Hölder's inequality and its reverse—A probabilistic point of view

Abstract In this article, we take a probabilistic look at Hölder's inequality, considering the ratio of terms in classical Hölder inequality for random vectors . We prove central limit theorem ratio, which then allows us to reverse up multiplicative constant with high probability. The models randomness include uniform distribution on balls and spheres. also provide Berry–Esseen–type result large moderate deviation principle suitably normalized ratio.