Research Problems in Function Theory — Problem 6.97

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 $\mathbb T=\partial\mathbb D$. For a nonconstant holomorphic function $f$ on a domain $D$, its valence is $v_D(f)=\sup_{w\in\mathbb C}n_D(w,f)\in\mathbb N\cup\{\infty\}$, where $n_D(w,f)$ counts the zeros of $f-w$ in $D$ with multiplicity. The function is $p$-valent if $v_D(f)\leq p$. Fix an integer $p\geq1$ and a holomorphic $p$-valent function $f(z)=\sum_{j=1}^{\infty}a_jz^j$ in $\mathbb D$; thus $f(0)=0$ but no condition $a_1=1$ is imposed.

Statement recorded in the supplied source.

For every integer $n>p$, does \[|a_n|\leq\sum_{k=1}^{p}\frac{2k(n+p)!}{(p+k)!(p-k)!(n-p-1)!(n^2-k^2)}|a_k|\] hold? Determine whether the bound is sharp with respect to $p,n$ and the initial coefficient magnitudes $|a_1|,\ldots,|a_p|$, as asserted in Goodman's conjecture. [S2]

Short English statement

Bound every later coefficient of a multivalent function by the stated sharp weighted sum of its first coefficients.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306097 (AMR-022-6097). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.97, complete problem and update. https://arxiv.org/abs/1809.07200.