Inequalities

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.

Chapter I. Introduction
1.1. Finite, infinite, and integral inequalities	p. 1
1.2. Notations	p. 2
1.3. Positive inequalities	p. 2
1.4. Homogeneous inequalities	p. 3
1.5. The axiomatic basis of algebraic inequalities	p. 4
1.6. Comparable functions	p. 5
1.7. Selection of proofs	p. 6
1.8. Selection of subjects	p. 8
Chapter II. Elementary Mean Values
2.1. Ordinary means	p. 12
2.2. Weighted means	p. 13
2.3. Limiting cases of m[subscript r] (a)	p. 14
2.4. Cauchy's inequality	p. 16
2.5. The theorem of the arithmetic and geometric means	p. 16
2.6. Other proofs of the theorem of the means	p. 18
2.7. Holder's inequality and its extensions	p. 21
2.8. Holder's inequality and its extensions (cont.)	p. 24
2.9. General properties of the means m[subscript r] (a)	p. 26
2.10. The sums G[subscript r] (a)	p. 28
2.11. Minkowski's inequality	p. 30
2.12. A companion to Minkowski's inequality	p. 32
2.13. Illustrations and applications of the fundamental inequalities	p. 32
2.14. Inductive proofs of the fundamental inequalities	p. 37
2.15. Elementary inequalities connected with Theorem 37	p. 39
2.16. Elementary proof of Theorem 3	p. 42
2.17. Tchebychef's inequality	p. 43
2.18. Muirhead's theorem	p. 44
2.19. Proof of Muirhead's theorem	p. 46
2.20. An alternative theorem	p. 49
2.21. Further theorems on symmetrical means	p. 49
2.22. The elementary symmetric functions of n positive numbers	p. 51
2.23. A note on definite forms	p. 55
2.24. A theorem concerning strictly positive forms	p. 57
Miscellaneous theorems and examples	p. 60
Chapter III. Mean Values with an Arbitrary Function and the Theory of Convex Functions
3.1. Definitions	p. 65
3.2. Equivalent means	p. 66
3.3. A characteristic property of the means m[subscript r]	p. 68
3.4. Comparability	p. 69
3.5. Convex functions	p. 70
3.6. Continuous convex functions	p. 71
3.7. An alternative definition	p. 73
3.8. Equality in the fundamental inequalities	p. 74
3.9. Restatements and extensions of Theorem 85	p. 75
3.10. Twice differentiable convex functions	p. 76
3.11. Applications of the properties of twice differentiable convex functions	p. 77
3.12. Convex functions of several variables	p. 78
3.13. Generalisations of Holder's inequality	p. 81
3.14. Some theorems concerning monotonic functions	p. 83
3.15. Sums with an arbitrary function: generalisations of Jensen's inequality	p. 84
3.16. Generalisations of Minkowski's inequality	p. 85
3.17. Comparison of sets	p. 88
3.18. Further general properties of convex functions	p. 91
3.19. Further properties of continuous convex functions	p. 94
3.20. Discontinuous convex functions	p. 96
Miscellaneous theorems and examples	p. 97
Chapter IV. Various Applications of the Calculus
4.1. Introduction	p. 102
4.2. Applications of the mean value theorem	p. 102
4.3. Further applications of elementary differential calculus	p. 104
4.4. Maxima and minima of functions of one variable	p. 106
4.5. Use of Taylor's series	p. 107
4.6. Applications of the theory of maxima and minima of functions of several variables	p. 108
4.7. Comparison of series and integrals	p. 110
4.8. An inequality of W. H. Young	p. 111
Chapter V. Infinite Series
5.1. Introduction	p. 114
5.2. The means m[subscript r]	p. 116
5.3. The generalisation of Theorems 3 and 9	p. 118
5.4. Holder's inequality and its extensions	p. 119
5.5. The means m[subscript r] (cont.)	p. 121
5.6. The sums G[subscript r]	p. 122
5.7. Minkowski's inequality	p. 123
5.8. Tchebychef's inequality	p. 123
5.9. A summary	p. 123
Miscellaneous theorems and examples	p. 124
Chapter VI. Integrals
6.1. Preliminary remarks on Lebesgue integrals	p. 126
6.2. Remarks on null sets and null functions	p. 128
6.3. Further remarks concerning integration	p. 129
6.4. Remarks on methods of proof	p. 131
6.5. Further remarks on method: the inequality of Schwarz	p. 132
6.6. Definition of the means m[subscript r] (f) when r [not equal] 0	p. 134
6.7. The geometric mean of a function	p. 136
6.8. Further properties of the geometric mean	p. 139
6.9. Holder's inequality for integrals	p. 139
6.10. General properties of the means m[subscript r] (f)	p. 143
6.11. General properties of the means m[subscript r] (f) (cont.)	p. 144
