Inequalities

Cover......Page 1
Cambridge University Press, 1934......Page 2
Preface......Page 6
Contents......Page 8
1.1. Finite, infinite, and integral inequalities......Page 14
1.3. Positive inequalities......Page 15
1.4. Homogeneous inequalities......Page 16
1.5. The axiomatic basis of algebraic inequalities......Page 17
1.6. Comparable functions......Page 18
1.7. Selection of proofs......Page 19
1.8. Selection of subjects......Page 21
2.1. Ordinary means......Page 25
2.2. Weighted means......Page 26
2.3. Limiting cases of R_r(a)......Page 27
2.5. The theorem of the arithmetic and geometric means......Page 29
2.6. Other proofs of the theorem of the means......Page 31
2.7. Holder's inequality and its extensions......Page 34
2.8. Holder's inequality and its extensions (continued)......Page 37
2.9. General properties of the means R_r(a)......Page 39
2.10. The sums S_r(a)......Page 41
2.11. Minkowski's inequality......Page 43
2.13. Illustrations and applications of the fundamental inequalities......Page 45
2.14. Inductive proofs of the fundamental inequalities......Page 50
2.15. Elementary inequalities connected with Theorem 37......Page 52
2.16. Elementary proof of Theorem 3......Page 55
2.17. Tchebychef's inequality......Page 56
2.18. Muirhead's theorem......Page 57
2.19. Proof of Muirhead's theorem......Page 59
2.21. Further theorems on symmetrical means......Page 62
2.22. The elementary symmetric functions of n positive numbers......Page 64
2.23. A note on definite forms......Page 68
2.24. A theorem concerning strictly positive forms......Page 70
Miscellaneous theorems and examples......Page 73
3.1. Definitions......Page 78
3.2. Equivalent means......Page 79
3.3. A characteristic property of the means R_r......Page 81
3.4. Comparability......Page 82
3.5. Convex functions......Page 83
3.6. Continuous convex functions......Page 84
3.7. An alternative definition......Page 86
3.8. Equality in the fundamental inequalities......Page 87
3.9. Restatements and extensions of Theorem 85......Page 88
3.10. Twice differentiable convex functions......Page 89
3.11. Applications of the properties of twice differentiable convex functions......Page 90
3.12. Convex functions of several variables......Page 91
3.13. Generalisations of Holder's inequality......Page 94
3.14. Some theorems concerning monotonic functions......Page 96
3.15. Sums with an arbitrary function: generalisationsof Jensen's inequality......Page 97
3.16. Generalisations of Minkowski's inequality......Page 98
3.17. Comparison of sets......Page 101
3.18. Further general properties of convex functions......Page 104
3.19. Further properties of continuous convex functions......Page 107
3.20. Discontinuous convex functions......Page 109
Miscellaneous theorems and examples......Page 110
4.2. Applications of the mean value theorem......Page 115
4.3. Further applications of elementary differentialcalculus......Page 117
4.4. Maxima and minima of functions of one variable......Page 119
4.5. Use of Taylor's series......Page 120
4.6. Applicationsofthe theory of maxima and minima offunctions of several variables......Page 121
4.7. Comparison of series and integrals......Page 123
4.8. An inequality of Young......Page 124
5.1. Introduction......Page 127
5.2. The means R_r......Page 129
5.3. The generalisation of Theorems 3 and 9......Page 131
5.4. Holder's inequality and its extensions......Page 132
5.5. The means R_r (continued)......Page 134
5.6. The sums S_r......Page 135
5.9. A summary......Page 136
Miscellaneous theorems and examples......Page 137
6.1. Preliminary remarks on Lebesgue integrals......Page 139
6.2. Remarks on nul sets and nul functions......Page 141
6.3. Further remarks concerning integration......Page 142
6.4. Remarks on methods of proo......Page 144
6.5. Further remarks on method: the inequality ofSchwarz......Page 145
6.6. Definition of the means R_r(f) when r \ne 0......Page 147
6.7. The geometric mean of a function......Page 149
6.9. Holder's inequality for integrals......Page 152
6.10. General properties of the means R_r(f)......Page 156
