Inequalities - A Journey into Linear Analysis
As is well-known, all animals are equal (but some are more equal). The character in George Orwell’s timeless book who said (and wrote) this was a crook. So perhaps, in fact, the opposite is true: there is a certain kind of inequality between any two animals (and, pretty much, between any two objects one can think of). Inequalities constitute a basic and fundamental concept in mathematics - and not only in mathematics. A need to compare two parameters of a certain phenomenon is everyday business of almost everybody. There exist several mathematical textbooks dedicated to inequalities, whether they be classical or modern or old or new, but there can never be enough of such books.
This book contains a wealth of inequalities, again both classical and contemporary, complemented with detailed recipes on how to use them. It has several quite favourable features. First, the content is not restricted solely to an array of inequalities. It also involves a broad variety of applications such as Lebesgue decomposition and density theorems and the martingale convergence theorem, as well as a rather detailed treatment of diverse topics such as singular integrals, the Hahn-Banach theorem and eigenvalues of distributions. Each chapter is well equipped with a collection of interesting and revealing notes and remarks and, in some cases, also with a number of nontrivial exercises. Many inequalities considered are of course classical but others are not generally known and it is good that they appear in a textbook (such as the Bonami inequality – it is pleasant and refreshing to meet an inequality that is called after a lady for a change). The author also brings back Muirhead’s maximal function, which is usually treated as a misnomer quoted to other authors. This book is a compulsory item on every teacher’s bookshelf and it should be strongly recommended to students. If nothing else, it is an endless source of very good problems for students’ theses of all levels.