Log-concavity and unimodality of compound polynomials

Let the polynomial g(x)=∑i=0kbixi with nonnegative coefficients be symmetric and log-concave. Given a nonnegative sequence {ai}i=0n, we present a sufficient condition insuring the unimodality of the polynomial ∑i=0naixig(x)n−i. In addition, if {ai}i=0n is nonnegative and non-increasing, then the polynomial ∑i=0naixi(1+x)2(n−i) is log-concave.