Sheil-Small T. — Complex polynomials :: Электронная библиотека попечительского совета мехмата МГУ

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Sheil-Small T. — Complex polynomials

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Название: Complex polynomials

Автор: Sheil-Small T.

Аннотация:

Complex Polynomials explores the geometric theory of polynomials and rational functions in the plane. Early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology, and analysis. Throughout the book, the author introduces a variety of ideas and constructs theories around them, incorporating much of the classical theory of polynomials as he proceeds. These ideas are used to study a number of unsolved problems. Several solutions to problems are given, including a comprehensive account of the geometric convolution theory.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 428

Добавлена в каталог: 07.12.2009

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Предметный указатель

$S(\alpha,\beta)$ class      296
class      271
$T(1,\beta)$ classification      273—275
class      302
$T(m,\beta)$ class      282
$T(\alpha,\beta)$ class      269 289
$T_{0}(\alpha,\beta)$ class      289
abc theorem      370
Abel      23
Affine mappings/transformations      5 73
Alander, M.      343
Algebraic function      20
Algebraic polynomial      1
Algebraic tract      103
Analytic continuation lemma      204
Analytic polynomial      1
Angular separation of critical points and univalence      241 243
Angular separation of zeros, convolution theorem      254
Angular separation of zeros, Kaplan class condition      248
Angular separation of zeros, sufficient condition      257
Angular separation of zeros, Suffridge condition      249
Angular separation of zeros, third condition      251
Annihilation property      129
Annular region for zeros      202
Apolar polynomials      180
Apolarity theorem      181
Apolarity, weak      182
Approximate identity      147
Approximation theorem      258
Argument      29
Argument and tangent of curve      387—390
Argument of polynomial with zeros on circle      243
Argument principle      23 27 44 84
Argument principle for Jordan domain      45
Argument principle for meromorphic functions      45
Asymptotic values      15—29
Asymptotic values of finitely valent mapping      90
Asymptotic values of Pinchuk mapping      93—100
Bernstein's inequality      152 186
Bernstein's inequality for trigonometric polynomials      155
Bezout's theorem      4 84
Bisector theorem      200 215 216
Blaschke product      60 193 231 233 351 373
Blaschke product and convex curve      382—387
Blaschke product and harmonic mappings      377—382
Blaschke product and Smale's conjecture      365
Blaschke product and step functions on convex curves      394
Blaschke product, Walsh theorem      377
Blow-up method      13
Bocher — Walsh theorem      192
Bojanov — Rahman — Szynal method      213 226
Bojanov — Rahman — Szynal theorem      215
Bolzano — Weierstrass property      128
Borsuk — Ulam theorem      79
Borwein — Erdelyi extensions      195—196
Borwein — Erdelyi inequality      193
Bound preserving g.c. operator      170
Bound preserving operator      140
Boundary inequality      224
Boundary rotation of curve      389
Bounded boundary rotation      298
Bounded boundary rotation and univalence      389
Bounded convolution operators      142
Bounded convolution operators, integral representation      168
Bounded operator      139
Bounded operator condition      140
Bounded variation      159 221
Branch of algebraic function      21
Branch of inverse function      57
Branch of logarithm      30
Branch of square root      31
Branch point      306
Brannan, D.A.      295
Brouwer's fixed point theorem      48—49
Brouwer's theorem on plane domains      40
Caratheodory's lemma      162
Cauchy — Riemann equations      266
Cauchy's integral formula      128
Cauchy's theorem      24 31 36
Cesaro mean      156—157
Cesaro mean of Kaplan class      301
Cesaro polynomial      300
Chain      36
Change in argument      29
Choquet's theorem      287
circle      73
Circular region      178
Circular region, zeros and critical points      187
Close-to-convex      405
Close-to-convex domain      242
Close-to-convex function      241—242
Closed convex hull      287
Closed under convolution      174
Closing gap problem      112—115
Clunie — Jack lemma      266
Clunie,J.G.      171 266
Cluster set      58
Coefficient      129
Coefficient and maximum modulus, zeros on unit circle      239
Coefficient bound, Kaplan class      294
Coefficient bound, Suffridge class      260
Cohn's reduction lemma      372
Coincidence theorem      47
Compact open topology      128
Comparison principle      19
complex curve      118
Complex form of Bezout's theorem      11
Complex polynomial      1
Composition remark      363
Conical surface      117
Conjugate pair      1
Conjugate polynomial      153
Connected set      30
Constant jacobian      87 105—124
Containment condition, Suffridge convolution theorem      257
Containment lemma      266
Containment property      335
Containment theorem for $T(m,\beta)$       286
Containment theorem for $T_{0}(\alpha,\beta)$       292
Continuation of inverse function      57
Continuity at infinity      83
Continuity of zeros      55
Continuous change in argument      29
Continuous logarithm      28
Convex curve      382
Convex curve and step function      392
Convex domain      78
Convex function      242
Convex function condition      275
Convex harmonic mapping, univalence theorem      391—392
Convex hull      25
Convex in direction of real axis      69
Convex mapping      260
Convex polygon: harmonic univalence criterion      400—401
Convex sequence      157
Convex set      24
Convexity lemma      162
Convexity preserving      141 174 279
Convexity preserving, functions      144
Convexity preserving, g.c. operator      170
Convexity preserving, operators and positive trigonometric polynomials      148
Convexity theory      286—288
Convexity, of level region      360
Convolution      172
Convolution condition, Suffridge angular separation      253
Convolution containment theorem      280
Convolution formula      127
Convolution inequality      279
Convolution inequality for $T(m,\beta)$       284
Convolution inequality for $T_{0}(\alpha,\beta)$       291

