Customer Reviews

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

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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.

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.

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.

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 .

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 :)

Really, you should own it.
It's such a beautiful book. The text, the content, the structure, the exercises, the solutions, the tricks and insight, and it's soft and handy.
What a book. It's modern, rigorous, gives you references to more advanced texts, and teaches you a lot! A pleasure.
If you're interested in inequalities you should really buy it, but even you if don't want to, it's available on the internet (try [..]
Have a very very nice reading.

This book's topic, mathematical inequalities, could be considered too esoteric to justify a book for more than a small niche audience of mathematicians. Yet this is an exceptionally well-written book that should appeal not only to those who might need inequalities in their work, but also to any student of mathematics who wants to learn how to discover and present elegant proofs. The only prerequisites are a familiarity with series, sequences, and standard notation in calculus and linear algebra.

In addition to its core content, the book does something that too few books in mathematics do: Provide a solution for every exercise. This makes it a precious resource for independent study.

it is amazing

i love maths but my math education in high school is a just a mess

they also discussed am-gm inequality, but the logic and insight is missing

i love this book. it really solidates my understanding of some basic inequalities and improves my ability to deal with more inequalities

it is worth the money

This is my first review ever of a technical book. I really love this book, and I highly recommend it. It is simultaneously useful, fun, and thought provoking.