On some constants of quasiconformal deformation and Zygmund class (Q1899638)

The necessary and sufficient condition for a real-valued continuous function \(f\) on \(\mathbb{R}\) to have an extension \(F\) in the class of quasiconformal deformations on the upper half plane: \(\text{Im } z> 0\) is that \(f\) belongs to the Zygmund class [see \textit{E. Reich} and the first author, Ann. Acad. Sci., Fenn. Ser. AI 16, 377-389 (1991; Zbl 0757.30023) and \textit{F. Gardiner} and \textit{D. Sullivan}, Am. J. Math. 114, No. 4, 683-736 (1992; Zbl 0778.30045)]. The authors obtain an upper bound for the \(\infty\)-norm of the generalized derivative of \(F\) in terms of the Zygmund norm of \(f\). An estimate of \(|f(x)|: 0\leq x\leq 1\) for \(f\) belonging to the Zygmund class with \(f(0)= f(1)= 0\) has been also obtained which improves the corresponding estimate obtained in \textit{F. Gardiner} and \textit{D. Sullivan}, loc. cit.