The difference property and higher-order convexity properties of difference operators

We prove that the class of delta-n-convex functions has the difference property, which is a generalization of Ger’s result on the difference property of the class of delta-convex functions [7]. Using this generalization, we prove that if for every non-negative number a the difference operator Δ ₐf is n-convex or n-concave, for some function f, then Δ ₐf is n-convex for all a or n-concave for every a. A connection of our results with Wright convexity of higher orders is also presented.