Schottky's theorem

In mathematical complex analysis, Schottky's theorem, introduced by is a quantitative version of Picard's theorem. It states that for a holomorphic function f in the open unit disk that does not take the values 0 or 1, the value of |f(z)| can be bounded in terms of z and f(0).

Schottky's original theorem did not give an explicit bound for f. gave some weak explicit bounds. gave a strong explicit bound, showing that if f is holomorphic in the open unit disk and does not take the values 0 or 1, then
 * $$\log |f(z)| \le \frac{1+|z|}{1-|z|}(7+\max(0,\log |f(0)|))$$.

Several authors, such as, have given variations of Ahlfors's bound with better constants: in particular gave some bounds whose constants are in some sense the best possible.