Sun's log-concavity conjecture on the Catalan-Larcombe-French sequence

Abstract: Let ${P_n}{n\geq 0}$ denote the Catalan-Larcombe-French sequence, which naturally came up from the series expansion of the complete elliptic integral of the first kind. In this paper, we prove the strict log-concavity of the sequence ${\sqrt[n]{P_n}}{n\geq 1}$, which was originally conjectured by Sun. We also obtain the strict log-concavity of the sequence ${\sqrt[n]{V_n}}{n\geq 1}$, where ${V_n}{n\geq 0}$ is the Fennessey-Larcombe-French sequence arising in the series expansion of the complete elliptic integral of the second kind.