Power majorization and majorization of sequences (Q1111718)

Let \(x,y\in R^ n_+\) be such that \(x_ 1\geq...\geq x_ n\), \(y_ 1\geq...\geq y_ n\) and \(\sum x_ i=\sum y_ i.\) We say that x is power majorized by y if \(\sum x^ p_ i\leq \sum y^ p_ i\) for all real \(p\not\in [0,1]\) and \(\sum x^ p_ i\geq \sum y^ p_ i\) for \(p\in [0,1]\). Let \(\phi\) : [0,\(\infty)\to R\) be a continuous function. Define \({\bar \phi}(x)=\sum \phi (x_ i).\) The author gives a classification of functions \(\phi\) for which \({\bar \phi}(x)\leq {\bar \phi}(y)\) when x is power majorized by y. He also answers a question posed by A. Clausing in 1984 by showing that there are vectors \(x,y\in R^ n\) of any dimension \(n\geq 4\) for which there is a convex function \(\phi\) such that x is power majorized by y and \({\bar \phi}(x)>{\bar \phi}(y).\)