2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139205We show that the $q$-derangement numbers satisfy a ratio monotone property, which is analogous to the log-concavity and is stronger than the spiral property and the unimodality.16 pages. log-concavity possibly except for one term is added as a corollaryCombinatoricstext