Inequalities

by G. H. Hardy , J. E. Littlewood , G. Pólya

• ISBN13: 9780521358804
• ISBN10: 0521358809
• Edition: 2nd
• Format: Paperback
• Copyright: 1988-02-26
• Publisher: Cambridge University Press

Summary

This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and lucidly both the statement and proof of all the standard inequalities of analysis. The authors were well-known for their powers of exposition and made this subject accessible to a wide audience of mathematicians.

Table of Contents

Introduction
Finite, infinite, and integral inequalities	1	(1)
Notations	2	(1)
Positive inequalities	2	(1)
Homogeneous inequalities	3	(1)
The axiomatic basis of algebraic inequalities	4	(1)
Comparable functions	5	(1)
Selection of proofs	6	(2)
Selection of subjects	8	(4)
Elementary Mean Values
Ordinary means	12	(1)
Weighted means	13	(1)
Limiting cases of Mr (a)	14	(2)
Cauchy's inequality	16	(1)
The theorem of the arithmetic and geometric means	16	(2)
Other proofs of the theorem of the means	18	(3)
Holder's inequality and its extensions	21	(3)
Holder's inequality and its extensions (cont.)	24	(2)
General properties of the means Mr (a)	26	(2)
The sums Gr (a)	28	(2)
Minkowski's inequality	30	(2)
A companion to Minkowski's inequality	32	(1)
Illustrations and applications of the fundamental inequalities	32	(5)
Inductive proofs of the fundamental inequalities	37	(2)
Elementary inequalities connected with Theorem 37	39	(3)
Elementary proof of Theorem 3	42	(1)
Tchebychef's inequality	43	(1)
Muirhead's theorem	44	(2)
Proof of Muirhead's theorem	46	(3)
An alternative theorem	49	(1)
Further theorems on symmetrical means	49	(2)
The elementary symmetric functions of n positive numbers	51	(4)
A note on definite forms	55	(2)
A theorem concerning strictly positive forms	57	(8)
Miscellaneous theorems and examples	60	(5)
Mean Values with an Arbitrary Function and the Theory of Convex Functions
Definitions	65	(1)
Equivalent means	66	(2)
A characteristic property of the means Mr	68	(1)
Comparability	69	(1)
Convex Functions	70	(1)
Continuous convex functions	71	(2)
An alternative definition	73	(1)
Equality in the fundamental inequalities	74	(1)
Restatements and extensions of theorem 85	75	(1)
Twice differentiable convex functions	76	(1)
Applications of the properties of twice differentiable convex functions	77	(1)
Convex functions of several variables	78	(3)
Generalisations of Holder's inequality	81	(2)
Some theorems concerning monotonic functions	83	(1)
Sums with an arbitrary function: generalisations of Jensen's inequality	84	(1)
Generalisations of Minkowski's inequality	85	(3)
Comparison of sets	88	(3)
Further general properties of convex functions	91	(3)
Further properties of continuous convex functions	94	(2)
Discontinuous convex functions	96	(6)
Miscellaneous theorems and examples	97	(5)
Various Applications of the calculus
Introduction	102	(1)
Applications of the mean value theorem	102	(2)
Further applications of elementary differential calculus	104	(2)
Maxima and minima of functions of one variable	106	(1)
Use of Taylor's series	107	(1)
Applications of the theory of maxima and minima of functions of several variables	108	(2)
Comparison of series and integrals	110	(1)
An inequality of W. H. Young	111	(3)
Infinite Series
Introduction	114	(2)
The means Mr	116	(2)
The generalisation of Theorems 3 and 9	118	(1)
Holder's inequality and its extensions	119	(2)
The means Mr (cont.)	121	(1)
The sums Gr	122	(1)
Minkowski's inequality	123	(1)
Tchebychef's inequality	123	(1)
A summary	123	(3)
Miscellaneous theorems and examples	124	(2)
Integrals
Preliminary remarks on Lebesgue integrals	126	(2)
Remarks on null sets and null functions	128	(1)
Further remarks concerning integration	129	(2)
Remarks on methods of proof	131	(1)
Further remarks on method: the inequality of Schwarz	132	(2)
Definition of the means Mr (f) when r ν 0	134	(2)
