The Cauchy-Schwarz Master Class

An Introduction to the Art of Mathematical Inequalities

J. Michael Steele

• Éditeur: The Mathematical Association of America
• Année: 2004
• ISBN:
  9780511207761
• Sujet: Mathematics
• Sous-sujet: Mathematical Analysis

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.

Table des matières

• Cover
• Half-title
• Series-title
• Title
• Copyright
• Contents
• Preface
• 1 Starting with Cauchy
• 2 Cauchy’s Second Inequality: The AM-GM Bound
• 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 Dificulties
• 11 Hardy’s Inequality and the Flop
• 12 Symmetric Sums
• 13 Majorization and Schur Convexity
• 14 Cancellation and Aggregation
• Solutions to the Exercises
• Chapter Notes
• References
• Index