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International Journal of Mathematics and Mathematical Sciences
Volume 9, Issue 1, Pages 65-70
http://dx.doi.org/10.1155/S016117128600008X

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

Department of Mathematics, Faculty of Science, Mansoura University, Egypt

Received 30 April 1985; Revised 2 July 1985

Copyright © 1986 Hindawi Publishing Corporation. 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(1b), k2 and b0 real, denotes the class of locally univalent analytic functions f(z)=z+n=2anzn in D={z:|z|<1} such that 02π|Re{1+1bzf(z)f(z)}|dθ<πk, z=reiθD. 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(1b) have been obtained.