Research Problems in Function Theory — Problem 3.14

Definitions and mathematical statement

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$, $\mathbb T=\partial\mathbb D$, let $F\in L^1(\mathbb T;{\mathbb{R}})$, and fix $z_0\in\mathbb D$. A subharmonic function is an upper-semicontinuous extended-real-valued function satisfying the submean inequality, with the usual exclusion of the identically $-\infty$ function.

The source asks to maximize $u(z_0)$ over subharmonic functions $u$ in $\mathbb D$ which equal $F$ on $\mathbb T$ and satisfy \[ \forall r\in(0,1),\qquad \inf_{\theta\in[0,2\pi)}u(re^{i\theta})\leq0. \] The infimum is retained; it is not replaced by an attained minimum or by an almost-everywhere-in-$r$ condition.

Because $F$ is only integrable and $u$ is initially defined in the open disk, the boundary equality requires an interpretation. It could mean radial or nontangential limits almost everywhere, convergence in an integral norm, a Perron boundary prescription, or an extension with specified regularity; these are not automatically equivalent. No boundary-growth or harmonic-majorant condition is supplied.

For any fixed precise trace convention, let $\mathcal U(F)$ denote the admissible class with that boundary condition and the displayed circle-infimum constraint. The extremal quantity would be \[ \sup\{u(z_0):u\in\mathcal U(F)\}. \] The task is to determine the sharp value and, insofar as "maximise" demands it, an attaining function or attainment criterion. Nonemptiness, finiteness and attainment are not assumed. The missing boundary/growth conventions must be recovered before this is a unique extremal problem.

Short English statement

Find the largest possible interior value of a subharmonic function with prescribed integrable boundary data, while every centered circle has infimum at most zero. The supplied assessment associates the question with Baernstein's symmetrisation program and speculates that star-function/spread methods may treat monotone boundary data, but supplies no verified theorem for that claim. It labels the general question OPEN-TRIAGE; this draft makes no independent resolution or current-status determination.

Sources

• UnsolvedMath permanent ID 2303014, source code catalogue entry, "Research Problems in Function Theory --- Problem 3.14." 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 3.14, attributed to A. Baernstein: 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.
• The supplied assessment names star-function/spread techniques but provides no specific reference or verified monotone-data solution. No missing bibliography is inferred.