A monotonicity result for discrete fractional difference operators

In this note we demonstrate that if y(t) ≥ 0, for each t in the domain of t → y(t), and if, in addition, Δ0y(t) ≥ 0, for each t in the domain of t → Δ0y(t), with 1 < ν < 2, then it holds that y is an increasing function of t. This demonstrates that, in some sense, the positivity of the νth order fractional difference has a strong connection to the monotonicity of y. Furthermore, we provide a dual result in case Δ0y(t) ≤ 0 and y is nonpositive on its domain. We conclude the note by mentioning some implications of these results. Mathematics Subject Classification (2010). Primary 39A12, 39A70; Secondary 26A48.