6.11. General properties of the means R_r(f) (continued)......Page 157
6.12. Convexity of log R_r^r......Page 158
6.13. Minkowski's inequality for integrals......Page 159
6.14. Mean values depending on an arbitrary function......Page 163
6.15. The definition of the Stieltjes integral......Page 165
6.16. Special cases of the Stieltjes integral......Page 167
6.17. Extensions of earlier theorems......Page 168
6.18. The means R_r(f;phi)......Page 169
6.19. Distribution functions......Page 170
6.20. Characterisation of mean values......Page 171
6.21. Remarks on the characteristic properties......Page 173
6.22. Completion of the proof of Theorem 215......Page 174
Miscellaneous theorems and examples......Page 176
7.1. Some general remarks......Page 185
7.2. Object of the present chapter......Page 187
7.3. Example of an inequality corresponding to anunattained extremum......Page 188
7.4. First proof of Theorem 254......Page 189
7.5. Second proof of Theorem 254......Page 191
7.6. Further examples illustrative of variational methods......Page 195
7.7. Further examples: Wirtinger's inequality......Page 197
7.8. An example involving second derivatives......Page 200
Miscellaneous theorems and examples......Page 206
8.2. An inequality for multilinear forms with positive variables and coefficients......Page 209
8.3. A theorem of W. H. Young......Page 211
8.4. Generalisations and analogues......Page 213
8.5. Applications to Fourier series......Page 215
8.6. The convexity theorem for positive multilinear forms......Page 216
8.7. General bilinear forms......Page 217
8.8. Definition of a bounded bilinear form......Page 219
8.9. Some properties of bounded forms in [p, q]......Page 221
8.10. The Faltung of two forms in [p, p']......Page 223
8.11. Some special theorems on forms in [2, 2]......Page 224
8.12. Application to Hilbert's forms......Page 225
8.13. The convexity theorem for bilinear forms with complex variables and coefficients......Page 227
8.14. Further properties of a maximal set (x, y)......Page 229
8.15. Proof of Theorem 295......Page 230
8.16. Applications of the theorem of M. Riesz......Page 232
8.17. Applications to Fourier series......Page 233
Miscellaneous theorems and examples......Page 235
9.1. Hilbert's double series theorem......Page 239
9.2. A general class of bilinear forms......Page 240
9.3. The corresponding theorem for integrals......Page 242
9.4. Extensions of Theorems 318 and 319......Page 244
9.5. Best possible constants: proof of Theorem 317......Page 245
9.6. Further remarks on Hilbert's theorems......Page 247
9.7. Applications of Hilbert's theorems......Page 249
9.8. Hardy's inequality......Page 252
9.9. Further integral inequalities......Page 256
9.10. Further theorems concerning series......Page 259
9.11. Deduction of theorems on series from theorems on integrals......Page 260
9.12. Carleman's inequality......Page 262
9.13. Theorems with 0 
9.14. A theorem with two parameters p and q......Page 266
Miscellaneous theorems and examples......Page 267
10.1. Rearrangements of finite sets of variables......Page 273
10.2. A theorem concerning the rearrangements of two sets......Page 274
10.3. A second proof of Theorem 368......Page 275
10.4. Restatement of Theorem 368......Page 277
10.5. Theorems concerning the rearrangements of three sets......Page 278
10.6. Reduction of Theorem 373 to a special case......Page 279
10.7. Completion of the proof......Page 281
10.8. Another proof of Theorem 371......Page 283
10.9. Rearrangements of any number of sets......Page 285
10.10. A further theorem on the rearrangement of any number of sets......Page 287
10.12. The rearrangement of a function......Page 289
10.13. On the rearrangement of two functions......Page 291
10.14. On the rearrangement of three functions......Page 292
10.15. Completion of the proof of Theorem 379......Page 294
10.16. An alternative proof......Page 298
10.17. Applications......Page 301
10.18. Another theorem concerning the rearrangementof a function in decreasing order......Page 304
10.19. Proof of Theorem 384......Page 305
Miscellaneous theorems and examples......Page 308
Bibliography......Page 313