Research Problems in Function Theory — Problem 6.76

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 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 $\mathbb D$. For $n\ge2$, let $\mathcal V_n$ consist of normalized polynomials $p(z)=z+\sum_{j=2}^n a_jz^j$ (degree at most $n$) which are injective on $\mathbb D$ and whose inverse germ continues holomorphically and injectively to $\mathbb D$.

Statement recorded in the supplied source.

For every $n\ge2$, determine both $\max_{p\in\mathcal V_n}|a_2|$ and $\max_{p\in\mathcal V_n}|a_n|$. [S2]

Short English statement

Find the sharp second and highest-degree coefficient bounds for normalized bi-univalent polynomials.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306076 (AMR-022-6076). 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 6.76 and Update 6.76; inverse-continuation clarification in Problem 6.82. https://arxiv.org/abs/1809.07200.