International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 3, Pages 435-438
doi:10.1155/S016117128600056X
On alpha-close-to-convex functions of order beta
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.
Abstract
Let Mβ(α)[α≥0 and β≥0] denote the class of all functionsf(z)=z+∑n=2∞anznanalytic in the unit disc U with f′(z)f(z)/z≠0 and which satisfy for z=reiθ∈U the condition ∫θ1θ2Re{(1−α)zf′(z)f(z)+α(1+zf″(z)f′(z))}dθ>−βπ for all θ2>θ1. In this note we show that each f∈Mβ(α) is close-to-star of order β when 0<β≤α.