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Numerical solution of the omitted area problem of univalent function theory. (English) Zbl 1007.30009

Summary: The omitted area problem was posed by Goodman in 1949: what is the maximum area \({\mathcal A}^*\) of the unit disk \(\mathbb{D}\) that can be omitted by the image of the unit disk under a univalent function normalized by \(f(0)= 0\) and \(f'(0)= 1\)? The previous best bounds were \(0.240005\pi<{\mathcal A}^*\leq .31\pi\). Here the problem is addressed numerically and it is found that these estimates are slightly in error. To ten digits, the correct value appears to be \({\mathcal A}^*= 0.2385813248\pi\).

MSC:

30C30 Schwarz-Christoffel-type mappings
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
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