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Zbl 1221.30077
Grahl, Jürgen
Differential polynomials with dilations in the argument and normal families.
(English)
[J] Monatsh. Math. 162, No. 4, 429-452 (2011). ISSN 0026-9255; ISSN 1436-5081/e

{\it W. K. Hayman} [Ann. Math. (2) 70, 9--42 (1959; Zbl 0088.28505)] proved that each function $f$ meromorphic in $\mathbb{C}$ and satisfying the condition $f^n (z) + af'(z) \neq b$ for all $z \in \mathbb{C}$ (where $n \geq 5$, $a, b \in \mathbb{C}$, and $a \neq 0$) is constant; if $f$ is entire, this holds also for $n \geq 3$ and for $n = 2$, $b = 0$. Counterexamples show that these bounds on $n$ are best possible. According to Bloch's principle, for every Picard type" theorem, there is the hope that a corresponding normality criterion holds. There are many extensions of such results admitting more general differential polynomials instead of $f^n + af'$. \par In this article, it is shown that a family $\mathcal{F}$ of analytic functions in the unit disk $\mathbb{D}$ which satisfies a condition of the form $$f^n(z)+P[f](xz)+b \neq 0$$ for all $f \in \mathcal{F}$ and $z \in \mathbb{D}$, where $n \geq 3$, $0 < |x| \leq 1$, $b \neq 0$, 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 $b=0$.
MSC 2000:
*30D45 Normal functions, etc.
30D35 Distribution of values (one complex variable)

Keywords: differential polynomials; normal families; Nevanlinna theory; Zalcman's Lemma

Citations: Zbl 0088.28505

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