We investigate relationships between the sign of the discrete fractional sequential difference (  1 ν + a−μ μ a f ) (t) and the convexity of the function t ↦ → f(t). In particular, we consider the case in which the bound

ν (  1 + a−μ μ a f ) (t) ≥ εf(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.