International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 3, Pages 435-438

On alpha-close-to-convex functions of order beta

M. A. Nasr

Faculty of Science, University of Mansoura, Mansoura, Egypt

Received 9 January 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.


Let Mβ(α)[α0andβ0] denote the class of all functionsf(z)=z+n=2anznanalytic in the unit disc U with f(z)f(z)/z0 and which satisfy for z=reiθU the condition θ1θ2Re{(1α)zf(z)f(z)+α(1+zf(z)f(z))}dθ>βπ for all θ2>θ1. In this note we show that each fMβ(α) is close-to-star of order β when 0<βα.