Research Problems in Function Theory — Problem 6.106

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Let $J$ be a Jordan curve, that is, the image of the circle under a continuous injective map, and let $D$ be its bounded complementary domain. Let $b:\mathbb T\to J$ be an orientation-preserving homeomorphism with respect to the positive boundary orientations. Its harmonic extension is the Poisson integral \[u(re^{i\theta})=\frac1{2\pi}\int_0^{2\pi} \frac{1-r^2}{1-2r\cos(\theta-t)+r^2}\,b(e^{it})\,dt,\qquad0\leq r<1.\]

Statement recorded in the supplied source.

If $u(\mathbb D)\subseteq D$, must $u:\mathbb D\to D$ be a homeomorphism onto $D$? No smoothness of the Jordan curve or boundary homeomorphism is imposed. [S2]

Short English statement

Does a harmonic extension of an oriented Jordan-boundary homeomorphism become a homeomorphism whenever its entire image stays inside the target domain?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306106 (AMR-022-6106). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.106, complete problem and update. https://arxiv.org/abs/1809.07200.