×

Normality and shared values. (English) Zbl 1079.30044

For \(f\) meromorphic on the unit disc \(\Delta\) and \(a\in \mathbb{C}\) define \(\overline E_f(a)=f^{-1}(\{a\})\cap\Delta=\{\,z\in\Delta:f(z)=a\,\}\). Two functions \(f\) and \(g\) on \(\Delta\) are said to share the value \(a\) if \(\overline E_f(a)=\overline E_g(a)\). A meromorphic function \(f\) on \( \mathbb{C}\) is called a normal function if there exists a positive number \(M\) such that \(f^\#(z)\leq M\), where \(f^\#(z)=| f'(z)| /(1+| f(z)| ^2)\) denotes the spherical derivative. The authors prove the following theorems: 1) Let \(\mathcal{F}\) be a family of meromorphic functions on the unit disc \(\Delta\), and let \(a\) and \(b\) be distinct complex numbers and \(c\) a nonzero complex number. If for every \(f\in \mathcal{F}\), \(\overline E_f(0)=\overline E_{f'}(a)\), \(\overline E_f(c)=\overline E_{f'}(b)\), then \(\mathcal{F}\) is normal on \(\Delta\). Earlier a similar result has been proved by W. Schwick [Arch. Math. 59, No. 1, 50–54 (1992; Zbl 0758.30028)]. 2) Let \(f\) be a meromorphic function on \(\mathbb{C}\) and \(a\) and \(b\) be distinct complex numbers. If \(f\) and \(f'\) share \(a\) and \(b\), then \(f\) is a normal function. This should be compared with the result of E. Mues and N. Steinmetz in [Manuscr. Math. 29, 195–206 (1979; Zbl 0416.30028)].

MSC:

30D45 Normal functions of one complex variable, normal families
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bergweiler, W. andEremenko, A., On the singularities of the inverse to a meromorphic function of finite order,Rev. Mat. Iberoamericana 11 (1995), 355–373. · Zbl 0830.30016
[2] Chen, H. andGu, Y., An improvement of Marty’s criterion and its applications,Sci. China Ser. A 36 (1993), 674–681.
[3] Clunie, J. andHayman, W. K., The spherical derivative of integral and meromorphic functions,Comment. Math. Helv. 40, (1966), 117–148. · Zbl 0142.04303 · doi:10.1007/BF02564366
[4] Frank, G. andWeissenborn, G., Rational deficient functions of meromorphic functions,Bull. London Math. Soc. 18 (1986), 29–33. · Zbl 0586.30025 · doi:10.1112/blms/18.1.29
[5] Hayman, W. K., Picard values of meromorphic functions and their derivatives,Ann. of Math. 70 (1959), 9–42. · Zbl 0088.28505 · doi:10.2307/1969890
[6] Hayman, W. K.,Meromorphic Functions, Clarendon Press, Oxford, 1964.
[7] Minda, D., Yosida functions, inLectures on Complex Analysis (Chuang, C.-T., ed.), pp. 197–213, World Scientific Publ., Singapore, 1988. · Zbl 0744.30022
[8] Mues, E. andSteinmetz, N., Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen,Manuscripta Math. 29 (1979), 195–206. · Zbl 0416.30028 · doi:10.1007/BF01303627
[9] Pang, X., On normal criterion of meromorphic functions,Sci. China Ser. A 33 (1990), 521–527. · Zbl 0706.30024
[10] Pang, X., Shared values and normal families, Preprint, 1998. · Zbl 1030.30030
[11] Pang, X. andZalcman, L., Normal families and shared values, to appear inBull. London Math. Soc.
[12] Schwick, W., Sharing values and normality,Arch. Math. (Basel) 59 (1992), 50–54. · Zbl 0758.30028
[13] Yang, L.,Value Distribution Theory, Springer-Verlag, Berlin-Heidelberg, 1993. · Zbl 0790.30018
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.