The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities
What reviewers are saying.
" This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics." --- Zentralblatt für Mathematik "The classic work in this field is Hardy, Littlewood, and Polya's Inequalities , but as much as I admire these authors for their other works, I never got much out of their inequalities book. Steele's book is different: extremely clear, erudite, and thorough, it almost makes everything obvious." --- Alan Stenger (Alamogordo, NM)
"... this is one of that handful of mathematics books that you can read almost like a novel." --- ktrimes (New York, NY)
"The proof of the Newton-Maclaurin inequalities presented here is really beautiful." --- Tamás Erdélyi ( full review on-line )
Have a Peek Inside The Cauchy-Schwarz Master Class...
You can get a useful view from Amazon's Search Inside which has the cool Surprise Me feature. The Preface will give you a sense of the over all design, but for a bread-and-butter accounting for the topics you'll do best with the Table of Contents , the Index, and the References .
Three Chapters --- Available On-Line!
While it's natural to begin by browsing Chapter 1: Starting with Cauchy , let me tell you up-front that this chapter is not typical. It does give some sense of the book's style and method, but, to give everyone a chance to get started, it has to cover some things that many readers may know. Still, it provides a decent quota of less common results that I hope will amuse and inform even the most experienced readers. The chapter also introduces the coaching voice that I try to use throughout the book.
Chapter 5: Consequences of Order is more typical, and --- in my view --- much more fun. It's organized around the simplest and most universal ingredient in the theory of inequalities: the ordered list of real numbers. Even after going through the chapter many times I am still surprised that so much useful and subtle information can be extracted from such basic ingredients..
Finally, for an example of a more sophisticated chapter, you might take a look at Chapter 10: Hilbert's Inequality and Compensating Difficulties. While the chapter is ostensibly (and honestly!) about a classic inequality of David Hilbert, it's also about a broader theme --- one that concerns every mathematician. Given a certain result, how can we improve it to give a better result --- one that is sharper, or more general, or which provides some new bit of information? There cannot be any sure-fire recipe, but there are many trustworthy suggestions that have worked well in the past.
Something Extra --- Easy Access to the Classics
Part of the fun of writing The Cauchy-Schwarz Master Class came from chasing down the original sources of essentially all of the classical mathematical inequalities. Almost every original paper had some surprise, and some were shocking. In due course, I hope to get these scanned and posted here. For the moment, I'll post two crown jewels of the collection, the 1859 Mémoire of V.V.Bunyakovsky and the 1885 paper of K.H.A. Schwarz . In both cases I had to work with copies provided by interlibrary loan. The copies are readable, but not beautiful.
Almost every mathematician (and certainly every Russian mathematician) knows that Bunyakovsky scooped Schwarz in the statement and proof of the integral inequality that is now universally known as Schwarz's inequality. Comparison of their papers goes a long way toward explaining why "Schwarz" became a household name while Bunyakovsky fell into the footnotes. The short version is that Bunyakovsky just took the obvious limit in Cauchy's inequality, but Schwarz made a real breakthrough by proving the inner product version of the inequality that we all know and love.
Incidentally, one can access essentially all of the works of Cauchy through the electronic collections of the Bibliothèque National de France. Roberto Dominijanni recently brought it to my attention that you can indeed download full documents, not just one page at a time. He writes:
Suppose, for example, that you go to http://gallica.bnf.fr/ark:/12148/bpt6k78986c for Cauchy's "Nouveaux exercices de mathématiques." Clicking on the word "Télécharger" near the top brings you to a page that asks for the starting and ending pages you want, and for the format (pdf or tiff). (The default seems to be first page to last page in pdf format.) Clicking the "OK" button brings you to a page that tells you it's working on assembling the document. When it's done, you'll get the message "Vous pouvez le télécharger en cliquant ici." This last link downloads the assembled document via ftp, so it's a little slow. For this example you get the document N0078986_PDF_1_284.pdf , 128 pages long.
Small Sidebar
In this book of Cauchy's one finds the interesting Article X. Sur la propagation de la lumière dans les milieux où sa vitesse reste la même pour toutes les couleurs , which offers a proof that the speed of light does not depend on the color (i.e. frequency) of the light.
Nowdays, this probably sounds "obvious" or "well known" --- but even if it is "known" to you, how would you go about proving it? Perhaps it does come "free" from the d'Alembert solution of the wave equation , but I can't quite decide if this is "cheating".
New and so lovely ... a tiling proof of Cauchy's inequality !
Roger Nelsen has written a charming piece that provides tiling proofs of many interesting results. To be sure, it covers the Pythagorean chestnuts, but there is also a stunning proof of the case d=2 of Cauchy's inequality.
The principle? Well, besides the usual (but nice) principle that "tilings of tilings yield identities," there is the natural observation that a rhombus of given edge lengths has less area than a rectangle with the same edge lengths.
Punch line? This "is" Cauchy --- after you see the right rhombus!
Building a Community
So far this page mainly introduces The Cauchy Schwarz Master Class and gently nudges you toward looking at a copy. Nevertheless, as I get the classic papers posted, I hope that that this page can evolve into one that provides a useful service for our community. Naturally, I have an errata page , and I've also begun a "Further Mathematical Inequalities Problems" page which I expect to grow over time.
Please let me know if you find a problem I should add.
Finally, if you have a web page that deals with inequalities (or problem solving, or mathematical coaching), please add a link to this page. I'll happily reciprocate. In the era of Google, one lives or dies by links. Also, by learning that someone actually reads these pages, I'll have more incentive to keep scanning and posting more classics.
