2. Non-normality and topological transitivity

In the sequel, for $\lambda > 0$ and $z_0 \in \mathbb C$ we set $D_{\lambda}(z_0) \;:\!=\; \{z \in \mathbb C\;:\; \left|{z - z_0}\right| < \lambda\}$ and denote by $\overline{D}_{\lambda}(z_0)$ the closure of $D_{\lambda}(z_0)$ in $\mathbb C$ . For $w_0 \in \mathbb C_\infty$ , we further set $D_{\lambda}^{\chi}(w_0) \;:\!=\; \{w \in \mathbb C_\infty\;:\; \chi(w, w_0) < \lambda\}$ . We say that a family $\mathcal F \subset M(\Omega)$ is (topologically) transitive with respect to a point $z_0 \in \Omega$ , if for every pair of non-empty open sets $U \subset \Omega$ and $V \subset \mathbb C_\infty$ with $z_0 \in U$ , there exists $f \in \mathcal F$ such that $f(U) \cap V \neq \emptyset$ . Note that in this case we have $f(U) \cap V \neq \emptyset$ for infinitely many $f \in \mathcal F$ . If $f(U) \cap V \neq \emptyset$ holds for cofinitely many $f \in \mathcal F$ , we say that $\mathcal F$ is (topologically) mixing with respect to $z_0$ . Furthermore, if for every non-empty open set $U \subset \Omega$ with $z_0 \in U$ and every pair of non-empty open sets $V_1, V_2 \subset \mathbb C_\infty$ , there exists $f \in \mathcal F$ such that $f(U) \cap V_i \neq \emptyset$ for $i = 1,2$ , we say that $\mathcal F$ is weakly mixing with respect to $z_0$ . Finally, we say that $\mathcal F$ is transitive (or (weakly) mixing) with respect to a relatively closed set $B \subset \Omega$ , if $\mathcal F$ is transitive (or (weakly) mixing) with respect to every $z_0 \in B$ .

With these notations, we obtain the following characterisation of (strong) non-normality.

Proof. (a) Let $\mathcal F$ be strongly non-normal at $z_0$ and suppose that $\mathcal F$ is not mixing with respect to $z_0$ . Then there exist non-empty open sets $U \subset \Omega$ and $V \subset \mathbb C_\infty$ with $z_0 \in U$ , and an infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ such that $f(U) \cap V = \emptyset$ for every $f \in \tilde{\mathcal F}$ . By Montel’s Theorem, we obtain that $\tilde{\mathcal F}$ is normal on U, hence also at $z_0$ , in contradiction to the strong non-normality of $\mathcal F$ at $z_0$ .

On the other hand, suppose that $\mathcal F$ is mixing with respect to $z_0 \in \Omega$ , but not strongly non-normal at $z_0$ . Then there exists an open neighbourhood U of $z_0$ and a sequence $(f_n) \subset \mathcal F$ , such that $(f_n)$ converges spherically uniformly on compact subsets of U to a function $f \in M(U)$ . Hence, for $\varepsilon > 0$ sufficiently small, we have that $\overline{D}_{\varepsilon}(z_0) \subset U$ and there exists $\delta > 0$ and $w_0 \in \mathbb C_\infty$ such that $f(\overline{D}_{\varepsilon}(z_0)) \cap D_{\delta}^{\chi}(w_0) = \emptyset$ . Since $(f_n)$ is mixing with respect to $z_0$ , we obtain that $f_n(D_{\varepsilon}(z_0)) \cap D_{\frac{\delta}{2}}^{\chi}(w_0) \neq \emptyset$ for all n sufficiently large, in contradiction to the spherically uniform convergence of $(f_n)$ to f on $\overline{D}_{\varepsilon}(z_0)$ .

(b) $\textrm{(i)} \Rightarrow \textrm{(ii)}:$ Since $\mathcal F$ is non-normal at $z_0$ , there exists an infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ that is strongly non-normal at $z_0$ . This subfamily is mixing with respect to $z_0$ according to the first statement of the theorem.

$\textrm{(ii)} \Rightarrow \textrm{(iii)}$ : This is clear, since a mixing family is also weakly mixing.

