Miller S.S., Mocanu P.T. — Differential subordinations: theory and applications :: Электронная библиотека попечительского совета мехмата МГУ

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Miller S.S., Mocanu P.T. — Differential subordinations: theory and applications

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Название: Differential subordinations: theory and applications

Авторы: Miller S.S., Mocanu P.T.

Аннотация:

Briefly defining differential subordination in the complex plane as the generalization of a differential inequality on the real line, Miller (mathematics, State U. of New York-Brockport) and Mocanu (complex analysis, Baves-Bolyai U., Cluj-Napoca, Romania) combine the basic concepts with recent results in such fields as differential equations, partial differential equations, meromorphic functions, harmonic functions, integral operators, Banach spaces, and functions of several complex variables. They describe the theory of first-order and second-order differential subordinations with some results from higher orders, simplify many of the proofs, extend the results of univalent function theory, and explain the relationship between differential subordinations and differential equations. The text is double spaced.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2000

Количество страниц: 459

Добавлена в каталог: 05.06.2005

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Предметный указатель

$<\frac{\partial \psi}{\partial \bar{w}}(w_{0})>^{\bot}$ , real orthogonal complement of $\frac{\partial \psi}{\partial \bar{w}}(w_{0})$       358
$B_{1} = B$ , open unit ball      342
$B_{2k}$ , Bernoulli numbers      415
$B_{2p}(r)$ , Reinhart domain      362
$B_{r}$ , open ball of radius r      342
$C_{n} = C_{n}(c)$ , open door constant      46
$D^{2}f(z)$ , Frechet derivative of the second order      343
$f \ast g$ , convolution of f and g      337
$f \prec F$ , subordination of f by F      4
$f_{n}(z)$ , functions with respect to symmetric points      313
$G_{\alpha, n}(z)$ , slit mapping function      251
$J(\alpha, f; z)$ , differential operator      10
$J_{n}(\alpha, f; z)$ , differential operator      314
$M(\alpha, n)$ , real constant      256
$M_{n}(\gamma)$ , real constant      279
$M_{n}(\tau)$ , real constant      296
$N(M, n, \gamma, \phi, \psi, w)$ , bound      207
$P^{n}_{r}$ , unit ball under supremum norm      342
$Q_{s}(g, \zeta)$ , subset of $\mathbb{C}^{n} \times \mathbb{C}^{n}$       369
$R_{c, n}(z)$ , open door mapping      45
$r_{\gamma}$ , radius      223
$\alpha$ -convex functions      10 173
$\alpha$ -convex integral operator      54 160 165 249—250 255 262 265
$\bar{A}$ , conjugate of matrix A      342
$\beta(x)$ , beta function      79
$\delta$ , constant      178
$\Delta$ , upper half-plane      397
$\delta_{0}$ , constant      178
$\delta_{n}(\alpha; \beta, \gamma)$ , order of starlikeness      112
$\dot{\mathbf{U}} \equiv \mathbf{U} \setminus \{0\}$ , deleted unit disk      382
$\frac{\partial f(z)}{\partial z}$ , derivative in $\mathbb{C}^{n}$       343
$\frac{\partial^{2}f(z)}{\partial z^{2}}$ , derivative in $\mathbb{C}^{n}$       343
$\frac{\partial^{2}f(z)}{\partial \bar{z}\partial z}$ , derivative in $\mathbb{C}^{n}$       343
$\lambda$ -spirallike functions      9 119 155—159
$\mathbb{C}$ , complex plane      3
$\mathbb{C}^{n}$ , space of n complex variables      342
