Research Problems in Function Theory — Problem 7.20

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.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, and let \[X=\{(z_1,z_2):|z_1|\le1,\ |z_2|\le2\}\setminus\{(\zeta,\zeta):|\zeta|=1\}. \] A plurisubharmonic function is upper semicontinuous on its open domain and subharmonic on each complex line, with the value $-\infty$ permitted. Consider a function $F:X\to[-\infty,\infty)$ whose restriction to $\mathbb D^2$ is plurisubharmonic. No additional boundary continuity is imposed. At $z_1=z_2$ interpret $\log(1/|z_1-z_2|)$ as $+\infty$.

Statement recorded in the supplied source.

Suppose that \[F(z_1,z_2)\le\log\frac1{|z_1-z_2|}\quad((z_1,z_2)\in X),\] and $F(z_1,z_2)\le0$ whenever $|z_1|=|z_2|=1$ and $z_1\ne z_2$. Must $F\le0$ throughout $\mathbb D^2$? [S2] The original domain includes $|z_2|\le2$; that printed domain is retained. Its boundary values and its interior plurisubharmonic restriction are stated separately. [S2]

Short English statement

Does the given logarithmic bound and the nonpositive values on the distinguished boundary force a plurisubharmonic function to be nonpositive inside the bidisc?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307020 (AMR-022-7020). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.20 and its complete update. https://arxiv.org/abs/1809.07200.