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Báo cáo hóa học: " Research Article Fractional Calculus and p-Valently Starlike Functions"

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 Fractional Calculus and p-Valently Starlike Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 683985 9 pages doi 10.1155 2009 683985 Research Article Fractional Calculus and p-Valently Starlike Functions Osman Altintas1 and Oznur Ozkan2 1 Department of Matematics Education Baậkent University Baglica TR-06530 Ankara Turkey 2 Department of Statistics and Computer Sciences Baậkent University Baglica TR 06530 Ankara Turkey Correspondence should be addressed to Oznur Ozkan oznur@baskent.edu.tr Received 18 November 2008 Accepted 28 February 2009 Recommended by Alberto Cabada In this investigation the authors prove coefficient bounds distortion inequalities for fractional calculus of a family of multivalent functions with negative coefficients which is defined by means of a certain nonhomogenous Cauchy-Euler differential equation. Copyright 2009 O. Altintas and O. Ozkan. 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 Definitions Let Tn p denote the class of functions f z of the form f z zp - akzk ak 0 n p e N 1 2 3 . 1.1 k n p which are analytic and multivalent in the unit disk U z z e C and z 1 . The fractional calculus are defined as follows e.g. 1 2 . Definition 1.1. The fractional integral of order 6 is defined by 1 r f if Cĩ -6 Dz f z f 6 0 z -i 1-6di 6 0 1.2 where f z is an analytic function in a simply-connected region of the z-plane containing the origin and the multiplicity of z - i 6-1 is removed by requiring log z - if to be real when z - i 0. 2 Journal of Inequalities and Applications Definition 1.2. The fractional derivative of order 6 is defined by D6zf z 1 d z f U r 1 - 6 dj 0 z - d 0 6 1 1.3 where f z is constrained and multiplicity of z - 6 is removed as in Definition 1.1. Definition 1.3. Under the hypotheses of Definition 1.1 the fractional derivative of order n 6 is .