Research Problems in Function Theory — Problem 7.23

Definitions and mathematical statement

For a locally univalent analytic function, its Schwarzian derivative is \[ \{f,z\}=\frac{f^{(3)}(z)}{f'(z)}-\frac32\left(\frac{f^{(2)}(z)}{f'(z)}\right)^2. \] The two-point expression $u_f(z,\zeta)$ is defined only by a cross-reference to Problem 6.57, absent from this record. The source says it is invariant under Möbius postcomposition and satisfies \[ \lim_{\zeta\to z}u_f(z,\zeta)=\tfrac16\{f,z\}. \] A familiar kernel with these properties is \[ \frac{f'(z)f'(\zeta)}{(f(z)-f(\zeta))^2}-\frac1{(z-\zeta)^2}, \] but the excerpt does not authenticate that formula as its definition. The two reported properties alone do not replace the missing definition.

A quasiconformal extension is an orientation-preserving quasiconformal map agreeing with $f$ on $\mathbb D$. In planar coordinates its Beltrami coefficient is $\mu_F=F_{\bar z}/F_z$, with $\|\mu_F\|_\infty<1$. The source calls the extension one to the whole plane; whether it means a plane self-homeomorphism or a map of the Riemann sphere is material and must be checked.

The record attributes to Kühnau the following necessary estimate: if $f$ has an extension with $|\mu_F|\leq q<1$ almost everywhere, then \[ |u_f(z,\zeta)|\leq \frac{q}{(1-|z|^2)(1-|\zeta|^2)}\qquad(z,\zeta\in\mathbb D), \tag{1} \] with diagonal values interpreted through the stated limit.

Sufficiency question (J. G. Krzyz). Show that (1) is also sufficient for an analytic $f$ on $\mathbb D$ to have a quasiconformal extension, retaining the source qualification "possibly with $1/3\leq q<1$." The intended scope of that qualification, and whether an extension must obey the same bound $q$ or merely some bound below one, are not specified. Nonconstancy/local univalence and the full off-diagonal domain of $u_f$ are also omitted.

There is a normalization issue in addition to the missing kernel: a Möbius-invariant criterion cannot distinguish bounded images from a Möbius transform with a boundary pole, whereas an extension as a homeomorphism ${\mathbb{C}}\to{\mathbb{C}}$ is finite at every finite boundary point. The plane-versus-sphere meaning must therefore be recovered rather than silently chosen.

The supplied summary says the item was open in the 2018 edition and no definitive resolution was located. The necessity theorem and current status remain unverified supplied claims.

Short English statement

Is the stated two-point Schwarzian-type bound sufficient for quasiconformal extension, including the suggested range $1/3\leq q<1$? The kernel definition, parameter scope, dilatation conclusion, and plane-versus-sphere extension convention are missing from the excerpt.

Sources

• Local UnsolvedMath registry, permanent ID 2307023, source code catalogue entry, "Research Problems in Function Theory --- Problem 7.23"; attribution: J. G. Krzyz. Supplied status open, classification OPEN-TRIAGE, updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018), Problem 7.23, with missing defining cross-reference to Problem 6.57, https://arxiv.org/abs/1809.07200. The record reports source-TeX extraction and access on 2026-07-29; the 2018 update was retained for status triage. Current status and rights require review; the record describes publicly available arXiv source TeX.
• The necessity estimate is attributed to Kühnau, expressly with no citation supplied. The source label C7.1 refers to that estimate; no original-paper reference is invented.