Log-concavity of the Fennessey-Larcombe-French Sequence

Abstract

We prove the log-concavity of the Fennessey-Larcombe-French sequence based on its three-term recurrence relation, which was recently conjectured by Zhao. The key ingredient of our approach is a sufficient condition for log-concavity of a sequence subject to certain three-term recurrence.

Log-behavior of two sequences related to the elliptic integrals

“…The log-convexity of {P n } n≥0 , conjectured by Sun [14], has been proved by Xia and Yao [15] and independently by Zhao [18]. The log-concavity of {V n } n≥1 , conjectured by Zhao [19], has been confirmed by Yang and Zhao [17]. The 2-log-convexity of {P n } n≥0 has been shown by Sun and Wu [13].…”

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

“…which was introduced by Yang and Zhao [13] in their study of the log-concavity of the sequence {V n } n≥1 . They [13] showed that h(n) is a lower bound for the ratio V n /V n−1 , precisely,…”

“…Remark. By employing a criterion due to Wang and Zhu [11, Theorem 2.1], Yang and Zhao [13] showed that the sequence { n √ V n+1 } n≥1 is strictly decreasing.…”