Positive superharmonic functions and the Hölder continuity of conformal mappings. (English) Zbl 0677.30013

Anderson, J. M.; Hinkkanen, A.

J. Lond. Math. Soc., II. Ser. 39, No. 2, 256-270 (1989).

The rate at which a positive superharmonic function of a domain D in the complex plane can tend to zero has been studied by Ü. Kuran [J. Lond. Math. Soc., II. Ser. 29, 269-275 (1984; Zbl 0558.31003)] under some conditions on \(\partial D\). In the paper under review the authors study a conjecture of R. Näkki and B. Palka [Comment. Math. Helv. 55, 485-498 (1980; Zbl 0447.30005)] concerning the Hölder-continuity of a conformal mapping of the unit disk onto a domain bounded by a k-circle and show that the conjecture is false. The authors make use of Kuran’s results and give a long example to show the sharpness of their results.
{Reviewer’s remarks: Some results extending Kuran’s work for other classes of functions or domains were given by D. A. Herron and the reviewer [Analysis 8, 187-206 (1988; Zbl 0661.31002)].}

Reviewer: M.Vuorinen

MSC:

30C62	Quasiconformal mappings in the complex plane
30C35	General theory of conformal mappings
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions

Keywords:

superharmonic functions; Hölder-continuity

Citations:

Zbl 0558.31003; Zbl 0447.30005; Zbl 0661.31002