The \(\alpha\)-normal functions (Q1886473)

Let \(D\) denote the unit disc in the complex plane, let \(f\) be a function meromorphic in \(D\), and define the spherical derivative \(f^{\#}(z)\) by \[ f^{\#}(z) = \frac {| f'(z)| } {1 + | f(z)| ^{2}} \;. \] The function \(f\) is called an \(\alpha\)-normal function if there exists a constant \(c_{\alpha}(f)\) such that \[ (1 - | z| ^{2})^{\alpha} f^{\#}(z) \leq c_{\alpha}(f) \;, \;\;z \in D \;. \] The special case \(\alpha = 1\) gives rise to the class of normal functions, which has been studied extensively. The author proves two results. Theorem 1. If \(0 < \alpha < \infty\) and if \(f\) is an \(\alpha\)-normal meromorphic function in \(D\), then for each positive integer \(n\) there exists a constant \(C_{n}(f)\) such that \[ (1 - | z| ^{2})^{\alpha n} \prod_{j=0}^{n-1} (f^{(j)})^{\#}(z) \leq C_{n}(f) \;, \;\;z \in D \;. \] This result extends a result about normal functions due to the reviewer [Ann. Acad. Sci. Fenn. Ser. A I 3, 301--310 (1977; Zbl 0387.30018)]. Theorem 2. (a) If \(\alpha \geq 1\), if \(f\) is a function meromorphic in \(D\) such that all the zeros of \(f\) are of multiplicity at least \(3\), and if there exist positive numbers \(\delta\) and \(M\) such that \[ (1 - | z| ^{2})^{\alpha - 1} f^{\#}(z)(f')^{\#}(z)((f'')^{\#}(z))^{1/\alpha} \leq M \] whenever both \(| f(z)| \leq \delta\) and \((1 - | z| ^{2})^{\alpha}| f'(z)| \leq \delta\), then \(f\) is an \(\alpha\)-normal function; (b) if all the zeros of \(f\) are of multiplicity at least two, and if there exist positive numbers \(\delta\) and \(M\) such that \[ (1 - | z| ^{2})^{\alpha}| f^{\#}(z)(f')^{\#}(z) \leq M \] whenever both \(| f(z)| \leq \delta\) and \((1 - | z| ^{2})^{\alpha}| f'(z)| \leq \delta\), then \(f\) is an \(\alpha\)-normal function. This result improves a result on normal functions due to \textit{H. Chen} and the reviewer [J. Austral. Math. Soc., Ser A 64, No. 2, 231--246 (1998; Zbl 0910.30026)]. Some examples are given illustrating the usefulness of the results.