Sufficient conditions for bounded turning of analytic functions

Abstract

Let function $f$ be analytic in the open unit disk and be normalized such that $f(0) = f\prime (0) 1 = 0$. In this paper methods from the theory of first order differential subordinations are used for obtaining sufficient conditions for $f$ to bewith bounded turning, i.e., the read part of its first derivative to map the unit disk onto the right half plane. In addition, several open problems are posed.