1. Introduction
For an open set
$\Omega \subset \mathbb C$
we denote by
$M(\Omega)$
the set of meromorphic functions on
$\Omega$
, by which we mean all functions whose restriction to a connected component of
$\Omega$
is either meromorphic or constant infinity. Endowed with the topology of spherically uniform convergence (i.e. uniform convergence with respect to the chordal metric
$\chi$
) on compact subsets of
$\Omega$
, the space
$M(\Omega)$
becomes a complete metric space (e.g. [Reference Conway12, Chapter VII]). As usual, we say that a family
$\mathcal F \subset M(\Omega)$
is normal in
$\Omega$
, if every sequence
$(f_n) \subset \mathcal F$
contains a subsequence
$(f_{n_k})$
that converges spherically uniformly on compact subsets of
$\Omega$
to a function
$f \in M(\Omega)$
. The family
$\mathcal F$
is called normal at a point
$z_0 \in \Omega$
, if there exists an open neighbourhood U of
$z_0$
, such that
$\mathcal F$
is normal in U. By
$J(\mathcal F)$
we denote the set of points in
$\Omega$
, at which the family
$\mathcal F$
is non-normal. If
$z_0 \in J(\mathcal F)$
, the family
$\mathcal F$
can still have infinite subfamilies
$\tilde{\mathcal F} \subset \mathcal F$
that are normal at
$z_0$
, in other words,
$z_0 \in J(\mathcal F)$
does in general not imply
$z_0 \in J(\tilde{\mathcal F})$
. We say that
$\mathcal F$
is strongly non-normal at a point
$z_0 \in \Omega$
, if we have
$z_0 \in J(\tilde{\mathcal F})$
for every infinite subfamily
$\tilde{\mathcal F} \subset \mathcal F$
. We further say that
$\mathcal F$
is strongly non-normal on a relatively closed set
$B \subset \Omega$
, if
$\mathcal F$
is strongly non-normal at every
$z_0 \in B$
, that is if
$B \subset J(\tilde{\mathcal F})$
for every infinite subfamily
$\tilde{\mathcal F} \subset \mathcal F$
. Moreover, we call
$\mathcal F$
hereditarily non-normal on B, if some infinite subfamily
$\tilde{\mathcal F} \subset \mathcal F$
is strongly non-normal on B. Note that on a single point set, hereditary non-normality is equivalent to non-normality, while this is in general not true for sets containing at least two points.
For a family
$\mathcal F \subset M(\Omega)$
and an open set
$U \subset \Omega$
, we write
$\limsup \mathcal F(U)$
for the intersection of all
$\bigcup_{f \in \tilde{\mathcal F}} f(U)$
, where
$\tilde{\mathcal F}$
ranges over the cofinite subsets of
$\mathcal F$
. Moreover, for
$z_0 \in \Omega$
we denote by
$\limsup_{z_0} \mathcal F$
the intersection of
$\limsup \mathcal F(U)$
taken over all open neighbourhoods
$U \subset \Omega$
of
$z_0$
. Similarly, we write
$\liminf \mathcal F(U)$
for the union of all
$\bigcap_{f \in \tilde{\mathcal F}} f(U)$
, where
$\tilde{\mathcal F}$
ranges over the cofinite subsets of
$\mathcal F$
and
$\liminf_{z_0} \mathcal F$
for the intersection of
$\liminf \mathcal F(U)$
taken over all open neighbourhoods
$U \subset \Omega$
of
$z_0$
. Obviously, we have that
$\liminf_{z_0}\mathcal F \subset \limsup_{z_0}\mathcal F$
, furthermore
$\liminf_{z_0}\mathcal F=\bigcap_{\tilde{\mathcal F} \subset \mathcal F \text{ infinite}} \limsup_{z_0} \tilde{\mathcal F}$
. For instance, if
$\mathcal F = \{f_n\;:\; n \in \mathbb N\}$
with
$f_n(z)=nz$
for even integers n and
$f_n(z) = z$
for odd n, then
$\limsup_0 \mathcal F = \mathbb C$
and
$\liminf_0 \mathcal F = \{0\}$
.
