Research Problems in Function Theory — Problem 6.25

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 $p\ge1$ be an integer. A nonconstant holomorphic $f(z)=\sum_{n\ge0}a_nz^n$ is $p$-valent if $f(z)=w$ has at most $p$ solutions in $\mathbb D$, counted with multiplicity, for every $w\in\mathbb C$. For $1\le k\le p<n$, put \[D(p,k,n)=\frac{2kn\prod_{j=1}^p(n^2-j^2)}{(p+k)!(p-k)!(n^2-k^2)}.\]

Statement recorded in the supplied source.

Is Goodman's coefficient bound \[|a_n|\le\sum_{k=1}^p|a_k|D(p,k,n)\qquad(n>p)\] valid for every $p$-valent $f$? Determine its sharpness for all $p,k,n$ if it holds. [S2]

Short English statement

Does Goodman’s proposed coefficient inequality hold sharply for all holomorphic p-valent disc functions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306025 (AMR-022-6025). 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 6.25, complete update and preceding notation. https://arxiv.org/abs/1809.07200.