Inclusion theorems of convolution operators associated with normalized hypergeometric functions

摘要

For the normalized Gaussian hypergeometric function zF(a,b;c;z) given byF(a,b;c;z)=∑n=0∞(a,n)(b,n)(c,n)(1,n)zn,|z|<1,the author aims at finding conditions on a,b and c such that the convolution operator zF(a,b;c;z)*f(z) satisfies some inclusion results in certain class of analytic functions and in particular when f∈Pγ(β), wherePγ(β)=f∈A:Reeiϕ(1-γ)f(z)z+γf′(z)-β>0,ϕ∈R,z∈Δ.A convolution result for the class Pγ(β) is also given. Analogous results for the confluent hypergeometric functions are also discussed.