不等式

內(nèi)容概要

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 given all references we can, but we have never undertaken systematic bibliographical research. We follow the common practice, when a particular inequality is habitually associated with a particular mathematician's name; we speak of the inequalities of Schwarz, HSlder, and Jensen, though all these inequalities can be traced further back; and we do not enumerate explicitly all the minor additions which are necessary for absolute completeness.   We have received a great deal of assistance from friends. Messrs G. A. Bliss, L. S. Bosanquet, R. Courant, B. Jessen, V. Levin, R. Rado, I. Schur, L. C. Young, and A. Zygmund have all helped us with criticisms or original contributions. Dr Bosanquet, Dr Jessen, and Prof. Zygmund have read tho proofs, and corrected many inaccuracies. In particular, Chapter III has been very largely rewritten as the result of Dr Jessen's suggestions. We hope that the book may now be reasonably free from error, in spite of the mass of detail which it contains.

書(shū)籍目錄

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 subjectsCHAPTERⅡ  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 examplesCHAPTER 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  INTEGRALSCHAPTERⅦ  SOME APPLICATIONS OF THE CALCULUS OF VARIATIONS     CHARTERⅧ  SOME THEOREMS CONCERNING BILINEAR AND MULTILINEAR FORMSCHAPTERⅨ  HILBERT'S INEQUALITY AND ITS ANALOGUES AND EXTENSIONS  CHAPTERⅩ  REARRANGEMENTSAPPENDIXⅠ  On strictly positive formsAPPENDIXⅡ  Thorin's proof and extension of Theorem 295     APPENDIXⅢ  On Hilbert's inequality BIBLIOGRAPHY

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