Analytic theory of polynomials. (English) Zbl 1072.30006

Rahman, Q. I.; Schmeisser, G.

London Mathematical Society Monographs. New Series 26. Oxford: Oxford University Press (ISBN 0-19-853493-0/hbk). xiv, 742 p. (2002).

This book is an excellent monograph about complex polynomials written by two very well-known specialists in this domain.
I am sure that from now it will be the first book which every mathematician and other specialists looking for some questions concerning polynomials can find an answer or indication and moreover inspiration for further research.
The book is written with great care about the reader giving him or her not only almost full knowledge about the topic under consideration but as well very detailed interesting historical background and development in ”Notes” after each chapter. These ”Notes” show how deeply the authors treat each topic trying to make everything very up-to-date. The extremely vast bibliography since the work of Chebyshev up to the positions from the year 2002 makes this book the best source for research work without looking for other references.
Starting with the basic knowledge about polynomials and topics from complex analysis the authors give a clear presentation of many different problems concerning polynomials like distribution of zeros and critical points, extremal problems, orthogonal expansions, inequalities, coefficient estimates etc.
A very important property of this book is its self-containment. The authors are giving all detailed proofs (several of them are new), sometimes even several of them, showing the reader the richness of the subject. From this point of view the book can be considered as excellent source of knowledge for some topics in complex and real analysis.
Being a vast material for seminars of graduate students, this book for sure will give new motivation for the research work concerning polynomials.
Moreover, completeness in covering of so many topics, underlying different connections, this book will find the main place on the desk of every mathematician in the neighborhood of such important monographs like G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities (1934; Zbl 0010.10703 and JFM 60.0169.01) and E. C. Titchmarsh, The theory of functions (1932; Zbl 0005.21004) (1939; JFM 65.0302.01).
Finally let us mention that this book contains 742 pages, 6 pages of Preface and is divided into Introduction and three parts:
I: Critical points in terms of zeros (Ch. 2–7) II: Zeros in terms of coefficients (Ch. 8–11) III: Extremal Properties (Ch. 12–16), plus References and List of notation and index.
The titles of chapters are as follows: 1. Introduction 2. Fundamental results on critical points 3. More sophisticated methods 4. More specific results on critical points 5. Applications to compositions of polynomials 6. Polynomials with real zeros 7. Conjectures and solutions 8. Inclusion of all zeros 9. Inclusion of some of the zeros 10. Number of zeros in an interval 11. Number of zeros in a domain 12. Growth estimates 13. Mean values 14. Derivative estimates on the unit disc 15. Derivative estimates on the unit interval 16. Coefficient estimates
Without questions this is the best book about polynomials since the book of M. Marden, Geometry of polynomials (Providence, AMS 1966; Zbl 0162.37101). Moreover, it can be suggested to other authors as an example how to write an excellent book.
I am sure that several years of the authors’ work will find great recognition in the mathematical community.

Reviewer: Jan Szynal (Lublin)

MSC:

30C10	Polynomials and rational functions of one complex variable
30-02	Research exposition (monographs, survey articles) pertaining to functions of a complex variable
00A05	Mathematics in general
11C08	Polynomials in number theory
12D10	Polynomials in real and complex fields: location of zeros (algebraic theorems)
30C15	Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
31-02	Research exposition (monographs, survey articles) pertaining to potential theory
41A05	Interpolation in approximation theory

Citations:

Zbl 0162.37101; Zbl 0010.10703; JFM 60.0169.01; JFM 65.0302.01; Zbl 0005.21004