Convexity for nabla and delta fractional differences

Abstract

In this paper we demonstrate thatTheorem AIf f:Na+1→R satisfies ∇aνf(t)≥0, for each t∈Na+1, with 2<ν<3, then ∇2f(t)≥0, for t∈Na+3.Theorem BIf f:Na→R satisfies Δaνf(t)≥0, for each t∈Na, with 2<ν<3, and f(a)≤0,Δf(a)≥0,Δ2f(a)≥0. Then Δf(t)≥0, for t∈Na.This demonstrates that, in some sense, the positivity of the νth order fractional difference has a strong connection to the convexity of f(t).In Section 3, by means of an example, we show that a recent result in {C. Goodrich, A convexity result for fractional differences, Appl. Math. Lett. 35 (2014), pp. 58–62.} is incorrect as stated.