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

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Ќазвание: Differential subordinations: theory and applications

јвторы: Miller S.S., Mocanu P.T.

јннотаци€:

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: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2000

 оличество страниц: 459

ƒобавлена в каталог: 05.06.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный 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 249Ч250 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 155Ч159
$\mathbb{C}$, 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 35Ч40 42Ч43 71
Admissibility condition for a disk of radius M      33Ч34 331Ч332
Admissibility condition for a right half-plane      34Ч35
Admissibility condition for Banach spaces      410Ч411
Admissibility condition for functions defined in the upper half-plane      405Ч407
Admissibility condition for functions in $\mathbb{C}^{n}$      365 370 373
Admissible functions      15 27Ч28 33 35 320 406 410
Admissible mappings      370
Alexander operator      11 174 299 307Ч313
Alexander's Theorem      375
Averaging integral operator      215 216Ч231 289Ч291 414
Averaging integral operator, nonlinear      227Ч231
Banach algebra      375
Banach spaces      408Ч414
Bernardi operator      11 67 278Ч282 299Ч301 305Ч306 317Ч318
Bernoulli functions      415 416Ч419
Bernoulli numbers      415 418
Best (a, n)-dominant      16 32 71Ч72 76Ч77 83
Best dominant      16 31Ч32 95 320
Best dominant for functions in $\mathbb{C}^{n}$      372 373 375Ч377
beta function      78Ч79
Biholomorphic mapping, locally biholomorphic      343 352 355 357Ч359 362
Biholormorphic mapping      343 355 369 371 373Ч375 377
Bilinear operator      343
Bilinear transformation      103 107 111 241
Bounded functions      36Ч37 189Ч195 207Ч214 331Ч333
Briot Ч Bouquet differential equations      80 81Ч91 95 179
Briot Ч Bouquet differential equations, Integral version      82
Briot Ч Bouquet differential equations, Univalent solutions      88Ч91 97
Briot Ч Bouquet differential subordinations      80 81Ч102
Briot Ч Bouquet differential subordinations for functions defined in the upper half-plane      407
Briot Ч Bouquet differential subordinations, applications for univalent functions      103Ч119
Briot Ч Bouquet differential subordinations, generalized      120Ч144
Briot Ч Bouquet differential subordinations, integral version      84 96 99
Close-to-convex functions      10Ч11 121 171Ч173
Clunie Ч Jack lemma      20
co E, convex hull      8
Complete circular domain      343Ч344
Confluent hypergeometric functions      5 232Ч239
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 70Ч73 76Ч78 81Ч85 89Ч92 95 97Ч98 101 103 105 108 115 117Ч118 120Ч127 130Ч139 142 144Ч145 166Ч173 183 195Ч200 216Ч217 231 234Ч235 237 240 242Ч243 278Ч282 289Ч291 415 417Ч419
Convex functions of order $\beta$      8
Convex functions, conditions for      283
Convex functions, odd      105
Convex hull      8 201 215Ч216
Convex null sequence      337
Convexity of Bernoulli functions      415Ч419
Convolution product      337 338Ч339
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      411Ч414
Differential inequalities for functions in $\mathbb{C}^{n}$      366Ч367
Differential subordinations      1Ч2 13Ч15 16 17Ч18
Differential subordinations for bounded functions      36Ч37 189Ч195 333
Differential subordinations for functions defined in the upper-half plane      397Ч408
Differential subordinations for functions in $\mathbb{C}^{n}$      371Ч372 373Ч377
Differential subordinations for functions with positive real part      38Ч43 188Ч189
Differential subordinations, first-order      69Ч145
Differential subordinations, first-order, linear      69Ч80 201Ч203
Differential subordinations, Goluzin's      70 75 135 174
Differential subordinations, nonlinear      76Ч78
Differential subordinations, nth-order      321
Differential subordinations, second-order      13Ч15 16 17Ч43 187Ч244
Differential subordinations, second-order, linear      43 187Ч201 203Ч206
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 323Ч334
