Majorization versus power majorization (Q912226)

Let \({\mathfrak x}=(x_ 1,...,x_ n)\) and \({\mathfrak y}=(y_ 1,...,y_ n)\) be n-tuples of positive real numbers and recall that \({\mathfrak x}\) is said to be majorized by \({\mathfrak y}\) provided that \((1)\quad \sum^{n}_{i=1}\phi (x_ i)\leq \sum^{n}_{i=1}\phi (y_ i)\) for every convex function \(\phi\) : (0,\(\infty)\to {\mathbb{R}}\). Several useful characterizations of majorization have been given and much of the general theory of inequalities may be derived from these ideas. On the other hand, in view of the central role of \(\ell^ p\)-means in this theory, the following weaker concept is also appealing. We say that \({\mathfrak x}\) is power majorized by \({\mathfrak y}\) provided that (2) \(\sum^{n}_{i=1}x^ p_ i\leq \sum^{n}_{i=1}y^ p_ i\), whenever \(p\in {\mathbb{R}}\) (with reversal of the inequality sign when \(0<p<1).\) It can be shown that (1) and (2) are equivalent for \(n\leq 3\), and it is natural to ask whether this equivalence holds for all positive integers n. The purpose of this note is to give a counterexample. Our construction is based on a non-homogeneous version of a familiar \(\ell^ p\)-space inequality.