Grunsky coefficient inequalities, Carathéodory metric and extremal quasiconformal mappings
The following results are obtained: I. There is given the complete solution of the question of how to characterize a class of functions f, for which the sharpened Grunsky inequality \[ (*)\quad | \sum^{\infty}_{m,n=1}\sqrt{mn}\alpha_{mn}(f)x_ mx_ n| \leq k\| x\|^ 2,\quad x=(x_ 1,x_ 2,...)\in I^ 2, \] is both necessary and sufficient to have a k-quasiconformal extension onto the whole sphere \({\bar {\mathbb{C}}}\). For example, in \(\Delta^*=\{z\in {\bar {\mathbb{C}}}:\) \(| z| >1\}\) that are only the functions which are the restrictions to \(\Delta^*\) of the quasiconformal automorphisms \(w^{\mu}\) of \({\bar {\mathbb{C}}}\) with complex dilatation \(\mu\) satisfying the equality \[ \sup \{| \iint_{\Delta}\mu \phi dz\wedge d\bar z|:\quad \| \phi \|_{A_ 1}=1\}=\| \mu \|_{\infty}, \] where \(A_ 1\) is the subspace of \(L_ 1(\Delta)\) formed by holomorphic functions and, in addition, \(\phi\) are the squares of functions holomorphic in \(\Delta\), \(\Delta =\{| z| <1\}\). The proof uses the properties of the Carathéodory metric of the universal Teichmüller space T. II. If \(\phi \in A_ 1\setminus \{0\}\) has zeros of even order only, then in the holomorphic disk in T, defined by the Beltrami differential t\({\bar \phi}\)/\(| \phi |\), \(t\in \Delta\), the Carathéodory metric coincides with the majorizing Teichmüller-Kobayashi metric. This is an analogue for T of the corresponding Kra's result for the Abelian- Teichmüller disks in finite-dimensional Teichmüller spaces. III. \textit{R. Kühnau} [Comment. Math. Helv. 61, 290-307 (1986; Zbl 0605.30023)] had established that if f is holomorphic in \({\bar \Delta}{}^*\) and in (*) the equality holds, then necessary \(f=w^{\mu}| \Delta^*\) with \(\mu =k{\bar \phi}/| \phi |\) where \(\phi\) has the zeros of even order only. In the paper the required order of the smoothness of the boundary curves is essentially decreasing and there is shown that such is valid at least for \(f\in C^{2,\alpha}({\bar \Delta}^*)\), \(0<\alpha <1\) (and even for f non-smooth on \(\partial \Delta)\). The proof employs Strebbel's frame mapping condition. On the other hand, the condition \(\mu =k{\bar \phi}/| \phi |\), \(\phi =\psi^ 2\), is not necessary in general.
scientific article; zbMATH DE number 4142391