The classical Montel Theorem suggests that the behaviour of families
$\mathcal F \subset M(\Omega)$
in neighbourhoods of points
$z_0 \in J(\mathcal F)$
consists in some sense of spreading points, since it asserts that for every
$z_0 \in J(\mathcal F)$
, the set
$E_{z_0}(\mathcal F) \;:\!=\; \mathbb C_\infty \setminus \limsup_{z_0} \mathcal F$
contains at most two points, where
$\mathbb C_\infty \;:\!=\; \mathbb C \cup \{\infty\}$
. Hence, for every neighbourhood U of
$z_0$
, every point
$a \in \mathbb C_\infty$
is covered by f(U) for infinitely many
$f \in \mathcal F$
, with at most two exceptions. In case that
$E_{z_0}(\mathcal F)$
contains two points and
$\mathcal F$
is strongly non-normal at
$z_0$
, a further consequence of Montel’s Theorem is that
$\liminf_{z_0}\mathcal F = \limsup_{z_0}\mathcal F$
, so that for every neighbourhood U of
$z_0$
, every point
$a \in \mathbb C_\infty \setminus E_{z_0}(\mathcal F)$
is covered by f(U) for cofinitely many
$f \in \mathcal F$
. Note, however, that Montel’s Theorem does not contain any information about the ‘size’ of the individual sets f(U), for instance, if U is any neighbourhood of a point
$z_0 \in J(\mathcal F)$
, it is in general not clear if for a given set
$A \subset \limsup_{z_0}\mathcal F$
we have
$A \subset f(U)$
for infinitely many
$f \in \mathcal F$
.
In this paper, we will further investigate the behaviour of (strongly) non-normal families near points of non-normality and show certain covering and ‘expanding’ properties that complement the statement of Montel’s Theorem. In particular, we will also derive different characterisations of (strong) non-normality in terms of these properties.
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.
Theorem 1. Let
$\Omega \subset \mathbb C$
be open,
$\mathcal F \subset M(\Omega)$
a family of meromorphic functions and
$z_0 \in \Omega$
. Then we have:
-
(a)
$\mathcal F$
is strongly non-normal at
$z_0$
if and only if
$\mathcal F$
is mixing with respect to
$z_0$
; -
(b) the following are equivalent:
-
(i)
$\mathcal F$
is non-normal at
$z_0$
; -
(ii) there exists an infinite subfamily
$\tilde{\mathcal F} \subset \mathcal F$
that is mixing with respect to
$z_0$
; -
(iii)
$\mathcal F$
is weakly mixing with respect to
$z_0$
.
-
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:
Proposition 1. Let
$\Omega \subset \mathbb C$
be open,
$\mathcal F \subset M(\Omega)$
a family of meromorphic functions and
$z_0 \in \Omega$
. Suppose that
$\mathcal F$
is transitive with respect to
$z_0$
and that
$z_0 \notin J(\mathcal F)$
. Then
$\{f(z_0)\;:\; f \in \mathcal F\}$
is dense in
$\mathbb C_\infty$
.
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
$z_0 \in J(\mathcal F)$
.
Example 1.
-
(i) Let f be an entire function that is neither constant nor a polynomial of degree 1, and let
$\mathcal F\;:\!=\;\{f^{\circ n}\;:\;n \in \mathbb N\}$
be the family of iterates of f. Then
$\mathcal F$
is strongly non-normal on the Julia set
$J=J(\mathcal F)$
, as follows e.g. from the facts that the repelling periodic points are dense in J and that J is the boundary of the escaping set (e.g. [Reference Carleson and Gamelin6,Reference Fatou14,Reference Morosawa, Nishimura, Taniguchi and Ueda26,Reference Schleicher30]). Here we have
$\liminf_{z_0}\mathcal F\supset \mathbb C \setminus E$
for each
$z_0 \in J$
, where E is the (empty or one-point) set of Fatou exceptional values of f, that is the set of points
$w \in \mathbb C$
whose backward orbit
