×

zbMATH — the first resource for mathematics

Normal families of meromorphic functions concerning shared values. (English) Zbl 1145.30013
Let \({\mathcal F}\) be a family of meromorphic functions in a domain \(D\). It is known that if every function in \({\mathcal F}\) omits three distinct values, then \({\mathcal F}\) 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 \(n\geq 4\) and for each pair of \(f\) and \(g\) in \({\mathcal F}\), \(f'- af^n\) and \(g'- ag^n\) share a value \(b\), then \({\mathcal F}\) 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:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D45 Normal functions of one complex variable, normal families
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hayman, W.K., Meromorphic function, (1964), Clarendon Press Oxford · Zbl 0115.06203
[2] Schiff, J., Normal families, (1993), Springer-Verlag Berlin · Zbl 0770.30002
[3] Yang, L., Value distribution theory, (1993), Springer-Verlag Berlin
[4] Yang, C.C.; Yi, H.-X., Uniqueness theory of meromorphic functions, (2003), Science Press/Kluwer Academic Beijing/New York
[5] Rubel, L.A., Four counterexamples to Block’s principle, Proc. amer. math. soc., 98, 257-260, (1986) · Zbl 0602.30040
[6] Drasin, D., Normal families and Nevanlinna theory, Acta math., 122, 231-263, (1969) · Zbl 0176.02802
[7] Schwick, W., Sharing values and normality, Arch. math., 59, 50-54, (1992) · Zbl 0758.30028
[8] Sun, D.C., On the normal criterion of shared values, J. Wuhan univ. natur. sci. ed., 3, 9-12, (1994), (in Chinese)
[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) · JFM 43.0509.05
[10] Fang, M.; Zalcman, L., A note on normality and shared values, J. aust. math. soc., 76, 141-150, (2004) · Zbl 1074.30032
[11] Pang, X.; Zalcman, L., Normal families and shared values, Bull. London math. soc., 32, 325-331, (2000) · Zbl 1030.30031
[12] Zhang, Q., Normal criteria concerning sharing values, Kodai math. J., 25, 8-14, (2002) · Zbl 1023.30036
[13] Hayman, W.K., Picard values of meromorphic functions and their derivatives, Ann. of math., 70, 9-42, (1959) · Zbl 0088.28505
[14] Mues, E., Über ein problem von Hayman, Math. Z., 164, 239-259, (1979) · Zbl 0402.30034
[15] Hayman, W.K., Research problems in function theory, (1967), Athlone Press of Univ. of London London · Zbl 0158.06301
[16] Li, S., On normality criterion of a class of the functions, J. Fujian normal univ., 2, 156-158, (1984), (in Chinese)
[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
[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 \(f^n f^\prime\), 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, suppl., 44-49, (1991), (in Chinese) · Zbl 0766.30029
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.