The Grunsky coefficient conditions
It is known that if f belongs to the class \(\Sigma\) (k) (the subset of the class \(\Sigma\), in which every function has a k-quasiconformal extension to the whole plane) then \[ (*)\quad | \sum^{\infty}_{m,n=1}\sqrt{mn}\alpha_{mn}x_ mx_ n| \leq k\sum^{\infty}_{n=1}| x_ n|^ 2 \] for every element \(x=(x_ 1,x_ 2,...)\) of the Hilbert space \(\ell^ 2\). Here \(\{\alpha_{mn}\}\) are the Grunsky coefficients of f. But the condition (*) is not sufficient for f belonging to \(\Sigma\) (k) [\textit{R. Kühnau}, Ann. Acad. Sci. Fenn., Ser. A I 6, 125-130 (1981; Zbl 0454.30016)]. In this present paper, the author gives a characterization of such functions for which the condition (*) is not only necessary but also sufficient for them to belong to \(\Sigma\) (k). As an application of such a characterization, a new example is given in which the function f satisfies the condition (*) but can only have k'- quasiconformal extensions to the whole plane with \(k'>k\).
scientific article; zbMATH DE number 4012978