6.12. Convexity of log m[subscript r superscript r]	p. 145
6.13. Minkowski's inequality for integrals	p. 146
6.14. Mean values depending on an arbitrary function	p. 150
6.15. The definition of the Stieltjes integral	p. 152
6.16. Special cases of the Stieltjes integral	p. 154
6.17. Extensions of earlier theorems	p. 155
6.18. The means m[subscript r] (f; [phis])	p. 156
6.19. Distribution functions	p. 157
6.20. Characterisation of mean values	p. 158
6.21. Remarks on the characteristic properties	p. 160
6.22. Completion of the proof of Theorem 215	p. 161
Miscellaneous theorems and examples	p. 163
Chapter VII. Some Applications of the Calculus of Variations
7.1. Some general remarks	p. 172
7.2. Object of the present chapter	p. 174
7.3. Example of an inequality corresponding to an unattained extremum	p. 175
7.4. First proof of Theorem 254	p. 176
7.5. Second proof of Theorem 254	p. 178
7.6. Further examples illustrative of variational methods	p. 182
7.7. Further examples: Wirtinger's inequality	p. 184
7.8. An example involving second derivatives	p. 187
7.9. A simpler problem	p. 193
Miscellaneous theorems and examples	p. 193
Chapter VIII. Some Theorems Concerning Bilinear and Multilinear Forms
8.1. Introduction	p. 196
8.2. An inequality for multilinear forms with positive variables and coefficients	p. 196
8.3. A theorem of W. H. Young	p. 198
8.4. Generalisations and analogues	p. 200
8.5. Applications to Fourier series	p. 202
8.6. The convexity theorem for positive multi-linear forms	p. 203
8.7. General bilinear forms	p. 204
8.8. Definition of a bounded bilinear form	p. 206
8.9. Some properties of bounded forms in [p, q]	p. 208
8.10. The Faltung of two forms in [p, p']	p. 210
8.11. Some special theorems on forms in [2, 2]	p. 211
8.12. Application to Hilbert's forms	p. 212
8.13. The convexity theorem for bilinear forms with complex variables and coefficients	p. 214
8.14. Further properties of a maximal set (x, y)	p. 216
8.15. Proof of Theorem 295	p. 217
8.16. Applications of the theorem of M. Riesz	p. 219
8.17. Applications to Fourier series	p. 220
Miscellaneous theorems and examples	p. 222
Chapter IX. Hilbert's Inequality and Its Analogues and Extensions
9.1. Hibert's double series theorem	p. 226
9.2. A general class of bilinear forms	p. 227
9.3. The corresponding theorem for integrals	p. 229
9.4. Extensions of Theorems 318 and 319	p. 231
9.5. Best possible constants: proof of Theorem 317	p. 232
9.6. Further remarks on Hilbert's theorems	p. 234
9.7. Applications of Hilbert's theorems	p. 236
9.8. Hardy's inequality	p. 239
9.9. Further integral inequalities	p. 243
9.10. Further theorems concerning series	p. 246
9.11. Deduction of theorems on series from theorems on integrals	p. 247
9.12. Carleman's inequality	p. 249
9.13. Theorems with 0 [ p [ 1	p. 250
9.14. A theorem with two parameters p and q	p. 253
Miscellaneous theorems and examples	p. 254
Chapter X. Rearrangements
10.1. Rearrangements of finite sets of variables	p. 260
10.2. A theorem concerning the rearrangements of two sets	p. 261
10.3. A second proof of Theorem 368	p. 262
10.4. Restatement of Theorem 368	p. 264
10.5. Theorems concerning the rearrangements of three sets	p. 265
10.6. Reduction of Theorem 373 to a special case	p. 266
10.7. Completion of the proof	p. 268
10.8. Another proof of Theorem 371	p. 270
10.9. Rearrangements of any number of sets	p. 272
10.10. A further theorem on the rearrangement of any number of sets	p. 274
10.11. Applications	p. 276
10.12. The rearrangement of a function	p. 276
10.13. On the rearrangement of two functions	p. 278
10.14. On the rearrangement of three functions	p. 279
10.15. Completion of the proof of Theorem 379	p. 281
10.16. An alternative proof	p. 285
10.17. Applications	p. 288
10.18. Another theorem concerning the rearrangement of a function in decreasing order	p. 291
10.19. Proof of Theorem 384	p. 292
Miscellaneous theorems and examples	p. 295
Appendix I. On strictly positive forms	p. 300
Appendix II. Thorin's proof and extension of Theorem 295	p. 305
Appendix III. On Hilbert's inequality	p. 308
Bibliography	p. 310

Mevcut:*

Bound With These Titles

On Order

Özet

Özet

Table of Contents