Research Problems in Function Theory — Problem 6.88

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 $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. For $f\in S$ whose image $\Omega=f(\mathbb D)$ has finite planar area, write $A(f)=\int_{\mathbb D}|f'|^2\,dx\,dy$. If the boundary has finite length, write $\ell(f)=\mathcal H^1(\partial\Omega)$, with $\mathcal H^1$ the one-dimensional Hausdorff measure. The constants in the question must be universal, independent of $f$.

Statement recorded in the supplied source.

Determine the largest constant $c$ for which $|a_2|\le2-cA(f)^{-1/2}$ holds for every such finite-area map, and identify the associated extremal function or functions. Equivalently determine $\inf_f (2-|a_2|)\sqrt{A(f)}$. Also, is there a universal $c_1>0$ for which $|a_2|\le2-c_1/\ell(f)$ for all such maps with finite boundary length? [S2]

Short English statement

Find the sharp area correction to the second-coefficient bound, and decide whether a corresponding positive perimeter correction always holds.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306088 (AMR-022-6088). 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 6.88 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.