Log-behavior of two sequences related to the elliptic integrals

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