Log-behavior of two sequences related to the elliptic integrals

Abstract: 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 ${V_n}{n\geq 1}$.