Inequalities

1. The Fundamental Inequalities and Related Matters.- § 1. Introduction.- § 2. The Cauchy Inequality.- § 3. The Lagrange Identity.- § 4. The Arithmetic-mean — Geometric-mean Inequality.- § 5. Induction — Forward and Backward.- § 6. Calculus and Lagrange Multipliers.- § 7. Functional Equations.- § 8. Concavity.- § 9. Majorization — The Proof of Bohr.- § 10. The Proof of Hurwitz.- § 11. A Proof of Ehlers.- § 12. The Arithmetic-geometric Mean of Gauss; the Elementary Symmetric Functions.- § 13. A Proof of Jacobsthal.- § 14. A Fundamental Relationship.- § 15. Young’s Inequality.- § 16. The Means Mt (x, ?) and the Sums St(x).- § 17. The Inequalities of Hölder and Minkowski.- § 18. Extensions of the Classical Inequalities.- § 19. Quasi Linearization.- § 20. Minkowski’s Inequality.- § 21. Another Inequality of Minkowski.- § 22. Minkowski’s Inequality for 0 < p < 1.- § 23. An Inequality of Beckenbach.- § 24. An Inequality of Dresher.- § 25. Minkowski-Mahler Inequality.- § 26. Quasi Linearization of Convex and Concave Functions.- § 27. Another Type of Quasi Linearization.- § 28. An Inequality of Karamata.- § 29. The Schur Transformation.- § 30. Proof of the Karamata Result.- § 31. An Inequality of Ostrowski.- § 32. Continuous Versions.- § 33. Symmetric Functions.- § 34. A Further Inequality.- § 35. Some Results of Whiteley.- § 36. Hyperbolic Polynomials.- § 37. Garding’s Inequality.- § 38. Examples.- § 39. Lorentz Spaces.- § 40. Converses of Inequalities.- § 41. Lp Case.- § 42. Multidimensional Case.- § 43. Generalizations of Favard-Berwald.- § 44. Other Converses of the Cauchy Theorem.- § 45. Refinements of the Cauchy-Buniakowsky-Schwarz Inequalities.- § 46. A Result of Mohr and Noll.- § 47. Generation of New Inequalities from Old.- § 48. Refinement of Arithmetic-mean — geometric-mean Inequality.- § 49. Inequalities with Alternating Signs.- § 50. Steffensen’s Inequality.- § 51. Brunk-Olkin Inequality.- § 52. Extensions of Steffensen’s Inequality.- Bibliographical Notes.- 2. Positive Definite Matrices, Characteristic Roots, and Positive Matrices.- § 1. Introduction.- § 2. Positive Definite Matrices.- § 3. A Necessary Condition for Positive Definiteness.- § 4. Representation as a Sum of Squares.- § 5. A Necessary and Sufficient Condition for Positive Definiteness.- § 6. Gramians.- § 7. Evaluation of an Infinite Integral.- § 8. Complex Matrices with Positive Definite Real Part.- § 9. A Concavity Theorem.- § 10. An Inequality Concerning Minors.- § 11. Hadamard’s Inequality.- § 12. Szász’s Inequality.- § 13. A Representation Theorem for the Determinant of a Hermitian Matrix.- § 14. Discussion.- § 15. Ingham-Siegel Integrals and Generalizations.- § 16. Group Invariance and Representation Formulas.- § 17. Bergstrom’s Inequality.- § 18. A Generalization.- § 19. Canonical Form.- § 20. A Generalization of Bergstrom’s Inequality.- § 21. A Representation Theorem for |A|1/n.- § 22. An Inequality of Minkowsei.- § 23. A Generalization due to Ky Fan.- § 24. A Generalization due to Oppenheim.- § 25. The Rayleigh Quotient.- § 26. The Fischer Min-max Theorem.- § 27. A Representation Theorem.- § 28. An Inequality of Ky Fan.- § 29. An Additive Version.- § 30. Results Connecting Characteristic Roots of A, AA*, and (A + A*)/2.- § 31. The Cauchy-Poincaré Separation Theorem.- § 32. An Inequality for ?n?tn-1...?k.- § 33. Discussion.- § 34. Additive Version.- § 35. Multiplicative Inequality Derived from Additive.- § 36. Further Results.- § 37. Compound and Adjugate Matrices.- § 38. Positive Matrices.- § 39. Variational Characterization of p (A).