  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.
