Inequalities

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Cover Title page LIST OF CONTRIBUTORS PREFACE Inequalities and the Principle of Nonsufficient Reason - George Polya 1. The Principle of Nonsufficient Reason 2. Examples 3. Restatement: The Principle of Symmetry 4. Symmetric Functions 5. Symmetric Functionals 6. Examples of Success 7. Examples of Failure 8. A Heuristic Remark on Inequalities as Side Conditions in Extremum Problems 9. Additional Examples References Inequalities in the Differentiai Geometry of Surfaces - E. F. Beckenbach 1. Introduction 2. Convex Functions 3. Geodesic Parallels 4. Geodesic Circles 5. Surfaces of Nonnegative Gaussian Curvature 6. Surfaces of Bounded Gaussian Curvature 7. Subharmonic Functions 8. Harmonic Surfaces 9. Isothermic Parameters and Functions of Class PL 10. Applications of Functions of Class PL References A "Workshop" on Minkowski's Inequality - E. F. Beckenbach 1. Introduction 2. Extension of the Minkowski Inequality 3. Kantorovich Inequalities 4. Other Inequalities References Uncertainty Principles in Fourier Analysis - N. G. De Bruijn 1. Introduction 2. The Musical Score of a Signal 3. The Heisenberg Inequality 4. Inequalities Concerning the Score References Inequalities Complementary to Cauchy's Inequality for Sums of Real Numbers - J. B. Diaz and F. T. Metcalf 1. The Four Known Complementary Inequalities 2. A Stronger Complementary Inequality References Variations of Wirtinger's Inequality - J. B. Diaz and F. T. Metcalf 1. Introduction 2. Variations References Inequalities for the Sum of Two M -Matrices - Ky Fan 1. Introduction 2. Definitions and Notation 3. Lemmas 4. Inequalities for Two M-Matrices Such That One Proportionally Dominates the Other 5. Inequalities for M-Matrices A, B Such That A < B References On Some Inequalities and Their Application to the Cauchy Problem - Avner Friedman 1. Two Inequalities 2. The Cauchy Problem 3. Additional Inequalities 4. The Cauchy Problem for Infinite Systems References General Subadditive Functions- R. P. Gosselin Introduction 1. Generalized Subadditivity 2. A Maximal Theorem 3. A Restriction Theorem References Chebyshevian Spline Functions - Samuel Karlin and Zvi Ziegler Introduction 1. Definitions and Preliminaries 2. The Basic Total Positivity Property and Applications 3. Best-Mean-Square Approximation and Smoothing Properties of Chebyshevian Spline Functions 4. Approximation of Linear Functionals References Some New Inequalities and Unsolved Problems - J. E. Littlewood Text References Lengths of Tensors - Marvin Marcus 1. Introduction 2. Symmetry Classes 3. Further Inequalities References Monotonicity of Ratios of Means and Other Applications of Majorization - Albert W. Marshall, lngram Olkin, and Frank Proschan 1. Introduction and Summary 2. Majorization 3. An Extension Using Total Positivity 4. Monotonicity of Ratios of Means 5. Statistical Applications 6. Condition Numbers 7. Inequalities for Absolute Deviations Appendix References Ratios of Means and Applications - B. Mond and O. Shisha Text References Algebraic Inequalities - T. S. Motzkin 1. Introduction 2. Problems 3. Simplest Paradigms 4. Characteristic Features 5. Solvability 6. Aigebraic Sets References The Arithmetic-Geometric Inequality - T. S. Motzkin 1. General Setting 2. Special Features 3. Power of the Average 4. Average of the Power 5. Additional Variable 6. Trinomial Inequality 7. Commutator of Power and Average References Signs of Minors - T. S. Motzkin 1. Introduction 2. The Minor Hypersurface Dissection 3. Restricted Elements 4. Diagonal Elements and Minors 5. Symmetric and Skew-Symmetric Matrices References A Priori Bounds in Problems of Steady Subsonic Flow - L. E. Payne 1. Introduction 2. Definitions and Notation 3. An Auxiliary Inequality 4. Bounds for the Solution of (2.2) 5. The Dirichlet Problem References On Spline Functions - I. J. Schoenberg Introduction I. The Role of Spline Functions in Elementary Numerical Analysis 1. Peano's Theorem 2. Spline Funétions 3. Spline Interpolation 4. The Best Approximation of Linear Functionals 5. The Newton-Cotes Procedure 6. Sard's Procedure 7. The Role of Spline Interpolation II. Spline Functions and Variation Diminishing Approximation Methods 8. The Bernstein Polynomials 9. The Variation Diminishing Properties of Spline Functions 10. A General Variation Diminishing Approximation Formula 11. The Entire Real Line with all Integers as Knots 12. The Finite Interval with Finitely Many Knots 13. Special Cases 14. The Half-Line with All Positive Integers as Knots 15. The Finite Interval with Equidistant Knots 16. Error Estimates Supplement: On the Normalization of the B-Splines and the Location of the Nodes for the Case of Unequally Spaced Knots, by T. N. E. Greville References Bounds on Differences of Means - O. Shisha and B. Mond Text References Positive-Definite Matrices - Olga Taussky Introduction 1. Pencils of Real Symmetric Matrices 2. Lyapunov's Theorem Selected References Inequalities of Chebyshev, Zolotareff, Cauer, and W. B. Jordan - John Todd Text References On the New Maximum-Minimum Theory of Eigenvalues - Alexander Weinstein 1. Introduction 2. The New Approach 3. The Main Theorem 4. Stenger's Counterexample 5. An Application 6. The Connection with the General Theory of Intermediate Problems References Generalizations of Ostrowski's Inequality for Matrices with Dominant Principal Diagonal - Y. K. Wong Introduction 1. Nonnegative Matrices 2. Complex Matrices Appendix References INDEX