Hypergeometric transforms of functions with derivative in a half plane

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:96

Article

Ponnusamy, S

关键词: univalent;    star-like;    convex;    close-to-convex;    hypergeometric functions;   

DOI  :  10.1016/S0377-0427(98)00090-9

来源: Elsevier

【 摘 要 】

Let A be the class of normalized analytic functions in the unit disk Delta, F(a, b; c; z) and Phi(a; c; z) denote respectively, the Gaussian and conffuent hypergeometric functions. Let R(beta) = {f is an element of A: There Exists eta is an element of R such that Re [e(i eta)(f'(z) - beta)] > 0, z is an element of Delta}. For f is an element of A, we define the hypergeometric transforms V-a,V-b;c(f) and U-a;c(f) by the convolution V-a,V-b;c(f):=zF(a, b; c;z)* f(z) and U-a;c(f):=z Phi(a;c;z) *f(z), respectively. The main aim of this paper is to find conditions on beta(1), beta(2) and the parameters (a, b,c) such that each of the operators V-a,V-b;c(f) and U-a;c(f) maps R(beta(1)) into R(beta(2)) We also find conditions such that the function (c/ab)[F(a, b; c; z) -1] or (c/a)[Phi(a; c;z)- 1] is in R(beta). (C) 1998 Elsevier Science B.V. All rights reserved.

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