Dominant      16 320
Dominant for functions in $\mathbb{C}^{n}$      372 375Ч377
Dominant, best dominant      16 31Ч33 95 320
Dominant, best dominant for functions in $\mathbb{C}^{n}$      372 375Ч377
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 351Ч354 356 365Ч367
Euler differential operator      39 334Ч335 336 337Ч340
F(a, b, c; z), Gaussian hypergeometric function      6
First-order differential subordinations      69Ч145
First-order linear differential subordinations      69Ч80 201Ч203
Frechet derivative      343
Functions analytic in the upper half-plane      397 398Ч408
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 293Ч313
Functions with positive imaginary part      398
Functions with positive real part      38Ч43 188Ч189 201Ч206
Fundamental Lemmas      18Ч26 322 325
Fundamental Lemmas for a disk of radius M      24Ч25 328Ч330
Fundamental Lemmas for a right half-plane      25Ч26 333Ч334
Fundamental Lemmas for Banach spaces      409Ч410
Fundamental Lemmas for functions defined in the upper half-plane      402Ч405
Fundamental Lemmas for functions in $\mathbb{C}^{n}$      344Ч365
G(X, Y), class of admissible functions      410
Gaussian curvature      355
Gaussian hypergeometric functions      6 231 239Ч243
Gaussian hypergeometric functions, convexity      240Ч242
Generalized Briot Ч Bouquet differential subordinations      120Ч144
Goluzin's differential subordination      70 75 135 174
Goluzin's Theorem      17 62 174
Hadamard product      337 338Ч339
Hahn Ч Banach theorem      349 409
Hallenbeck Ч Rusheweyh Theorem      18 71 175
Harmonic functions      379Ч382
Higher order differential subordinations      319Ч340
Hilbert space      351
Holomorphic mapping      342Ч343
Holomorphic vector-valued functions      408Ч414
Hydrodynamic normalization      397
Hypergeometric functions      5Ч7 109Ч113 231Ч243 289Ч291
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 50Ч51 52Ч56 85Ч88
Integral operators, analytic      10Ч12 82 96 145Ч185 201Ч206 214Ч231 278Ч282 289Ч291 298Ч313 317Ч318
Integral operators, analytic, $\alpha$-convex      54 160 165 249Ч250 255 262 265
Integral operators, analytic, Alexander      11 174 299 307Ч313 375
Integral operators, analytic, Averaging      215 216Ч231 289Ч291 415
Integral operators, analytic, Bernardi      11 67 278Ч282 299Ч301 305Ч306 317Ч318
Integral operators, analytic, Libera      11 63Ч66 106 116Ч117 145Ч146 173 279 299 301
Integral operators, meromorphic      389Ч396
Jack's Lemma      20
Koebe function      7
Kummer's differential equation      5 233
L(z, t), subordination (Loewner) chain      4
Lagrange multipliers      352Ч355
Libera operator      11 63Ч66 106 116Ч117 145Ч146 173 279 299 301
Linear functional      408Ч409 413
Local defining function      355
Locally biholomorphic      343 352 355 357Ч359 362
M(a, b), real constant      240
MacGregor's Theorem      115
Marx Ч Strohhaecker differential subordination system      263
Marx Ч Strohhaecker Theorem      9 44 56Ч61 68 104 115
Mean-value operator      215
Meromorphic functions      382 383Ч396
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      44Ч45 46Ч49 85 135 140 252Ч254 387Ч388
Open Door mapping      45Ч46
Order of starlikeness      106 112 115Ч116 118 163Ч171
Outward (outer) normal      23 93 123 133 355 359 362
Polydisk      342 345Ч347 374Ч377
Problem 1      15 17 320 321
Problem 2      15 17 320 321
Problem 3      15 17 95 131 320 321
Problem of S. Miller      335Ч340
Q(g), subset of $\mathbb{C}^{n} \times \mathbb{C}^{n}$      370
R(a), real constant      238
Radius of convexity      419
Radius of starlikeness      270Ч273
Real orthogonal complement      358
Regular value of a function      356
Reinhardt domain      362
Robertson's Theorem      284 285Ч288
Robinson's theorem      17Ч18 70 73 184 199
Schwarz lemma      19 127 208 283 344 348 368 399
Schwarzian derivative      244 245Ч247
Second-order differential subordinations      13Ч15 16 17Ч43 187Ч244
Second-order linear differential subordinations      43 187Ч201 203Ч206
Solution of a differential subordination      16 320
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