Inequalities / by G.H. Hardy, J.E. Littlewood, G. Pólya.

Format:

Book

Author/Creator:

Hardy, G. H. (Godfrey Harold), 1877-1947.

Contributor:

Littlewood, J. E.

Pólya, George, 1887-1985

Series:

Cambridge Mathematical Library

Language:

English

Subjects (All):

Calculus.

Physical Description:

1 online resource (340 p.)

Edition:

Second edition.

Place of Publication:

Cambridge [England] : University Press, 1952.

Summary:

This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and exhaustively both the statement and proof of all the standard inequalities of analysis. The authors were well known for their powers of exposition and were able here to make the subject accessible to a wide audience of mathematicians.

Contents:

Cover; Half-title; Title; Copyright; Preface to first edition; Preface to second edition; Contents; CHAPTER I. INTRODUCTION; 1.1. Finite, infinite, and integral inequalities; 1.2. Notations; 1.3. Positive inequalities; 1.4. Homogeneous inequalities; 1.5. The axiomatic basis of algebraic inequalities; 1.6. Comparable functions; 1.7. Selection of proofs; CHAPTER II. ELEMENTARY MEAN VALUES; 2.1. Ordinary means; 2.2. Weighted means; 2.3. Limiting cases of M[sub(r)](a); 2.4. Cauchy's inequality; 2.5. The theorem of the arithmetic and geometric means; 2.6. Other proofs of the theorem of the means

2.7. Hölder's inequality and its extensions2.8. Hölder's inequality and its extensions(continued); 2.9. General properties of the means M[sub(r)](a); 2.10. The sums C[sub(r)](a); 2.11. Minkowski's inequality; 2.12. A companion to Minkowski's inequality; 2.13. Illustrations and applications of the fundamental inequalities; 2.14. Inductive proofs of the fundamental inequalities; 2.15. Elementary inequalities connected with Theorem 37; 2.16. Elementary proof of Theorem 3; 2.17. Tchebychef's inequality; 2.18. Muirhead's theorem; 2.19. Proof of Muirhead's theorem; 2.20. An alternative theorem

2.21. Further theorems on symmetrical means2.22. The elementary symmetric functions of n positive numbers; 2.23. A note on definite forms; 2.24. A theorem concerning strictly positive forms; Miscellaneous theorems and examples; CHAPTER III. MEAN VALUES WITH AN ARBITRARY FUNCTION AND THE THEORY OF CONVEX FUNCTIONS; 3.1. Definitions; 3.2. Equivalent means; 3.3. A characteristic property of the means M[sub(r)]; 3.4. Comparability; 3.5. Convex functions; 3.6. Continuous convex functions; 3.7. An alternative definition; 3.8. Equality in the fundamental inequalities

3.9. Restatements and extensions of Theorem 853.10. Twice differentiate convex functions; 3.11. Applications of the properties of twice differentiable convex functions; 3.12. Convex functions of several variables; 3.13. Generalisations of Hölder's inequality; 3.14. Some theorems concerning monotonic functions; 3.15. Sums with an arbitrary function: generalisations of Jensen's inequality; 3.16. Generalisations of Minkowski's inequality; 3.17. Comparison of sets; 3.18. Further general properties of convex functions; 3.19. Further properties of continuous convex functions

3.20. Discontinuous convex functionsMiscellaneous theorems and examples; CHAPTER IV. VARIOUS APPLICATIONS OF THE CALCULUS; 4.1. Introduction; 4.2. Applications of the mean value theorem; 4.3. Further applications of elementary differential calculus; 4.4. Maxima and minima of functions of one variable; 4.5. Use of Taylor's series; 4.6. Applications of the theory of maxima and minima of functions of several variables; 4.7. Comparison of series and integrals; 4.8. An inequality of Young; CHAPTER V. INFINITE SERIES; 5.1. Introduction; 5.2. The means M[sub(r)]

5.3. The generalisation of Theorems 3 and 9

Notes:

Description based upon print version of record.

Bibliography: pages [310]-324.

Description based on print version record.

ISBN:

1-107-70147-3

1-107-68689-X

1-107-59776-5

1-107-70347-6

OCLC:

863203787