Goodman's conjecture on coefficients of p-valent functions

Goodman's conjecture. For every p-valent normalised function f on the unit disk and every n > p, the n-th coefficient is bounded by the Goodman bound: |b_n| ≤ Σ_k=1^p 2k (n+p)!/(p-k)! (p+k)! (n-p-1)! (n^2-k^2) |b_k|.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-goodman-conjecture,
  title        = {Goodman's conjecture on coefficients of p-valent functions},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/goodman-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Goodman's conjecture. For every $p$ -valent normalised function $f$ on the unit disk and every $n > p$ , the $n$ -th coefficient is bounded by the Goodman bound: $∣ b_{n} ∣ \leq k = 1 \sum p \frac{2 k ( n + p )!}{( p - k )! ( p + k )! ( n - p - 1 )! ( n ^{2} - k ^{2} )} ∣ b_{k} ∣.$

A regular (analytic) function is $p$ -valent on the unit disk if it assumes each value at most $p$ times there, and some value exactly $p$ times. For such a function $f (z) = \sum_{n \geq 1} b_{n} z^{n}$ (so $f (0) = 0$ ), Goodman's conjecture (1948) asserts that its coefficients satisfy $∣ b_{n} ∣ \leq k = 1 \sum p \frac{2 k ( n + p )!}{( p - k )! ( p + k )! ( n - p - 1 )! ( n ^{2} - k ^{2} )} ∣ b_{k} ∣ (n > p) .$ This generalises the classical coefficient bounds for univalent functions (the case $p = 1$ , where the bound reads $∣ b_{n} ∣ \leq n ∣ b_{1} ∣$ , i.e. $∣ b_{n} ∣ \leq n$ under the classical normalisation $b_{1} = 1$ ) to $p$ -valent functions. It has been verified in several special cases, including $p$ -valent typically-real functions.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.GoodmanConjecture.

theorem goodman_conjecture (f : ℂ → ℂ) (p n : ℕ) (hf : IsPValent f p)
    (hf' : IsNormalized f) (hn : p < n) :
    ‖coeff f n‖ ≤ goodmanBound f p n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
Goodman, A. W., On some determinants related to $p$ -valent functions, Trans. Amer. Math. Soc. 63 (1948), 175–192.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.