Log-Concavity under Hadamard Products

Definitions and mathematical statement

The supplied record does not determine an unambiguous reconstruction of the cited open question. The following makes its literal coefficientwise interpretation precise under an explicitly stated nonnegative real coefficient convention; that convention is not specified in the source excerpt.

Let $I=\{0,1,2,\ldots\}$. A sequence $a=(a_n)_{n\in I}$ of nonnegative real numbers is log-concave if \[ a_n^2\geq a_{n-1}a_{n+1}\qquad(n\geq1). \] Some conventions additionally require no internal zeros: whenever $i<j<k$ and $a_i a_k>0$, one requires $a_j>0$. The supplied record does not specify whether this additional condition is intended.

For formal power series \[ A(t)=\sum_{n\geq0}a_nt^n,\qquad B(t)=\sum_{n\geq0}b_nt^n, \] define their coefficientwise Hadamard product by \[ (A\odot B)(t)=\sum_{n\geq0}a_nb_nt^n. \] No analytic convergence is needed for this definition. Polynomials can be treated as such series by extending their coefficient sequences by zeros.

Literal question in the supplied cleaned statement. For every pair of nonnegative real log-concave sequences $a,b$, is the sequence $c_n=a_nb_n$ log-concave, that is, does \[ (a_nb_n)^2\geq(a_{n-1}b_{n-1})(a_{n+1}b_{n+1})\qquad(n\geq1) \] hold? The record also requests a characterization of the hypotheses under which preservation holds, but does not specify the larger class of coefficients, sequences, or operations within which that characterization is sought.

Unresolved source qualification. The original excerpt introduces sequences satisfying $a_n=p(n)$ for every $n\geq0$, where $p$ is a polynomial, and contains a corrupted generating-function display involving $W(p)$ and a denominator of exponent $\deg p+1$. This raises a materially different possible target: log-concavity of a generating-function numerator associated with polynomial evaluations. The excerpt does not reliably define $W$, its coefficient domain, or the relevant Hadamard operation on such numerators. In particular, coefficientwise multiplication of two numerators must not be identified without justification with the numerator associated with multiplying their evaluation sequences. The literal formulation above therefore cannot be certified as the intended statement of $[1,\text{ Question }6.1]$. The supplied open label is retained as source metadata, not asserted here as a verified mathematical fact.

Short English statement

The cleaned record asks whether multiplying corresponding terms of two log-concave coefficient sequences preserves log-concavity, and seeks the relevant hypotheses. Its original excerpt also discusses polynomial evaluation sequences and generating-function numerators, leaving the intended object of log-concavity unclear. This statement-only draft records a precise literal interpretation and preserves that unresolved distinction for human review.

Sources

• Local UnsolvedMath registry: supplied complete source record for permanent ID $30006219$, title Log-Concavity under Hadamard Products, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2025$, supplies no proposer, and labels the problem open. Its statement status is corrected_verified; the accompanying verification says that only formula delimiters, notation, grammar, and typography were repaired. These are supplied registry assertions.
• Mini-Workshop: Alcoved Polytopes in Physics and Optimization (2025), Oberwolfach Reports, DOI 10.4171/owr/2025/12, pp. 531--558. Supplied report and literature-source link: https://doi.org/10.4171/owr/2025/12. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Branden--Ferroni--Jochemko, Hadamard products, as identified by the supplied literature curation: https://arxiv.org/abs/2408.12386. The original excerpt refers to $[1,\text{ Question }6.1]$; its bibliography is not included, so the exact reference mapping and question wording require verification.
• Supplied literature assessment dated 2026-08-22: the primary paper is reported to prove preservation of ultra-log-concavity, $\gamma$-positivity, and related properties, while leaving preservation of log-concavity under the Hadamard product as Question 6.1; the curation reports that no later resolution was found. This is supplied dated curation derived from the OWR material, not independently verified here. The source explicitly requests independent verification of time-sensitive status and cited claims.