Preservation of inequalities under Hadamard products

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.