Conformality and semiconformality of a function holomorphic in the disk

Abstract:

Conformality and semiconformality at a boundary point, of a function f nonconstant and holomorphic in $\left | z \right | < 1$ are local properties. Therefore one would suspect the requirement of such global conditions on f as f is univalent in $\left | z \right | < 1$, or f is a member of a larger class which contains all univalent functions in $\left | z \right | < 1$. We shall prove some extensions and new results without any assumption on f, or with a local assumption on f at most. Our methods are, for the most part, different from the ones in the classical cases. One of the main tools is Theorem 8 on the angular limits of the real part of a holomorphic function and its derivative.