Computational Methods and Function Theory 1 (2001), No. 1, 259--273
Copyright Heldermann Verlag 2001
Lehel Banjai
lehelb@comlab.ox.ac.uk, Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, U.K.
Lloyd N. Trefethen
LNT@comlab.ox.ac.uk, Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, U.K.
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 $\mathcal{A}^*= 0.2385813248\pi$.
Keywords: Omitted area problem, numerical conformal mapping, univalent function theory.
MSC 2000: 30C30, 30C75, 65E05.
[FullText-pdf (496 K)] [FullText-ps (708 K)]