Research Problems in Function Theory — Problem 6.78

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 $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. Use the unit round sphere and stereographic coordinates, in which its area element is $4\,dx\,dy/(1+|w|^2)^2$. The spherical area of $f(\mathbb D)$ is therefore \[A_{\mathrm{sph}}(f)=\int_{\mathbb D}\frac{4|f'(z)|^2}{(1+|f(z)|^2)^2}\,dx\,dy.\]

Statement recorded in the supplied source.

Determine $\min_{f\in S}A_{\mathrm{sph}}(f)$ and identify the functions attaining this minimum. [S2]

Short English statement

Find the smallest spherical image area of a normalized univalent function and its extremal map.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306078 (AMR-022-6078). 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 6.78 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.