International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 2, Pages 241-258
doi:10.1155/S0161171287000310

On quasi-convex functions and related topics

Abstract

Let S be the class of functions f which are analytic and univalent in the unit disc E with f(0)=0, f′(0)=1. Let C, S* and K be the classes of convex, starlike and close-to-convex functions respectively. The class C* of quasi-convex functions is defined as follows:Let f be analytic in E and f(0), f′(0)=1. Then f ϵ C* if and only if there exists a g ϵ C such that, for z ϵ ERe(zf′(z))′g′(z)>0.In this paper, an up-to-date complete study of the class C* is given. Its basic properties, its relationship with other subclasses of S, coefficient problems, arc length problem and many other results are included in this study. Some related classes are also defined and studied in some detail.