$\textrm{(iii)} \Rightarrow \textrm{(i)}$ : Suppose that $\mathcal F$ is weakly mixing with respect to $z_0$ . Further consider two non-empty open sets $V_1, V_2 \subset \mathbb C_\infty$ such that $\inf_{z \in V_1, w \in V_2} \chi(z ,w) > \varepsilon$ for some $\varepsilon > 0$ . For $k \in \mathbb N$ , we set $U_k \;:\!=\; D_{\frac{1}{k}}(z_0) \cap \Omega$ . By assumption, for every $k \in \mathbb N$ there is a function $f_k \in \mathcal F$ such that $f_k(U_k) \cap V_1 \neq \emptyset$ and $f_k(U_k) \cap V_2 \neq \emptyset$ , and hence points $z_k^{(1)}, z_k^{(2)} \in U_k$ such that $f_k(z_k^{(1)}) \in V_1$ and $f_k(z_k^{(2)}) \in V_2$ . Note that $z_k^{(1)}, z_k^{(2)} \in U_k$ implies that $z_k^{(1)} \to z_0$ and $z_k^{(2)} \to z_0$ for $k \to \infty$ , furthermore we have that $\chi(f_k(z_k^{(1)}), f_k(z_k^{(2)})) > \varepsilon$ for every $k \in \mathbb N$ , and hence

\begin{equation*}\chi(f_k(z_0), f_k(z_k^{(1)})) > \frac{\varepsilon}{2} \quad \text{or} \quad \chi(f_k(z_0), f_k(z_k^{(2)})) > \frac{\varepsilon}{2}.\end{equation*}

Hence, we can find a sequence $(z_k)$ with $z_k \to z_0$ for $k \to \infty$ and $\chi(f_k(z_0), f_k(z_k)) > \frac{\varepsilon}{2}$ for every $k \in \mathbb N$ , implying that the family $\mathcal F$ is not spherically equicontinuous at $z_0$ , and thus also not normal.

By Montel’s Theorem, it is clear that $z_0 \in J(\mathcal F)$ implies that $\mathcal F$ is transitive with respect to $z_0$ . On the other hand, it is easily seen that transitivity of a family with respect to some point $z_0 \in \Omega$ is in general not sufficient for non-normality at $z_0$ . For instance, if $(z_n)$ is a sequence that is dense in $\mathbb C_\infty$ , the family $(f_n)$ of constant functions $f_n \equiv z_n$ is transitive with respect to any $z_0 \in \Omega$ , while at the same time we have $J(f_n) = \emptyset$ . However, the following proposition shows that this example is in some sense typical:

Proof. Suppose that $\{f(z_0)\;:\; f \in \mathcal F\}$ is not dense in $\mathbb C_\infty$ . Then there is $w \in \mathbb C_\infty$ and $\varepsilon > 0$ , such that $\{f(z_0)\;:\;f \in \mathcal F\} \cap D_{\varepsilon}^{\chi}(w) = \emptyset$ . Consider now for $k \in \mathbb N$ the sets $U_k \;:\!=\; D_{\frac{1}{k}}(z_0) \cap \Omega$ . Since $\mathcal F$ is transitive with respect to $z_0$ , for every $k \in \mathbb N$ there is $f_k \in \mathcal F$ such that $f_k(U_k) \cap D_{\frac{\varepsilon}{2}}^{\chi}(w) \neq \emptyset$ . In particular, there is a sequence $(z_k)$ with $z_k \in U_k$ , and hence $z_k \to z_0$ for $k \to \infty$ , such that $f_k(z_k) \in D_{\frac{\varepsilon}{2}}^{\chi}(w)$ for $k \in \mathbb N$ . On the other hand, we have $f_k(z_0) \notin D_{\varepsilon}^{\chi}(w)$ for $k \in \mathbb N$ . Finally, we obtain that

\begin{equation*}\chi(f_k(z_0), f_k(z_k)) > \frac{\varepsilon}{2} \qquad \text{for every } k \in \mathbb N,\end{equation*}

so that $\mathcal F$ is not spherically equicontinuous at $z_0$ , and thus also not normal, that is $z_0 \in J(\mathcal F)$ .

