Research Problems in Function Theory — Problem 7.73

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.

Fix $R>0$. For a plane domain $D\subset\{z:|z|<R\}$ let $\lambda_D(z)|dz|$ be its Poincare metric, normalized to curvature $-1$: if $\pi:\mathbb D\to D$ is a covering, $\lambda_D(\pi(w))|\pi'(w)|=2/(1-|w|^2)$. For $z\in D_1\cap D_2$ use the metric of the component of the intersection containing $z$. A uniform constant applies to every pair of such domains and every point of their intersection.

Statement recorded in the supplied source.

Determine the sharp constant $A(R)$ in \[\lambda_{D_1\cap D_2}(z)<A(R) \big(\lambda_{D_1}(z)+\lambda_{D_2}(z)\big). \] Equivalently determine the infimum of the positive constants for which this strict inequality always holds, with value $+\infty$ if none does, and whether that infimum itself is admissible. [S2]

Short English statement

Find the sharp uniform comparison between the hyperbolic metric on an intersection and the sum of the two original metrics.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307073 (AMR-022-7073). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.73, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.