An Equivalent Form of Young’s Inequality with Upper Bound

β1 ψ dx− ab+ α1β1. Young’s inequality [2, 1, 4] states that for every a ∈ [α1, α2] and b ∈ [β1, β2] (2) 0 ≤ F (a, b), where the equality holds iff φ(a) = b (or, equivalently, ψ(b) = a). Among the classical inequalities Young’s inequality is probably the most intuitive. Indeed, its meaning can be easily grasped once the integrals are regarded as areas below and on the left of the graph of φ (see, for instance, [5]). Despite its simplicity, it has profound consequences. For instance, the Cauchy, Holder and Minkowski inequalities can be easily derived from it [5]. In this work I am going to improve Young’s inequality as follows Theorem 1.1. Under the assumptions of Young’s inequality, we have for every a ∈ [α1, α2] and b ∈ [β1, β2], (3) 0 ≤ F (a, b) ≤ −(ψ(b)− a)(φ(a) − b). where the former equality holds if and only if the latter equality holds.