Research Problems in Function Theory — Problem 7.28

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 $D\subset\mathbb C$ be a domain and $f:D\to\mathbb C$ continuous. All circle integrals below are counterclockwise contour integrals with the complex differential $d\zeta$. The radii used at $z$ are small enough that the closed disc lies in $D$.

Statement recorded in the supplied source.

Does either of the following conditions imply that $f$ is holomorphic on $D$? (i) For every $z\in D$ there is $r(z)>0$ such that \[\int_{|\zeta-z|=r}f(\zeta)\,d\zeta=0\qquad(0<r\le r(z)).\] (ii) For every $z\in D$, \[\lim_{r\downarrow0}r^{-2}\int_{|\zeta-z|=r}f(\zeta)\,d\zeta=0.\] The second, weaker condition is a separate question. [S2]

Short English statement

Does vanishing of all sufficiently small circular contour integrals force analyticity, and is their scaled limit being zero already sufficient?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307028 (AMR-022-7028). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.28 and its complete update. https://arxiv.org/abs/1809.07200.