Integral operators on Ma–Minda type starlike and convex functions

Two integral operators on the classes consisting of normalized p -valent Ma–Minda type starlike and convex functions are considered. Functions in these classes have the form z f ′ ( z ) / f ( z ) ≺ p φ ( z ) and 1 + z f ″ ( z ) / f ′ ( z ) ≺ p φ ( z ) respectively, where φ is a convex function with φ ( 0 ) = 1 . It is shown that the first of these operators maps starlike functions into convex functions, while the convex mappings are shown to be closed under the second integral operator.