Constructing mappings onto radial slit domains (Q2478056)

A particular construction of a conformal mapping of the unit disc onto a planar domain with infinitely many radial slits is used in [\textit{R.~E.~Goodman}, Bull. Am. Math. Soc. 51, 234--239 (1945; Zbl 0063.01697)] to obtain an upper bound (Goodman's constant) for the univalent Bloch constant, and in [\textit{R.~Bañuelos} and \textit{T.~Carroll}, Duke Math. J. 75, No. 3, 575--602 (1994; Zbl 0817.58046); Addendum ibid. 82, No. 1, 227 (1996; Zbl 0847.58075)] to obtain an upper bound for the universal constant, the so-called Makai-Hayman (M-H) constant, which determines the lowest tone a membrane can produce if it contains no circular membrane with more than one specified radius. The goal of this article is to review this construction in order to obtain general formulas by which Goodman's constant and an upper bound for the M-H constant can be computed more easily and accurately than it was done in these two articles. Formulas are derived for computing a univalent mapping of the unit disc onto a radial slit domain with threefold circular symmetry. The method of working backwards from a target domain, i.e., successively reducing a radial slit domain to another radial slit domain with fewer slits requires some degree of symmetry of the target domain.