Abstract
The Minkowski and Hölder inequalities play an important role in many areas of pure and applied mathematics, such as Convex Analysis, Probabilities, Control Theory, Fixed Point theorems, and Mathematical Economics. Also, non-additive measures, non-additive integrals and set-valued integrals are useful tools in several areas of theoretical and applied mathematics. In this paper we present and prove some Hölder and Minkowski inequality (or reverse inequality) types obtained for Birkhoff weak integrable functions with respect to a non-additive measure. Then, we apply these results to the interval-valued case.