Differential polynomials with dilations in the argument and normal families (Q533698)

\textit{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 \(f^n + af'\). 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 \(b=0\).