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Normality and the differential polynomial of meromorphic functions

Normality and the differential polynomial of meromorphic functions Abstract Let ℱ be a family of meromorphic functions in D , M ( f ) = f n 0 ( f ' ) n 1 ⋯ ( f ( k ) ) n k $M(f)=f^{n_0}(f^{\prime })^{n_1}\cdots (f^{(k)})^{n_k}$ with Γ M ≥ 2 $\Gamma _M \ge 2$ , let H ( f ) be a differential polynomial of f satisfying Γ γ | H < Γ M γ M $\frac{\Gamma }{\gamma }|_H< \frac{\Gamma _M}{\gamma _M}$ , and let ϕ ( z ) $\varphi (z)$ ( ≠ 0 $\ne 0$ ) be an analytic function in D . If, for each function f ∈ ℱ $f \in \mathcal {F}$ , f ≠ 0, M ( f ) + H ( f ) - ϕ ( z ) $M(f)+H(f)- \varphi (z)$ has at most Γ M - 1 $\Gamma _M-1$ distinct zeros in D , then ℱ is normal in D . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

Normality and the differential polynomial of meromorphic functions

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References (10)

Publisher
de Gruyter
Copyright
Copyright © 2014 by the
ISSN
1072-947X
eISSN
1572-9176
DOI
10.1515/gmj-2014-0026
Publisher site
See Article on Publisher Site

Abstract

Abstract Let ℱ be a family of meromorphic functions in D , M ( f ) = f n 0 ( f ' ) n 1 ⋯ ( f ( k ) ) n k $M(f)=f^{n_0}(f^{\prime })^{n_1}\cdots (f^{(k)})^{n_k}$ with Γ M ≥ 2 $\Gamma _M \ge 2$ , let H ( f ) be a differential polynomial of f satisfying Γ γ | H < Γ M γ M $\frac{\Gamma }{\gamma }|_H< \frac{\Gamma _M}{\gamma _M}$ , and let ϕ ( z ) $\varphi (z)$ ( ≠ 0 $\ne 0$ ) be an analytic function in D . If, for each function f ∈ ℱ $f \in \mathcal {F}$ , f ≠ 0, M ( f ) + H ( f ) - ϕ ( z ) $M(f)+H(f)- \varphi (z)$ has at most Γ M - 1 $\Gamma _M-1$ distinct zeros in D , then ℱ is normal in D .

Journal

Georgian Mathematical Journal – de Gruyter

Published: Sep 1, 2014

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