Arkiv för Matematik

  • Ark. Mat.
  • Volume 50, Number 1 (2012), 89-110.

A normality criterion involving rotations and dilations in the argument

Jürgen Grahl

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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.


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

Article information

Ark. Mat., Volume 50, Number 1 (2012), 89-110.

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

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2011 © Institut Mittag-Leffler


Grahl, Jürgen. A normality criterion involving rotations and dilations in the argument. Ark. Mat. 50 (2012), no. 1, 89--110. doi:10.1007/s11512-011-0144-6.

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