Research Problems in Function Theory — Problem 3.4

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.

A subharmonic function is an upper semicontinuous function with values in $[-\infty,\infty)$, not identically $-\infty$ on a connected component, that satisfies the mean-value inequality on every closed ball in its domain. A real harmonic function is a twice continuously differentiable function $u$ satisfying $\Delta u=0$. Let $\mathbb D=\{|z|<1\}$ and let $E\subset[0,1]$ be a finite union of intervals. Write $A(r,u)=\inf_{|z|=r}u(z)$. Let $u_E$ be the Carleman--Milloux slit function: it is harmonic on $\mathbb D\setminus E$, has boundary values $0$ on $\partial\mathbb D\setminus E$ and $-1$ on $E\cap\mathbb D$, and is extended by $-1$ on the slit. For $0<r<1$ and $0<K<1$ set \[C_u(r,K)=\{\theta\in[0,2\pi):u(re^{i\theta})<-K\}.\] Length of this angular set means one-dimensional Lebesgue measure.

Statement recorded in the supplied source.

Among subharmonic $u\leq0$ on $\mathbb D$ with $A(r,u)\leq-1$ for every $r\in E\cap[0,1)$ (with $A(0,u)=u(0)$), is $|C_u(r,K)|$ minimized only when $u=u_E$? In particular, answer the question for the full slit $E=[0,1]$. [S2]

Short English statement

Is the Carleman--Milloux slit function the unique minimizer of the angular region below a prescribed negative level, especially for a full radial slit?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303004 (AMR-022-3004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 3.4. https://arxiv.org/abs/1809.07200.