BACK TO: J. Michael Steele's Home Page
Small Addendum : I made an attempt to get an html version of the references , but the Adobe tool I used is not ready for prime time. Still, I'll post a link just so I won't forget to close this loop. It also shows how lame even the great Adobe can be.
More Navigation? (1) Favorite Quotations , (2) Mild Manored Rants , or (3) Surprise Me!
Vornicu-Schur Inequality
The Vornicu-Schur Inequality is a generalization of Schur's Inequality discovered by the Romanian mathematician Valentin Vornicu .
- Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta ; GIL Publishing House; Zalau, Romania.
- Inequalities
Something appears to not have loaded correctly.
Click to refresh .
IMAGES
VIDEO
COMMENTS
Overview. Inequalities are arguably a branch of elementary algebra, and relate slightly to number theory.They deal with relations of variables denoted by four signs: .. For two numbers and : . if is greater than , that is, is positive.; if is smaller than , that is, is negative.; if is greater than or equal to , that is, is nonnegative.; if is less than or equal to , that is, is nonpositive.
Elementary Form. If are nonnegative real numbers and are nonnegative reals with sum of 1, then. Note that with two sequences and , and , this is the elementary form of the Cauchy-Schwarz Inequality. We can state the inequality more concisely thus: Let be several sequences of nonnegative reals, and let be a sequence of nonnegative reals such ...
communities, particularly the Mathlinks-Art of Problem Solving forums,1 as well as from various MOP lectures,2 Kiran Kedlaya's inequalities packet,3 and John Scholes' site.4 I have tried to take note of original sources where possible. This work in progress is distributed for personal educational use only.
Art of Problem Solving's Richard Rusczyk discusses what happens if we multiply both sides of an inequality by the same constant.
Art of Problem Solving's Richard Rusczyk discusses some basics tools for working with inequalities.
Art of Problem Solving's Richard Rusczyk explains how to graph a two-variable linear inequality on the Cartesian plane.
Uncertainty Principle, 205 union, 247. variable, 17 Venn diagrams, 248 volume, 160 of a box, 163 of a cone, 167 of a cube, 162 of a cylinder, 165 of a prism, 165 of a pyramid, 166 of a sphere, 161 of a tetrahedron, 168 of an octahedron, 169. without loss of generality, 252 word problems, 22-24 work problems, 23 working backwards, 91.
Without loss of generality, we assume that the increase as increases. Now for any , must have a root between and by Rolle's theorem if , and if , then is a root of times, so it must be a root of times. It follows that must have non-positive, real roots, i.e., for some non-negative reals , It follows that the symmetric sum for is , so the ...
AM-GM Inequality. In algebra, the AM-GM Inequality, also known formally as the Inequality of Arithmetic and Geometric Means or informally as AM-GM, is an inequality that states that any list of nonnegative reals' arithmetic mean is greater than or equal to its geometric mean. Furthermore, the two means are equal if and only if every number in ...
Now, by the triangle inequality, we have . Multiplying both sides of the inequality by and using equations and gives us , which is the desired inequality. Equality holds iff. , , and are collinear. But since the triangles and are similar, this would imply that the angles and are congruent, i.e., that is a cyclic quadrilateral. Outline for 3-D Case
The trick is to notice that the given inequality can be rewritten as. a2 + b2 + c2. a1/3b1/3c1/3(a + b + c) : Now the inequality is homogeneous. Observe that if we multiply a, b, c by any real number k > 0, all that happens is that both sides of the inequality are multiplied by k2, which doesn't change anything.
Art of Problem Solving's Richard Rusczyk introduces the Triangle Inequality.
one example where inequalities can be used to solve other types of problems. 2 Common identities and other means 2.1 Identities There are many identities that problem solvers use in order to prove in-equalities. They allow us to transform the inequality to another, equivalent inequality which is easier to prove. Here are some of the most used ...
The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean Inequality.. Inequality. For positive real numbers and positive real weights with sum , the power mean with exponent , where , is defined by . The Power Mean Inequality states that for all real numbers and , if .In particular, for nonzero and , and equal weights (i.e. ), if , then
Proofs. Here is a list of proofs of Cauchy-Schwarz. Consider the vectors and .If is the angle formed by and , then the left-hand side of the inequality is equal to the square of the dot product of and , or .The right hand side of the inequality is equal to .The inequality then follows from , with equality when one of is a multiple of the other, as desired.
Art of Problem Solving's Richard Rusczyk explains how to solve linear inequalities.
Pages in category "Olympiad Inequality Problems" The following 30 pages are in this category, out of 30 total. 1. 1960 IMO Problems/Problem 2; 1972 USAMO Problems/Problem 4; ... Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast Academy. AoPS Academy. About. About AoPS. Our Team. Our History. Jobs. AoPS ...
The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities What Reviewers Are Saying "This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics."--- Zentralblatt für Mathematik "The classic work in this field is Hardy, Littlewood ...
Pages in category "Inequalities" The following 27 pages are in this category, out of 27 total. A. Aczel's Inequality; AM-GM Inequality; C. ... Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast Academy. AoPS Academy. About. About AoPS. Our Team. Our History. Jobs. AoPS Blog. Site Info. Terms.
Art of Problem Solving's Richard Rusczyk solves a word problem with inequalities.
The Vornicu-Schur Inequality is a generalization of Schur's Inequality discovered by the Romanian mathematician Valentin Vornicu.. Statement. Consider real numbers such that and either or .Let be a positive integer and let be a function from the reals to the nonnegative reals that is either convex or monotonic.Then Schur's Inequality follows from Vornicu-Schur by setting , , , , and .
If you're a student who likes to ask "when am I ever gonna use this in real life?", then linear programming is the topic for you. Businesses actually do this...
Art of Problem Solving AoPS Online. Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12 ...