$\mathbf{A}[f]$ , Alexander operator      11
$\mathbf{A}_{\alpha}[f]$ , $\alpha$ -convex integral operator      54
$\mathbf{A}_{\beta, \gamma}[f]$ , integral operator      302
$\mathbf{I}[f, g]$ , integral operator      167
$\mathbf{I}[f]$ , integral operator      44
$\mathbf{I}_{w}[f]$ , integral operator      227
$\mathbf{I}_{\beta, \gamma}[f]$ , integral operator      52
$\mathbf{I}_{\beta/n, \gamma/n}[f]$ , integral operator      103
$\mathbf{I}_{\gamma}[f]$ , integral operator      389
$\mathbf{k}(z)$ , Koebe function      7
$\mathbf{L}[f]$ , Libera operator      11
$\mathbf{L}_{\gamma}[f]$ , Bernardi operator      11
$\mathbf{U}$ , unit disk      3
$\mathbf{U}_{r_{0}}$ , disk of radius $r_{0}$       18
$\mathcal{A} \equiv \mathcal{A}_{1}$ , class of normalized analytic functions in $\mathbf{U}$       7
$\mathcal{A}_{n}$       7
$\mathcal{A}_{n}$ , class of normalized analytic functions in $\mathbf{U}$       7
$\mathcal{C}$ , close-to-convex functions      10
$\mathcal{D}$ , special class of analytic functions      147
$\mathcal{D}_{n}$ , special class of analytic functions      201
$\mathcal{E}(g)$ , critical points of g      356
$\mathcal{E}_{\beta, \gamma}$ , special class of analytic functions      175
$\mathcal{F}_{\beta, \gamma, n}$ , special class of analytic functions      53
$\mathcal{G}_{\beta, \gamma, n}$ , special class of analytic functions      53
$\mathcal{H} = \mathcal{H}(\mathbf{U})$ , class of analytic functions in $\mathbf{U}$       3
$\mathcal{H}$       3
$\mathcal{H}(G)$       343
$\mathcal{H}(G)$ , class of holomorphic mappings on $G\subset \mathbb{C}^{n}$       343
$\mathcal{H}[a, n]$       3
$\mathcal{H}[a, n]$ , class of normalized analytic functions in $\mathbf{U}$       3
$\mathcal{H}[\Delta]$ , class of normalized analytic functions in $\Delta$       397
$\mathcal{H}_{0} \equiv \mathcal{H}[0, 1]$ , class of normalized analytic functions in $\mathbf{U}$       3
$\mathcal{H}_{0}$       3
$\mathcal{I}_{g, \beta, \gamma}[f]$ , integral operator      173
$\mathcal{I}_{g, \gamma}[f]$ , integral operator      155
$\mathcal{I}_{g}[f]$ , integral operator      149
$\mathcal{J}(\beta, h; z)$ , differential operator      178
$\mathcal{J}_{g}[f]$ , integral operator      151
$\mathcal{K}$ , class of convex functions      8
$\mathcal{K}(\beta)$ , class of convex function of order $\beta$       8
$\mathcal{K}[\Delta]$ , class of convex function in $\Delta$       398
$\mathcal{L}(\mathbb{C}^{n}, \mathbb{C}^{m})$ , space of linear operators from $\mathbb{C}^{n}$ into $\mathbb{C}^{m}$       342
$\mathcal{P}[\Delta]$ , functions with positive imaginary part      398
$\mathcal{P}_{1} \equiv \mathcal{P}$ , special class of analytic functions      201
$\mathcal{P}_{n}$ , special class of analytic functions      201
$\mathcal{Q}$ , class of univalent functions on $\bar{\mathbf{U}} \setminus \mathbf{E}(q)$       21
$\mathcal{Q}[\Delta]$ , class of univalent functions on $\bar{\Delta} \setminus \mathbf{E}(q)$       403
$\mathcal{R}$ , class of bounded turning functions      292
$\mathcal{R}(\alpha)$ , special class of analytic functions      300
$\mathcal{R}_{\alpha}$ , special class of analytic functions      307
$\mathcal{S}$ , class of univalent functions      7
$\mathcal{S}[\Delta]$ , class of univalent functions in $\Delta$       397
$\mathcal{S}^{*}$ , class of starlike functions      8
$\mathcal{S}^{*}(\beta)$ , class of starlike functions of order $\beta$       9
