The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (Maa Problem Books Series.) [Paperback]

J. Michael Steele (Author)

Book Description

ISBN-10: 052154677X | ISBN-13: 978-0521546775 | Publication Date: April 26, 2004

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.

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.

See all Editorial Reviews

Product Details

• Paperback: 318 pages
• Publisher: Cambridge University Press (April 26, 2004)
• Language: English
• ISBN-10: 052154677X
• ISBN-13: 978-0521546775
• Product Dimensions: 8.9 x 6.4 x 0.6 inches
• Shipping Weight: 13.6 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (9 customer reviews)
• Amazon Best Sellers Rank: #601,387 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

63 of 64 people found the following review helpful:

5.0 out of 5 stars Wonderful Book, August 20, 2004

By 

ktrmes "ktrmes" (New York, New York USA) - See all my reviews

This review is from: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (Maa Problem Books Series.) (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. I just wish that there were more books like this. A particular bonus to the experience has been Professor Steele's website relating to the book. [...]
If you wish to see sample chapters, examine some source materials, or continue the adventure, take a look at the site and you will get feel for some of Professor Steele's unique and wonderful aims in writing this book.

48 of 48 people found the following review helpful:

5.0 out of 5 stars Erudite and stimulating problem book in inequalities, September 10, 2005

By 

Allen Stenger (Alamogordo, NM USA) - See all my reviews
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This review is from: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (Maa Problem Books Series.) (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.

27 of 27 people found the following review helpful:

5.0 out of 5 stars A delicious smorgasbord of inequalities, June 24, 2006

By 

Peter Haggstrom (BONDI BEACH, NSW Australia) - See all my reviews
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This review is from: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (Maa Problem Books Series.) (Paperback)

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.