Closure properties of operators on the Ma–Minda type starlike and convex functions

A normalized univalent function f   is called Ma–Minda starlike or convex if zf′(z)/f(z)≺φ(z)zf′(z)/f(z)≺φ(z) or 1+zf″(z)/f′(z)≺φ(z)1+zf″(z)/f′(z)≺φ(z) where φφ is a convex univalent function with φ(0)=1φ(0)=1. The class of Ma–Minda convex functions is shown to be closed under certain operators that are generalizations of previously studied operators. Analogous inclusion results are also obtained for subclasses of starlike and close-to-convex functions. Connections with various earlier works are made.