zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Normal families of meromorphic functions concerning shared values. (English) Zbl 1145.30013
Let be a family of meromorphic functions in a domain D. It is known that if every function in omits three distinct values, then is normal. W. Schwick [Arch. Math. 59, No. 1, 50–54 (1992; Zbl 0758.30028)] obtained a normality criteria from the point of view of value distribution theory, in particular, shared values. The author considers the sharing conditions with differential polynomials. Let n be a positive integer, and a be a nonzero constant. If n4 and for each pair of f and g in , f ' -af n and g ' -ag n share a value b, then is normal. The author also considers a family of entire functions. Examples are given which imply that results in this paper are sharp. The methods for the proofs are the value distribution theory and Zalcman’s lemma.

MSC:
30D35Distribution of values (one complex variable); Nevanlinna theory
30D45Bloch functions, normal functions, normal families
References:
[1]Hayman, W. K.: Meromorphic function, (1964) · Zbl 0115.06203
[2]Schiff, J.: Normal families, (1993) · Zbl 0770.30002
[3]Yang, L.: Value distribution theory, (1993)
[4]Yang, C. C.; Yi, H. -X.: Uniqueness theory of meromorphic functions, (2003)
[5]Rubel, L. A.: Four counterexamples to block’s principle, Proc. amer. Math. soc. 98, 257-260 (1986) · Zbl 0602.30040 · doi:10.2307/2045694
[6]Drasin, D.: Normal families and Nevanlinna theory, Acta math. 122, 231-263 (1969) · Zbl 0176.02802 · doi:10.1007/BF02392012
[7]Schwick, W.: Sharing values and normality, Arch. math. 59, 50-54 (1992) · Zbl 0758.30028 · doi:10.1007/BF01199014
[8]Sun, D. C.: On the normal criterion of shared values, J. Wuhan univ. Natur. sci. Ed. 3, 9-12 (1994)
[9]Montel, P.: Sur LES familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine, Ann. sci. École norm. Sup. 29, 487-535 (1912) · Zbl 43.0509.05 · doi:numdam:ASENS_1912_3_29__487_0
[10]Fang, M.; Zalcman, L.: A note on normality and shared values, J. aust. Math. soc. 76, 141-150 (2004) · Zbl 1074.30032 · doi:10.1017/S1446788700008752
[11]Pang, X.; Zalcman, L.: Normal families and shared values, Bull. London math. Soc. 32, 325-331 (2000) · Zbl 1030.30031 · doi:10.1112/S002460939900644X
[12]Zhang, Q.: Normal criteria concerning sharing values, Kodai math. J. 25, 8-14 (2002) · Zbl 1023.30036 · doi:10.2996/kmj/1106171072
[13]Hayman, W. K.: Picard values of meromorphic functions and their derivatives, Ann. of math. 70, 9-42 (1959) · Zbl 0088.28505 · doi:10.2307/1969890
[14]Mues, E.: Über ein problem von Hayman, Math. Z. 164, 239-259 (1979)
[15]Hayman, W. K.: Research problems in function theory, (1967)
[16]Li, S.: On normality criterion of a class of the functions, J. fujian normal univ. 2, 156-158 (1984)
[17]Li, X.: Proof of Hayman’s conjecture on normal families, Sci. China ser. A 28, 596-603 (1985) · Zbl 0592.30035
[18]Langley, J.: On normal families and a result of drasin, Proc. roy. Soc. Edinburgh sect. A 98, 385-393 (1984) · Zbl 0556.30025 · doi:10.1017/S0308210500013548
[19]Pang, X.: On normal criterion of meromorphic functions, Sci. China ser. A 33, 521-527 (1990) · Zbl 0706.30024
[20]Chen, H.; Fang, M.: On the value distribution of fnf ' , Sci. China ser. A 38, 789-798 (1995) · Zbl 0839.30026
[21]L. Zalcman, On some questions of Hayman, unpublished manuscript, 1994
[22]Ye, Y.: A new criterion and its application, Chinese ann. Math. ser. A 12, No. suppl., 44-49 (1991)
[23]Bergweiler, W.; Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order, Rev. mat. Iberoamericana 11, 355-373 (1995) · Zbl 0830.30016
[24]Zalcman, L.: Normal families: new perspectives, Bull. amer. Math. soc. 35, 215-230 (1998) · Zbl 1037.30021 · doi:10.1090/S0273-0979-98-00755-1