Koebe 1/4 theorem

Koebe 1/4 theorem

The Koebe 1/4 theorem states that the image of an injective analytic function f:mathbb D omathbb C from the unit disk mathbb D onto a subset of the complex plane contains the disk whose center is f,(0) and whose radius is |f,'(0)|/4. The theorem is named after Paul Koebe, who conjectured the result in 1907. The theorem was proven by Ludwig Bieberbach in 1914. The Koebe function f(z)=z/(1-z)^2 shows that the constant 1/4 in the theorem cannot be improved.

References

*cite book| last=Rudin | first=Walter | authorlink=Walter Rudin | year=1987 | title=Real and Complex Analysis | series=Series in Higher Mathematics | publisher=McGraw-Hill | edition=3 | isbn=0070542341 | id=MR|924157


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