Inequalities

ISBN-13: 9783642649738 / Angielski / Miękka / 2011 / 198 str.

Edwin F. Beckenbach;R. Bellman

Opis

Since the elassie work on inequalities by HARDY, LITTLEWOOD, and P6LYA in 1934, an enonnous amount of effort has been devoted to the sharpening and extension of the elassieal inequalities, to the discovery of new types of inequalities, and to the application of inqualities in many parts of analysis. As examples, let us eite the fields of ordinary and partial differential equations, whieh are dominated by inequalities and variational prineiples involving functions and their derivatives; the many applications of linear inequalities to game theory and mathe- matieal economics, which have triggered a renewed interest in con- vexity and moment-space theory; and the growing uses of digital com- puters, which have given impetus to a systematie study of error esti- mates involving much sophisticated matrix theory and operator theory. The results presented in the following pages reflect to some extent these ramifications of inequalities into contiguous regions of analysis, but to a greater extent our concem is with inequalities in their native habitat. Since it is elearly impossible to give a connected account of the burst of analytic activity of the last twenty-five years centering about inequalities, we have d. eeided to limit our attention to those topies that have particularly delighted and intrigued us, and to the study of whieh we have contributed.

Spis treści

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.