  A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that
the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a
three-term recurrence for sequences, we develop techniques for dealing with the
ratio log-convexity of ratio sequences. As applications, we prove that the
ratio sequences of numbers, including the derangement numbers, the Motzkin
numbers, the Fine numbers, Franel numbers and the Domb numbers are ratio
log-convex, respectively. Finally, we not only prove that the sequence of
derangement numbers is asymptotically infinitely log-monotonic, but also show
some infinite log-monotonicity of some numbers related to the Gamma function,
in particular, implying two results of Chen {\it et al.} on the infinite
log-monotonicity of the Catalan numbers and the central binomial coefficients.
