Log-concavity and unimodality of compound polynomials

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