The book "Analytic Theory of Polynomials" by Q. I. Rahman and G. Schmeisser provides a comprehensive treatment of the analytic theory of polynomials, covering both fundamental results and advanced topics. The introduction sets the stage with key theorems such as the fundamental theorem of algebra, properties of symmetric and orthogonal polynomials, and tools from matrix analysis. The first part focuses on critical points in terms of zeros, exploring fundamental results, extensions, and more sophisticated methods. It delves into specific results, applications to compositions of polynomials, and polynomials with real zeros. The second part examines zeros in terms of coefficients, discussing inclusion bounds, number of zeros in intervals and domains, and general principles. The third part explores extremal properties, including growth estimates, mean values, derivative estimates, and coefficient estimates. The book concludes with references, a list of notation, and an index, making it a valuable resource for researchers and students in the field of polynomial theory.The book "Analytic Theory of Polynomials" by Q. I. Rahman and G. Schmeisser provides a comprehensive treatment of the analytic theory of polynomials, covering both fundamental results and advanced topics. The introduction sets the stage with key theorems such as the fundamental theorem of algebra, properties of symmetric and orthogonal polynomials, and tools from matrix analysis. The first part focuses on critical points in terms of zeros, exploring fundamental results, extensions, and more sophisticated methods. It delves into specific results, applications to compositions of polynomials, and polynomials with real zeros. The second part examines zeros in terms of coefficients, discussing inclusion bounds, number of zeros in intervals and domains, and general principles. The third part explores extremal properties, including growth estimates, mean values, derivative estimates, and coefficient estimates. The book concludes with references, a list of notation, and an index, making it a valuable resource for researchers and students in the field of polynomial theory.