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

This 2004 book presents a fascinating collection of problems related to the Cauchy-Schwarz inequality and coaches readers through solutions.

J. Michael Steele (Author)

9780521546775, Cambridge University Press

Paperback, published 26 April 2004

318 pages, 35 b/w illus. 161 exercises
22.7 x 15 x 1.6 cm, 0.43 kg

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

This lively, problem-oriented text, first published in 2004, 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.

1. Starting with Cauchy
2. The AM-GM inequality
3. Lagrange's identity and Minkowski's conjecture
4. On geometry and sums of squares
5. Consequences of order
6. Convexity - the third pillar
7. Integral intermezzo
8. The ladder of power means
9. Hölder's inequality
10. Hilbert's inequality and compensating difficulties
11. Hardy's inequality and the flop
12. Symmetric sums
13. Majorization and Schur convexity
14. Cancellation and aggregation
Solutions to the exercises
Notes
References.

Subject Areas: Combinatorics & graph theory [PBV], Probability & statistics [PBT], Calculus & mathematical analysis [PBK], Mathematical foundations [PBC]