Research Problems in Function Theory — Problem 7.1

Definitions and mathematical statement

Let $E\subset{\mathbb{C}}$ be compact. For $n\geq2$, define its $n$-diameter by \[ d_n(E)=\left(\max_{w_1,\ldots,w_n\in E} \prod_{1\leq\mu<\nu\leq n}|w_\mu-w_\nu|\right)^{2/(n(n-1))}. \] Its transfinite diameter, equal to its logarithmic capacity, is $d(E)=\lim_{n\to\infty}d_n(E)$. Assume $d(E)=1$. The source reports that $d_n$ decreases to one, hence $d_n(E)\geq1$.

The first question, literally as printed, asks whether \[ d_n(E)\geq n^{2/(n-1)}\qquad(n\geq2) \] holds for all such compact sets, and attributes this lower bound for connected $E$ to Pommerenke. There is a normalization inconsistency: at $n=2$ the displayed definition gives $d_2(E)=\operatorname{diam}(E)$, while the claimed connected lower bound would require diameter at least four for capacity one. The closed unit disk has capacity one and diameter two. Thus the quoted lower-bound formula cannot be accepted together with the supplied normalization. No exponent or normalization is silently repaired, and the intended general lower-bound question requires source verification.

The second question is independently well formed: is there an absolute constant $K>0$ such that \[ \forall E\subset{\mathbb{C}}\text{ compact and connected with }d(E)=1\ \forall n\geq2, \qquad d_n(E)^{(n-1)/2}\leq Kn? \] The source reports the weaker estimate \[ d_n(E)^{(n-1)/2}\leq\left(\frac4e\log n+4\right)n \] for connected $E$. The desired $K$ must be independent of both $E$ and $n$; connectedness is not replaced by a stronger domain or smoothness assumption.

Short English statement

Seek sharp universal finite-diameter estimates for capacity-one compact plane sets: a general lower bound and a linear upper bound for the indicated power when the set is connected. The printed lower-bound exponent conflicts with its own normalization, so the intended lower bound must be recovered. The logarithmic-factor upper estimate is retained as reported background. The supplied triage says the problem was open in Hayman's 2018 update and no definitive resolution was located; the consistency check does not claim a resolution of the intended research questions.

Sources

• UnsolvedMath permanent ID 2307001, source code catalogue entry, "Research Problems in Function Theory --- Problem 7.1." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018 source update), Problem 7.1, attributed to Ch. Pommerenke: https://arxiv.org/abs/1809.07200. Supplied source-TeX extraction accessed 2026-07-29. The metadata retains the update for status triage and marks status/release and rights review NEEDS_REVIEW.
• Pommerenke is named for the connected-set estimates without complete bibliographic entries. No corrected exponent, source theorem or missing reference is inferred.