Research Problems in Function Theory — Problem 6.108

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 $S$ be the class of holomorphic injective functions $f:\mathbb D\to\mathbb C$ with $f(0)=0$ and $f'(0)=1$. For a holomorphic univalent $f$ in $\mathbb D$, $0<r<1$ and $\lambda>0$, let \[I_\lambda(r,f')=\left(\frac1{2\pi}\int_0^{2\pi}|f'(re^{i\theta})|^\lambda\,d\theta\right)^{1/\lambda}.\] At $\lambda=0$ use the limiting geometric mean \[I_0(r,f')=\exp\left(\frac1{2\pi}\int_0^{2\pi}\log|f'(re^{i\theta})|\,d\theta\right).\] The derivative never vanishes. For normalized $f\in S$, put $k(z)=z/(1-z)^2$ and \[C_\lambda=\sup_{f\in S,\ 0<r<1}\frac{I_\lambda(r,f')}{I_\lambda(r,k')}.\]

Statement recorded in the supplied source.

Determine the maximal order of growth of $I_\lambda(r,f')$ as $r\uparrow1$ for $0\leq\lambda\leq2/5$, as $f$ ranges over univalent functions. Bounds for an individual unnormalized $f$ may depend on $f$. For $\lambda>2/5$, determine precisely when $C_\lambda=1$; in particular, is $C_\lambda=1$ for every $\lambda\geq2$? [S2]

Short English statement

Find the small-exponent growth of derivative integral means, and determine when the Koebe function gives the exact normalized bound at larger exponents.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306108 (AMR-022-6108). 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 6.108, complete problem and update. https://arxiv.org/abs/1809.07200.