Numerical solution of the omitted area problem of univalent function theory

Report

The omitted area problem was posed by Goodman in 1949: what is the maximum area $A^*$ of the unit disk 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$\pi$ < $A^*$ < .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 $A^*$ = 0.2385813248$\pi$.

Authors: Banjai, L; Trefethen, L

• Publisher: Unspecified
• Publication date: 2001-12-01