We investigate the relationship between the sign of the discrete fractional sequential difference ( ∆ 1+a− µ ν ∆ a µ f ) (t) and the monotonicity of the function t ↦ → f(t). More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound

( ∆ 1+a− µ ν ∆ a µ f ) (t) ≥ −εf(a),

for some ε > 0. We prove that even though the fractional difference can be negative, the monotonicity of the function f, nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges of a purely analytical approach, our analysis includes numerical simulation.