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The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (MAA Problem Books) Paperback – April 26, 2004

ISBN-13: 978-0521546775 ISBN-10: 052154677X

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Product Details

  • Series: MAA Problem Books
  • Paperback: 318 pages
  • Publisher: The Mathematical Association of America (April 26, 2004)
  • Language: English
  • ISBN-10: 052154677X
  • ISBN-13: 978-0521546775
  • Product Dimensions: 6 x 0.7 x 9 inches
  • Shipping Weight: 13.6 ounces (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (14 customer reviews)
  • Amazon Best Sellers Rank: #280,685 in Books (See Top 100 in Books)

Editorial Reviews

Review

"...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

Book Description

Michael Steele describes the fundamental topics in mathematical inequalities and their uses. Using the Cauchy-Schwarz inequality as a guide, Steele presents a fascinating collection of problems related to inequalities and coaches readers through solutions in a style reminiscent of George Polya, by teaching basic concepts and sharpening problem solving skills at the same time. Undergraduate and beginning graduate students in mathematics, theoretical computer science, statistics, engineering, and economics will find the book appropriate for self-study, and it can also be used as a supplement to courses in analysis, probability, and combinatorics.

More About the Author

J. Michael Steele teaches at the Wharton School of the University of Pennsylvania. His interests include probability theory, mathematical finance, financial time series, and, especially, mathematical inequalities. He is a fellow of the American Statistical Association and the Institute for Mathematical Statistics for which he also served as President.

Customer Reviews

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Such an approach throughout the book makes this book really enjoyable.
Kedar Hardikar
A wonderful expository book on building brain muscle for mathematical proof--obviously on inequalities.
Eunice Kim
I was tutoring a freshman in Linear Algebra around the time I bought the book.
Shubhendu Trivedi

Most Helpful Customer Reviews

73 of 74 people found the following review helpful By ktrmes on August 20, 2004
Format: Paperback
I rate this book with FIVE STARS *****
Somehow, the review rating software keeps changing the rating to two stars which is incorrect -- again I must emphasize it is FIVE STARS ****.
Get it now -- don't wait!

As might be expected from the title, Steele's book includes an in depth exploration of the Cauchy Schwarz. It, however, includes so much more -- for example, many, many useful inequalities are set forth in its pages. But even its richness in range and number of inequalities (and equalities) is 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 accrete after years of experience in the course of undergraduate and graduate mathematics courses are set forth one after another. On top of that, this is one of that handful of mathematics books that you can read almost like a novel. It's so readable and rewarding/interesting and engaging that when people have asked me what I have been reading lately, I can answer with a good deal of pride and satisfaction: "a book on the Cauchy Schwarz inequality" -- which I never said about Royden, etc. These techniques are vital for many types of research -- applied mathematics, CS, economics, statistics, (and competitions) to name a few -- in all of these areas finding bounds can play a central role in research. Well worth every penny.

A 2008 Addendum -- I have had this book well over three years now, and I continue to reread sections and refer to it regularly.
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56 of 56 people found the following review helpful By Allen Stenger on September 10, 2005
Format: Paperback
The classic work in this field is Hardy, Littlewood, and Polya's "Inequalities", but as much as I admire these authors for their other works, I have never gotten much out of their inequality book. Steele's book is different: extremely clear, erudite, and thorough, it almost makes everything obvious. The subject of inequalities is something of a hodge-podge, and Steele isn't able to change that, but he helps tie it together with lots of forward and backward references and with returns to problems after we have learned new methods. A good example is Carleman's inequality (easily the most startling result in the book); Steele provides three different proofs spread out through the book, plus a continuous analog.

Despite the title, the book is not primarily about the Cauchy-Schwarz inequality, although it (and the Arithmetic-Geometric Mean inequality and Jensen's inequality) do recur throughout the book.

The book is structured as a problem book. The body consists of a number of "challenges", each followed by an exploration of how to solve it. Each chapter ends with a copious selection of exercises; they are not as hard as the challenges, but they are hard enough and they will build your mastery of the material. All exercises are worked out in full in the back of the book.
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30 of 30 people found the following review helpful By Peter Haggstrom on June 24, 2006
Format: Paperback Verified Purchase
Professor Steele has done a wonderful job in developing the theory behind the Cauchy-Schwarz inequality. He starts off with the basic theory and then through the course of the book he teases out the limitless ways the inequality can be used. There is a breathtaking sweep of applications. What is interesting and valuable about his approach is that as he develops the building blocks he explains why or why not a particular approach might not work. I think there is quite a bit of Polya's inspiration in his approach. For instance, he gives Polya's proof of the Carleman inequality which, on it face, is almost outrageously unbelievable ( where does the "e" come from?) but by that stage you worked through the challenge problems and the other material and it is possible to see why the "e" makes sense.

The challenge problems are excellent and his solutions sometimes skip over some important steps which a teacher could get students to fill in so that they can demonstrate that they understand the material.

There is a lot to learn from this book and it should be read by everyone who is seriously interested in mathematics. The classic Hardy-Littlewood-Polya book on inequalities is a quite different beast but the two together provide the serious reader with a depth of understanding that is hard to surpass.
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21 of 22 people found the following review helpful By Gilles Benson on March 19, 2006
Format: Paperback Verified Purchase
this book deals in a friendly fashion with inequalities (and therefore) with the elementary use of convexity and integrals.
Famous inequalities bear the name of famous mathematicians, e.g: Tchebychev, Hilbert, Cauchy, Hardy, Rademacher...This is one way to understand their significance in maths. This book is about those ones and others such as 3/2 < a/(b+c) + b/(c+a) + c/(a+b) and the many ways to tackle with the fact of proving and using them. Study of this book should be seen as a good and rewarding path towards improving one's mathematical skills .
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9 of 10 people found the following review helpful By Kedar Hardikar on December 24, 2007
Format: Paperback Verified Purchase
Prof. Steele has done a great job in providing an "entertaining" (if I may say) book on inequalities. Along with Cauchy-Schwarz inequality the book provides very "lively and problem oriented" (adjectives from the first page of the book) chapters that are insightful and enjoyable. For example, the way you are introduced to Cauchy-Schwarz inequality involves attempting it as a "problem" - first looking at cases n=1 and n=2 trying to use induction. After that you get into fancy approach using quadratic expression. Such an approach throughout the book makes this book really enjoyable. Solutions provided make it ideal for self learning as well as a book to entertain yourself when you get bored :)
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