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