For the last 18 months or so, I have been writing a book on inequalities which I call The Cauchy-Schwarz Master Class: An Introduction to the Art of Inequalities. It will be published in the Fall of 2003 by Cambridge University Press and the Mathematical Association of America. The intention of the book is to provide students of mathematics with the kind of high level coaching that talented singers and pianists get in their famous "master classes." Such classes are an essential part of the path to excellence in the fine arts, and surely the "art" of mathematical inqualities deserves no less.
To get a flavor of the project, it would be natural to read Chapter 1:Starting with Cauchy, but one should keep in mind that this chapter is somewhat unusual; how one starts is not always the clearest indicator of how one ends. Chapter 5: Consequences of Order and Chapter 10: Hilbert's Inequality and Compensating Difficulties are in many ways more typical.
Feel free to download the current versions of these chapters, but please keep two points in mind. First, these are not the final versions, and, second, the completed volume will provide solutions for the majority of the exercises. The current draft is a bit more than 300 pages, and you may also want to review the table of contents or the index , though these will surely face major changes before they come to print. Finally, if you would like to be a reviewer of part (or all!) of this book, your help is certainly welcome. Just send me an e-mail note, and I will get you a copy of the full manuscript.
Over time, I hope that this page will provide a place where readers of The Cauchy-Schwarz Master Class will come to look for new links, observations, and perspectives on mathematical inequalities. In particular, I hope to offer easy access to some of the most important classic papers of mathematical inqualities --- especially those that would otherwise be difficult or impossible to find. The crown jewel of the collection is surely the original 1859 Mémoire of V.V.Bunyakovsky. Everyone knows that this article scooped the 1885 paper of K.H.A. Schwarz, but comparison of the papers goes a long way toward explaining how "Schwarz" became a household name while Bunyakovsky fell into the footnotes.
For those who find that the French and German of their school days has become rusty, I will soon provide a discussion of these papers that contains a parallel translations of the key passages. A chapter of this sort was originally part of The Cauchy-Schwarz Master Class, but a webpage with links to the originals is surely more effective. One can access almost all of the works of Cauchy and Lagrange through the electronic collections of the Bibliothèque National de France.