For a family of meromorphic functions $\mathcal F \subset M(\Omega)$ and $N \in \mathbb N$ , we consider the family $\mathcal F^{\times N} \;:\!=\; \{f^{\times N}\;:\;f \in \mathcal F\}$ , where $f^{\times N}\;:\;\Omega^N \to \mathbb C_\infty^N$ with $f^{\times N}(z_1,\dots,z_N) = (f(z_1), \dots, f(z_N))$ . We say that $\mathcal F^{\times N}$ is transitive with respect to $z \in \Omega^N$ , if for every pair of non-empty open sets $U \subset \Omega^N$ and $V \subset \mathbb C_\infty^N$ with $z \in U$ , there exists $f^{\times N} \in \mathcal F^{\times N}$ such that $f^{\times N}(U) \cap V \neq \emptyset$ . Furthermore, for a relatively closed set $B \subset \Omega$ , we say that $\mathcal F^{\times N}$ is transitive with respect to $B^N$ , if $\mathcal F^{\times N}$ is transitive with respect to every $z \in B^N$ . We then have the following characterisation of hereditary non-normality.

Proof. The equivalence of (i) and (ii) follows from Theorem 1.

$\textrm{(ii)} \Rightarrow \textrm{(iii)}$ : Without loss of generality consider $\tilde{\mathcal F}$ to be countable, $\tilde{\mathcal F} = \{f_n\;:\;n \in \mathbb N\}$ say. Let $N \in \mathbb N$ and consider non-empty open sets $U \subset \Omega^N$ and $V \subset \mathbb C_\infty^N$ with $B^N \cap U \neq \emptyset$ . Then there exist non-empty open sets $U_1, \ldots, U_N$ with $U_1\times \cdots \times U_N\subset U$ and $B \cap U_i \neq \emptyset$ for $i = 1,\dots,N$ , and non-empty open sets $V_1,\ldots,V_N\subset \mathbb C_\infty$ with $V_1\times \cdots \times V_N\subset V$ . According to the assumption, $\{f_n:n>m\}$ is transitive with respect to B, for all $m \in \mathbb N$ . Inductively, we can find a strictly increasing sequence $(n_k)$ in $\mathbb{N}$ with $f_{n_k}(U_1)\cap V_1\neq\emptyset$ for all $k\in\mathbb{N}$ . By assumption, the family $\{f_{n_k}:k\in\mathbb{N}\}$ is transitive with respect to B. Thus, the same argument as above yields the existence of a subsequence $(n^{(2)}_k)$ of $(n^{(1)}_k)\;:\!=\;(n_k)$ with $f_{n^{(2)}_k}(U_2)\cap V_2\neq\emptyset$ for all $k\in\mathbb{N}$ . Proceeding in the same way, for any j with $2 \le j \le N$ we find subsequences $(n_k^{(j)})$ of $(n_k^{(j-1)})$ with $f_{n^{(j)}_k}(U_j)\cap V_j\neq\emptyset$ for all $k\in\mathbb{N}$ . In particular, for $n\;:\!=\;n_1^{(N)}$ , we obtain that

\begin{equation*}(f_n(U_1)\times \cdots \times f_n(U_N)) \cap (V_1\times \cdots \times V_N) \neq \emptyset,\end{equation*}

hence also $f_n^{\times N}(U)\cap V \neq \emptyset$ , implying that $\mathcal F^{\times N}$ is transitive with respect to $B^N$ .

$\textrm{(iii)} \Rightarrow \textrm{(ii)}$ The proof follows along the same lines as the proof of the corresponding part of the Bès–Peris Theorem (e.g. [Reference Grosse-erdmann and Peris21, pp.76]).

3. Non-normality and expanding families

We define the following ‘expanding’ property of families $\mathcal F \subset M(\Omega)$ .

