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

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"This 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 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 Holder, 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.
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Editorial Reviews

From the Publisher
"...this book is a 'must have' for a university's library, and I recommend it highly to its 'ideal audience.' Many other readers are also bound to discover a satisfying number of attractive and less than familiar results."
MAA Reviews

"This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics."
Zentralblatt MATH

"The book is special...A large mathematics department with a functional graduate program could easily consider to offer a master course based on this book."
Tamas Erdelyi, Journal of Approximation Theory

"I believe George Polya would enjoy reading this book, and I recommend it to both the novice and the sophisticate. It is a nice read."
Ingram Olkin, Stanford University for SIAM Review

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Product Details

  • ISBN-13: 9780521546775
  • Publisher: Cambridge University Press
  • Publication date: 4/28/2004
  • Edition description: New Edition
  • Pages: 318
  • Sales rank: 1,157,183
  • Product dimensions: 5.98 (w) x 8.98 (h) x 0.67 (d)

Meet the Author

J. Michael Steele is C. F. Koo Professor of Statistics at Wharton School, University of Pennsylvania. He is the author of more than 100 mathematical publications including the books, Probability Theory and Combinatorial Optimization and Stochastic Calculus and Financial Applications. He is also the founding editor of the Annals of Applied Probability.
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Table of Contents

1 Starting with Cauchy 1
2 The AM-GM inequality 19
3 Lagrange's identity and Minkowski's conjecture 37
4 On geometry and sums of squares 51
5 Consequences of order 73
6 Convexity - the third pillar 87
7 Integral intermezzo 105
8 The ladder of power means 120
9 Holder's inequality 135
10 Hilbert's inequality and compensating difficulties 155
11 Hardy's inequality and the flop 166
12 Symmetric sums 178
13 Majorization and Schur convexity 191
14 Cancellation and aggregation 208
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  • Anonymous

    Posted June 11, 2004

    Great Book -- A Must Have

    Professor Steele's book includes an in depth exploration of the Cauchy Schwarz. Steele, however, also includes much more in this book, including for example, a wealth of useful inequalities. But even the richness in range and number of inequalities (and equalities) is itself secondary to Prof. Steele's method of explication. For the real fruit of this book is the techniques and confidence built by the exercises and exposure to the examples. The exercises feed and bolster confidence in approching or deriving familiar and more importantly, never-before-seen inequalities, a confidence which grows with each page and exercise. Techniques that might normally only acquired piecemeal in the course of undergraduate and graduate mathematics courses are set forth one after another. Moreover, due in no small part to the skillful presentation, this is one of that rare handful of mathematics books that you can just read as pleasure reading -- almost like reading a novel. Don't mistake this title for being narrowly focussed on competion problem solving. The techniques so skillfully set forth here are vital in many types of research -- applied mathematics, CS, economics, statistics, (and competitions) to name a few. Well worth every penny.

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