Majorization in IR_n

Abstract

As mentioned in Chapter 1, the name majorization appears first in HLP (1959). The idea had appeared earlier (HLP, 1929) although unchristened. Muirhead who dealt with Z ⁺_n (i.e. vectors of non-negative integers) already had identified the partial order defined in (1.1) (i.e. majorization). But he, when he needed to refer to it, merely called it “ordering.” Perhaps it took the insight of HLP to recognize that little of Muirhead’s work need necessarily be restricted to integers, but the key ideas including Dalton’s transfer principle were already present in Muirhead’s paper. If there was anything lacking in Muirhead’s development, it was motivation for the novel results he obtained. He did exhibit the arithmetic-geometric mean inequality as an example of his general results, but proofs of that inequality are legion. If that was the only use of his “inequalities of symmetric algebraic functions of n letters”, then they might well remain buried in the Edinburgh proceedings. HLP effectively rescued Muirhead’s work from such potential obscurity. In the present book theorems will be stated in generality comparable to that achieved by HLP and will be ascribed to those authors. Muirhead’s priority will not be repeatedly asserted. HLP restricted attention to IR ⁺_n , but the restriction to the positive orthant can and will be often dispensed with. First let us establish the relationship between majorization as defined by HLP (i.e., (1.1)) and averaging as defined by Schur.