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Grunsky inequalities and quasi-conformal extension. (English) Zbl 1126.30013

Let \( f=f(z)= z+b_0+b_1/z+\dots \) in \( \Delta^* = \{z \in \mathbb C \cup \{\infty \}: | z| >1\}\) be a univalent function of the known class \(\Sigma \). If \(f\) has a quasiconformal extension to \( \widehat {\mathbb C} = \mathbb C \cup \{\infty \} \) then there exists a number \(Q\geq 1\) so that \( f \) belongs to the subclass \(\Sigma(Q)\subset \Sigma \). The quantity \(k=k(f)\) is defined by the minimum of the norm \(\|f_{\overline z}(z)/f_z (z)\|_{\infty}\), where all quasiconformal extensions of a fixed \(f \in \Sigma(Q) \) to \( \widehat {\mathbb C}\) are considered. The number \(\kappa=\kappa (f)\) denotes the so-called Grunsky constant of \(f\). It was shown by the second author [see Math. Nachr. 48, 77–105 (1971; Zbl 0226.30021)] that \(\kappa\) and \(k\) satisfy \(\kappa (f) \leq k(f)\). In addition to the investigation of mappings \(f\) with equality in \(\kappa (f) \leq k(f)\) which are due to the second author [see Comment. Math. Helv. 61, 290-307 (1986; Zbl 0605.30023)] it is also interesting to analyze the properties of the other functions \(f\) with \(\kappa (f) < k(f)\). In the paper under review the authors give two different proofs of a theorem concerning the existence of a sequence \(\{f_n \}_{n=1,2,\dots }\) with \(f_n \in \Sigma(Q_{n})\) and \( Q_{n} \geq 1\) for \(n=1,2,\ldots \). They show that every function \(f \in \Sigma \) can be approximated by such a sequence \(\{f_n \}_{n=1,2,\dots }\) satisfying \(\kappa (f_{n}) < k(f_{n})\) for \(n=1,2,\dots \) and converging uniformly to \(f\) on compact sets in \( \Delta^* \).

MSC:

30C62 Quasiconformal mappings in the complex plane
30F60 Teichmüller theory for Riemann surfaces
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
30C35 General theory of conformal mappings
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References:

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