International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 3, Pages 497-501
doi:10.1155/S0161171288000584
Some classes of alpha-quasi-convex functions
Mathematics Department, College of Science Education for Girls, Sitteen Road, Malaz, Riyadh, Saudi Arabia
Received 13 April 1985; Revised 30 May 1986
Copyright © 1988 Khalida Inayat Noor. 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 C[C,D], −1≤D<C≤1 denote the class of functions g, g(0)=0, g′(0)=1, analytic in the unit disk E such that (zg′(z))′g′(z) is subordinate to 1+CZ1+DZ, z∈E. We investigate some classes of Alpha-Quasi-Convex Functions f, with f(0)=f′(0)−1=0 for which there exists a g∈C[C,D] such that (1−α)f′(z)g′(z)+α(zf′(z))′g′(z) is subordinate to 1+AZ1+BZ′, −1≤B<A≤1. Integral representation, coefficient bounds are obtained. It is shown that some of these classes are preserved under certain integral operators.