Inequalities from Complex Analysis.

Godfrey, Stephen

Inequalities from Complex Analysis

John P. D'Angelo Published by The Mathematical Association of America, 2002 Carus Mathematical Monographs series. Hardcover, 264 pp., ISBN 08-8385-0338 US$45.50

When does a polynomial p(x) of degree n with real coefficients assume positive values whenever the variable x is positive? A sufficient condition is that all of the n+1 coefficients be positive; but clearly this is not a necessary condition, as many polynomials are strictly positive while having some negative coefficients. A simple but unobvious necessary and sufficient condition was known to Poincare, namely that p(x) be expressible as a quotient of two polynomials, each having all coefficients positive.

A related issue is Hilbert's seventeenth problem, which asks whether a non-negative polynomial in n real variables can be written as a sum of squares of rational functions; in 1926, Artin showed that the answer is "yes".

The book discusses the complex variables analogue of Hilbert's seventeenth problem--which is to find the conditions under which certain kinds of complex polynomials are positive; this takes up the final two chapters. The proofs and concepts covered in these final two chapters are nontrivial and are the result of recent work.

The first five chapters of the book should be accessible to a good senior undergraduate with some knowledge in basic real and complex analysis. The remaining two chapters are much more demanding. There are many good exercises throughout the book.

The text starts with an introduction to complex numbers and leads the reader in topics such as Hilbert spaces, complex analysis in several variables, Compact integral operators, and gives some positivity conditions for real-valued functions. The final chapter is taken up by a discussion of stabilisation.

A nice touch at the end is that the author gives the reader an insight into where the current research in this topic is currently at, pointing the reader towards a number of currently open problems.

This book could have been significantly better if the author had used diagrams to help illustrate new concepts. This deficiency is easily overcome by the active reader with pen and paper in hand.

I am glad to have reviewed this book; it was a real joy to read because it brought together many topics from modern analysis into a single topic area. It is unfortunate that due to the advanced content, this book will have a limited readership.

Stephen Godfrey