A note on normality of meromorphic functions. (English) Zbl 1133.30009

Chang, Jianming

Proc. Japan Acad., Ser. A 83, No. 4, 60-62 (2007).

Author’s abstract: Let \(\mathcal F\) be the family of all functions \(f\) meromorphic in a domain \(D\subset \mathbb C\), for which, all zeros have multiplicity at least \(k\), and \(f(z)=0\Leftrightarrow f^{(k)}(z)=1 \Rightarrow | f^{(k+1)}(z)| \leq h\), where \(k\in \mathbb N\) and \(h\in \mathbb R^+\) are given. Examples show that \(\mathcal F\) is not normal in general (at least for \(k=1\) or \(k=2\)). The example we give for \(k=1\) shows that a recent result of [Y. Xu, Houston J. Math. 32, No. 3, 955–959 (2006; Zbl 1122.30023)] is not correct. However, we prove that for \(k\neq 2\), there exists a positive integer \(K\in \mathbb N\) such that the subfamily \(\mathcal G=\{ f\in \mathcal F: \text{all possible poles of }f\) in \(D\) have multiplicity at least

Reviewer: Lou Zengjian (Shantou Guangdong)

MSC:

30D45

Normal functions of one complex variable, normal families

Keywords:

holomorphic functions; normal family

Citations:

Zbl 1122.30023

References:

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