The relation between Nabla fractional differences and Nabla integer differences

In this paper we obtain two interrelated results. The first result is the following inequality: Theorem. Assume that f : Na → R satisfies va f(t)≥ 0, for each t  Na+1, v > 0,v  N1, and choose N  N1, such that N - 1 < v < N. Then for each k  Na+N, we have N-1 f(a+k)≥ - ΣN-2 i=0 H-v+i(a + k, a+i)i f(a+i+1)- Σk-1 i=N H-v+N-2(a + k, a+i-1)N-1 f(a + i), where H-v+N-2(a+k, a+i - 1) = (k-i + 1)-v+N-2 /Г(-v + N - 1) < 0. As an application of the above inequality we prove the following result: Theorem. Assume that f : Na → R satisfies va f(t)≥ 0, for each t  Na+1, where 5 < v < 6. Then 5 f(t)≥ 0, for t  Na+6. This demonstrates that, in some sense, the positivity of the v-th order fractional difference has a strong connection to the positivity of an integer-order difference of the function f.