CHAPTER Ⅰ.INTRODUCTION
1.1. Finite, infinite, and integral inequalities
1.2. Notations
1.3. Positive inequalities
1.4. Homogeneous inequalities
1.5. The axiomatic basis of algebraic inequalities
1.6. Comparable functions
1.7. Selection of proofs
1.8. Selection of subjects
CHAPTER Ⅱ.ELEMENTARY MEAN VALUES
2.1. Ordinary means
2.2. Weighted means
2.3. Limiting cases of Μr(a)
2.4. Cauchy's inequality
2.5. The theorem of the arithmetic and geometric means
2.6. Other proofs of the theorem of the means
2.7. Holder's inequality and its extensions
2.8. Holder's inequality and its extensiong (cont.)
2.9. General properties of the means Μr(a)
2.10. The sums □（無此符號）, (a)
2.11. Minkowski's inequality
2.12. A companion to Minkowski's inequality
2.13. Illustrations and applications of the fundamental inequalities
2.14. Inductive proofs of the fundamental inequalities
2.15. Elementary inequalities connected withTheorem 37
2.16. Elementary proof of Theorem 3
2.17. Tchebychef's inequality
2.18. Muirhead's theorem
2.19. Proof of Muirhead's theorem
2.20. An alternative theorem
2.21. Further theorems on aymmetrical means
2.22. The elementary symmetric funotions of n positive numbers
2.23. A note on definite forms
2.24. A theorem concerning strictly positive forms Miscellaneous theorems and examples
CHAPTER Ⅲ.MEAN VALUES WITH AN ARBITRARY FUNCTION AND THE THEORY OF CONVEX FUNCTIONS
3.1. Definitions
3.2. Equivalent meang
8.3. A characteristic property of the means Μr
3.4. Comparability
3.5. Convex functions
3.6. Continuous convex functions
3.7. An alternative definition
3.8. Equality in the fundamental inequalities
3.9. Restatements and extensions of Theorem 85
3.10. Twice differentiable convex functions
3.11. Applications of the properties of twice differentiable convex functions
3.12. Convex functions of several variables
3.13. Generalisations of Hlder's inequality
3.14. Some theorems concerning monotonic functions
3.15. Sums with an arbitrary function: generalisations of Jensen's inequality
3.16. Generalisations of Minkowski's inequality
3.17. Comparison of sets
3.18. Further general properties of convex functions
3.19. Further properties of continuous convex functions
3.20. Discontinuous convex functions
Miscellaneous theorems and examples
CHAPTER Ⅳ.VARIOUS APPLICATIONS OF THE CALCULUS
4.1. Introduotion
4.2. Applications of the mean value theorem
4.3. Further applications of elementary differential caloulus
4.4. Maxima and minima of functions of one variable
4.5. Use of Taylor's series
4.6. Applications of the theory of maxima and minima of functions of several variables
4.7. Comparison of series and integrals
4.8. An inequality of W.H.Young
CHAPTER Ⅴ.INFINITE SERIES
5.1. Introduction
5.2. The means Μr
5.3. The generalisation of Theorems 3 and 9
5.4. Holder's inequality and its extensions
5.5. The means Μr(cont.)
5.6. The sums □（無此符號）
5.7. Minkowski's inequality
5.8. Tchebychef's inequality
5.9. A summary
Miscellaneous theorems and examples
CHAPTER Ⅵ.INTEGRALS
6.1. Preliminary remarks on Lebesgue integrals
6.2. Remarks on null sets and null functions
6.3. Further remarks concerning integration
6.4. Remarks on methods of proof
6.5. Further remarks on method: the inequality of Schwarz
6.6. Definition of the means Μr(f)when r≠0
6.7. The geometric mean of a function
8.8. Further properties of the geometric mean
6.9. Holder's inequality for integrals
6.10. General properties of the means Μr(f)
6.11. General properties of the means Μr(f) (cont.)
6.12. Convexity of log Μrr
6.13. Minkowski's inequality for integrals
6.14. Mean values depending on an arbitrary function
6.15. The definition of the Stieltjes integral
6.16. Special cases of the Stieltjes integral
6.17. Extensions of earlier theorems
6.18. The means Μr(f;?)
6.19. Distribution functions
6.20. Characterisation of mean values
6.21. Remarks on the characteristic properties
6.22. Completion of the proof of Theorem 215
Miscellaneous theorems and examples
GBAPTER Ⅶ.SOME APPLICATIONS OF THE CALCULUS OF VARIATIONS
7.1. Some general remarks
7.2. Object of the present chapter
7.3. Example of an inequality corresponding to an unattained extremum
7.4. First proof of Theorem 254
7.5. Second proof of Theorem 254
7.6. Further examples illustrative of variational methods
7.7. Further examples: Wirtinger's inequality
7.8. An example involving second derivative
7.9. A simpler problem
Miscellaneous theorems and examples
CHAPTER Ⅷ.SOME THEOREMS CONCERNING BILINEAR AND MULTILINEAR FORMS
8.1. Introduction
8.2. An inequality for multilinear forms with positive variables and coefficients
8.3. A theorem of W.H.Young
8.4. Generalisations and analogues
8.5. Anplications to Fourier series
8.6. The convexity theorem for positive multilinear forms
8.7. General bilinear forms
8.8. Definition of a bounded bilinear form
8.9. Some properties of bounded formg in [p, q]
8.10. The Faltung of two forms in [p, p?]
8.11. Some special theorems on forms in [2, 2]
8.12. Application to Hilbert's formg
8.13. The convexity theorem for bilinear formg with complex variables and coefficients
8.14. Further properties of a maximal set (x, y)
8.15. Proof of Theorem 295
8.16. Applioations of the theorem of M.Riesz
8.17. Applications to Fourier series
Miscellaneous theorems and examples
CHAPTER Ⅸ.HILBERT'S INEQUALITY AND ITS ANALOGUES AND EXTENSIONS
9.1. Hilbert's double series theorem
9.2. A general olass of bilinear forms
9.3. The corresponding theorem for integrals
9.4. Extensions of Theorems 318 and 319
9.5. Best possible constants: proof of Theorem
9.6. Further remarks on Hilbert's theorems
9.7. Applications of Hilbert's theorems
9.8. Hardy's inequality
9.9. Further integral inequalities
9.10. Further theorems concerning series
9.11. Deduction of theorems on series from theorems on integrals
9.12. Carleman's inequality
9.13. Theorems with 0
9.14. A theorem with two parameters p and q
Miscellaneous theorems and examples
CHAPTER Ⅹ.REARRANGEMENTS
10.1. Rearrangements of finite sets of variables
10.2. A theorem concerning the rearrangements of two sets
10.3. A second proof of Theorem 368
10.4. Restatement of Theorem 368
10.5. Theorems concerning the rearrangements of three sets
10.6. Reduction of Theorem 373 to a special case
10.7. Completion of the proof
10.8. Another proof of Theorem 371
10.9. Rearangements of any number of sets
10.10. A further theorem on the rearrangement of any number of sets
10.11. Applications
10.12. The rearangement of a function
10.13. On the rearrangement of two functions
10.14. On the rearrangement of three funetions
10.15. Completion of the proof of Theorem 379
10.16. An alternative proof
10.17. Applications
10.18. Another theorem concerning the rearrangement of a function in decreasing order
10.19. Proof of Theorem 384
Miscellaneous theorems and examples
APPENDIX Ⅰ.On strictly positive forms
APPENDIX IⅡ.Thorin's proof and extension of Theorem 295
APPENDIX Ⅲ.On Hilbert's inequality
BIBLIOGRAPHY
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