Research Problems in Function Theory — Problem 3.7

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 a subharmonic function $u$ on the plane set \[B(r,u)=\sup_{|z|=r}u(z),\qquad T(r,u)=\frac1{2\pi}\int_0^{2\pi}\max\{u(re^{i\theta}),0\}\,d\theta.\] Assume $u$ is nonconstant and its order $\rho=\limsup_{r\to\infty}\log T(r,u)/\log r$ is finite and nonnegative; $T(r,u)>0$ for all sufficiently large $r$. This is the subharmonic version of $\log M$ and the Nevanlinna characteristic in Problem 1.17. The source update attributes the generalization to subharmonic functions in space to Dahlberg.

Statement recorded in the supplied source.

Does the finite-order Paley bound extend to all such subharmonic functions: is there a finite constant $C(\rho)$ depending only on the order such that \[1\leq\liminf_{r\to\infty}\frac{B(r,u)}{T(r,u)}\leq C(\rho)?\] Determine the sharp constants. In the two ranges explicitly stated in Problem 1.17, does one have \[C(\rho)=\frac{\pi\rho}{\sin(\pi\rho)}\quad(0<\rho<1/2),\qquad C(\rho)=\pi\rho\quad(\rho>1/2)?\] No value at $\rho=0$ or $\rho=1/2$ is specified by these two displayed formulas. [S2]

Short English statement

Does the sharp finite-order Paley inequality hold for arbitrary subharmonic functions, with the same constants as for logarithms of entire functions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303007 (AMR-022-3007). 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 3.7 and Update 3.7. https://arxiv.org/abs/1809.07200.