Abstract
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).
Citation
Jürgen Grahl. "A normality criterion involving rotations and dilations in the argument." Ark. Mat. 50 (1) 89 - 110, April 2012. https://doi.org/10.1007/s11512-011-0144-6
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