Sufficiency for hypergeometric transforms to be associated with conic regions

For a certain integral operator acting on the normalized Gaussian hypergeometric function z F ( 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 operator maps certain subclasses of analytic functions into some other classes of functions that have nice geometric properties related to certain conic regions.