The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical InequalitiesCambridge University Press, 26.04.2004 - 306 Seiten 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. |
Inhalt
Starting with Cauchy | 1 |
The AMGM Inequality | 19 |
Lagranges Identity and Minkowskis Conjecture | 37 |
On Geometry and Sums of Squares | 51 |
Consequences of Order | 73 |
Convexity The Third Pillar | 87 |
Integral Intermezzo | 105 |
The Ladder of Power Means | 120 |
Hardys Inequality and the Flop | 166 |
Symmetric Sums | 178 |
Majorization and Schur Convexity | 191 |
Cancellation and Aggregation | 208 |
Solutions to the Exercises | 226 |
Chapter Notes | 285 |
| 292 | |
| 302 | |
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The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical ... J. Michael Steele Keine Leseprobe verfügbar - 2004 |
Häufige Begriffe und Wortgruppen
a₁ Abel's inequality algebra AM-GM bound AM-GM inequality analog argument asserts b₁ calculation Carleman's inequality Cauchy Cauchy-Schwarz inequality Cauchy's inequality challenge problem complete complex numbers conjecture consider convex function defined doubly stochastic equality holds estimate example factor formula geometric mean given gives Hilbert's inequality Hölder's inequality hypothesis implies induction inner product space integral Jensen's inequality Lagrange's identity Lemma linear lower bound Math mathematical Minkowski's monotone Muirhead's inequality Newton's inequalities nonnegative polynomial nonnegative real numbers permutation Pólya proof prove provides quadratic rearrangement inequality relation representation result Riesz roots satisfies Schur convex Schwarz's inequality sequence side SOLUTION FOR EXERCISE suggests sum of squares summands symmetric target inequality theorem theory of inequalities triangle inequality variables vector Σ Σ ΣΣ
Verweise auf dieses Buch
The Mathematics of Preference, Choice and Order: Essays in Honor of Peter C ... Steven Brams,William V. Gehrlein,Fred S. Roberts Eingeschränkte Leseprobe - 2009 |