| Introduction | |
| Finite, infinite, and integral inequalities | p. 1 |
| Notations | p. 2 |
| Positive inequalities | p. 2 |
| Homogeneous inequalities | p. 3 |
| The axiomatic basis of algebraic inequalities | p. 4 |
| Comparable functions | p. 5 |
| Selection of proofs | p. 6 |
| Selection of subjects | p. 8 |
| Elementary Mean Values | |
| Ordinary means | p. 12 |
| Weighted means | p. 13 |
| Limiting cases of m[subscript r] (a) | p. 14 |
| Cauchy's inequality | p. 16 |
| The theorem of the arithmetic and geometric means | p. 16 |
| Other proofs of the theorem of the means | p. 18 |
| Holder's inequality and its extensions | p. 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 inequality | p. 30 |
| A companion to Minkowski's inequality | p. 32 |
| Illustrations and applications of the fundamental inequalities | p. 32 |
| Inductive proofs of the fundamental inequalities | p. 37 |
| Elementary inequalities connected with Theorem 37 | p. 39 |
| Elementary proof of Theorem 3 | p. 42 |
| Tchebychef's inequality | p. 43 |
| Muirhead's theorem | p. 44 |
| Proof of Muirhead's theorem | p. 46 |
| An alternative theorem | p. 49 |
| Further theorems on symmetrical means | p. 49 |
| The elementary symmetric functions of n positive numbers | p. 51 |
| A note on definite forms | p. 55 |
| A theorem concerning strictly positive forms | p. 57 |
| Miscellaneous theorems and examples | p. 60 |
| Mean Values with an Arbitrary Function and the Theory of Convex Functions | |
| Definitions | p. 65 |
| Equivalent means | p. 66 |
| A characteristic property of the means m[subscript r] | p. 68 |
| Comparability | p. 69 |
| Convex functions | p. 70 |
| Continuous convex functions | p. 71 |
| An alternative definition | p. 73 |
| Equality in the fundamental inequalities | p. 74 |
| Restatements and extensions of Theorem 85 | p. 75 |
| Twice differentiable convex functions | p. 76 |
| Applications of the properties of twice differentiable convex functions | p. 77 |
| Convex functions of several variables | p. 78 |
| Generalisations of Holder's inequality | p. 81 |
| Some theorems concerning monotonic functions | p. 83 |
| Sums with an arbitrary function: generalisations of Jensen's inequality | p. 84 |
| Generalisations of Minkowski's inequality | p. 85 |
| Comparison of sets | p. 88 |
| Further general properties of convex functions | p. 91 |
| Further properties of continuous convex functions | p. 94 |
| Discontinuous convex functions | p. 96 |
| Miscellaneous theorems and examples | p. 97 |
| Various Applications of the Calculus | |
| Introduction | p. 102 |
| Applications of the mean value theorem | p. 102 |
| Further applications of elementary differential calculus | p. 104 |
| Maxima and minima of functions of one variable | p. 106 |
| Use of Taylor's series | p. 107 |
| Applications of the theory of maxima and minima of functions of several variables | p. 108 |
| Comparison of series and integrals | p. 110 |
| An inequality of W. H. Young | p. 111 |
| Infinite Series | |
| Introduction | p. 114 |
| The means m[subscript r] | p. 116 |
| The generalisation of Theorems 3 and 9 | p. 118 |
| Holder's inequality and its extensions | p. 119 |
| The means m[subscript r] (cont.) | p. 121 |
| The sums G[subscript r] | p. 122 |
| Minkowski's inequality | p. 123 |
| Tchebychef's inequality | p. 123 |
| A summary | p. 123 |
| Miscellaneous theorems and examples | p. 124 |
| Integrals | |
| Preliminary remarks on Lebesgue integrals | p. 126 |
| Remarks on null sets and null functions | p. 128 |
| Further remarks concerning integration | p. 129 |
| Remarks on methods of proof | p. 131 |
| Further remarks on method: the inequality of Schwarz | p. 132 |
| Definition of the means m[subscript r] (f) when r [not equal] 0 | p. 134 |
| The geometric mean of a function | p. 136 |
| Further properties of the geometric mean | p. 139 |
| Holder's inequality for integrals | p. 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 integrals | p. 146 |
