Read e-book online Inequalities PDF

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

ISBN-10: 0521358809

ISBN-13: 9780521358804

This vintage of the mathematical literature varieties a finished research of the inequalities used all through arithmetic. First released in 1934, it offers in actual fact and lucidly either the assertion and facts of the entire normal inequalities of study. The authors have been recognized for his or her powers of exposition and made this topic available to a large viewers of mathematicians.

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Example text

W. We do not go into details because any systematic discussion of inequalities connected with determinants and quadratic forms would carry us beyond the limits which we have imposed on the book. 5. The theorem of the arithmetic and geometric means. We come now to the most famous theorem of the subject. a This is what is usually called Cauchy's inequality: see Cauchy (1,373). The corresponding inequality for integrals (Theorem 181) is usually called Schwarz's inequality, though it seems to have been stated first by Buniakowsky: see Buniakowsky (1, 4), Schwarz (2, 251).

For a, b, .... Theorem 24 may be given a very elegant symmetrical form c . 26. Suppose that m(l-'l denotes a mean taken with respect to the suffix fL, with weights PI-" and m("l one taken with respect to v with weights q,,;d and that O

A= I, then SJRr(ab ... l) < SJRr/rx(a) SJRrIP(b) ... (l) unless (a l/rx ), (bliP), ... ) are proportional or one of the factors on the right-hand side is zero. If r < 0, the inequality is reversed. It is to be observed that, when r> 0, the second exceptional • case occurs only if one of the sets (a), (b), ... is nul, whereas when r < 0 it occurs if any number of any set is zero. When r= 0 there is equality in any case. 1 ) k k ' = k-1' k being any real number except 1. 2) (k-1)(k'-1)=1, ~+:,=1 (the last form failing when k= 0, k' = 0).

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

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