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

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.