Bounds on the curvature for functions with bounded boundary rotation of order 1−b

Abstract

Let Vk(1−b), k≥2 and b≠0 real, denotes the class of locally univalent analytic functions f(z)=z+∑n=2∞anzn in D={z:|z|<1} such that ∫02π|Re{1+1bzf″(z)f′(z)}|dθ<πk, z=reiθ∈D. In this note sharp bounds on the curvature of the image of |z|=r, 0<r<1, under a mapping f belonging to the class Vk(1−b) have been obtained.