  We show that Stieltjes moment sequences are infinitely log-convex, which
parallels a famous result that (finite) P\'olya frequency sequences are
infinitely log-concave. We introduce the concept of $q$-Stieltjes moment
sequences of polynomials and show that many well-known polynomials in
combinatorics are such sequences. We provide a criterion for linear
transformations and convolutions preserving Stieltjes moment sequences. Many
well-known combinatorial sequences are shown to be Stieltjes moment sequences
in a unified approach and therefore infinitely log-convex, which in particular
settles a conjecture of Chen and Xia about the infinite log-convexity of the
Schr\"oder numbers. We also list some interesting problems and conjectures
about the log-convexity and the Stieltjes moment property of the (generalized)
Ap\'ery numbers.
