On Krzyż's Problem 7.23 in Hayman and Lingham: Kühnau's Two-Point Condition Implies Quasiconformal Extendability If and Only If q < 1/3

For \(f\) analytic in the unit disc \(\mathbb{D}\) let \(u(z,\zeta)\)\({}=f'(z)f'(\zeta)\)\(/(f(z)\)\({}-f(\zeta))^2\)\({}-(z\)\({}-\zeta)^{-2}\). Kühnau showed that a quasiconformal extension of \(f\) to the whole plane with \(|\mu|\le q<1\) implies \(|u(z,\zeta)|\)\({}\le q(1\)\({}-|z|^2)^{-1}(1\)\({}-|\zeta|^2)^{-1}\). J. G. Krzyż asked (Hayman–Lingham, Research Problems in Function Theory, Problem 7.23) to show that this condition is also sufficient for a quasiconformal extension, “possibly with \(1/3\le q<1\)”. We show that it is sufficient if and only if \(q<1/3\); in the range \(1/3\le q<1\) named in the problem it is not sufficient, for any \(q\). Here \(u\) carries the square on \(f(z)-f(\zeta)\) (missing in the printed Problem 6.57, but forced by the two properties of \(u\) stated in Problem 7.23), the extension is a quasiconformal self-map of the Riemann sphere that agrees with \(f\) on \(\mathbb{D}\), and “sufficient” means that some quasiconformal extension exists; the answer is the same for the exterior normalisation of Problem 6.57 and under both readings of the parenthesis, and for extensions to the plane it is the same for bounded \(f\) (unbounded \(f\) have none). The counterexample is the bounded Möbius image \(1\)\(/(\operatorname{artanh} z\)\({}-i)\) of the strip map: the supremum of \((1\)\({}-|z|^2)(1\)\({}-|\zeta|^2)|u(z,\zeta)|\) is exactly \(1/3\), and the map has no homeomorphic extension to the closed disc. For \(q<1/3\) the Ahlfors–Weill theorem gives an extension to the sphere with \(|\mu|\le3q\), and the best dilatation that the condition guarantees tends to \(1\) as \(q\to1/3^-\). For extensions with the same constant \(q\) the answer is negative for every \(q\in(0,1)\); this is due to Kühnau (1981), and we give a second proof. What is new is only the computation for the strip map and the resulting sharp threshold. No printed answer to the question in the main reading (some quasiconformal extension) was found in the sources searched; a search that finds nothing is not a proof of novelty. This is an unrefereed note.