General distortion theorem for univalent functions with quasiconformal extension (Q1683594)

Although the Bieberbach conjecture has been solved in the class \(S\) of univalent functions, the corresponding problem in the class \(S_k\) of univalent functions with quasiconformal extension is still open. Here \(S_k\) and \(S_k(\infty)\) are the subclasses of \(S\) whose functions admit quasiconformal extension with dilatation \(\kappa\) with \(|\kappa| \leq k < 1\) and the functions in the class \(S_k(\infty)\) fix \(\infty\). In [Ann. Acad. Sci. Fenn., Ser. A I, Math. 20, No. 2, 349--357 (1995; Zbl 0846.30014)] the author proved that if \(f \in S_k(\infty)\), then for \(k \leq k_n = 1/(n^2 + 1)\), \(|a_n| \leq 2k/(n-1)\) and the equality only holds for explicit functions. Now it is shown that for \(f \in S_k\) and \(k > k_n\), \(|a_n| \leq nk^{\alpha_n}\) with \(\alpha_n = 1 + \log(n(n-1)/2)/\log(n^2+1)\). The proof is based on the use of general distortion functionals, depending on the derivatives of a univalent function \(f\) in \(D^*\) at a finite set of distinct points in \(D^*\) under the hydrodynamical normalization, and \(D^*\) is an exterior domain of a quasicircle.