International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 2, Pages 241-258

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

Copyright © 1987 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.


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.