Research Problems in Function Theory — Problem 6.82

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 $\mathbb T=\partial\mathbb D$. The class $S$ consists of holomorphic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$ and $f'(0)=1$, so $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. The inverse in this question is the local inverse germ at $0$, analytically continued to all of $\mathbb D$. It is required to be holomorphic and injective there, but its image need not lie in $\mathbb D$. Let $\sigma$ be the subclass of $S$ whose inverse germs have the continuation just described. Put $a_2^*=\sup_{f\in\sigma}|a_2|$ and $d_n(f)=\left||a_{n+1}|-|a_n|\right|$.

Statement recorded in the supplied source.

Is $a_2^*=\sqrt2$? Is there an absolute $A$ with $0<A<1$ such that $d_n(f)\le A$ for every $f\in\sigma$ and every $n\ge2$? Alternatively, determine $\sup_{f\in\sigma}d_n(f)$ for each $n\ge2$. [S2] Update 6.82 reports verification of the second-coefficient conjecture for certain subclasses only. [S2]

Short English statement

Find the sharp second coefficient for bi-univalent maps, and decide whether all successive coefficient magnitudes have a uniform gap bound strictly below one.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306082 (AMR-022-6082). 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 6.82 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.