Note that if $\mathcal F$ is expanding at $z_0$ with respect to A, there exists an infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ that is strongly expanding at $z_0$ with respect to A. Moreover, in this case we have that A is contained in $\limsup_{z_0}\mathcal F$ . Also note that $\mathcal F$ is strongly expanding at $z_0$ with respect to A if and only if every infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ is expanding at $z_0$ with respect to A, and in this case, A is contained in $\liminf_{z_0}\mathcal F$ . On the other hand, we remark that $A \subset \liminf_{z_0}\mathcal F$ does in general not imply that $\mathcal F$ is (strongly) expanding at $z_0$ with respect to A. This can for instance be seen by considering the family $\mathcal F \;:\!=\; \{e^{nz} + (1 - {1}/{n})\;:\;n \in \mathbb N\}$ , for which we have $\liminf_{0}\mathcal F = \mathbb C$ (note that for each neighbourhood U of 0 and each $w \in \mathbb C$ , there is some N such that $w - 1 + {1}/{n} \in \exp(nU)$ for all $n \geq N$ ), but $\mathcal F$ is not expanding at 0 with respect to any set $A \subset \mathbb C$ with $1 \in A^{\circ}$ .

Our next result establishes a relationship between strong non-normality and the expanding property. Here and in the following, we denote by $\left|{E}\right| \in \mathbb N_0\cup\{\infty\}$ the number of elements of a set $E \subset \mathbb C_\infty$ .

Proof. (i) Suppose that $\mathcal F$ is strongly non-normal at $z_0$ and consider an infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ . Then $\tilde{\mathcal F}$ is strongly non-normal at $z_0$ and assuming that $\tilde{\mathcal F}$ is not expanding at $z_0$ with respect to $\mathbb C_\infty \setminus E$ for any $E \subset \mathbb C_\infty$ with $\left|{E}\right| \leq 2$ , we obtain that for every $E \subset \mathbb C_\infty$ with $\left|{E}\right| \leq 2$ , there is an open neighbourhood U of $z_0$ and a compact set $K \subset \mathbb C_\infty \setminus E$ , such that $K \setminus f(U) \neq \emptyset$ for cofinitely many $f \in \tilde{\mathcal F}$ . In particular, if $\tilde{\mathcal F}$ is not expanding at $z_0$ with respect to $\mathbb C_\infty$ , we can find an open neighbourhood $U_1$ of $z_0$ , a sequence $(f_n)$ in $\tilde{\mathcal F}$ , and a sequence $(a_n)$ in $\mathbb C_\infty$ with $a_n \to a \in \mathbb C_\infty$ for $n \to \infty$ , such that $a_n \notin f_n(U_1)$ for every $n \in \mathbb N$ . By assumption, $\tilde{\mathcal F}$ is not expanding at $z_0$ with respect to $\mathbb C_\infty \setminus \{a\}$ , hence, there is an open neighbourhood $U_2$ of $z_0$ and a compact set $K_2 \subset \mathbb C_\infty \setminus \{a\}$ , such that $K_2 \setminus f(U_2) \neq \emptyset$ for cofinitely many $f \in \tilde{\mathcal F}$ . In particular, there is a subsequence $(f_{n_k})$ of $(f_n)$ , and a sequence $(b_k)$ in $K_2$ with $b_k \to b \in K_2$ for $k \to \infty$ , such that $b_k \notin f_{n_k}(U_2)$ for every $k \in \mathbb N$ . Since $\tilde{\mathcal F}$ is not expanding at $z_0$ with respect to $\mathbb C_\infty \setminus \{a,b\}$ , a similar argumentation leads to an open neighbourhood $U_3$ of $z_0$ , a compact set $K_3 \subset \mathbb C_\infty \setminus \{a,b\}$ , a subsequence $(f_{n_{k_l}})$ of $(f_{n_k})$ and a sequence $(c_l)$ in $K_3$ with $c_l \to c \in K_3$ for $l \to \infty$ , such that $c_l \notin f_{n_{k_l}}(U_3)$ for every $l \in \mathbb N$ .

