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

In this paper, we mainly study the stability of iterated polynomials and linear transformations preserving the strong $q$-log-convexity of polynomials Let $[T_{n,k}]_{n,k\geq0}$ be an array of nonnegative numbers. We give some criteria for the linear transformation $$y_n(q)=\sum_{k=0}^nT_{n,k}x_k(q)$$ preserving the strong $q$-log-convexity (resp. log-convexity). As applications, we derive that some linear transformations (for instance, the Stirling transformations of two kinds, the Jacobi-Stirling transformations of two kinds, the Legendre-Stirling transformations of two kinds, the central factorial transformations, and so on) preserve the strong $q$-log-convexity (resp. log-convexity) in a unified manner. In particular, we confirm a conjecture of Lin and Zeng, and extend some results of Chen {\it et al.}, and Zhu for strong $q$-log-convexity of polynomials, and some results of Liu and Wang for transformations preserving the log-convexity. 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 type $A$ and $B$, and their $q$-analogs. In addition, we also prove that the generating functions of alternating runs of type $A$ and $B$, the longest alternating subsequence and up-down runs of permutations form a $q$-log-convex sequence, respectively.