THROUGHOUT the books treating of hyperbolic functions, although elaborate series for their determination are given, the possibility of calculating them directly from their definitions, by means of ...

Let f be an analytic and univalent function on a simply connected domain D, and let λD be the hyperbolic metric on D. We prove the sharp inequality $\left| {\frac{{{f^n}\left( w \right)}}{{f\left( w ...