Research Problems in Function Theory — Problem 6.15

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 $a_1=1$. Since $f^{\prime}$ is nowhere zero and $f^{\prime}(0)=1$, let $L_f$ be the analytic logarithm of $f^{\prime}$ with $L_f(0)=0$. For real $\alpha$ define the path-independent integral \[f_\alpha(z)=\int_0^z\exp(\alpha L_f(\zeta))\,d\zeta.\] For a locally univalent analytic function $h$ on $\mathbb D$, define the pre-Schwarzian norm \[\|h\|_{\mathrm{pre}}=\sup_{z\in\mathbb D}(1-|z|^2)\left|\frac{h^{\prime\prime}(z)}{h^{\prime}(z)}\right|.\]

Statement recorded in the supplied source.

Determine all real $\alpha$ for which $f_\alpha\in S$ for every $f\in S$. Separately determine the largest universal $c\ge0$ such that $\|h\|_{\mathrm{pre}}\le c$ forces every locally univalent analytic $h$ to be globally univalent on $\mathbb D$. The update reports that the second sharp constant is $c=1$. [S2]

Short English statement

Determine the universal real powers of derivatives whose integrals preserve normalized univalence, and record the separate sharp pre-Schwarzian univalence criterion.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306015 (AMR-022-6015). 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 6.15, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.