Finally, setting $U = U_1 \cap U_2 \cap U_3$ we obtain that

\begin{equation*}\{a_{n_{k_l}}, b_{k_l}, c_l\} \cap f_{n_{k_l}}(U) = \emptyset \quad \text{for every } l \in \mathbb N.\end{equation*}

Furthermore, since a,b,c are pairwise distinct, there exists $\varepsilon > 0$ such that

\begin{equation*}\chi(a_{n_{k_l}}, b_{k_l}) \, \chi(b_{k_l}, c_l) \, \chi(a_{n_{k_l}}, c_l) > \varepsilon,\end{equation*}

for $l \in \mathbb N$ sufficiently large, so that Carathéodory’s extension of Montel’s Theorem (e.g. [Reference Schiff29, p.104]) implies that $(f_{n_{k_l}}) \subset \tilde{\mathcal F}$ is normal in U, hence also at $z_0$ , in contradiction to the strong non-normality of $\tilde{\mathcal F}$ at $z_0$ .

To prove the second statement, suppose that $\mathcal F$ is not strongly expanding at $z_0$ with respect to $\mathbb C_\infty\setminus \mathcal E$ . Then there is an infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ that is not expanding at $z_0$ with respect to $\mathbb C_\infty \setminus \mathcal E$ , contradicting the fact that $\tilde{\mathcal F}$ is expanding at $z_0$ with respect to $\mathbb C_\infty \setminus E_{\tilde{\mathcal F}}$ for some set $E_{\tilde{\mathcal F}} \subset \mathbb C_\infty$ with $E_{\tilde{\mathcal F}} \subset \mathcal E$ .

(ii) Suppose that for some infinite subfamily $\tilde{\mathcal F} = \{f_n:n \in \mathbb N\}$ of $\mathcal F$ the sequence $(f_n)$ is spherically uniformly convergent on compact subsets of a neighbourhood of $z_0$ . Then $\limsup_{z_0} \tilde{\mathcal F}$ is a one-point set, and hence $|\liminf_{z_0}\mathcal F|\le 1$ . The second statement follows from the fact that in this case we have $A \subset \liminf_{z_0}\mathcal F$ .

From Theorem 2 we easily obtain the following characterisation of non-normality in terms of the expanding property, which in some sense complements the statement of Montel’s Theorem:

Proof. $\textrm{(i)} \Rightarrow \textrm{(ii)}$ Suppose that $\mathcal F$ is expanding at $z_0$ with respect to some $A \subset \mathbb C_\infty$ with $\left|{A}\right| \geq 2$ . Then there exists an infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ that is strongly expanding at $z_0$ with respect to A. By Theorem 2, the family $\tilde{\mathcal F}$ is strongly non-normal at $z_0$ , hence $\mathcal F$ is non-normal at $z_0$ .

$\textrm{(ii)} \Rightarrow \textrm{(iii)}$ If $\mathcal F$ is non-normal at $z_0$ , there exists an infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ that is strongly non-normal at $z_0$ . By Theorem 2, there then exists $E \subset \mathbb C_\infty$ with $\left|{E}\right| \leq 2$ such that $\tilde{\mathcal F}$ is expanding at $z_0$ with respect to $\mathbb C_\infty \setminus E$ . The same then holds for the family $\mathcal F$ .

$\textrm{(iii)} \Rightarrow \textrm{(i)}$ is obvious.

Let $\mathcal F \subset M(\Omega)$ be a family that is non-normal at a point $z_0 \in \Omega$ and consider the set $E_{z_0}(\mathcal F) = \mathbb C_\infty \setminus \limsup_{z_0} \mathcal F$ . If $\mathcal F$ is expanding at $z_0$ with respect to $\mathbb C_\infty \setminus E$ for some set $E \subset \mathbb C_\infty$ , we obviously have $E_{z_0}(\mathcal F) \subset E$ . If $\mathcal F$ is a family of holomorphic functions on $\Omega$ that is (strongly) non-normal at $z_0$ , we have $\infty \in E_{z_0}(\mathcal F)$ , so that in this case we obtain that the expanding property of $\mathcal F$ at $z_0$ in Theorem 2 and Corollary 1 holds with respect to $\mathbb C \setminus E$ for some set $E \subset \mathbb C$ with $\left|{E}\right| \leq 1$ .

A further consequence of Theorem 2 and the fact that we have $E_{z_0}(\mathcal F) \subset E$ if $\mathcal F \subset M(\Omega)$ is expanding at $z_0 \in \Omega$ with respect to $\mathbb C_\infty \setminus E$ is the following statement for the case $\left|{E_{z_0}(\mathcal F)}\right| = 2$ .

Proof. Suppose that $\mathcal F$ is non-normal at $z_0$ . By Corollary 1, there then exists $E \subset \mathbb C_\infty$ with $\left|{E}\right| \leq 2$ such that $\mathcal F$ is expanding at $z_0$ with respect to $\mathbb C_\infty \setminus E$ . Since $E_{z_0}(\mathcal F) \subset E$ , we obtain $E_{z_0}(\mathcal F) = E$ . If $\mathcal F$ is strongly non-normal at $z_0$ , every infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ is non-normal at $z_0$ with $E_{z_0}(\tilde{\mathcal F}) = E_{z_0}(\mathcal F)$ , hence expanding at $z_0$ with respect to $\mathbb C_\infty \setminus E_{z_0}(\mathcal F)$ .

4. Expanding families of derivatives

In the following, we show that under certain conditions, (strong) non-normality of a family $\mathcal F \subset M(\Omega)$ at a point $z_0 \in \Omega$ implies that the family of derivatives is (strongly) expanding at $z_0$ with respect to $\mathbb C \setminus \{0\}$ , hence in particular (strongly) non-normal at $z_0$ . Throughout this section, we denote by $\mathcal F^{(k)}$ the family of kth derivatives of the functions in $\mathcal F$ , that is $\mathcal F^{(k)} = \{f^{(k)}\;:\;f \in \mathcal F\}$ , where k is some natural number.

Proof. We first assume that $\mathcal F$ is strongly non-normal at $z_0$ . By assumption, $\mathcal F$ is not expanding at $z_0$ with respect to $\mathbb C$ , hence there exists an open neighbourhood $U_1$ of $z_0$ and a compact set $K_1 \subset \mathbb C$ such that $K_1 \setminus f(U_1) \neq \emptyset$ holds for cofinitely many $f \in \mathcal F$ .

Now assume that there exists $k \in \mathbb N$ , such that $\mathcal F^{(k)}$ is not strongly expanding at $z_0$ with respect to $\mathbb C \setminus \{0\}$ . Then there exists an open neighbourhood $U_2$ of $z_0$ and a compact set $K_2 \subset \mathbb C \setminus \{0\}$ such that $K_2 \setminus f^{(k)}(U_2) \neq \emptyset$ holds for infinitely many $f \in \mathcal F$ .

In particular, we can find a sequence $(f_n)$ in $\mathcal F$ , and sequences $(c^{(1)}_n)$ in $K_1$ and $(c^{(2)}_n)$ in $K_2$ , such that the equations $f_n(z) = c_n^{(1)}$ and $f_n^{(k)}(z) = c_n^{(2)}$ have no roots in $U \;:\!=\; U_1 \cap U_2$ for every $n \in \mathbb N$ . From [Reference Chuang10, theorem 3·17], which is an extension of Gu’s famous normality criterion (e.g. [Reference Gu17,Reference Schiff29]), we obtain that $(f_n)$ is normal in U, hence also at $z_0$ , in contradiction to the strong non-normality of $\mathcal F$ at $z_0$ .

If $\mathcal F$ is non-normal at $z_0$ , there exists an infinite subfamily $\tilde{\mathcal F} \subset \mathcal F$ that is strongly non-normal at $z_0$ . By assumption, $\mathcal F$ is not expanding at $z_0$ with respect to $\mathbb C$ , hence the same holds for $\tilde{\mathcal F}$ , so that by the above argumentation, $\tilde{\mathcal F}^{(k)}$ is strongly expanding at $z_0$ with respect to $\mathbb C \setminus \{0\}$ for every $k \in \mathbb N$ . Hence, $\mathcal F^{(k)}$ is expanding at $z_0$ with respect to $\mathbb C \setminus \{0\}$ for every $k \in \mathbb N$ .

Proof. Since it follows from the assumptions that $\mathcal F$ is not expanding at $z_0$ with respect to $\mathbb C$ , the statement follows from Theorem 3.

Note that the assumptions of Corollary 3 are fulfilled if $\mathcal F \subset M(\Omega)$ is (strongly) non-normal at $z_0 \in \Omega$ and for some $a \in \mathbb C$ we have $a \in E_{z_0}(\mathcal F)$ , hence in particular if $\left|{E_{z_0}(\mathcal F)}\right| = 2$ .

We mention that the statement of Corollary 3 remains valid to some extent, if instead of omitting a value $a_f$ in some neighbourhood of $z_0$ , cofinitely many functions $f \in \mathcal F$ have a value $a_f$ that they take with sufficiently high multiplicity in that neighbourhood.

Proof. Again, we first consider the case that $\mathcal F$ is strongly non-normal at $z_0$ . Assuming that $\mathcal F^{(k)}$ is not strongly expanding at $z_0$ with respect to $\mathbb C \setminus \{0\}$ , there exists an open neighbourhood $U_1$ of $z_0$ and a compact set $K \subset \mathbb C \setminus \{0\}$ such that $K \setminus f^{(k)}(U_1) \neq \emptyset$ for infinitely many $f \in \mathcal F$ . In particular, we can find a sequence $(c_n)$ in K with $c_n \to c$ for some $c \neq 0$ , and a sequence $(f_n)$ in $\mathcal F$ such that $c_n \notin f_n^{(k)}(U_1)$ for every $n \in \mathbb N$ . For $n \in \mathbb N$ sufficiently large, say $n > N$ , there is a point $a_{f_n} \in \mathbb C$ with $\left|{a_{f_n}}\right| < M$ , such that the $a_{f_n}$ -points of $f_n$ in U have multiplicity at least $k+2$ . Setting $g_n(z) = f_n(z) - a_{f_n}$ for $n > N$ , we obtain that the functions $g_n$ only have zeros of multiplicity at least $k+2$ in $U' \;:\!=\; U \cap U_1$ . Furthermore, since $c_n \notin g_n^{(k)}(U')$ for every $n > N$ , it follows from [Reference Cheng and Xu9, lemma 2·7] that the family $\{g_n\;:\;n > N\}$ is normal in U ^′, and as $\left|{a_{f_n}}\right| < M$ for every $n > N$ , the same holds for the family $\{f_n\;:\;n > N\}$ . This is in contradiction to the strong non-normality of $\mathcal F$ at $z_0$ .

If $\mathcal F$ is non-normal at $z_0$ , the statement follows as before from the fact that $\mathcal F$ contains a strongly non-normal subfamily.

In general, the number $k+2$ can not be replaced by $k+1$ in Proposition 3. Indeed, for fixed $k \in \mathbb N$ , the family $(f_n)$ with

\begin{equation*}f_n(z) = \frac{1}{k!} \, \frac{z^{k+1}}{(z - \frac{1}{n})}\end{equation*}

is strongly non-normal at the point 0 and has only zeros of multiplicity $k + 1$ (see also [Reference Wang and Fang31]). But as $f_n^{(k)}(z) \neq 1$ for every $n \in \mathbb N$ and every $z \in \mathbb C$ , the familiy $(f_n^{(k)})$ is obviously not expanding at 0 with respect to $\mathbb C \setminus \{0\}$ . Nevertheless, under certain additional conditions, $k+2$ can be replaced by $k+1$ :

Proof. Using [Reference Chen7, lemma 4] and [Reference Nevo, Pang and Zalcman27, lemma 6], respectively, the proofs of (i) and (ii) are similar to the proof of Proposition 3. In order to prove the third statement, we note that using [Reference Chen, Pang and Yang8, lemma 2·9], a similar argumentation as in the proof of Proposition 3 implies that the family $(g_n)$ with $g_n(z) = f_n(z) - a_{f_n}$ is quasinormal in some neighbourhood U of $z_0$ . Since $\left|{a_{f_n}}\right| < M$ for every $n \in \mathbb N$ , the same then holds for the family $(f_n)$ ([Reference Chuang10, lemma 5·2]). This contradicts the assumption that the set $\{z\;:\;\mathcal F \text{ is strongly non-normal at } z\}$ has an accumulation point in U.