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Báo cáo hóa học: " Research Article Some Applications of Srivastava-Attiya Operator to p-Valent 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 Some Applications of Srivastava-Attiya Operator to p-Valent Starlike Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 790730 11 pages doi 10.1155 2010 790730 Research Article Some Applications of Srivastava-Attiya Operator to p-Valent Starlike Functions E. A. Elrifai H. E. Darwish and A. R. Ahmed Faculty of Science Mansoura University Mansoura 35516 Egypt Correspondence should be addressed to H. E. Darwish darwish333@yahoo.com Received 25 March 2010 Accepted 14 July 2010 Academic Editor Ram N. Mohapatra Copyright 2010 E. A. Elrifai 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. We introduce and study some new subclasses of p-valent starlike convex close-to-convex and quasi-convex functions defined by certain Srivastava-Attiya operator. Inclusion relations are established and integral operator of functions in these subclasses is discussed. 1. Introduction Let A p denote the class of functions of the form f z zp Ệan pzn p p e N 1 2 3 . 1.1 which are analytic and p-valent in the open unit disc U z z e C and z 1 . Also let the Hadamard product or convolution of two functions TO fj z zp Ẹ an p izn p j 1 2 1.2 be given by f1 f2 z zp sn 1 an p 1an p 2Zn p f f1 z . A function f z e A p is said to be in the class Sp a of p-valent functions of order a if it satisfies a 0 a p z e U . 1.3 we write Sp 0 Sp the class of p-valent starlike in U. 2 Journal of Inequalities and Applications A function f e A p is said to be in the class Cp a of p-valent convex functions of order a if it satisfies A zf z . v Re 1 a 0 a p z e U . 1.4 f z - The class of p-valent convex functions in U is denoted by Cp Cp 0 . It follows from 1.3 and 1.4 that f z e Cp 0 iff zf z e Sp a 0 a p . 1.5 The classes Sp and Cp were introduced by Goodman 1 . Furthermore a function f z e A p is said to be p-valent close-to-convex of order p and type Y in U if there exists