The log-convexity of $r$-derangement numbers

Abstract

This paper focuses on the log-convexity of the sequence $\{D_r(n)\}_{n\ge r}$ of $r$-derangement numbers, where $r\ge 2$ is a positive integer. We mainly prove that $\{D_2(n)\}_{n\ge 2}$ and $\{D_3(n)\}_{n\ge 7}$ are log-convex. In addition, we also show that $\{\sqrt {D_2(n)}\}_{n\ge 2}$ and $\{\sqrt [3]{D_3(n)}\}_{n\ge 7}$ are log-balanced.