Shigeyoshi Owa Received 24 May 2007 Revised 13 July 2007 Accepted 28 July 2007 Recommended by Narendra K. Govil Integral means inequalities are obtained for the fractional derivatives of order p A 0 p n 0 A 1 of functions belonging to a unified subclass of prestarlike functions. Relevant connections with various known integral means inequalities are also pointed out. Copyright 2007 H. O. Guney and S. Owa. 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 y denote the class of normalized functions of the form 00 f z z anZn n 2 which are analytic and univalent in the open unit disk U z e C z 1 . Also let denote the subclass of y consisting of functions f of the form f z z - f anzn an 0 . n 2 The Hadamard product or convolution of two functions f given by and g given by O g z z bnzn n 2 2 Journal of Inequalities and Applications is defined by f g z z X anbnzn. n 2 We denote the subclass a f of y consisting of a-prestarlike functions of order f by a f f e y f Sa z e y f 0 a 1 0 f 1 where y f denotes the class of starlike functions of order f 0 f 1 and sa is the well-known extremal function for y a given by Sa z z 1 - z -2 1-a cf. 1 2 . Letting X n n 2 k - 2a cn a n 1 n 2 3 . sa can be written in the form 00 Sa z z Cn a zn. n 2 The class a f was investigated by Sheil-Small et al. 3 . We also denote the subclass a f of y which was investigated by Owa and Uralegaddi 4 by w f f e y zf z e a f . In particular the subclasses a f a f n a f a f n were considered .