Analytical and numerical convexity results for discrete fractional sequential differences with negative lower bound. Issue 3 (4th March 2021) - Goodrich, Christopher S.; Lyons, Benjamin; Scapellato, Andrea; Velcsov, Mihaela T.
Abstract : 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.