- § 40. A Modification due to Birkhoff and Varga.- § 41. Some Consequences.- § 42. Input-output Matrices.- § 43. Discussion.- § 44. Extensions.- § 45. Matrices and Hyperbolic Equations.- § 46. Nonvanishing of Determinants and the Location of Characteristic Values.- § 47. Monotone Matrix Functions in the Sense of Loewner.- § 48. Variation-diminishing Transformations.- § 49. Domains of Positivity.- Bibliographical Notes.- 3. Moment Spaces and Resonance Theorems.- § 1. Introduction.- § 2. Moments.- § 3. Convexity.- § 4. Some Examples of Convex Spaces.- § 5. Examples of Nonconvex Spaces.- § 6. On the Determination of Convex Sets.- § 7. Lp-Space — A Result of F. Riesz.- § 8. Bounded Variation.- § 9. Positivity.- § 10. Representation as Squares.- § 11. Nonnegative Trigonometric and Rational Polynomials.- § 12. Positive Definite Quadratic Forms and Moment Sequences.- § 13. Historical Note.- § 14. Positive Definite Sequences.- § 15. Positive Definite Functions.- § 16. Reproducing Kernels.- § 17. Nonconvex Spaces.- § 18. A “Resonance” Theorem of Landau.- § 19. The Banach-Steinhaus Theorem.- § 20. A Theorem of Minkowski.- § 21. The Theory of Linear Inequalities.- § 22. Generalizations.- § 23. The Min-max Theorem of vox Neumann.- § 24. The Neyman-Pearson Lemma.- § 25. Orthogonal Projection.- § 26. Equivalance of Minimization and Maximization Processes.- Bibliographical Notes.- 4. On the Positivity of Operators.- § 1. Introduction.- § 2. First-order Linear Differential Equations.- § 3. Discussion.- § 4. A Fundamental Result in Stability Theory.- § 5. Inequalities Of Bihari-Langenhop.- § 6. Matrix Analogues.- § 7. A Proof by Taussky.- § 8. Variable Matrix.- § 9. Discussion.- § 10. A Result of ?plygin.- § 11. Finite Intervals.- § 12. Variational Proof.- § 13. Discussion.- § 14. Linear Differential Equations of Arbitrary Order.- § 15. A Positivity Result for Higher-order Linear Differential Operators.- § 16. Some Results of Pólya.- § 17. Generalized Convexity.- § 18. Discussion.- § 19. The Generalized Mean-value Theorem of Hartman and Wintner.- § 20. Generalized Taylor Expansions.- § 21. Positivity of Operators.- § 22. Elliptic Equations.- § 23. Positive Reproducing Kernels.- § 24. Monotonicity of Mean Values.- § 25. Positivity of the Parabolic Operator.- § 26. Finite-difference Schemes.- § 27. Potential Equations.- § 28. Discussion.- § 29. The Inequalities of Haar-Westphal-Prodi.- § 30. Some Inequalities of Wendroff.- § 31. Results Of Weinberger-Bochner.- § 32. Variation-diminishing Transformations.- § 33. Quasi Linearization.- § 34. Stability of Operators.- § 35. Miscellaneous Results.- Bibliographical Notes.- 5. Inequalities for Differential Operators.- § 1. Introduction.- § 2. Some Inequalities of B. Sz.-Nagy.- § 3. Inequalities Connecting u, u?, and u?.- § 4. Inequalities Connecting u, u(k), and u(n).- § 5. Alternative Approach for u, u?, and u?.- § 6. An Inequality of Halperin and von Neumann and Its Extensions.- § 7. Results Analogous to Those of Nagy.- § 8. Carlson’s Inequality.- § 9. Generalizations of Carlson’s Inequality.- § 10. Wirtinger’s Inequality and Related Results.- § 11. Proof Using Fourier Series.- § 12. Sturm-Liouville Theory.- § 13. Integral Identities.- § 14. Colautti’s Results.- § 15. Partial Differential Equations.- § 16. Matrix Version.- § 17. Higher Derivatives and Higher Powers.- § 18. Discrete Versions of Fan, Taussky, and Todd.- § 19. Discrete Case — Second Differences.- § 20. Discrete Versions of Northcott-Bellman Inequalities.- § 21. Discussion.- Bibliographical Notes.- Name Index.