Research Problems in Function Theory — Problem 6.86

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 $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. Let $S_{\mathbb R}=\{f\in S:a_n\in\mathbb R\text{ for every }n\}$, equivalently $f(\bar z)=\overline{f(z)}$. For a fixed $z_0\in\mathbb D$, put $\rho=|z_0|$. The general estimate is $|z_0f''(z_0)/f'(z_0)-2\rho^2/(1-\rho^2)|\le4\rho/(1-\rho^2)$.

Statement recorded in the supplied source.

Can this inequality be sharpened under the hypothesis $f\in S_{\mathbb R}$? Equivalently, find the sharp upper bound \[\sup_{f\in S_{\mathbb R}}\left|z_0\frac{f''(z_0)}{f'(z_0)}-\frac{2|z_0|^2}{1-|z_0|^2}\right|\] for each fixed $z_0\in\mathbb D$, and determine when it is smaller than the unrestricted bound $4|z_0|/(1-|z_0|^2)$. [S2]

Short English statement

How does requiring real Taylor coefficients sharpen the displayed pointwise pre-Schwarzian inequality?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306086 (AMR-022-6086). 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 6.86 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.