In this paper, we introduce two new monotonicity concepts for a nonnegative or nonpositive valued function defined on a discrete domain. We give examples to illustrate connections between these new monotonicity concepts and the traditional ones. We then prove some monotonicity criteria based on the sign of the fractional difference operator of a function f , ∆ ν f with 0 < ν < 1. As an application, we state and prove the mean value theorem on discrete fractional calculus.