E. F. Beckenbach R. Bellman. Inequalities. Third Printing. '**" '»»..,,. t. Springer-Verlag Berlin Heidelberg New York 1971

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1 ) E. F. Beckenbach R. Bellman Inequalities Third Printing '**" '»»..,,. t Springer-Verlag Berlin Heidelberg New York 1971 I.!

Chapter 1. The Fundamental Inequalities and Related Matters 1 1. Introduction 1 2. The Cauchy Inequality 2 3. The Lagrange Identity 3 4. The Arithmetic-mean Geometric-mean Inequality 3 5.

2 Chapter 1. The Fundamental Inequalities and Related Matters 1 1. Introduction 1 2. The Cauchy Inequality 2 3. The Lagrange Identity 3 4. The Arithmetic-mean Geometric-mean Inequality 3 5. Induction Forward and Backward 4 6. Calculus and Lagrange Multipliers 5 7. Functional Equations 6 8. Concavity 6 9. Majorization The Proof of BOHR The Proof of HURWITZ A Proof of EHLERS The Arithmetic-geometric Mean of GAUSS; the Elementary Symmetrie Functions A Proof of JACOBSTHAL A Fundamental Relationship YOUNG'S Inequality The Means M, (x, a) and the Sums S, (x) The Inequalities of HOLDER and MINKOWSKI Extensions of the Classical Inequalities Quasi Linearization MINKOWSKIS Inequality Another Inequality of MINKOWSKI MINKOWSKIS Inequality for 0 < p < An Inequality of BECKENBACH An Inequality of DRESHER Minkowski-Mahler Inequality Quasi Linearization of Convex and Concave Functions Another Type of Quasi Linearization An Inequality of KARAMATA The Schur Transformation Proof of the Karamata Result An Inequality of OSTROWSKI Continuous Versions Symmetrie Functions A Further Inequality Some Results of WHITELEY Hyperbolic Polynomials GARDING'S Inequality Examples Lorentz Spaces Converses of Inequalities L»Case Multidimensional Case Generalizations of FAVARD-BERWALD 43

IX 44. Other Converses of the Cauchy Theorem 44 45. Refinements of the Cauchy-Buniakowsky-Schwarz Inequalities... 45 46. A Result of MOHR and Noix 46 47. Generation of New Inequalities from Old 46 48.

3 IX 44. Other Converses of the Cauchy Theorem Refinements of the Cauchy-Buniakowsky-Schwarz Inequalities A Result of MOHR and Noix Generation of New Inequalities from Old Refinement of Arithmetic-mean geometric-mean Inequality Inequalities with Alternating Signs STEFFENSEN'S Inequality BRUNK-OLKIN Inequality Extensions of STEFFENSEN'S Inequality 49 Bibliographical Notes 50 Chapter 2. Positive Definite Matrices, Characteristic Roots, and Positive Matrices Introduction Positive Definite Matrices A Necessary Condition for Positive Definiteness Representation as a Sum of Squares A Necessary and Sufficient Condition for Positive Definiteness Gramians Evaluation of an Infinite Integral Complex Matrices with Positive Definite Real Part A Concavity Theorem An Inequality Concerning Minors HADAMARD'S Inequality Szlsz's Inequality A Representation Theorem for the Determinant of a Hermitian Matrix Discussion Ingham-Siegel Integrals and Generalizations Group Invariance and Representation Formulas BERGSTROM'S Inequality A Generalization Canonical Form A Generalization of BERGSTROM'S Inequality A Representation Theorem for \A\ llu An Inequality of MINKOWSKI A Generalization due to KY FAN A Generalization due to OPPENHEIM The Rayleigh Quotient The Fischer Min-max Theorem A Representation Theorem An Inequality of KY FAN An Additive Version Results Connecting Characteristic Roots of A, AA*, and (A + A*)ß The Cauchy-Poincarö Separation Theorem An Inequality for A^At,-!... A* Discussion AdditiveVersion Multiplicative Inequality Derived from Additive Further Results Compound and Adjugate Matrices Positive Matrices Variational Characterization of p(a) A Modification due to BIRKHOFF and VARGA 82

X Contents 41. Some Consequences 83 42. Input-output Matrices 83 43. Discussion 84 44. Extensions 84 45. Matrices and Hyperbolic Equations 85 46.

