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DIFFERENTIAL INEQUALITIES AND A MARTY-TYPE CRITERION FOR QUASI-NORMALITY
Published online by Cambridge University Press: 18 June 2018
Abstract
We show that the family of all holomorphic functions $f$ in a domain
$D$ satisfying
$$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$
$k$ is a natural number and
$C>0$) is quasi-normal. Furthermore, we give a general counterexample to show that for
$\unicode[STIX]{x1D6FC}>1$ and
$k\geq 2$ the condition
$$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|^{\unicode[STIX]{x1D6FC}}}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$
MSC classification
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- Research Article
- Information
- Copyright
- © 2018 Australian Mathematical Publishing Association Inc.
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