Research Problems in Function Theory — Problem 7.40

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 $\overline B(z,r)=\{w\in\mathbb C:|w-z|\le r\}$. For $N$ pairwise disjoint closed discs of the same positive radius, contained in $\{|z|\le1/2\}$, define \[\Omega=\mathbb D\setminus\bigcup_{j=1}^N\overline B(z_j,r).\] Let $h$ be its harmonic-measure function: the continuous solution on $\overline\Omega$ of $\Delta h=0$ in $\Omega$, with boundary values one on $\partial\mathbb D$ and zero on the inner circles, extended by zero over the removed discs. Write $\lfloor x\rfloor$ for the greatest integer at most $x$.

Statement recorded in the supplied source.

Is there an absolute $\delta\in(0,1/4)$ such that, whenever \[0<r\le\delta,\qquad N\ge\lfloor1/r\rfloor^{\,2-\delta},\] at least one index $j$ satisfies \[\int_0^{2\pi}h(z_j+2re^{i\theta})\,d\theta\le r^{2+\delta}?\] [S2] Taking $\delta<1/4$ ensures that all circles being sampled stay inside the unit disc and is harmless for this existential small-radius assertion. Update7.40 records Lewis and Wu's proof and its role in their solutions of Problems1.6 and4.18. [S2]

Short English statement

When sufficiently many equal small discs are removed, must at least one have an exceptionally small surrounding circular integral of the outer-boundary harmonic measure?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307040 (AMR-022-7040). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.40 and its complete update. https://arxiv.org/abs/1809.07200.