On α-convex functions of order β with m-fold symmetry

Abstract

This note is a continuation of the previous work [1,2,3]. First we get a new subordination for α-convex functions of order β when α=1−2β, which implies the rotation theorem for (1−2β)/m-convex functions of order β with m-fold symmetry. Then we extend the known results on α-convex functions of order β to the functions with m-fold symmetry. In particular, we give the sharp order of convexity of α-convex functions of order β with m-fold symmetry for α≥1, which is analogous in sharpness to a result given by Miller, Mocanu and Reade [1].