Analytic Theory of Polynomials Q. I. Rahman Universite de Montreal and G. Schmeisser Universitat Erlangen -Niirnberg CLARENDON PRESS-OXFORD 2002
1 Introduction 1 1.1 The fundamental theorem of algebra 2 1.2 Symmetric polynomials 6 1.3 The continuity theorem 9 1.4 Orthogonal polynomials: general properties 13 1.5 The classical orthogonal polynomials 19 1.6 Harmonic and subharmonic functions 30 1.7 Tools from matrix analysis 50 1.8 Notes 61 I CRITICAL POINTS IN TERMS OF ZEROS 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 Average 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 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 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 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
xii 5.4 Linear combination of derivatives: recursive approach 151 5.5 Multiplicative composition: Schur-Szego approach 158 5.6 Multiplicative composition: Laguerre's approach 164 5.7 Multipliers preserving the reality of zeros 172 5.8 Notes 177 6 Polynomials with real zeros 184 6.1 The span of a polynomial 184 6.2 Largest zero and largest critical point 189 6.3 Interlacing and the Hermite-Biehler theorem 196 6.4 Consecutive zeros and critical points 201 6.5 Refinement of Rolle's theorem 203 6.6 Notes 209 7 Conjectures and solutions 212 7.1 A conjecture of Popoviciu 212 7.2 A conjecture of Smale 214 7.3 The conjecture of Sendov 224 7.4 Notes 237 II ZEROS IN TERMS OF COEFFICIENTS 8 Inclusion of all zeros 243 8.1 The Cauchy bound and its estimates 243 8.2 Various refinements 249 8.3 Multipliers and the Enestrom-Kakeya theorem 252 8.4 More general expansions 255 8.5 Orthogonal expansions with real coefficients 259 8.6 Alternative approach by matrix methods 263 8.7 Notes 270 9 Inclusion of some of the zeros 275 9.1 Inclusions in terms of a norm 275 9.2 Pellet's theorem and its consequences 284 9.3 Bounds in terms of some of the coefficients 290 9.4 Orthogonal expansions with real coefficients 294 9.5 The Landau-Montel problem 304 9.6 Notes 309 10 Number of zeros in an interval 315 10.1 The Budan-Fourier theorem and Descartes' rule 315 10.2 Exact count under a side condition 320 10.3 Extensions to pairs of conjugate zeros 323 10.4 More general expansions 330 10.5 Exact count by Sturm sequences 335 10.6 Exact count via quadratic forms 339 10.7 Notes 350
xiii 11 Number of zeros in a domain 357 11.1 General principles 357 11.2 Number of zeros in a sector 359 11.3 Number of zeros in a half-plane 362 11.4 The Routh-Hurwitz problem 366 11.5 Number of zeros in a disc 374 11.6 Distribution of zeros 384 11.7 Notes 392 III EXTREMAL PROPERTIES 12 Growth estimates 403 12.1 The Bernstein-Walsh lemma 403 12.2 The convolution method 408 12.3 The method of functionals 416 12.4 Various refinements 428 12.5 Local behaviour 447 12.6 Extensions to functions of exponential type 454 12.7 Notes 456 13 Mean values. 460 13.1 Mean values on circles 460 13.2 A class of linear operators 468 13.3 Mean values on the unit interval 486 13.4 Notes 504 14 Derivative estimates on the unit disc 508 14.1 Bernstein's inequality and generalizations 508 14.2 Refinements 515 14.3 Conditions on the coefficients 526 14.4 Conditions on the zeros 532 14.5 Some special operators 538 14.6 Inequalities involving mean values 552 14.7 Notes 557 15 Derivative estimates on the unit interval 566 15.1 Inequalities of S. Bernstein and A. Markov 566 15.2 Extensions to higher-order derivatives 568 15.3 Two other extensions 577 15.4 Dependence of the bounds on the zeros 585 15.5 Some special classes 601 15.6 L p analogues of Markov's inequality 611 15.7 Notes 622
xiv 16 Coefficient estimates 636 16.1 Polynomials on the unit circle 636 16.2 Coefficients of real trigonometric polynomials 653 16.3 Polynomials on the unit interval 672 16.4 Notes 677 References 681 List of notation 729 Index 733