Research Problems in Function Theory — Problem 6.40

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 $\mathbb T=\partial\mathbb D$. Let $S$ consist of holomorphic injective functions $f(z)=z+\sum_{n\ge2}a_nz^n$ on $\mathbb D$; set $a_0=0$, $a_1=1$. A function in $S$ is starlike if its image is starlike about $0$, and convex if its image is a convex domain.

Statement recorded in the supplied source.

For every $f\in S$ and every integer $n\ge1$, must $1+\sum_{j=2}^{n}|a_{2j-1}|\ge|a_n|^2$? For real coefficients the source records the related signed inequality $1+\sum_{j=2}^{n}a_{2j-1}\ge a_n^2$; the question is the displayed modulus version for the full complex-coefficient class. [S2]

Short English statement

Does the sum of odd coefficient moduli up to degree $2n-1$ bound the square of the nth coefficient modulus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306040 (AMR-022-6040). 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 6.40, complete update and preceding notation. https://arxiv.org/abs/1809.07200.