Inequalities

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