Bloch's Theorem, in complex analysis, describes the behavior of holomorphic functions defined on the unit disk by giving a universal lower bound on the size of a disk within the function's image on which a well-defined holomorphic inverse exists. Named for the French mathematician André Bloch, the theorem guarantees this bound uniformly for every non-constant holomorphic function on the disk with a normalized derivative at the origin, independent of the particular function chosen. It gave rise to the Bloch constant, the best possible value of this universal bound, whose exact value remains unknown.
Facts
StatementFor a holomorphic function f on the unit disk with f'(0) = 1, there is a disk S on which f is biholomorphic and f(S) contains a disk of radius 1/72. 1 Classification
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Source Bloch's theorem (complex analysis) (Wikipedia)
Sources
1. Bloch's theorem (complex variables), Wikipedia
Statement
Bloch's theorem states that there is a disk S ⊂ D on which f is biholomorphic and f(S) contains a disk with radius 1/72.
References
Bloch, André (1925).
View the SourceBloch's theorem (complex analysis) (Wikipedia)
In Branch: Complex Analysis, Lead sentenceQuote, In Branch: Complex Analysis, Lead sentence
In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit
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