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
:$\log , f(z), \le \frac(7+\max(0,\log , f(0), ))$.
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.

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Theorems in complex analysis

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