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Inequalities: Theory of Majorization and Its Applications I Theory of Majorization 1 Introduction 3 A Motivation and Basic Definitions . . . . . . . . . . 3 B Majorization as a Partial Ordering . . . . . . . . . 18 C Order-Preserving Functions . . . . . . . . . . . . . 19 D Various Generalizations of Majorization . . . . . . . 21 2 Doubly Stochastic Matrices 29 A Doubly Stochastic Matrices and Permutation Matrices . . . . . . . . . . . . . . . . . . . . . . . . 29 B Characterization of Majorization Using Doubly StochasticMatrices . . . . . . . . . . . . . . . . . . 32 C Doubly Substochastic Matrices and Weak Majorization . . . . . . . . . . . . . . . . . . . . . . 36 D Doubly Superstochastic Matrices and Weak Majorization . . . . . . . . . . . . . . . . . . . . . . 42 E Orderings on D . . . . . . . . . . . . . . . . . . . . 45 F Proofs of Birkhoff’s Theorem and Refinements . . . 47 G Classes of Doubly Stochastic Matrices . . . . . . . . 52 xvii xviii Contents H More Examples of Doubly Stochastic and Doubly Substochastic Matrices . . . . . . . . . . . . . . . . 61 I Properties of Doubly Stochastic Matrices . . . . . . 67 J Diagonal Equivalence of Nonnegative Matrices . . . 76 3 Schur-Convex Functions 79 A Characterization of Schur-Convex Functions . . . . 80 B Compositions Involving Schur-Convex Functions . . 88 C Some General Classes of Schur-Convex Functions . 91 D Examples I. Sums of Convex Functions . . . . . . . 101 E Examples II. Products of Logarithmically Concave (Convex) Functions . . . . . . . . . . . . . 105 F Examples III. Elementary Symmetric Functions . . 114 G Muirhead’s Theorem . . . . . . . . . . . . . . . . . 120 H Schur-Convex Functions on D and Their Extension to Rn . . . . . . . . . . . . . . . . . . . 132 I Miscellaneous Specific Examples . . . . . . . . . . . 138 J Integral Transformations Preserving Schur-Convexity . . . . . . . . . . . . . . . . . . . . 145 K Physical Interpretations of Inequalities . . . . . . . 153 4 Equivalent Conditions