The Koebe 1/4 theorem states that the image of an injective analytic function from the unit disk onto a subset of the complex plane contains the disk whose center is and whose radius is . 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 shows that the constant in the theorem cannot be improved.

Proof

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There is a proof based on the area theorem and some power series calculations. Following is a proof based on the notion and properties of extremal length.

We start by assuming that   and  . Since every point   has a neighborhood in which   can be defined as an analytic function, the monodromy theorem implies that there is an analytic function   such that   for every  . Fix such a   satisfying  . Note that since   is injective, also   must be injective, and moreover,  . This implies that for all   sufficiently small so that  , the extremal distance in   from   to   is at least twice the extremal distance from   to the boundary of  .