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Inequalities

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ISBN-10: 0521358809

ISBN-13: 9780521358804

Edition: 2nd

Authors: J. E. Littlewood, G. P�lya, G. H. Hardy

List price: $83.99
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Description:

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.
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Book details

List price: $83.99
Edition: 2nd
Publisher: Cambridge University Press
Publication date: 2/25/1988
Binding: Paperback
Pages: 340
Size: 6.25" wide x 9.25" long x 1.25" tall
Weight: 1.254
Language: English

Introduction
Finite, infinite, and integral inequalities
Notations
Positive inequalities
Homogeneous inequalities
The axiomatic basis of algebraic inequalities
Comparable functions
Selection of proofs
Selection of subjects
Elementary Mean Values
Ordinary means
Weighted means
Limiting cases of m[subscript r] (a)
Cauchy's inequality
The theorem of the arithmetic and geometric means
Other proofs of the theorem of the means
Holder's inequality and its extensions
Holder's inequality and its extensions (cont.)
General properties of the means m[subscript r] (a)
The sums G[subscript r] (a)
Minkowski's inequality
A companion to Minkowski's inequality
Illustrations and applications of the fundamental inequalities
Inductive proofs of the fundamental inequalities
Elementary inequalities connected with Theorem 37
Elementary proof of Theorem 3
Tchebychef's inequality
Muirhead's theorem
Proof of Muirhead's theorem
An alternative theorem
Further theorems on symmetrical means
The elementary symmetric functions of n positive numbers
A note on definite forms
A theorem concerning strictly positive forms
Miscellaneous theorems and examples
Mean Values with an Arbitrary Function and the Theory of Convex Functions
Definitions
Equivalent means
A characteristic property of the means m[subscript r]
Comparability
Convex functions
Continuous convex functions
An alternative definition
Equality in the fundamental inequalities
Restatements and extensions of Theorem 85
Twice differentiable convex functions
Applications of the properties of twice differentiable convex functions
Convex functions of several variables
Generalisations of Holder's inequality
Some theorems concerning monotonic functions
Sums with an arbitrary function: generalisations of Jensen's inequality
Generalisations of Minkowski's inequality
Comparison of sets
Further general properties of convex functions
Further properties of continuous convex functions
Discontinuous convex functions
Miscellaneous theorems and examples
Various Applications of the Calculus
Introduction
Applications of the mean value theorem
Further applications of elementary differential calculus
Maxima and minima of functions of one variable
Use of Taylor's series
Applications of the theory of maxima and minima of functions of several variables
Comparison of series and integrals
An inequality of W. H. Young
Infinite Series
Introduction
The means m[subscript r]
The generalisation of Theorems 3 and 9
Holder's inequality and its extensions
The means m[subscript r] (cont.)
The sums G[subscript r]
Minkowski's inequality
Tchebychef's inequality
A summary
Miscellaneous theorems and examples
Integrals
Preliminary remarks on Lebesgue integrals
Remarks on null sets and null functions
Further remarks concerning integration
Remarks on methods of proof
Further remarks on method: the inequality of Schwarz
Definition of the means m[subscript r] (f) when r [not equal] 0
The geometric mean of a function
Further properties of the geometric mean
Holder's inequality for integrals
General properties of the means m[subscript r] (f)
General properties of the means m[subscript r] (f) (cont.)
Convexity of log m[subscript r superscript r]
Minkowski's inequality for integrals
Mean values depending on an arbitrary function
The definition of the Stieltjes integral
Special cases of the Stieltjes integral
Extensions of earlier theorems
The means m[subscript r] (f; [phis])
Distribution functions
Characterisation of mean values
Remarks on the characteristic properties
Completion of the proof of Theorem 215
Miscellaneous theorems and examples
Some Applications of the Calculus of Variations
Some general remarks
Object of the present chapter
Example of an inequality corresponding to an unattained extremum
First proof of Theorem 254
Second proof of Theorem 254
Further examples illustrative of variational methods
Further examples: Wirtinger's inequality
An example involving second derivatives
A simpler problem
Miscellaneous theorems and examples
Some Theorems Concerning Bilinear and Multilinear Forms
Introduction
An inequality for multilinear forms with positive variables and coefficients
A theorem of W. H. Young
Generalisations and analogues
Applications to Fourier series
The convexity theorem for positive multi-linear forms
General bilinear forms
Definition of a bounded bilinear form
Some properties of bounded forms in [p, q]
The Faltung of two forms in [p, p']
Some special theorems on forms in [2, 2]
Application to Hilbert's forms
The convexity theorem for bilinear forms with complex variables and coefficients
Further properties of a maximal set (x, y)
Proof of Theorem 295
Applications of the theorem of M. Riesz
Applications to Fourier series
Miscellaneous theorems and examples
Hilbert's Inequality and Its Analogues and Extensions
Hibert's double series theorem
A general class of bilinear forms
The corresponding theorem for integrals
Extensions of Theorems 318 and 319
Best possible constants: proof of Theorem 317
Further remarks on Hilbert's theorems
Applications of Hilbert's theorems
Hardy's inequality
Further integral inequalities
Further theorems concerning series
Deduction of theorems on series from theorems on integrals
Carleman's inequality
Theorems with 0 [ p [ 1
A theorem with two parameters p and q
Miscellaneous theorems and examples
Rearrangements
Rearrangements of finite sets of variables
A theorem concerning the rearrangements of two sets
A second proof of Theorem 368
Restatement of Theorem 368
Theorems concerning the rearrangements of three sets
Reduction of Theorem 373 to a special case
Completion of the proof
Another proof of Theorem 371
Rearrangements of any number of sets
A further theorem on the rearrangement of any number of sets
Applications
The rearrangement of a function
On the rearrangement of two functions
On the rearrangement of three functions
Completion of the proof of Theorem 379
An alternative proof
Applications
Another theorem concerning the rearrangement of a function in decreasing order
Proof of Theorem 384
Miscellaneous theorems and examples
On strictly positive forms
Thorin's proof and extension of Theorem 295
On Hilbert's inequality
Bibliography