Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties (London Mathematical Society Monographs New Series)

This text presents easy to understand proofs of some of the most difficult results about polynomials. It encompasses a self-contained account of the properties of polynomials as analytic functions of a special kind. The zeros of compositions of polynomials are also investigated along with their growth, and some of these considerations lead to the study of analogous questions for trigonometric polynomials and certain transcendental entire functions. The strength of methods are fully explained and demonstrated by means of applications. Content: 1. Introduction -- I. Critical Points in Terms of Zeros. 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 -- II. Zeros in Terms of Coefficients. 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 -- III. Extremal Properties. 12. Growth estimates. 13. Mean values. 14. Derivative estimates on the unit disc. 15. Derivative estimates on the unit interval. 16. Coefficient estimates.