$\mathcal{S}^{*}[\beta]$ , stongly starlike functions of order $\beta$       263
$\mathcal{S}^{*}[\Delta]$ , class of starlike functions in $\Delta$       399
$\mathcal{S}^{*}_{n} \equiv \mathcal{S}^{*} \cap \mathcal{A}_{n}$ , special class of starlike functions      112
$\mathcal{S}^{*}_{n}(\alpha) \equiv \mathcal{S}^{*}(\alpha) \cap \mathcal{A}_{n}$ , special class of starlike functions      112
$\mathcal{S}^{s}_{n}$ , starlike with respect to n-symmetric points      313
$\mathcal{S}^{\lambda}$ , $\lambda$ -spirallike functions      9
$\mathcal{S}^{\lambda}(\rho)$ , class of $\lambda$ -spirallike functions of order $\rho$       155
$\mathfrak{M}_{\alpha}$ , class of $\alpha$ -convex functions      10
$\nu^{'}$ , transpose of vector $\nu$       342
$\omega(\rho)$ , order of spirallikeness      156
$\Phi(a, c; z)$ , confluent hypergeometric function      5
$\psi(r, s, t; z)$ , admissible function      14
$\Psi[M]$ , class of admissible functions      365
$\Psi_{n}[h, q]$ , class of admissible functions      27
$\Psi_{n}[M, a]$ , class of admissible functions      33
$\Psi_{n}[\Omega, M, a]$ , class of admissible functions      33
$\Psi_{n}[\Omega, q]$ , class of admissible functions      27
$\Psi_{n}\{a\}$ , class of admissible functions      35
$\Psi_{n}\{\Omega, a\}$ , class of admissible functions      35
$\Psi_{\Delta}[\Omega, q]$ , class of admissible functions      406
$\Sigma = \Sigma_{0}$ , class of meromorphic functions      382
$\Sigma^{*}(\alpha) = \Sigma^{*}_{0}(\alpha)$ , class of meromorphic functions      383
$\Sigma^{*}_{n} = \Sigma^{*}_{n}(0)$ , class of meromorphic functions      383
$\Sigma^{*}_{n}(\alpha)$ , class of meromorphic functions      383
$\Sigma_{n}$ , class of meromorphic functions      382
$\widehat{\mathcal{S}}$ , class of spirallike functions      9
$\|z\| = <z, z>^{1/2}$ , Euclidean norm      347
$\|\cdot\|$ , arbitrary norm      342
$_{1}F_{1}(a, c; z)$ , confluent hypergeometric function      5
$_{2}F_{1}(a, b, c; z)$ , Gaussian hypergeometric function      6
(a, n)-dominant      16 320
(a, n)-dominant, dominant      16
(a, n)-solution      16 320
(a, n)-solution, solution      16
<z, w>, inner product      347
A', transpose of matrix A      342
Admissibility condition      27 29 35—40 42—43 71
Admissibility condition for a disk of radius M      33—34 331—332
Admissibility condition for a right half-plane      34—35
Admissibility condition for Banach spaces      410—411
Admissibility condition for functions defined in the upper half-plane      405—407
Admissibility condition for functions in $\mathbb{C}^{n}$       365 370 373
Admissible functions      15 27—28 33 35 320 406 410
Admissible mappings      370
Alexander operator      11 174 299 307—313
Alexander's Theorem      375
Averaging integral operator      215 216—231 289—291 414
Averaging integral operator, nonlinear      227—231
Banach algebra      375
Banach spaces      408—414
Bernardi operator      11 67 278—282 299—301 305—306 317—318
Bernoulli functions      415 416—419
Bernoulli numbers      415 418
Best (a, n)-dominant      16 32 71—72 76—77 83
Best dominant      16 31—32 95 320
Best dominant for functions in $\mathbb{C}^{n}$       372 373 375—377
beta function      78—79
Biholomorphic mapping, locally biholomorphic      343 352 355 357—359 362
Biholormorphic mapping      343 355 369 371 373—375 377
Bilinear operator      343

