Inequalities

, by G. H. Hardy , J. E. Littlewood , G. PÃ³lya

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ISBN: 9780521358804 | 0521358809
Cover: Paperback
Copyright: 2/26/1988

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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.

Introduction

Finite, infinite, and integral inequalities

(1)

Notations

(1)

Positive inequalities

(1)

Homogeneous inequalities

(1)

The axiomatic basis of algebraic inequalities

(1)

Comparable functions

(1)

Selection of proofs

(2)

Selection of subjects

(4)

Elementary Mean Values

Ordinary means

(1)

Weighted means

(1)

Limiting cases of Mr (a)

(2)

Cauchy's inequality

(1)

The theorem of the arithmetic and geometric means

(2)

Other proofs of the theorem of the means

(3)

Holder's inequality and its extensions

(3)

Holder's inequality and its extensions (cont.)

(2)

General properties of the means Mr (a)

(2)

The sums Gr (a)

(2)

Minkowski's inequality

(2)

A companion to Minkowski's inequality

(1)

Illustrations and applications of the fundamental inequalities

(5)

Inductive proofs of the fundamental inequalities

(2)

Elementary inequalities connected with Theorem 37

(3)

Elementary proof of Theorem 3

(1)

Tchebychef's inequality

(1)

Muirhead's theorem

(2)

Proof of Muirhead's theorem

(3)

An alternative theorem

(1)

Further theorems on symmetrical means

(2)

The elementary symmetric functions of n positive numbers

(4)

A note on definite forms

(2)

A theorem concerning strictly positive forms

(8)

Miscellaneous theorems and examples

(5)

Mean Values with an Arbitrary Function and the Theory of Convex Functions

Definitions

(1)

Equivalent means

(2)

A characteristic property of the means Mr

(1)

Comparability

(1)

Convex Functions

(1)

Continuous convex functions

(2)

An alternative definition

(1)

Equality in the fundamental inequalities

(1)

Restatements and extensions of theorem 85

(1)

Twice differentiable convex functions

(1)

Applications of the properties of twice differentiable convex functions

(1)

Convex functions of several variables

(3)

Generalisations of Holder's inequality

(2)

Some theorems concerning monotonic functions

(1)

Sums with an arbitrary function: generalisations of Jensen's inequality

(1)

Generalisations of Minkowski's inequality

(3)

Comparison of sets

(3)

Further general properties of convex functions

(3)

Further properties of continuous convex functions

(2)

Discontinuous convex functions

(6)

Miscellaneous theorems and examples

(5)

Various Applications of the calculus

Introduction

102

(1)

Applications of the mean value theorem

102

(2)

Further applications of elementary differential calculus

104

(2)

Maxima and minima of functions of one variable

106

(1)

Use of Taylor's series

107

(1)

Applications of the theory of maxima and minima of functions of several variables

108

(2)

Comparison of series and integrals

110

(1)

An inequality of W. H. Young

111

(3)

Infinite Series

Introduction

114

(2)

The means Mr

116

(2)

The generalisation of Theorems 3 and 9

118

(1)

Holder's inequality and its extensions

119

(2)

The means Mr (cont.)

121

(1)

The sums Gr

122

(1)

Minkowski's inequality

123

(1)

Tchebychef's inequality

123

(1)

A summary

123

(3)

Miscellaneous theorems and examples

124

(2)

Integrals

Preliminary remarks on Lebesgue integrals

126

(2)

Remarks on null sets and null functions

128

(1)

Further remarks concerning integration

129

(2)

Remarks on methods of proof

131

(1)

Further remarks on method: the inequality of Schwarz

132

(2)

Definition of the means Mr (f) when r ν 0

134

(2)

The geometric mean of a function

136

(3)

Further properties of the geometric mean

139

(1)

Holder's inequality for integrals

139

(4)

General properties of the means Mr (f)

143

(1)

General properties of the means Mr (f) (cont.)

