Research Problems in Function Theory — Problem 6.17

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 $a_1=1$. The image area is the extended quantity \[A(f)=\operatorname{area}(f(\mathbb D))=\pi\sum_{n=1}^{\infty}n|a_n|^2\in[\pi,\infty].\] For a prescribed feasible second coefficient $b\in\mathbb C$, let $S_b=\{f\in S:a_2=b\}$.

Statement recorded in the supplied source.

Determine $\inf_{f\in S_b}A(f)$ for every feasible $b$, and the minimizing functions when the infimum is attained. The elementary bound is $A(f)\ge\pi(1+2|b|^2)$; the source notes that it is not sharp for $|b|>1/2$. [S2]

Short English statement

Determine the smallest image area of a normalized univalent disk function when its second Taylor coefficient is prescribed.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306017 (AMR-022-6017). 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 6.17, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.