International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 1, Pages 65-70
doi:10.1155/S016117128600008X
Bounds on the curvature for functions with bounded boundary rotation of order 1−b
Department of Mathematics, Faculty of Science, Mansoura University, Egypt
Received 30 April 1985; Revised 2 July 1985
Copyright © 1986 M. A. Nasr. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.