The Cauchy-Schwarz Master Class has been in print for more than a year now, so, as it it were a law of nature, beautiful problems and proofs start turning up that I never noticed before.
I plan to post these new discoveries here from time to time. I have three to get the ball rolling.
I'll separate the problems from the solutions to avoid accidental spoilers. Still, the linking text will usually give a pretty big hint that other problem solvers would have been left without!
Please do send me a note if you have a problem in mathematical inequalities that you think I should add to the collection. I am delighted to give attributions and links.
I certainly like this one that just showed up on my screen. It's from 2003, but it is amazingly slick -- and instructive. It begins with a simple fractional inequality without a name. How about the "baby fractional Cauchy Inequality "? For the whole piece, the most appropriate might be "The Cauchy-Schwarz Inequality and Convexity of the Linear/Quadratic quotient." Still, that is a bit fancy, for such a down to Earth proof of Cauchy's inequality.
There is also a real "proof from the book" of the case d=2 of Cauchy's inequality by tilings! Do look at the lovely piece by Roger Nelsen that covers this (on page 4). The principle? Well, besides the usual (but nice) principle that "tilings of tilings" yield identities, there is the clever observation that a rhombus of given edge lengths has less area than a rectangle with the same edge lengths. Punch line? This "is" Cauchy --- if you see the right rhombus! (on page 4)!
This leads to the question: How many proofs are there of the Cauchy-Schwarz inequality? If you have looked into this, or have a view for any reason at all --- please let me know.
1. A Problem Relating Linear Fractions of Sums and Summands and it's lovely Cauchy-Schwarz inequality based solution. This one is from the Telecom Australian Mathematical Competition and appeared in R. E. Woodrow's Olympiad Corner. (By the way, there are several problems in the CSMC that get close to this one, such as Exercises 1.2 and 5.8, but the Telecom problem adds something to the story.)
2. A Problem Providing a Necessary and Sufficient Condition for a Quadratic Polynomial of a Matrix to Be Positive Definite. It's simple, but nice and perhaps even useful. The solution? --- well it's not a big hint to say it uses Cauchy-Schwarz, but I already said it was easy so this does not give too much away.
3. A Problem Providing A Hardy-Style Convolution Inequality. Convolution inequalities are important in many areas, and they are perhaps under represented in the Cauchy-Schwarz Master Class. I'll try to add a few here, and perhaps even do an on-line chapter on convolution inequalities. Let me know if you think this would be useful --- or if you have nice candidate problems!
I've always found Schur's inequality to be the most mysterious of the named elementary inequalities, and I put together a little piece both to share and to resolve some of that mystery.
Small Paradox in the AMGM Inequality There is a nice bound that is an "obvious generalization" of the AMGM inequality, yet which is itself a not-too-hard consequence of the AMGM inequality. Is it paradoxical to think of one as stronger? Take a look and see what you think.
Many contest problems call for knowledge special algebraic identities --- or at least if you have the knowledge you get a pass on a substantial chunk of brilliance. The identities with three letters seem to be particularly common --- not too trivial, and yet not too messy. I've started summary page of three letter identities where I plan to organize some of this material..
Please let me know if you have any favorite three letter identities you would like to share.
More generally, I would love to find material that offers coaching about how to think about algebraic identities. Almost everything I have seen suffers from a "three bears problem" --- either too trivial or too advanced.
For a starters, it would be nice to have an exposition about symmetric polynomials and their applications. There are some decent resources for symmetric functions on the web, but what are your favorites?
Geometric Duality and The Solids of Theaeteus There are many "duals" in mathematics, and we may find the freedom to discover new ones of our own by recalling the oldest. Read a page that reminds you of the first duality of them all.