Preservation of inequalities under Hadamard products

TL;DR

This paper investigates how certain combinatorial properties like ultra log-concavity and gamma-positivity are preserved under Hadamard products, and it disproves a conjecture about real-rootedness of Hadamard powers.

Contribution

It extends the understanding of property preservation under Hadamard products and refutes a previous conjecture on real-rootedness of Hadamard powers.

Findings

Ultra log-concavity is preserved under Hadamard products.

Gamma-positivity and interlacing symmetric decompositions are preserved.

A conjecture on the real-rootedness of Hadamard powers is disproved.

Abstract

Wagner (1992) proved that the Hadamard product of two P\'olya frequency sequences that are interpolated by polynomials is again a P\'olya frequency sequence. We study whether related combinatorial properties are preserved under Hadamard products. In particular, we show that ultra log-concavity, $\gamma$-positivity, and interlacing symmetric decompositions are preserved. Furthermore, we disprove a conjecture by Fischer and Kubitzke (2014) concerning the real-rootedness of Hadamard powers.