$O^{-}(w) \;:\!=\; \bigcup_{n \geq 1}\{z\;:\;f^{\circ n}(z) = w\}$
is finite. Indeed, consider
$z_0 \in J$
and an infinite subfamily
$\tilde{\mathcal F} = \{f^{\circ n_k}\;:\;k \in \mathbb N\}$
with
$n_k > 2$
. It follows from Picard’s Theorem that if
$a \in \mathbb C$
is not Fatou exceptional, there are points
$a_1, a_2 \in \mathbb C$
with
$a_1 \neq a_2$
and
$f^{\circ 2}(a_1) = a = f^{\circ 2}(a_2)$
. Since
$\mathcal F$
is strongly non-normal at
$z_0$
, Montel’s Theorem implies that the set
$\mathbb C \setminus \limsup_{z_0} \tilde{\mathcal F}^-$
contains at most one point, where
$\tilde{\mathcal F}^- \;:\!=\; \{f^{\circ (n_k - 2)}\;:\;k \in \mathbb N\}$
. Hence,
$\{a_1,a_2\} \cap \limsup_{z_0} \tilde{\mathcal F}^- \neq \emptyset$
, which implies
$a \in \limsup_{z_0}{\tilde{\mathcal F}}$
. -
(ii) Let M denote the Mandelbrot set and let, with
$p_0\;:\!=\;\mathrm{id}_\mathbb C$
, the family
$(p_n)$
of polynomials of degree
$2^n$
be recursively defined by
$p_n\;:\!=\;p_{n-1}^2 + \mathrm{ id}_\mathbb C$
. Since
$p_n \to \infty$
pointwise on
$\mathbb C \setminus M$
for
$n \to \infty$
and
$|p_n|\le 2$
on M (e.g. [Reference Carleson and Gamelin6]), we have
$\partial M \subset J(\mathcal F)$
, where
$\mathcal F\;:\!=\;\{p_n\;:\;n \in \mathbb N_0\}$
, and no infinite subfamily of
$\mathcal F$
can be normal at any point of
$\partial M$
. Hence,
$\mathcal F$
is strongly non-normal and thus mixing on
$\partial M$
. -
(iii) A function
$f \in M(\mathbb C)$
is called Yosida function, if it has bounded spherical derivative
$${f^\# }$$
(e.g. [Reference Minda and Chuang24,Reference Yosida32]). Hence, if f is not a Yosida function, there exists a sequence
$(z_n)$
in
$\mathbb C$
with
$z_n \to \infty$
and
${f^\# }(z_n) \to \infty$
for
$n \to \infty$
. Marty’s Theorem (e.g. [Reference Schiff29, p.75]) implies that the family
$(f_n)$
with
$f_n(z) \;:\!=\; f(z + z_n)$
is strongly non-normal at 0, hence by Theorem 1, we obtain that
$(f_n)$
is mixing with respect to 0. Note that it is easily seen that if
$f \in M(\mathbb C)$
is a Yosida function, then its order of growth is at most 2, while entire Yosida functions are necessarily of exponential type (e.g. [Reference Clunie and Hayman11,Reference Minda and Chuang24]).
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.
Proposition 2. Let
$\Omega \subset \mathbb C$
be open,
$\mathcal F \subset M(\Omega)$
a family of meromorphic functions and
$B \subset \Omega$
closed in
$\Omega$
. Then the following are equivalent:
-
(i)
$\mathcal F$
is hereditarily non-normal on B; -
(ii) there exists an infinite subfamily
$\tilde{\mathcal F} \subset \mathcal F$
that is mixing with respect to B; -
(iii) for all
$N \in \mathbb N$
the family
$\mathcal F^{\times N}$
is transitive with respect to
$B^N$
.
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]).
Remark 1.
-
(i) Let
$\mathcal{K}(A)$
denote the hyperspace of
$A \subset \mathbb C$
, that is the space of all non-empty compact subsets of A endowed with the Hausdorff metric, and suppose that B as in Proposition 2 has non-empty interior. Then [Reference Beise, Meyrath and Müller2, corollay 1·2] shows that, under the conditions of Proposition 2, for each
$\mathbb C$
-closed set
$A \subset B$
which coincides with the closure of its interior, the family
$\mathcal F|_E$
is dense in
$C(E,\mathbb C_\infty)$
for generically many sets
$E \in \mathcal{K}(A)$
. -
(ii) We mention that Proposition 2 is an extension of Theorem 3·7 from the recent paper [Reference Bernal-gonzález, Jung and Müller4].
Example 2.
