q-log-convexity from linear transformations and polynomials with only real zeros

The stability property of iterated polynomials implies the q-log-convexity. By applying the method of interlacing of zeros, we also present two criteria for the stability of the iterated Sturm sequences and q-log-convexity of polynomials. As consequences, we get the stabilities of iterated Eulerian polynomials of types A and B, and their q-analogs. In addition, we also prove that the generating functions of alternating runs of types A and B, the longest alternating subsequence and up-down runs of permutations form a q-log-convex sequence, respectively.