Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain
Let \(G\) be the interior of a closed Jordan curve \(\Gamma\) and let \(f\) be a conformal mapping of the unit disk \(D= \{| z| <1\}\) onto \(G\), extended continuously to a homeomorphism of the closures of \(D\) and \(G\). The author is interested in estimating the modulus of continuity of \(f\) on \(\overline D\) when \(\Gamma\) is not ``smooth''. For example when \(\Gamma\) is only known to be a quasicircle, or when \(\Gamma\) has cusps. In particular he assumes that \(\Gamma\) satisfies a general chord-arc condition or chord-diameter condition, and establishes corresponding moduli of continuity for \(f\). In particular, if \(\Gamma\) satisfies a bounded diameter-arc condition with constant \(Q\) (Ahlfors' condition for a quasicircle), then \(f\) is Hölder continuous with exponent \(k=(1/8) \pi^2 (Q+1)^{-2}\). Another corollary of his main result establishes a modulus of continuity if \(\Gamma\) has cusps of a certain sharpness. (The author seems to be unaware of work done on this topic by the reviewer [Indiana Univ. Math. J. 31, 341-354 (1982; Zbl 0497.30013) and Proc. Am. Math. Soc. 93, 483-488 (1985; Zbl 0566.30006) and by \textit{R. Näkki} and \textit{B. Palka} [Indiana Univ. Math. J. 31, 377-401 (1982; Zbl 0496.30010)] [Complex Variables, Theory Appl. 5, 165-180 (1986; Zbl 0593.30011) and Comment. Math. Helv. 61, 389-414 (1986; Zbl 0597.30011)]).
scientific article; zbMATH DE number 1158884