Research Problems in Function Theory — Problem 6.8

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.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, $\mathbb T=\partial\mathbb D$, and $\mathbb D^*=\{z\in\mathbb C:|z|>1\}$. Let $S$ be the class of analytic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$, $f^{\prime}(0)=1$, with expansion $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $a_1=1$. Also let $\Sigma$ be the class of analytic injective $F$ on $|z|>1$ with $F(z)=z+\sum_{n\ge1}b_nz^{-n}$. For $\lambda>0$ and a circle in the function’s domain set \[I_\lambda(r,h)=\left(\frac1{2\pi}\int_0^{2\pi}|h(re^{i\theta})|^{\lambda}\,d\theta\right)^{1/\lambda}.\] Thus $0<r<1$ for $S$ and $r>1$ for $\Sigma$. The derivative means use $h=f^{\prime}$ or $F^{\prime}$.

Statement recorded in the supplied source.

Determine the exact upper bounds for the function and derivative means $I_\lambda(r,f)$ and $I_\lambda(r,f^{\prime})$ over $S$, and their exterior analogues over $\Sigma$, for all $\lambda>0$. Include the bounded subclass $|f|<M$ of $S$ and the sharp orders as the relevant boundary is approached: $r\uparrow1$ in $S$, $r\downarrow1$ in $\Sigma$. The source update identifies the Koebe function as the extremal for the classical function means in $S$, and distinguishes the exterior questions. [S2]

Short English statement

Determine sharp integral-mean bounds for normalized univalent functions and their derivatives, inside and outside the disk and for bounded disk functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306008 (AMR-022-6008). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 6.8, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.