Abstract
We consider conditions under which the positivity of a fractional difference implies either positivity, monotonicity, or convexity, and we consider both the non‐sequential and sequential cases. The particular difference we study is the discrete nabla ABC‐type difference, and it possesses a Mittag–Leffler kernel; this leads to some different results when compared to other types of kernels. We conclude by applying our results to certain classes of initial value problems in discrete fractional calculus.