MONOTONICITY AND CONVEXITY FOR NABLA FRACTIONAL q-DIFFERENCES

In this paper, we examine the relation between monotonicity and convexity for nabla fractional q-differences. In particular we prove that Theorem A. Assume f : q(N0) -> R, del(v)(q)f(t) >= 0 for each t is an element of q(N0), with 1 < v < 2, then del(q)f(t) >= 0 for t is an element of q(N1). Theorem B. Assume f : q(N0) -> R, del(v)(q)f(t) >= 0 for each t is an element of q(N1), with 2 < v < 3, then del(2)(q)f(t) >= 0 for t is an element of q(N2). This shows that, in some sense, the positivity of the with order q-fractional difference has a strong connection to the monotonicity and convexity of f(t).