-
(i) Consider a function
$f(z) = \sum_{\nu = 0}^{\infty} a_{\nu} z^{\nu}$
that is holomorphic on the unit disk
$\mathbb D$
. Suppose that f has at least one singularity on
$\partial \mathbb D$
and denote by
$D \subset \partial \mathbb D$
the set of all singularities. Then, denoting by
$s_n(z) \;:\!=\; (s_n f)(z) \;:\!=\; \sum_{\nu = 0}^{n} a_{\nu} z^{\nu}$
the nth partial sum of f, the family
$(s_n)$
is non-normal on
$\partial \mathbb D$
and strongly non-normal on D. Moreover, in case
$D \neq \partial\mathbb D$
, Vitali’s Theorem implies that a subsequence of
$(s_n)$
forms a normal family at a point
$z_0 \in \partial \mathbb D \setminus D$
if and only if it converges to an analytic continuation of f in some neighbourhood of
$z_0$
. From refined versions of Ostrowski’s results on overconvergence ([Reference Gehlen16, Theorems 3 and 4]), it follows that a subsequence
$(s_{n_k})$
is strongly non-normal at
$z_0 \in \partial \mathbb D \setminus D$
if and only if
$(s_{n})$
has no Hadamard–Ostrowski gaps relative to
$(n_k)$
, that is, if and only if there is a sequence
$(\delta_k)$
of positive numbers tending to 0 with
as
\begin{equation*}\sup_{(1-\delta_k)n_k\le\nu \le {n_k}} |a_\nu|^{1/\nu} \to 1\end{equation*}
$k \to \infty$
. In this case, the sequence
$(s_{n_k})$
is already strongly non-normal at all
$z \in \partial \mathbb D$
. Since the non-normality of
$(s_n)$
on
$\partial \mathbb D$
implies that, given
$z_0 \in \partial \mathbb D \setminus D$
, some subsequence of
$(s_n)$
is strongly non-normal at
$z_0$
, we finally obtain that the family
$(s_n)$
is always hereditarily non-normal on
$\partial \mathbb D$
. According to a result of Gardiner ([Reference Gardiner15, corollary 3]), for each f that is holomorphic on
$\mathbb D$
and analytically continuable to some domain U such that
$\mathbb C \setminus U$
is thin at some
$z_0 \in \partial \mathbb D$
but not continuable to the point
$z_0$
, the sequence
$(s_n)$
has no Hadamard–Ostrowski gaps with respect to any
$(n_k)$
, hence
$(s_n)$
is strongly non-normal on
$ \partial \mathbb D$
. In particular, this holds for each f that has an isolated singularity at some point
$z_0 \in \partial \mathbb{D}$
.
-
(ii) We write
$H_0$
for the space of functions holomorphic on
$\mathbb{C} \setminus \{1\}$
that vanish at
$\infty$
. For
$f(z)=1/(1-z)$
, the sequence
$(s_n f)$
is the geometric series which tends to
$\infty$
spherically uniformly on compact subsets of
$\mathbb C \setminus \overline{\mathbb{D}}$
. From [Reference Beise, Meyrath and Müller3, theorem 1·1] it can be deduced that generically many functions
$f \in H_0$
enjoy the property that some subsequence of the sequence
$((f-s_{n} f)(z)/z^n)$
converges to
$1/(1-z)$
spherically uniformly on compact subsets of
$\mathbb C_\infty \setminus \{1\}$
. This implies that the corresponding subsequence of
$(s_n f)$
converges to
$\infty$
spherically uniformly on compact subsets of
$\mathbb C \setminus \overline{\mathbb{D}}$
and thus forms a normal family on
$\mathbb C \setminus \overline{\mathbb{D}}$
. In particular,
$(s_n f)$
is not strongly non-normal at any point
$z_0 \in \mathbb C \setminus \overline{\mathbb{D}}$
. On the other hand, if A is a countable and dense subset of
$\mathbb C \setminus \mathbb{D}$
, from [Reference Melas23, theorem 2] it follows that for generically many functions
$f \in H_0$
, a subsequence
$(s_{n_k}f)$
of
$(s_{n}f)$
converges to 0 pointwise on A. Since a result from [Reference Kalmes, Müller and Niess22] implies that for
$f \in H_0$
, normality of a subsequence of
$(s_nf)$
at a point
$z_0 \in \mathbb C \setminus \overline{\mathbb D}$
forces the subsequence to tend to
$\infty$
spherically uniformly on compact subsets of some neighbourhood of
$z_0$
, it follows that no subsequence of
$(s_{n_k} f)$
can form a normal family at any point of
$\mathbb C \setminus \overline{\mathbb D}$
. By the previous example,
$(s_n f)$
is strongly non-normal on
$\partial \mathbb{D}$
for
$f \in H_0$
, thus we obtain that for generically many
$f \in H_0$
, the family
$(s_n f)$
is hereditarily non-normal on
$\mathbb C \setminus \mathbb D$
. By Remark 1, for generically many
$f \in H_0$
, the sequence
$(s_nf|_E)$
is dense in
$C(E, \mathbb C_\infty)$
for generically many
$E \in \mathcal{K}(\mathbb C\setminus \mathbb D)$
(see also [Reference Beise, Meyrath and Müller1, theorem 2]).
