Second and Third Order Forward Difference Operator: What Is in Between?

In this article, we present a new geometrical notion for a real-valued function defined in a discrete domain that depends on a parameter α≥ 2. We give examples to illustrate connections between convexity and this new concept. We then prove two criteria based on the sign of the discrete fractional operator of a function u, Δ αu with 2 ≤ α< 4. Two examples show that the given criteria are optimal with respect to the established geometrical notion. © 2021, The Royal Academy of Sciences, Madrid.