Research Problems in Function Theory — Problem 7.30

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 $u:\mathbb D\to\mathbb R$ be bounded and continuous. Harmonicity means $\Delta u=0$ on $\mathbb D$.

Statement recorded in the supplied source.

Suppose that for every $z\in\mathbb D$ there is a radius $r(z)$ with $0<r(z)<1-|z|$ such that \[u(z)=\frac1{2\pi}\int_0^{2\pi}u(z+r(z)e^{i\theta})\,d\theta.\] Must $u$ be harmonic? The averaging here is along the circumference, rather than over its interior. [S2] Update7.30 identifies this with Problem3.8. Update3.8 distinguishes circular from disc averaging and records a negative answer for circumference averaging. [S2]

Short English statement

Does a bounded continuous function become harmonic if each point has at least one circle on which its average equals the value at the centre?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307030 (AMR-022-7030). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.30 and its complete update; Problem3.8 and Update3.8. https://arxiv.org/abs/1809.07200.