144

(1)

Convexity of log Mrr

145

(1)

Minkowski's inequality for integrals

146

(4)

Mean values depending on an arbitrary function

150

(2)

The definition of the Stieltjes integral

152

(2)

Special cases of the Stieltjes integral

154

(1)

Extensions of earlier theorems

155

(1)

The means Mr (f; &thetas;)

156

(1)

Distribution functions

157

(1)

Characterisation of mean values

158

(2)

Remarks on the characteristic properties

160

(1)

Completion of the proof of Theorem

161

(11)

Miscellaneous theorems and examples

163

(9)

Some Applications of the Calculus of Variations

Some general remarks

172

(2)

Object of the present chapter

174

(1)

Example of an inequality corresponding to an unattained extremum

175

(1)

First proof of theorem 254

176

(2)

Second proof of theorem 254

178

(4)

Further examples illustrative of variational methods

182

(2)

Further examples: Wirtinger's inequality

184

(3)

An example involving second derivatives

187

(6)

A simpler problem

193

(3)

Miscellaneous theorems and examples

193

(3)

Some theorems concerning Bilinear and Multilinear forms

Introduction

196

(1)

An inequality for multilinear forms with positive variables and coefficients

196

(2)

A theorem of W. H. Young

198

(2)

Generalisations and analogues

200

(2)

Applications to fourier series

202

(1)

The convexity theorem for positive multilinear forms

203

(1)

General bilinear forms

204

(2)

Definition of a bounded bilinear form

206

(2)

Some properties of bounded forms in [p,q]

208

(2)

The Faltung of two forms in [p, p']

210

(1)

Some special theorems on forms in [2, 2]

211

(1)

Application to Hilbert's forms

212

(2)

The convexity theorem for bilinear forms with complex variables and coefficients

214

(2)

Further properties of a maximal set (x, y)

216

(1)

Proof of theorem 295

217

(2)

Applications of the theorem of M. Riesz

219

(1)

Applications to Fourier series

220

(6)

Miscellaneous theorems and examples

222

(4)

Hilbert's Inequality and its Analogues and Extensions

Hilbert's double series theorem

226

(1)

A general class of bilinear forms

227

(2)

The corresponding theorem for integrals

229

(2)

Extensions of theorems 318 and 319

231

(1)

Best possible constants: proof of theorem 317

232

(2)

Further remarks on Hilbert's theorems

234

(2)

Applications of Hilbert's theorems

236

(3)

Hardy's inequality

239

(4)

Further integral inequalities

243

(3)

Further theorems concerning series

246

(1)

Deduction of theorems on series from theorems on integrals

247

(2)

Carleman's inequality

249

(1)

Theorems with 0 <p<1

250

(3)

A theorem with two parameters p and q

253

(7)

Miscellaneous theorems and examples

254

(6)

Rearrangements

Rearrangements of finite sets of variables

260

(1)

A theorem concerning the rearrangements of two sets

261

(1)

A second proof of Theorem 368

262

(2)

Restatement of theorem 368

264

(1)

Theorems concerning the rearrangements of three sets

265

(1)

Reduction of theorem 373 to a special case

266

(2)

Completion of the proof

268

(2)

Another proof of theorem 371

270

(2)

Rearrangements of any number of sets

272

(2)

A further theorem on the rearrangement of any number of sets

274

(2)

Applications

276

(1)

The rearrangement of a function

276

(2)

On the rearrangement of two functions

278

(1)

On the rearrangement of three functions

279

(2)

Completion of the proof of theorem 379

281

(4)

An alternative proof

285

(3)

Applications

288

(3)

Another theorem concerning the rearrangement of a function in decreasing order

291

(1)

Proof of theorem 384

292

(8)

Miscellaneous theorems and examples

295

(5)

Appendix I. On strictly positive forms

300

(5)

Appendix II. Thorin's proof and extension of theorem 295

305

(3)

Appendix III. On Hilbert's inequality

308

(2)

Bibliography

310

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Inequalities

Summary

Table of Contents

Supplemental Materials