On the Grunsky coefficient conditions (Q2769234)

In 1939, H. Grunsky discovered a necessary and sufficient condition for univalence. It is evident that \(\log{w(z)- w(\zeta)\over z-\eta}=-\sum^\infty_{k,l=1} {a_{kl}\over z^k\zeta^l}\) is analytic in \(\{|z|>1\), \(|\zeta |>1\}\) if and only if \(w(z)\) belongs to the well-known class \(\Sigma\). The Grunsky inequality states that NEWLINE\[NEWLINE\left|\sum^n_{k,l=1} a_{kl} x_kx_l\right |\leq\sum^n_{k=1} {|x_k|^2 \over k},NEWLINE\]NEWLINE for all \(x_k\in \mathbb{C}\) and \(n\geq 1\). It was proved by \textit{R. Kühnau} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 22/24, 105-111 (1972; Zbl 0248.30021)] that NEWLINE\[NEWLINE\left|\sum^n_{k,l=1} a_{kl}x_kx_l \right|\leq\kappa \sum^n_{k=1} {|x_k|^2 \over k},\tag{1}NEWLINE\]NEWLINE for all \(n\geq 1\) and all \(x_k\) if \(w(z)\) has a \(Q\)-quasiconformal extension, \(Q={1+ \kappa\over 1-\kappa}\), \(\kappa<1\), to \(\{|z|<1\}\). It is also known [\textit{Ch. Pommerenke}, Univalent functions (1975; Zbl 0298.30014)] that from (1) follows it exists a quasiconformal extension to \(\{|z|<1\}\). A still open problem is to determine \(Q^* (\kappa)= \inf Q\) such that all \(w(z)\) which satisfy (1) have a \(Q\)-quasiconformal extension to \(\{|z|<1\}\). The aim of this paper is to give an upper bound for \(Q^*\). The proof based on results of \textit{R. Kühnau} [Ann. Acad. Sci. Fenn., Math. 25, No. 2, 413-415 (2000; Zbl 0968.30010)] and \textit{M. Lehtinen} [Ann. Acad. Sci. Fenn., Ser. A I 9, 133-139 (1984; Zbl 0539.30010)].