文件名称：Inequalities：theory of majorization and its applications
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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 for Majorization 155 A Characterization by Linear Transformations . . . . 155 B Characterization in Terms of Order-Preserving Functions . . . . . . . . . . . . . . . . . . . . . . . . 156 C A Geometric Characterization . . . . . . . . . . . . 162 D A Characterization Involving Top Wage Earners . . 163 5 Preservation and Generation of Majorization 165 A Operations Preserving Majorization . . . . . . . . . 165 B Generation of Majorization . . . . . . . . . . . . . . 185 C Maximal and Minimal Vectors Under Constraints . 192 D Majorization in Integers . . . . . . . . . . . . . . . 194 E Partitions . . . . . . . . . . . . . . . . . . . . . . . 199 F Linear Transformations That Preserve Majorization 202 6 Rearrangements and Majorization 203 A Majorizations from Additions of Vectors . . . . . . 204 B Majorizations from Functions of Vectors . . . . . . 210 C Weak Majorizations from Rearrangements . . . . . 213 D L-Superadditive Functions—Properties and Examples . . . . . . . . . . . . . . . . . . . . . 217 Contents xix E Inequalities Without Majorization . . . . . . . . . . 225 F A Relative Arrangement Partial Order . . . . . . . 228 II Mathematical Applications 7 Combinatorial Analysis 243 A Some Preliminaries on Graphs, Incidence Matrices, and Networks . . . . . . . . . . . . . . . . 243 B Conjugate Sequences . . . . . . . . . . . . . . . . . 245 C The Theorem of Gale and Ryser . . . . . . . . . . . 249 D Some Applications of the Gale–Ryser Theorem . . . 254 E s-Graphs and a Generalization of the Gale–Ryser Theorem . . . . . . . . . . . . . . . . . 258 F Tournaments . . . . . . . . . . . . . . . . . . . . . . 260 G Edge Coloring in Graphs . . . . . . . . . . . . . . . 265 H Some Graph Theory Settings in Which Majorization Plays a Role . . . . . . . . . . . . . . 267 8 Geometric Inequalities 269 A Inequalities for the Angles of a Triangle . . . . . . . 271 B Inequalities for the Sides of a Triangle . . . . . . . 276 C Inequalities for the Exradii and Altitudes . . . . . . 282 D Inequalities for the Sides, Exradii, and Medians . . 284 E Isoperimetric-Type Inequalities for Plane Figures . 287 F Duality Between Triangle Inequalities and Inequalities Involving Positive Numbers . . . . . . . 294 G Inequalities for Polygons and Simplexes . . . . . . . 295 9 MatrixTheory 297 A Notation and Preliminaries . . . . . . . . . . . . . . 298 B Diagonal Elements and Eigenvalues of a Hermitian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 C Eigenvalues of a Hermitian Matrix and Its Principal Submatrices . . . . . . . . . . . . . . . . . 308 D Diagonal Elements and Singular Values . . . . . . . 313 E Absolute Value of Eigenvalues and Singular Values 317 F Eigenvalues and Singular Values . . . . . . . . . . . 324 G Eigenvalues and Singular Values of A, B, and A + B . . . . . . . . . . . . . . . . . . . . . . . 329 H Eigenvalues and Singular Values of A, B, and AB . 338 I Absolute Values of Eigenvalues and Row Sums . . . 347