Inequalities for sums of multipowers
The author offers generalizations of Hölder's inequality in the form \[ \prod^{m}_{j=0}(\sum^{n}_{i=1}x_{(i)}^{a_{(j)}}/n)^{\gamma _ j}\geq 1, \] where \(\gamma_ j\in {\mathbb{R}}\); \(a_{(j)}\in {\mathbb{R}}^ k\), \(x_{(i)}\in {\mathbb{R}}^ k_+\); \(i=1,...,n\); \(j=0,1,...,m\), and \[ x^ a=(x_ 1,...,x_ k)^{(a_ 1,...,a_ k)}:=\prod^{k}_{\ell =1}x_{\ell}^{a_{\ell}}. \] Note: \(x_ n\) in the line 4 of p. 165 should be \(x_ k\), \({\mathbb{R}}_+\) at the end of line 12 of the same page should be \({\mathbb{R}}^ k_+\). More importantly, the condition, under which the above inequality holds, follows considerably later, in the Corollary on p. 168, later quoted as ``Corollary 1'': \(\sum^{m}_{j=0}\gamma_ jf(a_{(j)})\geq 0\) for all convex f with \(f(0)=0\). If \(\sum^{m}_{j=0}\gamma_ j=0\) and \(k=1\) then this strong condition is weakened to \(\sum^{n}_{j=0}\gamma_ j| a_ j-a_ r| \geq 0\quad (r=0,...,n).\) Several known inequalities are sketchily deduced from this result. The nice idea would deserve a more careful exposition.
scientific article; zbMATH DE number 4192051