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Differential inequalities and quasi-normal families

Differential inequalities and quasi-normal families We show that a family $$\mathcal {F}$$ F of meromorphic functions in a domain $$D$$ D satisfying $$\begin{aligned} \frac{|f^{(k)}|}{1+|f^{(j)}|^\alpha }(z)\ge C \qquad \text{ for } \text{ all } z\in D \text{ and } \text{ all } f\in \mathcal {F}\end{aligned}$$ | f ( k ) | 1 + | f ( j ) | α ( z ) ≥ C for all z ∈ D and all f ∈ F (where $$k$$ k and $$j$$ j are integers with $$k>j\ge 0$$ k > j ≥ 0 and $$C>0$$ C > 0 , $$\alpha >1$$ α > 1 are real numbers) is quasi-normal. Furthermore, if all functions in $$\mathcal {F}$$ F are holomorphic, the order of quasi-normality of $$\mathcal {F}$$ F is at most $$j-1$$ j - 1 . The proof relies on the Zalcman rescaling method and previous results on differential inequalities constituting normality. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Differential inequalities and quasi-normal families

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Publisher
Springer Journals
Copyright
Copyright © 2013 by Springer Basel
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-013-0064-7
Publisher site
See Article on Publisher Site

Abstract

We show that a family $$\mathcal {F}$$ F of meromorphic functions in a domain $$D$$ D satisfying $$\begin{aligned} \frac{|f^{(k)}|}{1+|f^{(j)}|^\alpha }(z)\ge C \qquad \text{ for } \text{ all } z\in D \text{ and } \text{ all } f\in \mathcal {F}\end{aligned}$$ | f ( k ) | 1 + | f ( j ) | α ( z ) ≥ C for all z ∈ D and all f ∈ F (where $$k$$ k and $$j$$ j are integers with $$k>j\ge 0$$ k > j ≥ 0 and $$C>0$$ C > 0 , $$\alpha >1$$ α > 1 are real numbers) is quasi-normal. Furthermore, if all functions in $$\mathcal {F}$$ F are holomorphic, the order of quasi-normality of $$\mathcal {F}$$ F is at most $$j-1$$ j - 1 . The proof relies on the Zalcman rescaling method and previous results on differential inequalities constituting normality.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Nov 6, 2013

References