Analytic Theory of PolynomialsThis 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. |
Contents
Introduction | 1 |
Fundamental results on critical points | 71 |
More sophisticated methods | 96 |
More specific results on critical points | 117 |
Applications to compositions of polynomials | 141 |
Polynomials with real zeros | 184 |
Conjectures and solutions | 212 |
Inclusion of all zeros | 243 |
Number of zeros in a domain | 357 |
Growth estimates | 403 |
Mean values | 460 |
Derivative estimates on the unit disc | 508 |
Derivative estimates on the unit interval | 566 |
Coefficient estimates | 636 |
References | 681 |
List of notation | 729 |
Common terms and phrases
algebraic analytic apply Bojanov Cauchy bound Cauchy index Chebyshev polynomial closed unit disc conjecture constant contains convex Corollary critical points deduce defined denote do(x entire function equality holds equation estimate f and g f of degree f₁ following result formula Furthermore Gauss-Lucas theorem half-plane Hence Hurwitz implies inequality integral interval Lemma Let f Let f(z linear matrix max f(z modulus monic polynomial multiplicity n₁ nomial non-negative number of zeros obtain open unit disc orthogonal polynomials point of f polynomial f polynomial of degree polynomials with real positive number Proof Let proved Rahman rational function real coefficients real numbers real zeros right-hand side Rolle's theorem satisfies Schmeisser sector sequence shows statement suppose Szegő trigonometric polynomial unit circle upper half-plane vanish zeros of f λη Σαν ドー レー