Grunsky's theorem

In mathematics, Grunsky's theorem, due to the German mathematician Helmut Grunsky, is a result in complex analysis concerning holomorphic univalent functions defined on the unit disk in the complex numbers. The theorem states that a univalent function defined on the unit disc, fixing the point 0, maps every disk |z| < r onto a starlike domain for r ≤ tanh π/4.

The radius of starlikeness of an univalent function f satisfying f(0) = 0 is the largest radius r for which the function f maps the open disk |z| < r into a starlike domain with respect to the origin.

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