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Introduction | |

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Finite, infinite, and integral inequalities | |

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Notations | |

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Positive inequalities | |

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Homogeneous inequalities | |

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The axiomatic basis of algebraic inequalities | |

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Comparable functions | |

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Selection of proofs | |

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Selection of subjects | |

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Elementary Mean Values | |

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Ordinary means | |

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Weighted means | |

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Limiting cases of m[subscript r] (a) | |

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Cauchy's inequality | |

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The theorem of the arithmetic and geometric means | |

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Other proofs of the theorem of the means | |

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Holder's inequality and its extensions | |

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Holder's inequality and its extensions (cont.) | |

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General properties of the means m[subscript r] (a) | |

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The sums G[subscript r] (a) | |

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Minkowski's inequality | |

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A companion to Minkowski's inequality | |

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Illustrations and applications of the fundamental inequalities | |

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Inductive proofs of the fundamental inequalities | |

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Elementary inequalities connected with Theorem 37 | |

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Elementary proof of Theorem 3 | |

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Tchebychef's inequality | |

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Muirhead's theorem | |

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Proof of Muirhead's theorem | |

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An alternative theorem | |

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Further theorems on symmetrical means | |

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The elementary symmetric functions of n positive numbers | |

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A note on definite forms | |

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A theorem concerning strictly positive forms | |

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Miscellaneous theorems and examples | |

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

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Definitions | |

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Equivalent means | |

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A characteristic property of the means m[subscript r] | |

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Comparability | |

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Convex functions | |

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Continuous convex functions | |

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An alternative definition | |

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Equality in the fundamental inequalities | |

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Restatements and extensions of Theorem 85 | |

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Twice differentiable convex functions | |

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Applications of the properties of twice differentiable convex functions | |

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Convex functions of several variables | |

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Generalisations of Holder's inequality | |

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Some theorems concerning monotonic functions | |

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

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Generalisations of Minkowski's inequality | |

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Comparison of sets | |

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Further general properties of convex functions | |

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Further properties of continuous convex functions | |

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Discontinuous convex functions | |

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Miscellaneous theorems and examples | |

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Various Applications of the Calculus | |

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Introduction | |

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Applications of the mean value theorem | |

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Further applications of elementary differential calculus | |

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Maxima and minima of functions of one variable | |

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Use of Taylor's series | |

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

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Comparison of series and integrals | |

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An inequality of W. H. Young | |

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Infinite Series | |

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Introduction | |

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The means m[subscript r] | |

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The generalisation of Theorems 3 and 9 | |

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Holder's inequality and its extensions | |

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The means m[subscript r] (cont.) | |

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The sums G[subscript r] | |

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Minkowski's inequality | |

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Tchebychef's inequality | |

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A summary | |

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Miscellaneous theorems and examples | |

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Integrals | |

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Preliminary remarks on Lebesgue integrals | |

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Remarks on null sets and null functions | |

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Further remarks concerning integration | |

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Remarks on methods of proof | |

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Further remarks on method: the inequality of Schwarz | |

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Definition of the means m[subscript r] (f) when r [not equal] 0 | |

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The geometric mean of a function | |

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Further properties of the geometric mean | |

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Holder's inequality for integrals | |

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General properties of the means m[subscript r] (f) | |

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General properties of the means m[subscript r] (f) (cont.) | |

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Convexity of log m[subscript r superscript r] | |

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Minkowski's inequality for integrals | |

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Mean values depending on an arbitrary function | |

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The definition of the Stieltjes integral | |

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Special cases of the Stieltjes integral | |

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Extensions of earlier theorems | |

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The means m[subscript r] (f; [phis]) | |

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Distribution functions | |

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Characterisation of mean values | |

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Remarks on the characteristic properties | |

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Completion of the proof of Theorem 215 | |

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Miscellaneous theorems and examples | |

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Some Applications of the Calculus of Variations | |

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Some general remarks | |

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Object of the present chapter | |

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Example of an inequality corresponding to an unattained extremum | |

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First proof of Theorem 254 | |

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Second proof of Theorem 254 | |

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Further examples illustrative of variational methods | |

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Further examples: Wirtinger's inequality | |

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An example involving second derivatives | |

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A simpler problem | |

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Miscellaneous theorems and examples | |

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Some Theorems Concerning Bilinear and Multilinear Forms | |

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Introduction | |

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An inequality for multilinear forms with positive variables and coefficients | |

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A theorem of W. H. Young | |

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Generalisations and analogues | |

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Applications to Fourier series | |

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The convexity theorem for positive multi-linear forms | |

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General bilinear forms | |

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Definition of a bounded bilinear form | |

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Some properties of bounded forms in [p, q] | |

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

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Some special theorems on forms in [2, 2] | |

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Application to Hilbert's forms | |

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The convexity theorem for bilinear forms with complex variables and coefficients | |

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Further properties of a maximal set (x, y) | |

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Proof of Theorem 295 | |

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Applications of the theorem of M. Riesz | |

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Applications to Fourier series | |

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Miscellaneous theorems and examples | |

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Hilbert's Inequality and Its Analogues and Extensions | |

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Hibert's double series theorem | |

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A general class of bilinear forms | |

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The corresponding theorem for integrals | |

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Extensions of Theorems 318 and 319 | |

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Best possible constants: proof of Theorem 317 | |

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Further remarks on Hilbert's theorems | |

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Applications of Hilbert's theorems | |

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Hardy's inequality | |

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Further integral inequalities | |

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Further theorems concerning series | |

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Deduction of theorems on series from theorems on integrals | |

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Carleman's inequality | |

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Theorems with 0 [ p [ 1 | |

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A theorem with two parameters p and q | |

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Miscellaneous theorems and examples | |

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Rearrangements | |

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Rearrangements of finite sets of variables | |

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A theorem concerning the rearrangements of two sets | |

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A second proof of Theorem 368 | |

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Restatement of Theorem 368 | |

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Theorems concerning the rearrangements of three sets | |

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Reduction of Theorem 373 to a special case | |

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Completion of the proof | |

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Another proof of Theorem 371 | |

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Rearrangements of any number of sets | |

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A further theorem on the rearrangement of any number of sets | |

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Applications | |

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The rearrangement of a function | |

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On the rearrangement of two functions | |

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On the rearrangement of three functions | |

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Completion of the proof of Theorem 379 | |

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An alternative proof | |

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Applications | |

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Another theorem concerning the rearrangement of a function in decreasing order | |

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Proof of Theorem 384 | |

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Miscellaneous theorems and examples | |

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On strictly positive forms | |

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Thorin's proof and extension of Theorem 295 | |

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On Hilbert's inequality | |

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Bibliography | |