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

Khalida Inayat Noor

 Department of Mathematics, Girls College for Science Education, Sitteen Road, Al-Malaz, Riyadh, Saudi Arabia
Received 28 January 1985; Revised 20 March 1985

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.