Research Problems in Function Theory — Problem 6.51

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 $D\subset\mathbb C$ be a connected open set containing $0$, of finite connectivity $n\ge1$, meaning that $\widehat{\mathbb C}\setminus D$ has $n$ connected components. Let $S(D)$ consist of holomorphic injective maps with $f(0)=0$, $f'(0)=1$. Define $E_D(f)=\int_D|f'(z)|^2\,dA(z)\in[0,\infty]$, equal to the area of $f(D)$.

Statement recorded in the supplied source.

For each such $D$, determine $\inf_{f\in S(D)}E_D(f)$ and characterize all functions attaining the infimum, including cases where the infimum is not finite or is not attained. [S2]

Short English statement

Find the normalized conformal maps of a finitely connected plane domain having smallest image area.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306051 (AMR-022-6051). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 6, Problem 6.51, complete update and preceding notation. https://arxiv.org/abs/1809.07200.