Research Problems in Function Theory — Problem 3.11

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.

A subharmonic function is an upper semicontinuous function with values in $[-\infty,\infty)$, not identically $-\infty$ on a connected component, that satisfies the mean-value inequality on every closed ball in its domain. A real harmonic function is a twice continuously differentiable function $u$ satisfying $\Delta u=0$. For $m\geq2$ and $n\geq1$, let $\mathcal P_{n,m}$ be the nonzero real homogeneous harmonic polynomials of degree $n$ on $\mathbb R^m$. Homogeneity means $u(tx)=t^n u(x)$. Put $A(r,u)=\inf_{|x|=r}u(x)$ and $B(r,u)=\sup_{|x|=r}u(x)$. The zero spherical mean gives $A(r,u)<0<B(r,u)$ when $u(0)=0$ and $u$ is nonconstant. For homogeneous polynomials, $-A(r,u)/B(r,u)$ is independent of $r>0$. A set $E\subset[1,\infty)$ has finite logarithmic measure when $\int_E dr/r<\infty$. The source update attributes an extremal-ratio solution for homogeneous harmonic polynomials in dimensions $m\geq3$ to Armitage.

Statement recorded in the supplied source.

Determine the optimal lower and upper bounds \[L_{n,m}=\inf_{u\in\mathcal P_{n,m}}\frac{-A(1,u)}{B(1,u)},\qquad U_{n,m}=\sup_{u\in\mathcal P_{n,m}}\frac{-A(1,u)}{B(1,u)}.\] For odd $n$ both bounds are $1$. Also determine the sharp comparison of $-A(r,u)$ and $B(r,u)$ for nonpolynomial harmonic $u$ on $\mathbb R^m$ with $u(0)=0$: can the source's bound \[-A(r,u)<B(r,u)\left(\log B(r,u)\right)^{m-1+\varepsilon}\] be improved? Here $\varepsilon>0$ is arbitrary, the bound applies for all sufficiently large $r$ outside a set of finite logarithmic measure that may depend on $u$ and $\varepsilon$, and the displayed logarithm is taken where $B(r,u)>1$. Include the planar comparison $A(r,u)\sim-B(r,u)$ outside such a set as the known benchmark. [S2]

Short English statement

Find the sharp extrema ratio for homogeneous harmonic polynomials and the sharp growth comparison between negative and positive extrema for nonpolynomial harmonic functions, allowing an exceptional set of finite logarithmic measure.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303011 (AMR-022-3011). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 3.11 and Update 3.11. https://arxiv.org/abs/1809.07200.