Normal Families of Meromorphic Functions

by Chuang, Chi-Tai

• ISBN13: 9789810212575
• ISBN10: 9810212577
• Format: Hardcover
• Copyright: 1993-04-01
• Publisher: World Scientific Pub Co Inc

Table of Contents

Preface
Basic Notions and Theorems	p. 1
Spherical distance	p. 1
Sequences of meromorphic functions	p. 4
Families of meromorphic functions	p. 13
Criterions of Normality of Families of Holomorphic Functions and Applications	p. 27
Montel's theorem	p. 27
Miranda's theorem	p. 40
Bloch's theorem	p. 51
Criterions of Normality of Families of Meromorphic Functions and Applications	p. 57
Montel's theorem	p. 57
Zalcman's theorem	p. 73
Gu's theorem	p. 86
Closed Families of Meromorphic Functions	p. 93
Closed families of holomorphic functions	p. 93
Examples	p. 101
Closed families of holomorphic functions	p. 106
Examples	p. 118
A covering theorem	p. 123
Quasi-normal Families of Meromorphic Functions	p. 131
Some preliminary definitions and theorems	p. 131
Criterions of quasi-normality of families of meromorphic functions	p. 136
Some applications of criterions of quasi-normality	p. 166
Quasi-normal families of meromorphic functions of finite total order	p. 183
Applications	p. 203
Further Applications	p. 219
Uniform convergence of sequences of meromorphic functions	p. 219
Generalizations of Bloch's theorem and some theorems of Valiron	p. 238
Univalent and multivalent functions	p. 249
Extensions of Some Criterions of Normality and Quasi-normality	p. 273
Case of holomorphic functions	p. 273
Case of meromorphic functions	p. 281
Case of univalent and multivalent functions	p. 303
Q[subscript m]-normal Families of Meromorphic Functions	p. 309
Some notions and theorems	p. 309
An existence theorem	p. 314
Necessary and sufficient conditions for Q[subscript m]-normality	p. 318
Conditions and criterions of Q[subscript m]-normality	p. 330
Other criterions of Q[subscript m]-normality	p. 341
Applications of the Theory of Q[subscript m]-normal Families of Meromorphic Functions	p. 347
Uniform convergence of sequences of meromorphic functions	p. 347
Distribution of the value of meromorphic functions in the neighborhood of an essential singularity	p. 354
Limitation of the modulus of meromorphic functions	p. 359
Limitation of the number of the zeros of meromorphic functions	p. 379
Domains covered by the values of meromorphic functions	p. 383
Appendix A. A General Theorem on Holomorphic Functions in the Unit Circle	p. 393
A theorem on convex functions	p. 393
A general theorem on holomorphic functions in the unit circle	p. 401
Generalization of theorem 2	p. 435
Appendix B. Some Theorems on Meromorphic Functions	p. 443
A theorem on meromorphic functions in the plane	p. 443
A theorem on meromorphic functions in the unit circle	p. 445
References	p. 465
Index	p. 469
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