Inclusion theorems of convolution operators associated with normalized hypergeometric functions

For the normalized Gaussian hypergeometric function zF ( a , b ; c ; z ) given by F ( a , b ; c ; z ) = ∑ n = 0 ∞ ( a , n ) ( b , n ) ( c , n ) ( 1 , n ) z n , | 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 ∈ P γ ( β ) , where P γ ( β ) = f ∈ A : Re e i ϕ ( 1 - γ ) f ( z ) z + γ f ′ ( z ) - β > 0 , ϕ ∈ R , z ∈ Δ . A convolution result for the class P γ ( β ) is also given. Analogous results for the confluent hypergeometric functions are also discussed.