The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities

by J. Michael Steele

• ISBN13: 9780521837750
• ISBN10: 0521837758
• Format: Hardcover
• Copyright: 2004-04-26
• Publisher: Cambridge University Press

Summary

This lively, problem-oriented text is designed to coach readers toward mastery of the most fundamental mathematical inequalities. With the Cauchy-Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered - either by one of history's famous mathematicians or by the reader. The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of Hölder, Hilbert, and Hardy. The text is accessible to anyone who knows calculus and who cares about solving problems. It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics.

Table of Contents

Preface	ix
1 Starting with Cauchy	1	(18)
2 The AM-GM Inequality	19	(18)
3 Lagrange's Identity and Minkowski's Conjecture	37	(14)
4 On Geometry and Sums of Squares	51	(22)
5 Consequences of Order	73	(14)
6 Convexity - The Third Pillar	87	(18)
7 Integral Intermezzo	105	(15)
8 The Ladder of Power Means	120	(15)
9 Holder's Inequality	135	(20)
10 Hilbert's Inequality and Compensating Difficulties	155	(11)
11 Hardy's Inequality and the Flop	166	(12)
12 Symmetric Sums	178	(13)
13 Majorization and Schur Convexity	191	(17)
14 Cancellation and Aggregation	208	(18)
Solutions to the Exercises	226	(58)
Chapter Notes	284	(7)
References	291	(10)
Index	301