| Mean values depending on an arbitrary function | p. 150 |
| The definition of the Stieltjes integral | p. 152 |
| Special cases of the Stieltjes integral | p. 154 |
| Extensions of earlier theorems | p. 155 |
| The means m[subscript r] (f; [phis]) | p. 156 |
| Distribution functions | p. 157 |
| Characterisation of mean values | p. 158 |
| Remarks on the characteristic properties | p. 160 |
| Completion of the proof of Theorem 215 | p. 161 |
| Miscellaneous theorems and examples | p. 163 |
| Some Applications of the Calculus of Variations | |
| Some general remarks | p. 172 |
| Object of the present chapter | p. 174 |
| Example of an inequality corresponding to an unattained extremum | p. 175 |
| First proof of Theorem 254 | p. 176 |
| Second proof of Theorem 254 | p. 178 |
| Further examples illustrative of variational methods | p. 182 |
| Further examples: Wirtinger's inequality | p. 184 |
| An example involving second derivatives | p. 187 |
| A simpler problem | p. 193 |
| Miscellaneous theorems and examples | p. 193 |
| Some Theorems Concerning Bilinear and Multilinear Forms | |
| Introduction | p. 196 |
| An inequality for multilinear forms with positive variables and coefficients | p. 196 |
| A theorem of W. H. Young | p. 198 |
| Generalisations and analogues | p. 200 |
| Applications to Fourier series | p. 202 |
| The convexity theorem for positive multi-linear forms | p. 203 |
| General bilinear forms | p. 204 |
| Definition of a bounded bilinear form | p. 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 forms | p. 212 |
| The convexity theorem for bilinear forms with complex variables and coefficients | p. 214 |
| Further properties of a maximal set (x, y) | p. 216 |
| Proof of Theorem 295 | p. 217 |
| Applications of the theorem of M. Riesz | p. 219 |
| Applications to Fourier series | p. 220 |
| Miscellaneous theorems and examples | p. 222 |
| Hilbert's Inequality and Its Analogues and Extensions | |
| Hibert's double series theorem | p. 226 |
| A general class of bilinear forms | p. 227 |
| The corresponding theorem for integrals | p. 229 |
| Extensions of Theorems 318 and 319 | p. 231 |
| Best possible constants: proof of Theorem 317 | p. 232 |
| Further remarks on Hilbert's theorems | p. 234 |
| Applications of Hilbert's theorems | p. 236 |
| Hardy's inequality | p. 239 |
| Further integral inequalities | p. 243 |
| Further theorems concerning series | p. 246 |
| Deduction of theorems on series from theorems on integrals | p. 247 |
| Carleman's inequality | p. 249 |
| Theorems with 0 [ p [ 1 | p. 250 |
| A theorem with two parameters p and q | p. 253 |
| Miscellaneous theorems and examples | p. 254 |
| Rearrangements | |
| Rearrangements of finite sets of variables | p. 260 |
| A theorem concerning the rearrangements of two sets | p. 261 |
| A second proof of Theorem 368 | p. 262 |
| Restatement of Theorem 368 | p. 264 |
| Theorems concerning the rearrangements of three sets | p. 265 |
| Reduction of Theorem 373 to a special case | p. 266 |
| Completion of the proof | p. 268 |
| Another proof of Theorem 371 | p. 270 |
| Rearrangements of any number of sets | p. 272 |
| A further theorem on the rearrangement of any number of sets | p. 274 |
| Applications | p. 276 |
| The rearrangement of a function | p. 276 |
| On the rearrangement of two functions | p. 278 |
| On the rearrangement of three functions | p. 279 |
| Completion of the proof of Theorem 379 | p. 281 |
| An alternative proof | p. 285 |
| Applications | p. 288 |
| Another theorem concerning the rearrangement of a function in decreasing order | p. 291 |
| Proof of Theorem 384 | p. 292 |
| Miscellaneous theorems and examples | p. 295 |
| On strictly positive forms | p. 300 |
| Thorin's proof and extension of Theorem 295 | p. 305 |
| On Hilbert's inequality | p. 308 |
| Bibliography | p. 310 |
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