Preface

Summary

In the fine arts, a master class is a small class where students and coaches work together to support a high level of technical and creative excellence. This book tries to capture the spirit of a master class while providing coaching for readers who want to refine their skills as solvers of problems, especially those problems dealing with mathematical inequalities.

The most important prerequisite for benefiting from this book is the desire to master the craft of discovery and proof. The formal requirements are quite modest. Anyone with a solid course in calculus is well prepared for almost everything to be found here, and perhaps half of the material does not even require calculus. Nevertheless, the book develops many results which are rarely seen, and even experienced readers are likely to find material that is challenging and informative.

With the Cauchy–Schwarz inequality as the initial guide, the reader is led through a sequence of interrelated 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 one finds systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and all of the so-called classical inequalities, including those of Hölder, Hilbert, and Hardy.

To solve a problem is a very human undertaking, and more than a little mystery remains about how we best guide ourselves to the discovery of original solutions.