Research Problems in Function Theory — Problem 7.9

Definitions and mathematical statement

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$ and $m$ be planar Lebesgue measure. For $K\geq1$, use the standard orientation-preserving convention: a $K$-quasiconformal map $f:\mathbb D\to\mathbb D$ is a homeomorphism onto the disc, belongs to $W^{1,2}_{\mathrm{loc}}$, and satisfies \[ \|Df(z)\|_{\mathrm{op}}^2\leq KJ_f(z) \quad\text{for almost every }z\in\mathbb D, \] where $J_f=\det Df$.

F. W. Gehring's requested estimate is \[ \forall K\geq1\ \exists b(K)<\infty\ \forall f\ \forall E\subseteq\mathbb D \text{ measurable}:\quad m(f(E))\leq b(K)m(E)^{1/K}, \] with $f$ ranging over the intended normalized class of such maps and the bound independent of both $f$ and $E$. The supplied statement, however, gives no normalization at all. That omission is material: already at $K=1$ disk automorphisms can have arbitrarily large local area distortion as their normalization points approach the boundary. Thus the full unnormalized class cannot carry a single bound of the displayed form. A condition such as $f(0)=0$ is a plausible missing normalization, but is not inserted as a verified source assumption.

The source says an estimate of this type was known with exponent $a(K)$ in place of $1/K$, but its citation and accompanying normalization are absent. Its intended target is the precise exponent $1/K$, not merely another positive exponent. The supplied summary labels the problem open as of the 2018 edition and reports no definitive resolution located; those status claims do not resolve the formulation's normalization defect and are not independently verified here.

Short English statement

The source asks for an area-distortion bound with exponent $1/K$ and a constant depending only on quasiconformal distortion. A map normalization is missing: disk automorphisms already prevent such a uniform statement in the unnormalized conformal case. The intended normalized theorem must be recovered from the source.

Sources

• Local UnsolvedMath registry, permanent ID 2307009, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 7.9"; status open, classification OPEN-TRIAGE.
• Hayman, Research Problems in Function Theory (2018), Problem 7.9, attributed to F. W. Gehring, https://arxiv.org/abs/1809.07200. The record identifies extraction from public arXiv source TeX, accessed July 29, 2026; the 2018 update was retained for status triage.
• The statement says the weaker exponent $a(K)$ is known, but its "see" citation is empty in the supplied extraction. No precise bound, normalization, or authoritative bibliographic reference is provided for that result.
• The supplied research summary reports openness as of the 2018 edition and no definitive resolution located, with no additional reference or precise search date.