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 Delta(2)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.