Über eine Vermutung von Vainio. (On a conjecture of Vainio) (Q1820270)

In this paper [Ann. Acad. Sci. Fenn., Ser. A I 53, 43 p. (1985; Zbl 0549.30033)] \textit{J. V. Vainio} made two conjectures on conformal sewing, one of which reads as follows: Let \(\phi_ 1\) and \(\phi_ 2\) be two increasing continuous functions of an interval (-a,a) onto itself such that \(\phi_ 1(0)=\phi_ 2(0)\) and \(\phi_ 1(x)<\phi_ 2(x)\) for \(x\neq 0\). Then there exists a homeomorphic mapping of (-a,a) onto itself such that \(\phi_ 1<\psi <\phi_ 2\) for \(x\neq 0\), which allows a conformal sewing on each interval (-a,0) and (0,a) but not in (-a,a). \textit{A. Huber} proves this conjecture to hold true by a construction he uses in a similar situation [Math. Z. 151, 25-28 (1976; Zbl 0319.30012)].