It is often really difficult to trace the origin of a familiar inequality. It is quite likely to occur first as an auxiliary proposition, often without explicit statement, in a memoir on geometry or astronomy; it may have been rediscovered, many years later, by half a dozen different authors; and no accessible statement of it may be quite complete. We have almost always found, even with the most famous inequalities, that we have a little new to add. We have done our best to be accurate and have

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
CHAPTERⅡ ELEMENTARY MEAN VALUES
2.1 Ordinary means
2.2 Weighted means
2.3 Limiting cases of 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 extensions cont
2.9 General properties of the means a
2.10 The sums r 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 with Theorem 37
2.16 Elementary proof of Theorem 3
2.17 Tchebyehef's inequality
2.18 Muirhead's theorem
2.19 Proof of Muirhead's theorem
2.20 An alternative theorem
2.21 Further theorems on symmetrical means
2.22 The elementary symmetric functions 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 means
3.3 A characteristic property of the means
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 Applieations of the properties of twice differentiable convex functions
3.12 Convex functions of several variables
3.13 Generalisations of Holder''''s inequality
3.14 Some theorems concerning monotonic functions
3.15 Sums with an arbitrary function: generalisa. tions of Jensen''''s inequality
3.16 Generalisations of Minkowski''''s inequality
3.17 Comparison of sets
3.18 Fur ther general properties of convex functions
3.19 Further properties of continuous convex functions
3.20 Discontinuous convex functions Miscellaneous theorems and examples
……
CHAPTERⅣ VARIOUS APPLICATIONS OF THE CALCULUS
CHAPTERⅤ INFINITE SERIES
CHAPTERⅥ INTEGRALS
CHAPTERⅦ SOME APPLICATIONS OF THE CALCULUS OF VARIATIONS
CHARTERⅧ SOME THEOREMS CONCERNING BILINEAR AND MULTILINEAR FORMS
CHAPTERⅨ HILBERT'S INEQUALITY AND ITS ANALOGUES AND EXTENSIONS
CHAPTERⅩ REARRANGEMENTS
APPENDIXⅠ On strictly positive forms
APPENDIXⅡ Thorin's proof and extension of Theorem 295
APPENDIXⅢ On Hilbert's inequality
BIBLIOGRAPHY