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April 2012 A normality criterion involving rotations and dilations in the argument
Jürgen Grahl
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Ark. Mat. 50(1): 89-110 (April 2012). DOI: 10.1007/s11512-011-0144-6


We show that a family $\mathcal{F}$ of analytic functions in the unit disk ${\mathbb{D}}$ all of whose zeros have multiplicity at least k and which satisfy a condition of the form $$f^n(z)f^{(k)}(xz)\ne1$$ for all $z\in{\mathbb{D}}$ and $f\in\mathcal{F}$ (where n≥3, k≥1 and 0<|x|≤1) is normal at the origin. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman–Pang rescaling method. Furthermore we prove the corresponding Picard-type theorem for entire functions and some generalizations.

Funding Statement

Part of this work was supported by the German Israeli Foundation for Scientific Research and Development (No. G 809-234.6/2003).


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Jürgen Grahl. "A normality criterion involving rotations and dilations in the argument." Ark. Mat. 50 (1) 89 - 110, April 2012.


Received: 7 January 2010; Revised: 5 February 2011; Published: April 2012
First available in Project Euclid: 31 January 2017

zbMATH: 1285.30019
MathSciNet: MR2890346
Digital Object Identifier: 10.1007/s11512-011-0144-6

Rights: 2011 © Institut Mittag-Leffler

Vol.50 • No. 1 • April 2012
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