Research Problems in Function Theory — Problem 3.26

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\ge2$, let $B_n=\{x\in\mathbb R^n:|x|<1\}$. A subharmonic function is an upper semicontinuous function satisfying the sub-mean inequality on every ball compactly contained in its domain. Here require $u:B_n\to[0,1)$ and $u(0)>0$. An admissible path is a rectifiable continuous curve $\gamma:[0,1]\to\overline B_n$ with $\gamma(0)=0$, $\gamma(1)\in\partial B_n$, $\gamma(t)\in B_n$ and $u(\gamma(t))>0$ for $t<1$. Its Euclidean arc length is denoted by $\ell(\gamma)$. No boundary value of $u$ is required.

Statement recorded in the supplied source.

For each fixed $n\ge4$, find a continuous function $f_n:(0,1)\to(0,\infty)$ such that every such $u$ has an admissible path with \[\ell(\gamma)\le f_n(u(0)).\] For $n=2$, determine the smallest exponent $\beta$ for which one can use a uniform bound $f_2(t)=Ct^{-\beta}$, $0<t<1$, with a constant $C$ independent of $u,t$; determine the optimal infimum of the exponents and whether it is attained. [S2]

Short English statement

Bound the length of a positive path from the centre to the boundary for bounded subharmonic functions in dimensions at least four, and find the optimal power exponent for the planar case.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303026 (AMR-022-3026). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 3.26, its complete Update 3.26, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.