Preservation of inequalities under Hadamard products

Wagner [J. Math. Anal. Appl. 163 (1992), pp. 459–483] proved that the Hadamard product of two Pólya frequency sequences that are interpolated by polynomials is again a Pólya frequency sequence. We study whether related combinatorial properties are preserved under Hadamard products. In particular, we show that ultra log-concavity, γ \gamma -positivity, and interlacing symmetric decompositions are preserved. Furthermore, we disprove a conjecture by Fischer and Kubitzke [Adv. in Appl. Math. 56 (2014), pp. 1–19] concerning the real-rootedness of Hadamard powers.