3. Non-normality and expanding families
We define the following ‘expanding’ property of families
$\mathcal F \subset M(\Omega)$
.
Definition 1. Let
$\Omega \subset \mathbb C$
be open,
$\mathcal F \subset M(\Omega)$
a family of meromorphic functions and
$z_0 \in \Omega$
. Consider further a set
$A \subset \mathbb C_\infty$
. We say that
$\mathcal F$
is expanding at
$z_0$
with respect to A, if for every open neighbourhood
$U \subset \Omega$
of
$z_0$
and every compact set
$K \subset A$
we have
$K \subset f(U)$
for infinitely many
$f \in \mathcal F$
. If
$K \subset f(U)$
holds for cofinitely many
$f \in \mathcal F$
, we say that
$\mathcal F$
is strongly expanding at
$z_0$
with respect to A. Finally, we say that
$\mathcal F$
is (strongly) expanding on a set
$B \subset \Omega$
with respect to A, if
$\mathcal F$
is (strongly) expanding with respect to A at every
$z_0 \in B$
.
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
$E \subset \mathbb C_\infty$
.
Theorem 2. Let
$\Omega \subset \mathbb C$
be open,
$\mathcal F \subset M(\Omega)$
a family of meromorphic functions and
$z_0 \in \Omega$
. Then we have:
-
(i) if
$\mathcal F$
is strongly non-normal at
$z_0$
, then for each infinite subfamily
$\tilde{\mathcal F} \subset \mathcal F$
there 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
$\mathbb C_\infty \setminus E$
. Moreover,
$\mathcal F$
is strongly expanding at
$z_0$
with respect to
$\mathbb C_\infty \setminus \mathcal E$
, where
$\mathcal E \;:\!=\; \bigcup_{\tilde{\mathcal F} \subset \mathcal F \text{ infinite}} E_{\tilde{\mathcal F}}$
with
$E_{\tilde{\mathcal F}} \subset \mathbb C_\infty$
being some set such that
$\tilde{\mathcal F}$
is expanding at
$z_0$
with respect to
$\mathbb C_\infty \setminus E_{\tilde{\mathcal F}}$
; -
(ii) if
$|\liminf_{z_0}\mathcal F|\ge 2$
, then
$\mathcal F$
is strongly non-normal at
$z_0$
. In particular, this holds if
$\mathcal F$
is strongly expanding at
$z_0$
with respect to some
$A \subset \mathbb C_\infty$
with
$\left|{A}\right| \geq 2$
.
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
$A \subset \liminf_{z_0}\mathcal F$
.
Remark 2. Note that if
$\mathcal F$
is strongly non-normal at
$z_0$
,
$\mathcal F$
does not need to be strongly expanding at
$z_0$
with respect to any open set
$A \subset \mathbb C_\infty$
. Indeed, let
$(q_n)$
be an enumeration of the Gaussian rational numbers with
$q_n^2/n \to 0$
as
$n \to \infty$
and consider the family
$(f_n)$
with
$f_n(z)\;:\!=\;e^{nz}+q_n$
for
$z \in \mathbb C$
. From Marty’s Theorem, it is easily seen that
$(f_n)$
is strongly non-normal on the imaginary axis
$i\mathbb R$
, but for a point
$z_0 \in i\mathbb R$
and an open neighbourhood U of
$z_0$
, there is no
$N \in \mathbb N$
such that
$K \subset f_n(U)$
holds for all
$n \geq N$
for any compact set
$K \subset \mathbb C$
with
$K^{\circ} \neq \emptyset$
.
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:
Corollary 1. Let
$\Omega \subset \mathbb C$
be open,
$\mathcal F \subset M(\Omega)$
a family of meromorphic functions and
$z_0 \in \Omega$
. Then the following are equivalent:
-
(i) there exists
$A \subset \mathbb C_\infty$
with
$\left|{A}\right| \geq 2$
such that
$\mathcal F$
is expanding at
$z_0$
with respect to A; -
(ii)
$\mathcal F$
is non-normal at
$z_0$
; -
(iii) there exists
$E \subset \mathbb C_\infty$
with
$\left|{E}\right| \leq 2$
such that
$\mathcal F$
is expanding at
$z_0$
with respect to
$\mathbb C_\infty \setminus E$
.
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
$\left|{E}\right| \leq 1$
.
Example 3.
