×

On the normality criteria of Montel and Bergweiler-Langley. (English) Zbl 1357.30025

The paper deals with some normality criteria for families of meromorphic functions.
First, the following generalization of the P. Montel [Ann. Sci. Éc. Norm. Supér. (3) 33, 223–302 (1916; JFM 46.0519.01)] and J. Grahl and S. Nevo [Isr. J. Math. 202, 21–34 (2014; Zbl 1300.30068)] results to the case of spherical derivatives is proved:
Theorem 1. Let \({\mathcal F}\) be a family of meromorphic functions in a domain \(D \subset {\mathbb C}\), and denote by \(\sigma\) the spherical metric on \(\mathbb C\). Assume that for each compact subset \(K \subset D\), there are
i) positive integers (or \(+\infty\)) \(l_1, \ldots, l_q\) satisfying \[ \sum\limits_{j=1}^{q} \frac{1}{l_j} < q - 2, \]
ii) meromorphic functions \(a_{1f}, \ldots, a_{qf}\) in \(D\), with \(f \in {\mathcal F}\), and positive constants \(\epsilon\) and \(M\) such that \(\sigma(a_{if}(z), a_{jf}(z)) \geq \epsilon\) for all \(z \in D\), \(1 \leq i, j \leq q\), \(i \not= j\), and \[ \sup\limits_{z\in K: f(z)=a_{jf}(z) \not=+\infty} \left(f^{(k)}\right)^{\sharp}(z) \leq M,\quad \sup\limits_{z\in K: f(z)=a_{jf}(z) =+\infty} \left(\left(\frac{1}{f}\right)^{(k)}\right)^{\sharp}(z) \leq M, \] for all \(f\in {\mathcal F}\), \(j=1,\ldots,l_j-2\).
Then \({\mathcal F}\) is a normal family.
Definition. Let \(k\) be a positive integer, \(0 < M \leq +\infty\), \(0 < N \leq +\infty\), and let \(D\) be a plain domain.
1. Let \({\mathcal F}(k, M, N, D)\) be the set of all meromorphic functions \(f\) on \(D\) such that all zeros of \(f\) have multiplicity at least \(M\), while all zeros of \(f^{(k)} - 1\) have multiplicity at least \(N\).
2. Let \(c\) be a positive constant. Let \(\widetilde{\mathcal F}(k, M, N, c, D)\) be the set of all meromorphic functions \(f\) on \(D\) such that \[ \sup \big\{(f^{(i)})^{\sharp}(z) : z\in f^{-1}(0)\big\} \leq c, \] for all \(i = 1, \ldots, M - 2\), and \[ \sup \{(f^{(k+l)})^{\sharp}(z) : z\in (f^{(k)})^{-1}(1)\} \leq c, \] for all \(l = 1, \ldots, N - 2\).
The authors prove the following theorem which is a generalization of a theorem by W. Bergweiler and J. K. Langley [J. Aust. Math. Soc. 78, No. 1, 37–57 (2005; Zbl 1074.30028)].
Theorem. If \({\mathcal F}(k, M, N, {\mathbb C})\) consists of constants only, then \(\widetilde{\mathcal F}(k, M, N, c, D)\) is a normal family.

MSC:

30D45 Normal functions of one complex variable, normal families
30D30 Meromorphic functions of one complex variable (general theory)
Full Text: DOI

References:

[1] Bergweiler, W.; Langley, J. K., Multiplicities in Hayman’s alternative, J. Aust. Math. Soc., 78, 37-57 (2005) · Zbl 1074.30028
[2] Carathéodory, C., Theory of Functions of Complex Variable, vol. II (1960), Chelsea Publ. Co.: Chelsea Publ. Co. New York
[3] Fujimoto, H., On families of meromorphic maps into the complex projective space, Nagoya Math. J., 54, 21-51 (1974) · Zbl 0267.32005
[4] Grahl, J.; Nevo, S., Exceptional functions wandering on the sphere and normal families, Israel J. Math., 202, 21-34 (2014) · Zbl 1300.30068
[5] Grahl, J.; Nevo, S., An extension of one direction in Marty’s normality criterion, Monatsh. Math., 174, 205-217 (2014) · Zbl 1300.30056
[6] Gu, Y. X., A criterion for normality of meromorphic functions, Sci. Sin., 1, 267-274 (1979), Special Issue on Math. (Chinese) · Zbl 1171.30308
[7] Hayman, W. K., Picard values of meromorphic functions and their derivatives, Ann. of Math., 70, 9-42 (1959) · Zbl 0088.28505
[8] Lappan, P., A criterion for a meromorphic function to be normal, Comment. Math. Helv., 49, 492-495 (1974) · Zbl 0292.30029
[9] Lappan, P., A uniform approach to normal families, Rev. Roumaine Math. Pures Appl., 39, 691-702 (1994) · Zbl 0843.30031
[10] Lehto, O.; Virtanen, K. L., Boundary behaviour and normal meromorphic functions, Acta Math., 97, 47-65 (1957) · Zbl 0077.07702
[11] Montel, P., Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine, Ann. Sci. Éc. Norm. Supér., 29, 487-535 (1912) · JFM 43.0509.05
[12] Montel, P., Sur les familles normales de fonctions analytiques, Ann. Éc. Norm. Supér., 33, 223-302 (1916) · JFM 46.0519.01
[13] Tan, T. V.; Thin, N. V., On Lappan’s five point theorem, Comput. Methods Funct. Theory (2016)
[14] Valiron, G., Families normales et quasi-normales de fonctions meromorphes, Mém. Sci. Math., Fasc., 38 (1929) · JFM 55.0762.02
[15] Zalcman, L., Normal families: new perspective, Bull. Amer. Math. Soc., 35, 215-230 (1998) · Zbl 1037.30021
[16] Zalcman, L., Variations on Montel’s theorem, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform., 59, 25-36 (2009) · Zbl 1230.30018
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.