Research Problems in Function Theory — Problem 6.77

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 $\mathbb T=\partial\mathbb D$. For $n\in\mathbb N$, let $\mathcal P_n$ consist of polynomials $p(z)=z+\sum_{j=2}^n a_jz^j$ of degree at most $n$, injective on $\mathbb D$. Their boundary values are ordinary polynomial values on $\mathbb T$, and integration is with respect to angular Lebesgue measure.

Statement recorded in the supplied source.

For every $n\in\mathbb N$ and $0<q<\infty$, determine \[\max_{p\in\mathcal P_n}\int_0^{2\pi}|p(e^{it})|^q\,dt.\] [S2]

Short English statement

Find the largest boundary integral mean of every positive exponent among normalized univalent polynomials of bounded degree.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306077 (AMR-022-6077). 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 6.77 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.