Two Important Generalizations of Schwarz Lemma

A proof of Carathéodory inequality by Schwarz lemma is given, then the relation between \lim\limits _r \rightarrow \infty \fracA(r)r^n and \infty is obtained for a class of complex functions f which are analytic on \mathbbC and take \infty as the natural singular point. Secondly, the radius of the univalent disk of a class of analytic functions f_a which satisfy some certain conditions was obtained by using Schwarz lemma. Finally, this result is extended to a class of analytic functions f_a^r which satisfy f(0)=0, f^\prime(0)=a>0 and maps B(0, r) to itself and obtain the radius of univalent disk of functions belong to f_a^r.