The geometric mean of a function	136	(3)
Further properties of the geometric mean	139	(1)
Holder's inequality for integrals	139	(4)
General properties of the means Mr (f)	143	(1)
General properties of the means Mr (f) (cont.)	144	(1)
Convexity of log Mrr	145	(1)
Minkowski's inequality for integrals	146	(4)
Mean values depending on an arbitrary function	150	(2)
The definition of the Stieltjes integral	152	(2)
Special cases of the Stieltjes integral	154	(1)
Extensions of earlier theorems	155	(1)
The means Mr (f; &thetas;)	156	(1)
Distribution functions	157	(1)
Characterisation of mean values	158	(2)
Remarks on the characteristic properties	160	(1)
Completion of the proof of Theorem	161	(11)
Miscellaneous theorems and examples	163	(9)
Some Applications of the Calculus of Variations
Some general remarks	172	(2)
Object of the present chapter	174	(1)
Example of an inequality corresponding to an unattained extremum	175	(1)
First proof of theorem 254	176	(2)
Second proof of theorem 254	178	(4)
Further examples illustrative of variational methods	182	(2)
Further examples: Wirtinger's inequality	184	(3)
An example involving second derivatives	187	(6)
A simpler problem	193	(3)
Miscellaneous theorems and examples	193	(3)
Some theorems concerning Bilinear and Multilinear forms
Introduction	196	(1)
An inequality for multilinear forms with positive variables and coefficients	196	(2)
A theorem of W. H. Young	198	(2)
Generalisations and analogues	200	(2)
Applications to fourier series	202	(1)
The convexity theorem for positive multilinear forms	203	(1)
General bilinear forms	204	(2)
Definition of a bounded bilinear form	206	(2)
Some properties of bounded forms in [p,q]	208	(2)
The Faltung of two forms in [p, p']	210	(1)
Some special theorems on forms in [2, 2]	211	(1)
Application to Hilbert's forms	212	(2)
The convexity theorem for bilinear forms with complex variables and coefficients	214	(2)
Further properties of a maximal set (x, y)	216	(1)
Proof of theorem 295	217	(2)
Applications of the theorem of M. Riesz	219	(1)
Applications to Fourier series	220	(6)
Miscellaneous theorems and examples	222	(4)
Hilbert's Inequality and its Analogues and Extensions
Hilbert's double series theorem	226	(1)
A general class of bilinear forms	227	(2)
The corresponding theorem for integrals	229	(2)
Extensions of theorems 318 and 319	231	(1)
Best possible constants: proof of theorem 317	232	(2)
Further remarks on Hilbert's theorems	234	(2)
Applications of Hilbert's theorems	236	(3)
Hardy's inequality	239	(4)
Further integral inequalities	243	(3)
Further theorems concerning series	246	(1)
Deduction of theorems on series from theorems on integrals	247	(2)
Carleman's inequality	249	(1)
Theorems with 0 <p<1	250	(3)
A theorem with two parameters p and q	253	(7)
Miscellaneous theorems and examples	254	(6)
Rearrangements
Rearrangements of finite sets of variables	260	(1)
A theorem concerning the rearrangements of two sets	261	(1)
A second proof of Theorem 368	262	(2)
Restatement of theorem 368	264	(1)
Theorems concerning the rearrangements of three sets	265	(1)
Reduction of theorem 373 to a special case	266	(2)
Completion of the proof	268	(2)
Another proof of theorem 371	270	(2)
Rearrangements of any number of sets	272	(2)
A further theorem on the rearrangement of any number of sets	274	(2)
Applications	276	(1)
The rearrangement of a function	276	(2)
On the rearrangement of two functions	278	(1)
On the rearrangement of three functions	279	(2)
Completion of the proof of theorem 379	281	(4)
An alternative proof	285	(3)
Applications	288	(3)
Another theorem concerning the rearrangement of a function in decreasing order	291	(1)
Proof of theorem 384	292	(8)
Miscellaneous theorems and examples	295	(5)
Appendix I. On strictly positive forms	300	(5)
Appendix II. Thorin's proof and extension of theorem 295	305	(3)
Appendix III. On Hilbert's inequality	308	(2)
Bibliography	310