This book was planned and begun in 1 929．Our original inten．tion was that it should be ono of the Cambridge Tracts，but it soon became plain that a tract would be much too short for our purpose．

TABLE OF CONTENTS

CHAPTEB Ⅰ．INTRODUCTl0N

 1．1．Finite，infinite，and integral inequalities

 1．2．NotatiOIlS

 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 

CHAPTEB Ⅱ．ELEMENTARY MEAN VALUES

 2．1．Ordinary mealls

 2．2．Weighted means

 2．3．Limiting eases of ■

 2．4．Cauchy’S inequality 

 2．5．The theorem of the arithmetic and geometric means    

 2．6．Other proofs of the theorem of the meails 

 2．7．H61der’S inequality and its extensions 

 2．8．HSlder’S inequality and its extensions(cont)

 2．9．General properties of the means■

 2．10．The sums ■

 2．11．Minkowsld’S inequality ．

 2．12．A companion to Minkowski’S inequality

 2．13．Illustrations and applications of the funda- mental inequalities 

 2．14．Inductive proofs of the fundamental in equalities

 2．15．Elementary inequalities connected with Theorem 37 

 2．16．Elementary proof of Theorem

 2．17．Tchebychef’S inequality 

 2．18．Muirhead’B theorem

 2．19．Proof of Muirhead’8 theorem．

 2．20．An alternative theorem

 2．21．Further theorems on symmetrical means

 2．22．The elementary symmetric functions of positive numbers

 2．23．A note 0n definite forms

 2．24．A theorem concerning strictly positive forms Miscellaneous theorems and examples

CXAPTER Ⅲ．MEAN VALUES WITH AN ARBITRARY FUNCTION AND THE THE RY OFCONVEX FUNCTI0NS

 3．1．Deftnitions

 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 altornative deftn／tion  

 3．8．Equality in the fundamental inequahties

 3．9．Restatements and extensions of Theorem 85

 3．10．Twice differentiable convex functions 

 3．11．Applications of the properties of twice differ- entiable convex functions 

 3．12．Convex functions of several variables 

 3．13．Generalisations of H61der’s inequality

 3．14．Some theorems concerning monotonie func tions   

 3．15．Sums with an arbitrary function：generalisations of Jcnsen,s inequality  

 3．16．Generalisations of Minkowsld’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 Miscellaneotis theorems and examples 

CHAPTER Ⅳ．VARIOUS APPLICATIONS OF THE CALCULUS

 4．1．Introduction

 4．2．Applications of the mean value theorem

 4．3．Further applications of elementary differential ealculus

 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 

 47．Comparison of series and integrals

 48An inequality of W．H．Young

CHAPTER Ⅴ．INFINITE SERIES

 5．1．Intr Oduetion   

 5．2．The means吼 

 5．3．The generalisation of Theorems 3 and 9

 6．4．HSlder’s inequality and its extensions

 5．5．The means ■

 5．6．The sums ■

 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 functio

 6．3．Further remarks concerning ingration

 6．4．Remarks on methods of proof 

 6．5．Further remarks on method：the inequality of Schwarz   

 6．6．Definition of the means ■

 6．7．The geometric inean of a function

 6．8．Further properties of the geometric mean 

 6．9．HSlder’s inequality for integrals 

 6．10．General properties of the means■ 

 6．11．General properties of the means■

 6．12．Convexity of log■’

 6．13．Minkowski’s inequality for integrals 

 6．14．Tean values depending on aD arbitrary function   

 6．15．The definition of the Stieltjes integral

 6．16．Special cases of the Stieltjes integral

 6．17．Extensions of earlior theorems 

 6．18．The means■ 

 6．19．Distrlbution functions  

 6．20．Characterisation of mean values 

 6．21．Remarks 0n the characteristic properties

 6．22．Completion of the proof of Theorem 215 Miscellaneous theorems and examples

CHAPTER Ⅶ．SOME APPLICATIONS OF THE CALCULUS OF VARIATION8

 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 derivatives

 7．9．A simpler problem Miscellaneous theorems and examples 

CHAPTER Ⅶ．SOME THEOREMS CONCERNING BILINEAR AND MULTILINEAR FORMS

 8．1．Introduction

 8．2．An inequaHty for multilinear forms with positive variables and coefficients 

 8．3．A theorem of W．H．Young

 8．4 Generalisations and analogues

 8．5．Applications to Fourier series 

 8．6．Th0 convexity theorem for posilive linear forms

 8．8．DeFmltion of a bounded bilinear form 

 8．9．Some properties of bounded forms in[p，g]

 8．10．The Faltung of two forms in [p，p’]

 8．11．Some special theorems on forms in[2，2]

 8．12．Application t0丑nbert’S forms 

 8．13．ThO eonvexity theorem for bilinear forms with complex variables and coefficients 

 8．14．Further properties of a maximal set(x，Y) 

 8．15．Proof of Theorem 295

 8．16．Applications of the theorem of M．Riesz

 8．17．Applications to Fourier series  Miscellaneous theorems and examples 

CEAPTEB Ⅸ．HILBERT’8 INEQUALITY AND ITS ANALOGUES AND EXTENSl0NS

 9．1．Hilbert’S double series theorem

 9．2．A general class of bihnear forms 

 9．3．The corresponding theorem for integrals

 g．4．Extensions of Theorems 318 and 319 

 9．5．Best possible constants：proof of Theorem 317

 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．Deduetion of theorems on series from theorems on integrals

 9．12．Carleman’S inequality

 9．13．Theorems with 0<P<1 

 9．14．A theorem with two parameters p and g  Miscellaneous theorems and examples

CIIAPTER Ⅹ．REARRANGEMENTS

 lO．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．ReductiOil of Theorem 373 to a special case

 10．7．Completion of the proof 

 10．8．Another proof of Theorem 371  

 10．9．Rearrangements of any number of sets 

 10．10．A further theorem on the rearrangement of any number of sets  

 10．II．Applications

 10．12．The rearrangement of a function 

 10．13．0n the rearrangement of two functions

 10．14．On the rearrangement of three functions 

 10．15．Completion of the proof of Theorem 379 

 10．16．An alternative proof

 10．17．Applications．

 10．18．Another theorem concerning the rearrange． ment of a function in decreasing order

 10．19．Proof of Theorem 384

Miscellaneous theorems and examples 

APPENDIX Ⅰ．On strictly positive forms

APPRNDIX Ⅱ．Thorin’S proof and extension of

Theorem 295

APPENDIX Ⅲ．0n Hilbert’S inequality

BIBLIEQRAPHY