Research Problems in Function Theory — Problem 3.3

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 $\Delta u=0$. Let $H=\{z\in\mathbb C:\operatorname{Re}z>0\}$, and fix $K>0$. Let $\mathcal U_K$ consist of subharmonic functions $u<0$ on $H$ satisfying \[\inf_{|\theta|<\pi/2}u(re^{i\theta})\leq-K\qquad(r>0).\] The source update gives counterexamples to the constant $1/2$ and asks for the best possible replacement.

Statement recorded in the supplied source.

Must every $u\in\mathcal U_K$ satisfy $u(r)\leq-K/2$ for every $r>0$? More generally, determine the sharp universal constant \[\alpha_* =\sup\{\alpha\geq0: u(r)<-\alpha K\text{ for every }K>0,\ u\in\mathcal U_K,\ r>0\}.\] [S2]

Short English statement

Does reaching level minus K somewhere on every right-half-plane semicircle force level minus K/2 on the positive axis, and what is the sharp universal constant?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303003 (AMR-022-3003). 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 3.3 and Update 3.3. https://arxiv.org/abs/1809.07200.