Monotonicity results for sequential fractional differences of mixed orders with negative lower bound

We investigate the relationship between the sign of the discrete fractional sequential difference (Δ1+a−μνΔaμf)(t) and the monotonicity of the function t↦f(t) in the case where the orders of the fractional difference are mixed – i.e. μ∈(0,1) but ν∈(1,2) with μ+ν∈(1,2). We consider the special case in which the bound (Δ1+a−μνΔaμf)(t)≥−εf(a),for some ε>0 and where f(a)≥0, is satisfied. Even though the fractional difference is negative, we demonstrate that there still exists a relationship between this quantity and the monotonicity of f. We augment our analytical results with numerical simulations.