Let A n denote the nth-cycle index polynomial, in the variables X j , for the symmetric group on n letters. We show that if the variables X j are assigned nonnegative real values which are log-concave, then the resulting quantities A n satisfy the two inequalities A n&1 A n+1 A 2 n ((n+1)Γn)A n&1 A n+1 . This implies that the coefficients of the formal power series exp(g(u)) are log-concave whenever those of g(u) satisfy a condition slightly weaker than log-concavity. The latter includes many familiar combinatorial sequences, only some of which were previously known to be log-concave. To prove the first inequality we show that in fact the difference A 2 n &A n&1 A n+1 can be written as a polynomial with positive coefficients in the expressions X j and X j X k &X j&1 X k+1 , j k. The second inequality is proven combinatorially, by working with the notion of a marked permutation, which we introduce in this paper. The latter is a permutation each of whose cycles is assigned a subset of available markers [M i, j ]. Each marker has a weight, wt(M i, j )=x j , and we relate the second inequality to properties of the weight enumerator polynomials. Finally, using asymptotic analysis, we show that the same inequalities hold for n sufficiently large when the X j are fixed with only finite many nonzero values, with no additional assumption on the X j .