Abstract
We show that a family \({\mathcal{F}}\) of analytic functions in the unit disk \({\mathbb{D}}\) which satisfy a condition of the form
for all \({f\in\mathcal{F}}\) and all \({z\in\mathbb{D}}\) (where n ≥ 3, 0 < |x| ≤ 1, b ≠ 0 and P is an arbitrary differential polynomial of degree at most n − 2 with constant coefficients and without terms of degree 0) is normal at the origin. Under certain additional assumptions on P the same holds also for b = 0. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman–Pang rescaling method. Furthermore we prove some corresponding results of Picard type for functions meromorphic in the plane.
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Grahl, J. Differential polynomials with dilations in the argument and normal families. Monatsh Math 162, 429–452 (2011). https://doi.org/10.1007/s00605-009-0186-z
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DOI: https://doi.org/10.1007/s00605-009-0186-z