On the omitted area problem (Q1821224)

Let \(S\) denote the class of holomorphic and univalent functions \(f\) in \(\Delta =\{z:| z| <1\}\) with \(f(0)=f'(0)-1=0\). The omitted area problem was first posed by \textit{A. W. Goodman} [Bull. Am. Math. Soc. 55, 363-369 (1949; Zbl 0033.176)]. It was next the object of interest of other mathematicians, among others, of \textit{J. L. Lewis} [Indiana Univ. Math. J. 34, 631-661 (1985; Zbl 0579.30007)]. This problem consists in finding the maximum area of the region omitted from \(\Delta\) by \(f(\Delta)\) as \(f\) varies over the family \(S\). If we let \[ \beta =\sup_{f\in S}area[\Delta -f(\Delta)], \] then it is known that \(.24\pi \leq \beta <.38\pi\). The main aim of this paper is to obtain a sharper estimate from above of the quantity \(\beta\).