# 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

, Volume 4 (2) – Nov 6, 2013
9 pages

/lp/springer-journals/differential-inequalities-and-quasi-normal-families-vR0vDWj849
Publisher
Springer Journals
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