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 $$\alpha \ge 2.$$ 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,  $$\Delta ^{\alpha }u$$ with $$2 \le \alpha < 4.$$ Two examples show that the given criteria are optimal with respect to the established geometrical notion.