Bilinear transformation      103 107 111 241
Bounded functions      36—37 189—195 207—214 331—333
Briot — Bouquet differential equations      80 81—91 95 179
Briot — Bouquet differential equations, Integral version      82
Briot — Bouquet differential equations, Univalent solutions      88—91 97
Briot — Bouquet differential subordinations      80 81—102
Briot — Bouquet differential subordinations for functions defined in the upper half-plane      407
Briot — Bouquet differential subordinations, applications for univalent functions      103—119
Briot — Bouquet differential subordinations, generalized      120—144
Briot — Bouquet differential subordinations, integral version      84 96 99
Close-to-convex functions      10—11 121 171—173
Clunie — Jack lemma      20
co E, convex hull      8
Complete circular domain      343—344
Confluent hypergeometric functions      5 232—239
Confluent hypergeometric functions, convexity      234
Confluent hypergeometric functions, univalence      234
Continuous linear operators      342
Convex domain      8 93 121 133
Convex domain in $\mathbb{C}^{n}$       375 377
Convex functions      8 70—73 76—78 81—85 89—92 95 97—98 101 103 105 108 115 117—118 120—127 130—139 142 144—145 166—173 183 195—200 216—217 231 234—235 237 240 242—243 278—282 289—291 415 417—419
Convex functions of order $\beta$       8
Convex functions, conditions for      283
Convex functions, odd      105
Convex hull      8 201 215—216
Convex null sequence      337
Convexity of Bernoulli functions      415—419
Convolution product      337 338—339
Critical point      356
Curvature      23 355
Df(z), Frechet derivative      343
Differentiable boundary $\partial E$ of class $C^{2}$       355 356
Differential inequalities      2 13 44 195
Differential inequalities for Banach spaces      411—414
Differential inequalities for functions in $\mathbb{C}^{n}$       366—367
Differential subordinations      1—2 13—15 16 17—18
Differential subordinations for bounded functions      36—37 189—195 333
Differential subordinations for functions defined in the upper-half plane      397—408
Differential subordinations for functions in $\mathbb{C}^{n}$       371—372 373—377
Differential subordinations for functions with positive real part      38—43 188—189
Differential subordinations, first-order      69—145
Differential subordinations, first-order, linear      69—80 201—203
Differential subordinations, Goluzin's      70 75 135 174
Differential subordinations, nonlinear      76—78
Differential subordinations, nth-order      321
Differential subordinations, second-order      13—15 16 17—43 187—244
Differential subordinations, second-order, linear      43 187—201 203—206
Differential subordinations, solution of      16 320
Differential subordinations, solution of, for functions in $\mathbb{C}^{n}$       371
Differential subordinations, strong      145
Differential subordinations, third-order      322 323—334
Dominant      16 320
Dominant for functions in $\mathbb{C}^{n}$       372 375—377
Dominant, best dominant      16 31—33 95 320
Dominant, best dominant for functions in $\mathbb{C}^{n}$       372 375—377
Dual space      408
E, set in complex plane      222
Erf(z), error function      6
Error function      6 222 231 258 273
Euclidean inner product      347 351
Euclidean norm      347 348 351—354 356 365—367
Euler differential operator      39 334—335 336 337—340
F(a, b, c; z), Gaussian hypergeometric function      6
First-order differential subordinations      69—145
First-order linear differential subordinations      69—80 201—203
Frechet derivative      343
Functions analytic in the upper half-plane      397 398—408
Functions analytic in the upper half-plane, convexity      398
Functions analytic in the upper half-plane, starlikeness      399
