In mathematical
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, Schottky's theorem, introduced by is a quantitative version of
Picard's theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
The theorems
Little Picard Theorem: If a function f: \mathbb \to\mathbb is ...
. It states that for a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
''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
:
.
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.
References
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Theorems in complex analysis
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