International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 2, Pages 397-398
doi:10.1155/S0161171295000494

Note on Hölder inequalities

Abstract

In this note, we show that if m,n are positive integers and xij≥0, for i=1,…,n, for j=1,…,m, then (∑i=1nxi1⋯xim)m≤(∑i=1nxi1m)⋯(∑i=1nximm) with equality, in case (x11,⋯,xn1)≠0 if and only if each vector (x1j,⋯,xnj), j=1,⋯,m, is a scalar multiple of (x11,⋯,xn1). The proof is a straight-forward application of Hölder inequalities Conversely, we show that Hölder inequalities can be derived from the above result.