-
(i) Consider a compact set
$K \subset \mathbb C$
with connected complement and let f be a function that is continuous on K and holomorphic in
$K^{\circ}$
. Further assume that f has at least one singularity on
$\partial K$
and denote by
$D \subset \partial K$
the set of all singularities. Let
$(p_n)$
be a sequence of polynomials converging uniformly on K to f (such a sequence exists by Mergelian’s Theorem). Then,
$(p_n)$
is strongly non-normal on D, hence also expanding at every point
$z_0 \in D$
with respect to
$\mathbb C \setminus E$
for some set
$E \subset \mathbb C$
with
$\left|{E}\right| \leq 1$
. Indeed, since otherwise there exists a point
$z_0 \in D$
, an open neighbourhood U of
$z_0$
, and a subsequence
$(p_{n_k})$
of
$(p_n)$
that converges uniformly on compact subsets of U to a function holomorphic in U, contradicting that f does not have an analytic continuation across
$z_0 \in D$
. -
(ii) Consider the function
$f(z) = \left|{z}\right|$
on the interval
$[\!-\!1,1]$
and denote by
$(p_n^{\star})$
the sequence of polynomials of best uniform approximation to f on
$[\!-\!1,1]$
. Then, according to the previous example,
$(p_n^{\star})$
is strongly non-normal at the point 0. However, since
$p_n^{\star}(z) \to \infty$
for
$n \to \infty$
spherically uniformly on compact subsets of
$\mathbb C \setminus [\!-\!1,1]$
(e.g. [Reference Saff and Stahl28]), the family
$(p_n^{\star})$
is strongly non-normal on
$[\!-\!1,1]$
, hence expanding at every point
$z_0 \in [\!-\!1,1]$
with respect to
$\mathbb C \setminus E$
for some set
$E \subset \mathbb C$
with
$\left|{E}\right| \leq 1$
. (Note that the strong non-normality on
$[\!-\!1,1]$
also holds for several specific ray sequences of best uniform rational approximants to f on
$[\!-\!1,1]$
([Reference Saff and Stahl28, corollary 1·3]).) In fact, [Reference Blatt, Blatt and Luh5, corollary 2] implies that
$(p_n^{\star})$
is expanding on
$[\!-\!1,1]$
with respect to
$\mathbb C$
, as it shows the existence of a subsequence
$(p_{n_k}^{\star})$
of
$(p_n^{\star})$
that is strongly expanding on
$[\!-\!1,1]$
with respect to
$\mathbb C$
. -
(iii) Consider again a function
$f(z) = \sum_{\nu = 0}^{\infty} a_{\nu} z^{\nu}$
that is holomorphic on
$\mathbb D$
and has at least one singularity on
$\partial \mathbb D$
. Then the family of partial sums
$(s_n)$
is non-normal on
$\partial \mathbb D$
, hence,
$(s_n)$
is expanding at every
$z_0 \in \partial \mathbb D$
with respect to
$\mathbb C \setminus E$
for some set
$E \subset \mathbb C$
with
$\left|{E}\right| \leq 1$
. In fact,
$(s_n)$
is expanding on
$\partial \mathbb D$
with respect to
$\mathbb C$
, as results in [Reference Blatt, Blatt and Luh5,Reference Dvoretzky13] show that if
$(a_{n_k})$
is a sequence such that
$\lim_{k \to \infty} \left|{a_{n_k}}\right|^{\frac{1}{n_k}} = 1$
, the subfamily
$(s_{n_k})$
is strongly expanding on
$\partial \mathbb D$
with respect to
$\mathbb C$
.
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
$\left|{E_{z_0}(\mathcal F)}\right| = 2$
.
Corollary 2. Let
$\Omega \subset \mathbb C$
be open and
$\mathcal F \subset M(\Omega)$
be a family of meromorphic functions. Consider
$z_0 \in \Omega$
and suppose that
$\mathcal F$
is (strongly) non-normal at
$z_0$
with
$\left|{E_{z_0}(\mathcal F)}\right| = 2$
. Then
$\mathcal F$
is (strongly) expanding at
$z_0$
with respect to
$\mathbb C_\infty \setminus E_{z_0}(\mathcal F)$
.
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
$\mathbb C_\infty \setminus E_{z_0}(\mathcal F)$
.
Example 4.
