Convex Functions on Discrete Time Domains

In this paper, we introduce the definition of a convex real valued function f defined on the set of integers, ℤ. We prove that f is convex on Z if and only if Δ2 f ≥ 0 on ℤ. As a first application of this new concept, we state and prove discrete Hermite–Hadamard inequality using the basics of discrete calculus (i.e., the calculus on Z). Second, we state and prove the discrete fractional Hermite–Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale.