On the behaviour of the successive derivatives of meromorphic functions on the final set (Q393413)

Let \(H\left(\mathbb{C}\right) \) and \(M\left(\mathbb{C}\right) \) denote respectively the set of holomorphic and meromorphic functions in \(\mathbb{C}.\) For \(\phi \in M\left(\mathbb{C}\right) \), let \(P_{\phi }\) denote the set of poles of \(\phi \) and let \(M^{\ast }\left(\mathbb{C}\right) :=\left\{ f\in M\left(\mathbb{C}\right) ,\text{ }P_{f}\neq \emptyset \right\}\). The final set \(\mathfrak{F }\left( \phi \right) \) of a function \(\phi \in M\left(\mathbb{C}\right) \) is the set of points \(z_{0}\in\mathbb{C}\) such that there exists a sequence \(\left\{ z_{k}\right\} \subset\mathbb{C}\) with \(z_{k}\rightarrow z_{0}\) and a sequence \(\left\{ n_{k}\right\} \subset\mathbb{N}\) with \(n_{k}\rightarrow \infty \) satisfying \(\phi ^{\left( n_{k}\right) }\left( z_{k}\right) \) \(=0\) for every \(k\in\mathbb{N}\). \noindent The set \(\mathfrak{F}\left( \phi \right) \) was first studied by \textit{G. Pólya} in [Math. Z. 12, 36--60 (1922; JFM 48.0370.02)], where he proved the following. Let \(\phi \in M^{\ast }\left(\mathbb{C}\right) \). Then the set \(\mathfrak{F}\left( \phi \right) \) is completely determined by the location of the poles of \(\phi \). Furthermore, \(z_{0}\in \) \(\mathfrak{F}\left( \phi \right) \) if and only if the circle \(\left\{ z:\left| z-z_{0}\right| =\mathrm{dist}(z_{0},P_{\phi })\right\} \) contains at least two poles of \(\phi \). If \(P_{\phi }\) is a singleton, then \(\mathfrak{F} \left( \phi \right) =\emptyset \). Later, \textit{W. K. Hayman} [Meromorphic functions. Oxford: At the Clarendon Press (1964; Zbl 0115.06203)] gave a new proof of Pólya's theorem and proved additionally that, for \(\phi \in M^{\ast }\left(\mathbb{C}\right) \) and every point \(z_{0}\not\in\mathfrak{F}\left( \phi \right) \), \(\phi ^{\left( n\right) }\left( z\right) \rightarrow \infty \) uniformly in \( \left\{ z:\left| z-z_{0}\right| \leq \delta \right\} \) for sufficiently small \(\delta \) as \(n\rightarrow \infty \). \noindent In this paper, the authors study the behaviour of the sequence of derivatives \(\left\{ \phi ^{\left( n\right) }\right\} \) on \( \mathfrak{F}\left( \phi \right) \) and obtain the following result. Suppose that \(\phi \in M^{\ast }\left(\mathbb{C}\right) \) has at least two poles. Let \(z_{0}\in \) \(\mathfrak{F}\left( \phi \right) \), and denote by \(\zeta _{1},\cdots ,\zeta _{n}\) the poles of \(\phi \) on the circle \(\left\{ z:\left| z-z_{0}\right| =\mathrm{dist}(z_{0},P_{\phi })\right\} \). Let \(\zeta _{1},\cdots ,\zeta _{m}\) be the poles of highest order and their corresponding principal parts defined by \[ g_{i}\left( z\right) =\underset{j=1}{\overset{k}{\sum }}\dfrac{a_{j}^{\left( i\right) }}{\left( z-\zeta _{i}\right) ^{j}} \] for \(i\in \left\{ 1,\cdots ,m\right\} \), where \(1\leq m\leq n\). Furthermore, suppose that there exists \(i\in \left\{ 1,\cdots ,m\right\} \) such that \[ \left| a_{k}^{\left( i\right) }\right| >\underset{p\in \left\{ 1,\cdots ,m\right\} \backslash \left\{ i\right\} }{\sum }\left| a_{k}^{\left( p\right) }\right| . \] Then \(\left| \phi ^{\left( l\right) }\left( z_{0}\right) \right| \rightarrow \infty \) as \(l\rightarrow \infty \).