-
(i) Consider again the family
$\mathcal F \;:\!=\; \{e^{nz} + (1 - {1}/{n})\;:\;n \in \mathbb N\}$
, which is strongly non-normal at the point 0. It is easily seen that
$\mathcal F$
is strongly expanding at 0 with respect to
$\mathbb C_\infty \setminus \{1,\infty\}$
, but since
$E_{0}(\mathcal F) = \{\infty\}$
, this can not be derived from Corollary 2. On the other hand, the family
$\mathcal F \;:\!=\; \{e^{nz} + (1 - {1}/{n!})\;:\;n \in \mathbb N\}$
is strongly non-normal at the point 0 with
$E_{0}(\mathcal F) = \{1, \infty\}$
(note that for each neighbourhood U of 0 and each
$1 \not = w \in \mathbb C$
, there is some N with
$w - 1 + {1}/{n!} \in \exp(nU)$
for all
$n\ge N$
, but
${1}/{n!} \not\in \exp(n\mathbb D)$
for sufficiently large n). So, in this case Corollary 2 can be applied. -
(ii) Consider again a power series
$f(z) = \sum_{\nu = 0}^{\infty} a_{\nu} z^{\nu}$
with radius of convergence 1 and denote by
$(s_n)$
its partial sums. As mentioned in Example 3, the family
$\mathcal F = \{s_n\;:\;n \in \mathbb N\}$
is expanding on
$\partial \mathbb D$
with respect to
$\mathbb C$
, so that for every
$z_0 \in \partial \mathbb D$
we have
$E_{z_0}(\mathcal F) = \{\infty\}$
(note that this is also easily derived from the classical Jentzsch Theorem ([Reference Jentzsch19]) stating that for every
$a \in \mathbb C$
, every
$z_0 \in \partial \mathbb D$
is a limit point of a-points of the partial sums). However, a further result of Jentzsch ([Reference Jentzsch20]) states that there exist power series with radius of convergence 1, such that the zeros of some subsequence
$(s_{n_k})$
of the partial sums do not have a finite limit point. Hence, in this case Corollary 2 shows that the family
$\tilde{\mathcal F} = \{s_{n_k}\;:\;k \in \mathbb N\}$
is strongly expanding with respect to
$\mathbb C \setminus \{0\}$
at every point
$z_0 \in \partial \mathbb D$
at which the function does not admit an analytic continuation (there must be at least one such point), since
$\tilde{\mathcal F}$
is strongly non-normal at such
$z_0$
with
$E_{z_0}(\tilde{\mathcal F}) = \{0, \infty\}$
. In a similar vein, it was shown in [Reference Ivanov, Saff and Totik18, theorem 1] that there exists a function f holomorphic on
$\mathbb D$
and continuous on
$\overline{\mathbb D}$
with at least one singularity on
$\partial \mathbb D$
, for which the zeros of some subsequence
$(p_{n_k}^{\star})$
of the sequence
$(p_n^{\star})$
of polynomials of best uniform approximation do not have a finite limit point. Hence, as before, Corollary 2 can be applied to the family
$\mathcal F = \{p_{n_k}^{\star}\;:\;k \in \mathbb N \}$
at every singular point
$z_0 \in \partial \mathbb D$
of f, since
$\mathcal F$
is strongly non-normal at
$z_0$
(see Example 3 (i)) and we have
$E_{z_0}(\mathcal F) = \{0, \infty\}$
. Moreover, [Reference Ivanov, Saff and Totik18, theorem 2] shows the existence of a function f that is holomorphic on
$\mathbb D$
and continuous on
$\overline{\mathbb D}$
with at least one singularity on
$\partial \mathbb D$
, for which there is a sequence
$(q_n)$
of polynomials of near-best uniform approximation that has no finite limit point of zeros. Hence, in this case Corollary 2 implies that the family
$\mathcal F = \{q_n\;:\;n \in \mathbb N \}$
is strongly expanding with respect to
$\mathbb C \setminus \{0\}$
at every singular point
$z_0 \in \partial \mathbb D$
of 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.
Theorem 3. Let
$\Omega \subset \mathbb C$
be open and
$\mathcal F \subset M(\Omega)$
be a family of meromorphic functions. Consider
$z_0 \in \Omega$
and suppose that
$\mathcal F$
is (strongly) non-normal at
$z_0$
. Further assume that
$\mathcal F$
is not expanding at
$z_0$
with respect to
$\mathbb C$
. Then, for every
$k \in \mathbb N$
, the family
$\mathcal F^{(k)}$
is (strongly) expanding at
$z_0$
with respect to
$\mathbb C \setminus \{0\}$
.
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
$k \in \mathbb N$
.
