On quasi-convex functions and related topics (Q1098965)

Let S be the class of functions f univalent and analytic in the unit disk \(E=\{z:| z| <1\}\) with \(f(0)=0=f'(0)-1\). Let C, S *, K denote the subclasses of S consisting of convex, starlike and close-to-convex functions respectively. A function f analytic in E with \(f(0)=0=f'(0)-1\) is called quasi-convex if there exists a \(g\in C\) such that \[ Re\{(zf'(z))')/(g'(z))\}>0,\text{ for }z\in E. \] The class of all such functions is denoted by \(C^*.\) In this paper a comprehensive study of \(C^*\) is given with reference to coefficient problems, arc length problem and its relation to other subclassesd of S. Some results are proved and for the proofs of other results the reader is referred to other papers. The paper is partly expository in nature.