Convolution lemma      269
Convolution of harmonic functions      126
Convolution operator      132
Convolution operator, extension property      141—143
Convolution operator, extension theorem      142
Convolution theorem      279
Convolution theorem for $S(\alpha,\beta)$       291
Convolution theorem for $T(m,\beta)$       285
Convolution theorem for $T_{0}(\alpha,\beta)$       292
Convolution theorem, zeros on unit circle      254
Cordova — Ruscheweyh method      361
Craven — Csordas — Smith conjecture      306
Craven — Csordas — Smith conjecture, proof when all critical points real      311
Craven — Csordas — Smith conjecture, proof when exactly 2 non-real zeros      308—310
Craven, T.      306
Critical circle      225
Critical point      24 186 187
Critical point inequality      359
Critical point of Blaschke product      374
Critical point of logarithmic derivative      306
Critical point of logarithmic existence of non-real      307
Critical point of rational function      190
Critical point of real polynomial      305
Critical point of real rational functions, theorem on number of non-real      323—324
Critical point of strongly real rational function      321
Critical point on unit circle, univalence theorem      243
Critical Point Theorem      334
Critical set      66
Cross-cap      123
Csordas, G.      306
CYCLE      36
Cylindrical surface      117
De la Vallee Poussin means      160 301
Degree estimate, Jacobian conjecture      102—104
Degree of function on curve      29
Degree of function on cycle      36
Degree of function on rectilinear polygon      35
Degree of harmonic polynomial      3
Degree of Pinchuk mapping      93
Degree of polynomial      1
Degree of real analytic polynomial      4 83
Degree of trigonometric polynomial      144
Degree principle      31 37
Degree principle for circle      34
Degree principle for homotopic curves      41
Degree principle for rational function      37
Degree principle for simply-connected domain      39
Degree principle for starlike domain      33
Degree principle for triangle      32
Descartes' Rule of Signs      317
Descartes' rule of signs, polynomials with all real zeros      318
Dirichlet means      158
Domain      30
Dual      172
Dual of T(1,1)      271
Duality      172
Duality principle      286
Duality theorem for $T(1,\beta)$       275
Duality theorem for $T(m,\beta)$       282
Duality theorem for $T_{0}(\alpha,\beta)$       289
Duality, sums and products      23
Elementary mappings      109
ellipse      73
Enestroem — Kakeya theorem      281
Enestroem — Kakeya theorem, Ruscheweyh generalisation      281
Entire functions of finite order      327
Entire functions with real zeros      343
Erdoes, P.      407
Essential singularity      44
Euler gamma function      239
Extending boundary inequalities      224
Extension property      141
Exterior of curve      34
Extremal distance theorem      221
Extremal polynomial      223
Extremal polynomial, Suffridge      251
Extreme point      287
Extreme point of Kaplan class      293
Fejer means      147
Fejer — Riesz inequality      129
Fermat's last theorem for polynomials      371
Finite limiting value and asymptotic value      19
Finite product duality lemma      175 265 270 282
Finite valence      84
Finite valence and N-fold mapping      390
Finite valence condition      390
Fixed point      48
Fourier coefficient      129
Fourier coefficient of N-fold mapping      391
Fourier series      129
Function of bounded boundary rotation      298
Fundamental group      42
Fundamental group of punctured plane      43
Fundamental theorem of algebra      1 22 26 49—50
g.c. kernel      169
g.c. operator      168
Galois      23
Gamma function      239
Gauss — Lucas Theorem      25 185 189 201
Generalised convolution operator      168 279
Genus of entire function      344
Global homeomorphism condition      58
Global homeomorphism, Jordan domain condition      59
Goodman — Rahman — Ratti theorem      207
Goursat method      32
Grace class      174 253
Grace theorem for rational functions      263
Grace — Heawood theorem      198
Grace — Szegoe theorem      173
Grace — Szegoe theorem, circular regions      179
Grace — Szegoe theorem, linear functionals      177
Grace — Szegoe theorem, proof      176—177
Grace's apolarity theorem      181
Grace's theorem      216 263 282
Grace's theorem, additional forms      183—184
Grace's theorem, applied to Ilieff — Sendov conjecture      208—211
Graph of surface      101
Greatest common divisor      106
Hadamard representation of entire function      344
Harmonic extension      129
Harmonic extension of step function      395
Harmonic extension, containment theorem      396
Harmonic extension, univalence criterion      399
Harmonic function      126
Harmonic function at non-isolated zero      53
Harmonic function, convolution      126
Harmonic function, locally 1-1      64
Harmonic mapping and Blaschke product      377—382
Harmonic multiplication operator      135
Harmonic multiplication operator, class $\mathcal{M}$       137
Harmonic multiplication representation      135
Harmonic polynomial      3 50 82
Harmonic polynomial on critical set      66—72
Harmonic polynomial, degree 2      73—78
Harmonic polynomial, theorem on valence      72
Harmonic polynomial, valence near critical set      70—71
Harmonic polynomial, zeros      51 54
Harmonic series      126
Hayman, W.K.      206
Heine — Borel theorem      58
Helly selection theorem      146
Highest common factor 2      110—111
Highest common factor 3      115—117
Hilbert's inequality      130
Homeomorphism      57 81
Homeomorphism of sphere      80
Homeomorphism of unit disc      60
Homeomorphism, condition at infinity      57
Homologous      36
Homology group      42

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