On relative log-concavity and stochastic comparisons

In this paper, we investigate the equivalence among stochastic orderings between random variables under the scenario where the ratio of density functions or survival functions is log-concave. Some sufficient and necessary conditions that may be easily checked for the usual stochastic order, the hazard rate order, the mean residual lifetime order, the harmonic mean residual lifetime order and the mean inactivity time order are presented. As for illustration, we also provide several numerical examples and three applications.