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 □（无此符号）, (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 □（无此符号）
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
