Rounding corners of gearlike domains and the omitted area problem (Q5966450)

A function \(g(z)=a_ 1z+a_ 2z^ 2+...\), analytic and univalent in \({\mathcal D}=\{z:\) \(| z| <1\}\) is called gearlike if \(\partial g({\mathcal D})\) consists of arcs of circles centred at 0 and segments of rays through 0. \textit{A. W. Goodman} [Bull. Am. Math. Soc. 55, 363-369 (1949; Zbl 0033.176)] asked the question: If \(a_ 1=1\), what is \(A^*\), the minimum area of g(\({\mathcal D})\cap {\mathcal D} ?\) The authors construct a structural formula for zg'/g; and, extending work of \textit{P. Henrici} [Applied and computational complex analysis. Vol. 1 (1974; Zbl 0313.30001)], give formulae for rounding or smoothing each corner of \(\partial g({\mathcal D})\). Using a modified gearlike function they show that \(A^*\simeq 0.769\pi\) (to within 0.01\(\pi)\).