Functions analytic in the upper half-plane, univalence      397
Functions with bounded turning      292 293—313
Functions with positive imaginary part      398
Functions with positive real part      38—43 188—189 201—206
Fundamental Lemmas      18—26 322 325
Fundamental Lemmas for a disk of radius M      24—25 328—330
Fundamental Lemmas for a right half-plane      25—26 333—334
Fundamental Lemmas for Banach spaces      409—410
Fundamental Lemmas for functions defined in the upper half-plane      402—405
Fundamental Lemmas for functions in $\mathbb{C}^{n}$       344—365
G(X, Y), class of admissible functions      410
Gaussian curvature      355
Gaussian hypergeometric functions      6 231 239—243
Gaussian hypergeometric functions, convexity      240—242
Generalized Briot — Bouquet differential subordinations      120—144
Goluzin's differential subordination      70 75 135 174
Goluzin's Theorem      17 62 174
Hadamard product      337 338—339
Hahn — Banach theorem      349 409
Hallenbeck — Rusheweyh Theorem      18 71 175
Harmonic functions      379—382
Higher order differential subordinations      319—340
Hilbert space      351
Holomorphic mapping      342—343
Holomorphic vector-valued functions      408—414
Hydrodynamic normalization      397
Hypergeometric functions      5—7 109—113 231—243 289—291
Hypersurface      355 357 359 362
I, identity operator in $\mathcal{L}(\mathbb{C}^{n}, \mathbb{C}^{n})$       343
Incomplete beta function      242
Integral Existence Theorem      44 50—51 52—56 85—88
Integral operators, analytic      10—12 82 96 145—185 201—206 214—231 278—282 289—291 298—313 317—318
Integral operators, analytic, $\alpha$ -convex      54 160 165 249—250 255 262 265
Integral operators, analytic, Alexander      11 174 299 307—313 375
Integral operators, analytic, Averaging      215 216—231 289—291 415
Integral operators, analytic, Bernardi      11 67 278—282 299—301 305—306 317—318
Integral operators, analytic, Libera      11 63—66 106 116—117 145—146 173 279 299 301
Integral operators, meromorphic      389—396
Jack's Lemma      20
Koebe function      7
Kummer's differential equation      5 233
L(z, t), subordination (Loewner) chain      4
Lagrange multipliers      352—355
Libera operator      11 63—66 106 116—117 145—146 173 279 299 301
Linear functional      408—409 413
Local defining function      355
Locally biholomorphic      343 352 355 357—359 362
M(a, b), real constant      240
MacGregor's Theorem      115
Marx — Strohhaecker differential subordination system      263
Marx — Strohhaecker Theorem      9 44 56—61 68 104 115
Mean-value operator      215
Meromorphic functions      382 383—396
N(a), real constant      234
N(A, M, P, Q, w), bound      213
N(M, n, A, B, C, D), bound      195
N(M, n, B, C, D), bound      191
Odd convex function      105
Open Door lemma      44—45 46—49 85 135 140 252—254 387—388
Open Door mapping      45—46
Order of starlikeness      106 112 115—116 118 163—171
Outward (outer) normal      23 93 123 133 355 359 362
Polydisk      342 345—347 374—377
Problem 1      15 17 320 321
Problem 2      15 17 320 321
Problem 3      15 17 95 131 320 321
Problem of S. Miller      335—340
Q(g), subset of $\mathbb{C}^{n} \times \mathbb{C}^{n}$       370
R(a), real constant      238
Radius of convexity      419
Radius of starlikeness      270—273
Real orthogonal complement      358
Regular value of a function      356
Reinhardt domain      362
Robertson's Theorem      284 285—288
Robinson's theorem      17—18 70 73 184 199
Schwarz lemma      19 127 208 283 344 348 368 399
Schwarzian derivative      244 245—247
Second-order differential subordinations      13—15 16 17—43 187—244
Second-order linear differential subordinations      43 187—201 203—206
Solution of a differential subordination      16 320

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