Research Problems in Function Theory — Problem 7.64

Definitions and mathematical statement

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$, let $\Gamma$ be a Fuchsian group of holomorphic automorphisms of $\mathbb D$, and let $i(z)=z$ be its identity. The source does not assume torsion-freeness. For a disk automorphism $\gamma$, $|\gamma'(0)|=1-|\gamma(0)|^2$.

For a literal nonnegative interpretation, define the sum in the extended sense and the product as the limit over finite subsets: \[ A(\Gamma)=\sum_{\gamma\in\Gamma}|\gamma'(0)|\in[0,\infty],\qquad B(\Gamma)=\prod_{\substack{\gamma\in\Gamma\\\gamma\ne i}}|\gamma(0)|^2\in[0,1]. \] A zero factor makes $B=0$, and a divergent nonnegative sum is assigned $+\infty$. If only positive convergent products or finite sums are intended, extra hypotheses are missing.

Recorded question (N. Suita, communicated by Ch. Pommerenke). Is \[ \sum_{\gamma\in\Gamma}|\gamma'(0)|\ \geq\ \prod_{\substack{\gamma\in\Gamma\\\gamma\ne i}}|\gamma(0)|^2 \] true? The statement says this is an equivalent formulation of a problem connecting the Bergman kernel and capacity function of a Riemann surface.

There is a substantive transcription/formulation concern: the left side includes the identity contribution $|i'(0)|=1$, while each factor on the right lies in $[0,1)$. Thus under the displayed nonnegative conventions the bound is automatic and cannot by itself identify the substantive kernel--capacity comparison alluded to. The exact absolute-value placement, normalization, and intended group-series expression must be checked; no alternative signed, complex, or renormalized formula is invented.

The registry classification is SOLVED-IN-LITERATURE, but its research summary itself labels the precise situation partial/cautionary: it says the underlying Suita conjecture has been proved and the specific normalized Fuchsian-group form still needs checking. The summary's description refers to a boundary point, whereas the displayed expression is evaluated at the interior point $0$ of $\mathbb D$ and no kernel or capacity definitions are provided. The intended correspondence and point conventions require review. The solved label is therefore not treated as authentication of this exact formula.

Short English statement

The recorded Fuchsian-group inequality is attributed to Suita and linked to a Bergman-kernel/capacity problem. Its literal absolute-value form is automatic from the identity term and unit-disc factors, so the intended substantive formula needs checking. The summary expressly cautions that the exact group formulation is unverified despite the solved Suita-conjecture label.

Sources

• Local UnsolvedMath registry, permanent ID 2307064, source code catalogue entry, "Research Problems in Function Theory --- Problem 7.64"; attribution: N. Suita; communicated by Ch. Pommerenke. Supplied status solved, classification SOLVED-IN-LITERATURE, updated 2026-08-06; the research summary explicitly qualifies the exact formulation as needing review.
• Hayman, Research Problems in Function Theory (2018), Problem 7.64, 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 supplied summary reports proof of the underlying Suita conjecture but gives no proof reference or precise identification of this Fuchsian-group formula with that theorem.