Research Problems in Function Theory — Problem 3.6

Definitions and mathematical statement

The source records a planar result attributed to Wolf: if an entire-plane function $u$ in the intended class has an angular majorant $f$, \[ u(re^{i\theta})\le f(\theta)\qquad(0<r<\infty), \qquad \int_0^{2\pi}\log^+f(\theta)\,{\,\mathrm{d}}\theta<\infty, \] then $u$ is bounded above and hence constant. For a nonnegative majorant, $\log^+t=\log\max(1,t)$. The excerpt does not state the class of $u$, measurability/pointwise conventions for $f$, or the complete Wolf reference.

The requested three-dimensional analogue would replace directions $e^{i\theta}$ by $\omega\in S^2$, with a directional bound \[ u(r\omega)\le F(\omega)\qquad(0<r<\infty), \] and seek an appropriate condition on $F$, relative to surface measure $\sigma$, giving the intended boundedness or constancy conclusion. A direct candidate condition would be $\int_{S^2}\log^+F\,{\,\mathrm{d}}\sigma<\infty$, but the question asks for an analogue rather than asserting this particular condition sufficient. No unprovided exponent or integrability criterion is invented.

A crucial missing datum is whether $u$ is harmonic ($\Delta u=0$) or subharmonic (upper semicontinuous with $\Delta u\ge0$ in the distributional sense), or belongs to another class. The inference from boundedness above to constancy, which is valid for entire-plane subharmonic functions, does not have the same scope for subharmonic functions on ${\mathbb{R}}^3$. Thus a precise three-dimensional theorem must specify both the class and whether its target is boundedness, constancy, or a modified conclusion. The supplied OPEN-TRIAGE summary reports the question open in the 2018 edition and no recent resolution located.

Short English statement

Find a three-dimensional counterpart of a planar theorem controlling a function by an angular majorant with integrable positive logarithm. The missing function class and the intended boundedness-versus-constancy conclusion are essential.

Sources

• Local UnsolvedMath registry, permanent ID $2303006$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 3.6."
• Hayman, Research Problems in Function Theory (2018), Problem 3.6; https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29.
• The source attributes the planar theorem to Wolf without a complete reference. The supplied OPEN-TRIAGE summary gives no independent current-status citation.