Analytical and numerical convexity results for discrete fractional sequential differences with negative lower bound.

We investigate relationships between the sign of the discrete fractional sequential difference ( Δ 1 + a − μ ν Δ a μ f) (t) and the convexity of the function t ↦ f (t). In particular, we consider the case in which the bound ( Δ 1 + a − μ ν Δ a μ f) (t) ≥ ε f (a) , for some ε > 0 and where f (a) < 0 , is satisfied. Thus, we allow for the case in which the sequential difference may be negative, and we show that even though the fractional difference can be negative, the convexity of the function f can be implied by the above inequality nonetheless. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. We use a combination of both hard analysis and numerical simulation. [ABSTRACT FROM AUTHOR]