On the Product of Log-concave Polynomials

A real polynomial is called log-concave if its coefficients form a log-concave sequence. We give a new elementary proof of the fact that a product of log-concave polynomials with nonnegative coefficients and no internal zero coefficients is again log-concave. In addition, we show that if the coefficients of the polynomial ∏ m∈M(x + m) form a monotone sequence where M is a finite multiset of positive real numbers, so do the coefficients of ∏ r∈N(x + r) for any submultiset N ⊂ M .