Hermite-Hadamard Inequality on Time Scales

Recently, new developments of the theory and applications of dynamic derivatives on time scales were made. The study provides an unification and an extension of traditional differential and difference equations and, in the same time, it is a unification of the discrete theory with the continuous theory, from the scientific point of view. Moreover, it is a crucial tool in many computational and numerical applications. Based on the wellknown Δ delta and ∇ nabla dynamic derivatives, a combined dynamic derivative, socalled α diamond-α dynamic derivative, was introduced as a linear combination of Δ and ∇ dynamic derivatives on time scales. The diamond-α dynamic derivative reduces to the Δ derivative for α 1 and to the ∇ derivative for α 0. On the other hand, it represents a “weighted dynamic derivative” on any uniformly discrete time scale when α 1/2. See 1–5 for the basic rules of calculus associated with the diamond-α dynamic derivatives. The classical Hermite-Hadamard inequality gives us an estimate, from below and from above, of the mean value of a convex function. The aim of this paper is to establish a full analogue of this inequality if we compute the mean value with the help of the delta, nabla, and diamond-α integral. The left-hand side of the Hermite-Hadamard inequality is a special case of the Jensen inequality. Recently, it has been proven a variant of diamond-α Jensen’s inequality see 6 .