  Two interesting sequences arose in the study of the series expansions of the
complete elliptic integrals, which are called the Catalan-Larcombe-French
sequence $\{P_n\}_{n\geq 0}$ and the Fennessey-Larcombe-French sequence
$\{V_n\}_{n\geq 0}$ respectively. In this paper, we prove the log-convexity of
$\{V_n^2-V_{n-1}V_{n+1}\}_{n\geq 2}$ and $\{n!V_n\}_{n\geq 1}$, the ratio
log-concavity of $\{P_n\}_{n\geq 0}$ and the sequence $\{A_n\}_{n\geq 0}$ of
Ap\'{e}ry numbers, and the ratio log-convexity of $\{V_n\}_{n\geq 1}$.
