On the log-convexity of the difference sequence of a log-convex sequence

Abstract

Let $\{z_n\}_{n\ge 0}$ be a log-convex sequence, where $z_{n+1}-z_n>0$ for $n\ge 0$. In this paper, we give several sufficient conditions for the log-convexity of the sequence $\{z_{n+1}-z_n\}_{n\ge 0}$. As applications, we discuss log-convexity for a series of sequences involving derangement numbers, Motzkin numbers, Fine numbers, numbers of counting directed animals, and so on.