Bulletin of the Belgian Mathematical Society - Simon Stevin

Normal families of holomorphic functions and multiple values

Lijuan Zhao and Xiangzhong Wu

Full-text: Open access

Abstract

Let $\mathcal{F}$ be a family of holomorphic functions defined in $D \subset C$, and let $ k, m, n, p $ be four positive integers with $ \frac{k+p+1}{m}+\frac{p+1}{n} < 1 $. Let $\psi (\not \equiv 0, \infty )$ be a meromorphic function in $ D $ and which has zeros only of multiplicities at most $ p $. Suppose that, for every function $ f \in \mathcal{F} $, (i) $ f $ has zeros only of multiplicities at least $ m $; (ii) all zeros of $ f^{(k)}-\psi(z) $ have multiplicities at least $ n $; (iii) all poles of $ \psi $ have multiplicities at most $ k $, and (iv) $ \psi(z) $ and $ f(z) $ have no common zeros, then $\mathcal{F}$ is normal in $ D $.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3 (2012), 535-547.

Dates
First available in Project Euclid: 14 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1347642381

Digital Object Identifier
doi:10.36045/bbms/1347642381

Mathematical Reviews number (MathSciNet)
MR3027359

Zentralblatt MATH identifier
1267.30088

Subjects
Primary: 30D35: Distribution of values, Nevanlinna theory 30D45: Bloch functions, normal functions, normal families

Keywords
holomorphic functions normal family multiplicity

Citation

Zhao, Lijuan; Wu, Xiangzhong. Normal families of holomorphic functions and multiple values. Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 3, 535--547. doi:10.36045/bbms/1347642381. https://projecteuclid.org/euclid.bbms/1347642381


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