Hypergeometric transforms of functions with derivative in a half plane

✍ Scribed by S. Ponnusamy

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 724 KB
• Volume: 96
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(98)00090-9

✦ Synopsis

Let ~¢ he the class of normalized analytic functions in the unit disk A, F (a,b;c;z) and ~(a;c;z) denote respectively, the Gaussian and confluent hypergeometfic functions. Let ~(fl) = {f E ~1:3 r/C • such that Re [ein(f'(z) -fl)] >0, z C A}. For f E sO, we define the hypergeometric transforms Va, b;c(f) and UQ;c(f) by the convolution V~,b:c(f):=zF(a,b;c;z)* f(z) and Ua;c(f):=z4~(a;c;z)* f(z), respectively. The main aim of this paper is to find conditions on ill, r2 and the parameters (a,b,c) such that each of the operators V~,b;c(f) and Ua;c(f) maps ~(fll ) into ~(f12). We also find conditions such that the function (c/ab)[F(a, b; c; z)-1] or (c/a)[~(a;c;z) -1] is in :~(fl).