Research Problems in Function Theory — Problem 7.37

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 $m$ be planar Lebesgue measure and $\Delta(R)=\{z\in\mathbb C:|z|<R\}$. For arbitrary $n\in\mathbb N$, $z_1,\ldots,z_n\in\mathbb C$ and positive real weights summing to one, put \[g(z)=\sum_{j=1}^n\frac{\lambda_j}{(z-z_j)^2},\qquad \sum_{j=1}^n\lambda_j=1.\] Integrals of $|g|$ are Lebesgue integrals, with the value $+\infty$ allowed.

Statement recorded in the supplied source.

Does there exist an absolute constant $C$ such that for each such $g$ there is a measurable set $S=S(g)\subset\mathbb C$ with $m(S)=\pi$ satisfying, simultaneously for every real $R>1$, \[\int_{\Delta(R)\setminus S}|g(z)|\,dm(z)\le2\pi\log R+C?\] [S2] The original connects the proposed estimate with the sharp area-distortion inequality of Problem7.36. The set $S$ may depend on $g$, but does not depend on $R$. [S2]

Short English statement

Can one discard a set of area exactly pi from each weighted rational function so that a single universal logarithmic integral bound holds for every radius?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307037 (AMR-022-7037). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.37 and its complete update. https://arxiv.org/abs/1809.07200.