Research Problems in Function Theory — Problem 6.30

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 $S$ consist of holomorphic injective functions $f(z)=z+\sum_{n\ge2}a_nz^n$ on $\mathbb D$; set $a_0=0$, $a_1=1$. A function in $S$ is starlike if its image is starlike about $0$, and convex if its image is a convex domain.Let $k(z)=z/(1-z)^2$ be the Koebe function. For $p>0$, define $J_p(r,h)=\int_0^{2\pi}|h(re^{it})|^p\,dt$.

Statement recorded in the supplied source.

Determine the exponents $p>0$ for which $J_p(r,f')\le J_p(r,k')$ for every $f\in S$ and every $0<r<1$. In particular, examine the possible range $p\ge1/3$. Also determine the sharp asymptotic orders as $r\uparrow1$ where an exact inequality is unavailable. [S2]

Short English statement

For which powers does the Koebe derivative dominate every univalent derivative’s integral mean, and what are the corresponding sharp asymptotic orders?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306030 (AMR-022-6030). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 6, Problem 6.30, complete update and preceding notation. https://arxiv.org/abs/1809.07200.