Research Problems in Function Theory — Problem 8.18

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 simply connected domain containing $0$. For analytic $f$ on $D$ with $f(0)=0$, define \[E_D(f)=\int_D|f^{\prime}|^2\,dA,\qquad N_D(f)=\int_D|f|^2\,dA,\] permitting the value $+\infty$. The analytic Poincaré inequality means that there is $K_D<\infty$ such that $N_D(f)\le K_DE_D(f)$ whenever $E_D(f)<\infty$; functions with infinite energy give no restriction.

Statement recorded in the supplied source.

Give necessary and sufficient geometric conditions on $D$ for this inequality to hold for every such analytic $f$. [S2]

Short English statement

Characterize the simply connected domains that satisfy the analytic Poincaré inequality for functions vanishing at the origin.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308018 (AMR-022-8018). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 8.18, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.