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Tables of Contents for Inequalities
Chapter/Section Title
Page #
Page Count
Introduction
Finite, infinite, and integral inequalities
1
1
Notations
2
1
Positive inequalities
2
1
Homogeneous inequalities
3
1
The axiomatic basis of algebraic inequalities
4
1
Comparable functions
5
1
Selection of proofs
6
2
Selection of subjects
8
4
Elementary Mean Values
Ordinary means
12
1
Weighted means
13
1
Limiting cases of Mr (a)
14
2
Cauchy's inequality
16
1
The theorem of the arithmetic and geometric means
16
2
Other proofs of the theorem of the means
18
3
Holder's inequality and its extensions
21
3
Holder's inequality and its extensions (cont.)
24
2
General properties of the means Mr (a)
26
2
The sums Gr (a)
28
2
Minkowski's inequality
30
2
A companion to Minkowski's inequality
32
1
Illustrations and applications of the fundamental inequalities
32
5
Inductive proofs of the fundamental inequalities
37
2
Elementary inequalities connected with Theorem 37
39
3
Elementary proof of Theorem 3
42
1
Tchebychef's inequality
43
1
Muirhead's theorem
44
2
Proof of Muirhead's theorem
46
3
An alternative theorem
49
1
Further theorems on symmetrical means
49
2
The elementary symmetric functions of n positive numbers
51
4
A note on definite forms
55
2
A theorem concerning strictly positive forms
57
8
Miscellaneous theorems and examples
60
5
Mean Values with an Arbitrary Function and the Theory of Convex Functions
Definitions
65
1
Equivalent means
66
2
A characteristic property of the means Mr
68
1
Comparability
69
1
Convex Functions
70
1
Continuous convex functions
71
2
An alternative definition
73
1
Equality in the fundamental inequalities
74
1
Restatements and extensions of theorem 85
75
1
Twice differentiable convex functions
76
1
Applications of the properties of twice differentiable convex functions
77
1
Convex functions of several variables
78
3
Generalisations of Holder's inequality
81
2
Some theorems concerning monotonic functions
83
1
Sums with an arbitrary function: generalisations of Jensen's inequality
84
1
Generalisations of Minkowski's inequality
85
3
Comparison of sets
88
3
Further general properties of convex functions
91
3
Further properties of continuous convex functions
94
2
Discontinuous convex functions
96
6
Miscellaneous theorems and examples
97
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