4 X Contents 41. Some Consequences Input-output Matrices Discussion Extensions Matrices and Hyperbolic Equations Nonvanishing of Determinants and the Location of Characteristic Values Monotone Matrix Functions in the Sense of LOEWNER Variation-diminishing Transformations Domains of Positivity 87 Bibliographical Notes 88 Chapter 3. Moment Spaces and Eesonance Theorems Introduction Moments Convexity Some Examples of Convex Spaces Examples of Nonconvex Spaces On the Determination of Convex Sets Z»-Space A Result of F. RIESZ Bounded Variation Positivity Representation as Squares Nonnegative Trigonometrie and Rational Polynomials Positive Definite Quadratic Forms and Moment Sequences Historical Note Positive Definite Sequences Positive Definite Functions Reproducing Kernels Nonconvex Spaces A "Resonance" Theorem of LANDAU The Banach-Steinhaus Theorem A Theorem of MINKOWSKI The Theory of Linear Inequalities Generalizations The Min-max Theorem of VON NEUMANN The Neyman-Pearson Lemma Orthogonal Projection Equivalance of Minimization and Maximization Processes 124 Bibliographical Notes 125 Chapter 4. On the Positivity of Operators Introduction First-order Linear Differential Equations Discussion A Fundamental Result in Stability Theory Inequalities of BIHARI-LANGENHOP Matrix Analogues A Proof by TAUSSKV Variable Matrix Discussion A Result of CAPLYGIN Finite Intervals 140

XI 12. Variational Proof 141 13. Discussion 142 14. Linear Differential Equations of Arbitrary Order 142 15. A Positivity Result for Higher-order Linear Differential Operators.. 143 16.

5 XI 12. Variational Proof Discussion Linear Differential Equations of Arbitrary Order A Positivity Result for Higher-order Linear Differential Operators Some Results of P6LYA Generalized Convexity Discussion The Generalized Mean-value Theorem of HARTMAN and WINTNER Generalized Taylor Expansions Positivity of Operators Elliptic Equations Positive Reproducing Kernels Monotonicity of Mean Values Positivity of the Parabolic Operator Finite-difference Schemes ' Potential Equations Discussion The Inequalities of HAAR-WESTPHAL-PRODI Some Inequalities of WENDROFF Results of WEINBERGER-BOCHNER Variation-diminishing Transformations Quasi Linearization Stability of Operators Miscellaneous Results 157 Bibliographical Notes 157 Chapter 5. Inequalities for Differential Operators Introduction Some Inequalities of B. SZ.-NAGY Inequalities Connecting u, u', and u" Inequalities Connecting u, w (l), and w ( "> Alternative Approach for u, u', and u" An Inequality of HALPERIN and VON NEUMANN and Its Extensions Results Analogous to Those of NAGY CARLSON'S Inequality Generalizations of CARLSON'S Inequality WIRTINGER'S Inequality and Related Results Proof Using Fourier Series Sturm-Liouville Theory Integral Identities COLAUTTI'S Results Partial Differential Equations Matrix Version Higher Derivatives and Higher Powers Discrete Versions of FAN, TAUSSKY, and TODD Discrete Case Second Differences Discrete Versions of Northcott-Bellman Inequalities Discussion 184 Bibliographical Notes 185 Name Index 189 Subject Index 195

Matrix Differential Calculus with Applications in Statistics and Econometrics

Matrix Differential Calculus with Applications in Statistics and Econometrics Revised Edition JAN. R. MAGNUS CentERjor Economic Research, Tilburg University and HEINZ NEUDECKER Cesaro, Schagen JOHN WILEY