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Inequalities : Cambridge Mathematical Library - LITTLEWOOD HARDY

Inequalities

Cambridge Mathematical Library

Paperback

Published: 25th April 1988
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This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and lucidly both the statement and proof of all the standard inequalities of analysis. The authors were well-known for their powers of exposition and made this subject accessible to a wide audience of mathematicians.

In retrospect one sees that Hardy, Littlewood and P has been one of the most important books in analysis in the last few decades. It had an impact on the trend of research and is still influencing it. In looking through the book now one realises how little one would like to change the existing text. A. Zygmund, Bulletin of the AMS

Introduction
Finite, infinite, and integral inequalitiesp. 1
Notationsp. 2
Positive inequalitiesp. 2
Homogeneous inequalitiesp. 3
The axiomatic basis of algebraic inequalitiesp. 4
Comparable functionsp. 5
Selection of proofsp. 6
Selection of subjectsp. 8
Elementary Mean Values
Ordinary meansp. 12
Weighted meansp. 13
Limiting cases of m[subscript r] (a)p. 14
Cauchy's inequalityp. 16
The theorem of the arithmetic and geometric meansp. 16
Other proofs of the theorem of the meansp. 18
Holder's inequality and its extensionsp. 21
Holder's inequality and its extensions (cont.)p. 24
General properties of the means m[subscript r] (a)p. 26
The sums G[subscript r] (a)p. 28
Minkowski's inequalityp. 30
A companion to Minkowski's inequalityp. 32
Illustrations and applications of the fundamental inequalitiesp. 32
Inductive proofs of the fundamental inequalitiesp. 37
Elementary inequalities connected with Theorem 37p. 39
Elementary proof of Theorem 3p. 42
Tchebychef's inequalityp. 43
Muirhead's theoremp. 44
Proof of Muirhead's theoremp. 46
An alternative theoremp. 49
Further theorems on symmetrical meansp. 49
The elementary symmetric functions of n positive numbersp. 51
A note on definite formsp. 55
A theorem concerning strictly positive formsp. 57
Miscellaneous theorems and examplesp. 60
Mean Values with an Arbitrary Function and the Theory of Convex Functions
Definitionsp. 65
Equivalent meansp. 66
A characteristic property of the means m[subscript r]p. 68
Comparabilityp. 69
Convex functionsp. 70
Continuous convex functionsp. 71
An alternative definitionp. 73
Equality in the fundamental inequalitiesp. 74
Restatements and extensions of Theorem 85p. 75
Twice differentiable convex functionsp. 76
Applications of the properties of twice differentiable convex functionsp. 77
Convex functions of several variablesp. 78
Generalisations of Holder's inequalityp. 81
Some theorems concerning monotonic functionsp. 83
Sums with an arbitrary function: generalisations of Jensen's inequalityp. 84
Generalisations of Minkowski's inequalityp. 85
Comparison of setsp. 88
Further general properties of convex functionsp. 91
Further properties of continuous convex functionsp. 94
Discontinuous convex functionsp. 96
Miscellaneous theorems and examplesp. 97
Various Applications of the Calculus
Introductionp. 102
Applications of the mean value theoremp. 102
Further applications of elementary differential calculusp. 104
Maxima and minima of functions of one variablep. 106
Use of Taylor's seriesp. 107
Applications of the theory of maxima and minima of functions of several variablesp. 108
Comparison of series and integralsp. 110
An inequality of W. H. Youngp. 111
Infinite Series
Introductionp. 114
The means m[subscript r]p. 116
The generalisation of Theorems 3 and 9p. 118
Holder's inequality and its extensionsp. 119
The means m[subscript r] (cont.)p. 121
The sums G[subscript r]p. 122
Minkowski's inequalityp. 123
Tchebychef's inequalityp. 123
A summaryp. 123
Miscellaneous theorems and examplesp. 124
Integrals
Preliminary remarks on Lebesgue integralsp. 126
Remarks on null sets and null functionsp. 128
Further remarks concerning integrationp. 129
Remarks on methods of proofp. 131
Further remarks on method: the inequality of Schwarzp. 132
