Research Problems in Function Theory — Problem 6.93

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 $\mathbb T=\partial\mathbb D$. The class $S$ consists of holomorphic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$ and $f'(0)=1$, so $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. For $\Omega=f(\mathbb D)$ let $F:\Omega\to\mathbb D$ be the conformal inverse. The intersection $\Omega\cap\mathbb R$ can have multiple interval components; its integral means the one-dimensional Lebesgue integral over their union.

Statement recorded in the supplied source.

For every $f\in S$ and every real $p$ with $1\le p<2$, is \[\int_{\Omega\cap\mathbb R}|F'(x)|^p\,dx<\infty?\] [S2] The source records the affirmative endpoint $p=1$ and slit-domain counterexamples at $p=2$. [S2]

Short English statement

Is the inverse derivative integrable on the real part of every normalized univalent image for every exponent from one up to, but excluding, two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306093 (AMR-022-6093). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 6, Problem 6.93 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.