The Cauchy-Schwarz Master Class
The Cauchy-Schwarz Master Class
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.
- Well-written: lively and user-friendly
- Over 100 worked exercises with coaching and hints
- Standard and non-standard topics
Reviews & endorsements
'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
'… pleaseant reading for everyone with a solid real analysis background at undergraduate level, even before reading Pólya-Szegö. In fact, even researchers working on topics close to those in this book can find much to add to their repertoire.' Tamás Erdélyi, Department of Mathematics, Texas A&M University
'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
Product details
November 2004Paperback
9780521546775
318 pages
227 × 150 × 16 mm
0.43kg
35 b/w illus. 161 exercises
Available
Table of Contents
- 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.