Definition of the means m[subscript r] (f) when r [not equal] 0p. 134
The geometric mean of a functionp. 136
Further properties of the geometric meanp. 139
Holder's inequality for integralsp. 139
General properties of the means m[subscript r] (f)p. 143
General properties of the means m[subscript r] (f) (cont.)p. 144
Convexity of log m[subscript r superscript r]p. 145
Minkowski's inequality for integralsp. 146
Mean values depending on an arbitrary functionp. 150
The definition of the Stieltjes integralp. 152
Special cases of the Stieltjes integralp. 154
Extensions of earlier theoremsp. 155
The means m[subscript r] (f; [phis])p. 156
Distribution functionsp. 157
Characterisation of mean valuesp. 158
Remarks on the characteristic propertiesp. 160
Completion of the proof of Theorem 215p. 161
Miscellaneous theorems and examplesp. 163
Some Applications of the Calculus of Variations
Some general remarksp. 172
Object of the present chapterp. 174
Example of an inequality corresponding to an unattained extremump. 175
First proof of Theorem 254p. 176
Second proof of Theorem 254p. 178
Further examples illustrative of variational methodsp. 182
Further examples: Wirtinger's inequalityp. 184
An example involving second derivativesp. 187
A simpler problemp. 193
Miscellaneous theorems and examplesp. 193
Some Theorems Concerning Bilinear and Multilinear Forms
Introductionp. 196
An inequality for multilinear forms with positive variables and coefficientsp. 196
A theorem of W. H. Youngp. 198
Generalisations and analoguesp. 200
Applications to Fourier seriesp. 202
The convexity theorem for positive multi-linear formsp. 203
General bilinear formsp. 204
Definition of a bounded bilinear formp. 206
Some properties of bounded forms in [p, q]p. 208
The Faltung of two forms in [p, p']p. 210
Some special theorems on forms in [2, 2]p. 211
Application to Hilbert's formsp. 212
The convexity theorem for bilinear forms with complex variables and coefficientsp. 214
Further properties of a maximal set (x, y)p. 216
Proof of Theorem 295p. 217
Applications of the theorem of M. Rieszp. 219
Applications to Fourier seriesp. 220
Miscellaneous theorems and examplesp. 222
Hilbert's Inequality and Its Analogues and Extensions
Hibert's double series theoremp. 226
A general class of bilinear formsp. 227
The corresponding theorem for integralsp. 229
Extensions of Theorems 318 and 319p. 231
Best possible constants: proof of Theorem 317p. 232
Further remarks on Hilbert's theoremsp. 234
Applications of Hilbert's theoremsp. 236
Hardy's inequalityp. 239
Further integral inequalitiesp. 243
Further theorems concerning seriesp. 246
Deduction of theorems on series from theorems on integralsp. 247
Carleman's inequalityp. 249
Theorems with 0 [ p [ 1p. 250
A theorem with two parameters p and qp. 253
Miscellaneous theorems and examplesp. 254
Rearrangements
Rearrangements of finite sets of variablesp. 260
A theorem concerning the rearrangements of two setsp. 261
A second proof of Theorem 368p. 262
Restatement of Theorem 368p. 264
Theorems concerning the rearrangements of three setsp. 265
Reduction of Theorem 373 to a special casep. 266
Completion of the proofp. 268
Another proof of Theorem 371p. 270
Rearrangements of any number of setsp. 272
A further theorem on the rearrangement of any number of setsp. 274
Applicationsp. 276
The rearrangement of a functionp. 276
On the rearrangement of two functionsp. 278
On the rearrangement of three functionsp. 279
Completion of the proof of Theorem 379p. 281
An alternative proofp. 285
Applicationsp. 288
Another theorem concerning the rearrangement of a function in decreasing orderp. 291
Proof of Theorem 384p. 292
Miscellaneous theorems and examplesp. 295
On strictly positive formsp. 300
Thorin's proof and extension of Theorem 295p. 305
On Hilbert's inequalityp. 308
Bibliographyp. 310
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521358804
ISBN-10: 0521358809
Series: Cambridge Mathematical Library
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 340
Published: 25th April 1988
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 2.2
Weight (kg): 0.5
Edition Number: 2