Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties

Introduction 
1.1 The fundamental theorem of algebra 
1.2 Symmetric polynomials 
1.3 The continuity theorem 
1.4 Orthogonal polynomials: general properties 
1.5 The classical orthogonal polynomials 
1.6 Harmonic and subharmonic functions 
1.7 Tools from matrix analysis 
1.8 Notes 

I CRITICAL POINTS IN TERMS OF ZEROS 
2 Fundamental results on critical points 71 
2.1 Convex hulls and the Gauss-Lucas theorem 71 
2.2 Extensions of the Gauss-Lucas theorem 75 
2.3 A verage distances from a line or a point 78 
2.4 Real polynomials and Jensen's theorem 85 
2.5 Extensions of Jensen's theorem 88 
2.6 Notes 91 

3 More sophisticated methods 96 
3.1 Circular domains and polar derivative 96 
3.2 Laguerre's theorem, its variants, and applications 98 
3.3 Apolarity 102 
3.4 Grace's theorem and equivalent forms 107 
3.5 Notes 114 

4 More specific results on critical points 117 
4.1 Products and quotients of polynomials 117 
4.2 Derivatives of reciprocals of polynomials 121 
4.3 Complex analogues of Rolle's theorem 125 
4.4 Bounds for some of the critical points 129 
4.5 Converse results 132 
4.6 Notes 137 

5 Applications to compositions of polynomials 141 
5.1 Linear combination of rational functions 142 
5.2 Complex analogues of the intermediate-value theorem 143 
5.3 Linear combination of derivatives: Walsh's approach 148
Linear combination of derivatives: recursive approach 
Multiplicative composition: Schur-Szego approach 
Multiplicative composition: Laguerre's approach 
Multipliers preserving the reality of zeros 
Notes 

6 Polynomials with real zeros 
6.1 The span of a polynomial 
6.2 Largest zero and largest critical point 
6.3 Interlacing and the Hermite-Biehler theorem 
6.4 Consecutive zeros and critical points 
6.5 Refinement of Rolle's theorem 
6.6 Notes 

7 Conjectures and solutions 
7.1 A conjecture of Popoviciu 
7.2 A conjecture of Smale 
7.3 The conjecture of Sendov 
7.4 Notes 

II ZEROS IN TERMS OF COEFFICIENTS 
8 Inclusion of all zeros 
8.1 The Cauchy bound and its estimates 
8.2 Various refinements 
8.3 Multipliers and the Enestrom-Kakeya theorem 
8.4 More general expansions 
8.5 Orthogonal expansions with real coefficients 
8.6 Alternative approach by matrix methods 
8.7 Notes 

9 Inclusion of some of the zeros 
9.1 Inclusions in terms of a norm 
9.2 Pellet's theorem and its consequences 
9.3 Bounds in terms of some of the coefficients 
9.4 Orthogonal expansions with real coefficients 
9.5 The Landau-Montel problem 
9.6 Notes 

10 Number of zeros in an interval 
10.1 The Budan-Fourier theorem and Descartes' rule 
10.2 Exact count under a side condition 
10.3 Extensions to pairs of conjugate zeros 
10.4 More general expansions 
10.5 Exact count by Sturm sequences 
10.6 Exact count via quadratic forms 
10.7 Notes 

11 Number of zeros in a domain 
11.1 General principles 
11.2 Number of zeros in a sector 
11.3 Number of zeros in a half-plane 
11.4 The Routh-Hurwitz problem 
11.5 Number of zeros in a disc 
11.6 Distribution of zeros 
11.7 Notes 

III EXTREMAL PROPERTIES
12 Growth estimates 
12.1 The Bernstein-Walsh lemma 
12.2 The convolution method 
12.3 The method of functionals 
12.4 Various refinements 
12.5 Local behaviour 
12.6 Extensions to functions of exponential type 
12.7 Notes 

13 Mean values 
13.1 Mean values on circles 
13.2 A class of linear operators 
13.3 Mean values on the unit interval 
13.4 Notes

14 Derivative estimates on the unit disc 
14.1 Bernstein's inequality and generalizations 
14.2 Refinements 
14.3 Conditions on the coefficients 
14.4 Conditions on the zeros 
14.5 Some special operators 
14.6 Inequalities involving mean values 
14.7 Notes 

15 Derivative estimates on the unit interval 
15.1 Inequalities of S. Bernstein and A. Markov 
15.2 Extensions to higher-order derivatives 
15.3 Two other extensions 
15.4 Dependence of the bounds on the zeros 
15.5 Some special classes 
15.6 LP analogues of Markov's inequality 
15.7 Notes

16 Coefficient estimates 
16.1 Polynomials on the unit circle 
16.2 Coefficients of real trigonometric polynomials 
16.3 Polynomials on the unit interval 
16.4 Notes  

References 

List of notation 

Index