This book, "Analytic Theory of Polynomials" by Q. I. Rahman and G. Schmeisser, provides a comprehensive overview of the analytic theory of polynomials, covering fundamental concepts, critical points, zeros, and their relationships with coefficients. The book is divided into two main parts: "Critical Points in Terms of Zeros" and "Zeros in Terms of Coefficients," with additional sections on extremal properties. The first part focuses on critical points of polynomials, including the Gauss–Lucas theorem, Jensen's theorem, Laguerre's theorem, and Grace's theorem. It also discusses complex analogues of Rolle's theorem, bounds for critical points, and the relationship between zeros and critical points. The second part explores the relationship between zeros and coefficients, including the Cauchy bound, the Eneström–Kakeya theorem, and the Landau–Montel problem. The book also covers the number of zeros in intervals and domains, as well as the distribution of zeros. The text includes detailed discussions on polynomials with real zeros, conjectures and solutions related to zero distribution, and applications to compositions of polynomials. It also addresses extremal properties of polynomials, such as growth estimates, mean values, and derivative estimates on the unit disc and interval. The book concludes with references, a list of notation, and an index. The authors provide a thorough treatment of the analytic theory of polynomials, making it a valuable resource for researchers and students in mathematics and related fields. The content is well-organized, with clear explanations and detailed proofs, and it covers both classical and modern results in the field.This book, "Analytic Theory of Polynomials" by Q. I. Rahman and G. Schmeisser, provides a comprehensive overview of the analytic theory of polynomials, covering fundamental concepts, critical points, zeros, and their relationships with coefficients. The book is divided into two main parts: "Critical Points in Terms of Zeros" and "Zeros in Terms of Coefficients," with additional sections on extremal properties. The first part focuses on critical points of polynomials, including the Gauss–Lucas theorem, Jensen's theorem, Laguerre's theorem, and Grace's theorem. It also discusses complex analogues of Rolle's theorem, bounds for critical points, and the relationship between zeros and critical points. The second part explores the relationship between zeros and coefficients, including the Cauchy bound, the Eneström–Kakeya theorem, and the Landau–Montel problem. The book also covers the number of zeros in intervals and domains, as well as the distribution of zeros. The text includes detailed discussions on polynomials with real zeros, conjectures and solutions related to zero distribution, and applications to compositions of polynomials. It also addresses extremal properties of polynomials, such as growth estimates, mean values, and derivative estimates on the unit disc and interval. The book concludes with references, a list of notation, and an index. The authors provide a thorough treatment of the analytic theory of polynomials, making it a valuable resource for researchers and students in mathematics and related fields. The content is well-organized, with clear explanations and detailed proofs, and it covers both classical and modern results in the field.