Remark 3. Note that the assumption that
$\mathcal F$
is not expanding at
$z_0$
with respect to
$\mathbb C$
turns out to be necessary, as is seen e.g. by considering the sequence of polynomials
$f_n(z) = nz$
and
$z_0 = 0$
. Moreover, it is easily seen that a similar argumentation as in the proof of the theorem leads to the following result: Let
$\Omega \subset \mathbb C$
be open and
$\mathcal F \subset M(\Omega)$
be a family of meromorphic functions. Consider
$z_0 \in \Omega$
and suppose that
$\mathcal F$
is (strongly) non-normal at
$z_0$
. Further assume that for some
$k \in \mathbb N$
, the family
$\mathcal F^{(k)}$
is not expanding at
$z_0$
with respect to
$\mathbb C \setminus \{0\}$
. Then, the family
$\mathcal F$
is (strongly) expanding at
$z_0$
with respect to
$\mathbb C$
.
Corollary 3. Let
$\Omega \subset \mathbb C$
be open and
$\mathcal F \subset M(\Omega)$
be a family of meromorphic functions. Consider
$z_0 \in \Omega$
and suppose that
$\mathcal F$
is (strongly) non-normal at
$z_0$
. Suppose further that there exists an open neighbourhood U of
$z_0$
and a number
$M > 0$
, such that for cofinitely many
$f \in \mathcal F$
there is a point
$a_f \in \mathbb C$
with
$\left|{a_f}\right| < M$
and
$a_f \notin f(U)$
. Then, for every
$k \in \mathbb N$
, the family
$\mathcal F^{(k)}$
is (strongly) expanding at
$z_0$
with respect to
$\mathbb C \setminus \{0\}$
.
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
$\left|{E_{z_0}(\mathcal F)}\right| = 2$
.
Example 5.
-
(i) In Example 4 (ii) we considered strongly non-normal families
$\mathcal F$
of polynomials for which
$E_{z_0}(\mathcal F) = \{0, \infty\}$
, hence we obtain that the corresponding families of derivatives
$\mathcal F^{(k)}$
are strongly expanding at
$z_0$
with respect to
$\mathbb C \setminus \{0\}$
for every
$k \in \mathbb N$
. -
(ii) Consider the family
$(f_n)$
with
$f_n \;:\!=\; \exp^{\circ n}$
, the nth iterate of
$e^z$
. Then
$J(f_n)$
coincides with the Julia set of
$e^z$
, which is known to equal
$\mathbb C$
([Reference Misiurewicz25]). According to Example 1 (i), we thus have that
$(f_n)$
is strongly non-normal on
$\mathbb C$
. Furthermore, we obviously have
$0 \in E_{z_0}(f_n)$
for every
$z_0 \in \mathbb C$
, so that Corollary 3 implies that for every
$k \in \mathbb N$
, the family
$(f_n^{(k)})$
is strongly expanding on
$\mathbb C$
with respect to
$\mathbb C \setminus \{0\}$
.
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.
Proposition 3. Let
$\Omega \subset \mathbb C$
be open and
$\mathcal F \subset M(\Omega)$
be a family of meromorphic functions. Consider
$z_0 \in \Omega$
and suppose that
$\mathcal F$
is (strongly) non-normal at
$z_0$
. Suppose further that there exists an open neighbourhood U of
$z_0$
, a number
$M > 0$
and some
$k \in \mathbb N$
, such that for cofinitely many
$f \in \mathcal F$
there is a point
$a_f \in \mathbb C$
with
$\left|{a_f}\right| < M$
, such that the
$a_f$
-points of f in U have multiplicity at least
$k+2$
. Then the family
$\mathcal F^{(k)}$
is (strongly) expanding at
$z_0$
with respect to
$\mathbb C \setminus \{0\}$
.
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
$k+1$
:
Proposition 4. Under each of the following additional conditions, the statement of Proposition 3 remains valid if
$k+2$
is replaced by
$k+1$
:
-
(i) the functions
$f \in \mathcal F$
are holomorphic in
$\Omega$
; -
(ii) the functions
$f \in \mathcal F$
only have multiple poles; -
(iii) there exists a sequence
$(z_n)$
in
$\Omega$
with
$z_n \to z_0$
and
$\mathcal F$
is strongly non-normal at
$z_n$
for every
$n \in \mathbb N$
.
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.
Acknowledgements
The authors would like to thank the anonymous reviewer for his careful reading of the manuscript and his valuable comments and suggestions.
