Note on Hölder inequalities (Q1805142)

Summary: We show that if \(m\), \(n\) are positive integers and \(x_{ij}\geq 0\), for \(i= 1,\dots, n\), and \(j= 1,\dots, m\), then \[ \left(\sum^ n_{i= 1} x_{i1}\cdots x_{im}\right)^ m\leq \left(\sum^ n_{i= 1} x_{i1}^ m\right)\cdots \left(\sum^ n_{i= 1} x_{im}^ m\right) \] with equality in case \((x_{11},\dots, x_{n1})\neq 0\) if and only if each vector \((x_{1j},\dots, x_{nj})\), \(j= 1,\dots, m\), is a scalar multiple of \((x_{11},\dots, x_{n1})\). The proof is a straight- forward application of Hölder inequalities. Conversely, we show that Hölder inequalities can be derived from the above result.