Univalent harmonic mappings of annuli and a conjecture of J. C. C. Nitsche (Q5951499)

Let \(\rho\in[0,1)\) and \(A_\rho:= \{z\in\mathbb{C} \mid\rho <|z |<1\}\). The author considers functions \(f=u+iv: A_\rho\to A_\sigma\) such that \(f\) is bijective and \(u\) and \(v\) are real harmonic functions. \textit{J. C. C. Nitsche} [Am. Math. Mon. 69, 781-782 (1962; Zbl 0109.30503)] asked for the least upper bound \(\sigma_0(\rho)\) of the set of those \(\sigma\) for which a function \(f\) as above exists. For \(\rho\) close to 1 the author improves a known upper bound for \(\sigma_0(p)\) by the estimate \(\sigma_0(\rho)\leq{1\over 1+ {\rho^2 \over 2}(\log \rho)^2}\).