The Bloch–Nevanlinna Problem Under Lineability

Abstract

The original version of the celebrated Bloch–Nevanlinna problem asked whether there exists or not a holomorphic function in the unit disc of bounded characteristic whose derivative is not of bounded characteristic. This problem has been solved in the affirmative by a number of mathematicians. Starting from a result on topological genericity of this class of functions due to Hahn, our work intends—under an algebraic as well as a topological point of view—to contribute to this topic. Specifically, the family of solutions of the Bloch–Nevanlinna problem is proved to be residual in appropriate topological spaces, and it contains, except for the zero function, dense maximal dimensional vector subspaces, large linear algebras and infinite-dimensional Banach spaces.

1 Introduction, Notation, Preliminaries and Aim of This Paper

Throughout this paper, we shall use the following (mostly standard) notation. If $\mathbb {D}$ is the open unit disc of the complex plane $\mathbb {C}$, then $H(\mathbb {D})$ will stand for the vector space of all holomorphic functions $\mathbb {D}\rightarrow \mathbb {C}$. It becomes a Fréchet space when endowed with the topology of uniform convergence on compacta. We shall deal with several linear subspaces of $H(\mathbb {D})$. For the study of these subspaces, the reader is referred to a number of books such as [18, 26, 29, 31, 35].

We shall consider, specially, the disk algebra and the Nevanlinna class. If $\mathbb {T}$ denotes the unit circle $\{z \in \mathbb {C}: |z| = 1\}$, then $A(\mathbb {D})$ will represent the disk algebra, that is, the vector space of all holomorphic functions in $\mathbb {D}$ admitting continuous extension on the closed unit disc ${\overline{\mathbb {D}}} = \mathbb {D}\cup \mathbb {T}$. In other words, $A(\mathbb {D}) = H(\mathbb {D}) \cap C({\overline{\mathbb {D}}})$. It is well known that, under the supremum norm $\Vert f\Vert _\infty := \sup _{|z|<1} |f(z)|$, the set $A(\mathbb {D})$ is a (separable) Banach space.

The characteristic function Tf of a function $f \in H(\mathbb {D})$ is defined as

$$\begin{aligned} (Tf)(r) = \frac{1}{2\pi }\int _{0}^{2\pi } \log ^+ |f(r e^{i\theta })| \, \textrm{d}\theta \quad (0< r < 1). \end{aligned}$$

where $\log ^+ t = \max \{0, \log t \}$ for $t > 0$. The Nevanlinna class N is defined as the class of holomorphic functions f in $\mathbb {D}$ having bounded characteristic, that is, satisfying

$$\begin{aligned} \displaystyle \sup _{0<r<1} (Tf)(r) < +\infty . \end{aligned}$$

The class N is a linear subspace, and even a linear algebra. It is known that $f \in N$ if and only if $f = g/h$ for some pair of functions $g,h \in H^\infty (\mathbb {D})$, where h is zero-free and $H^\infty (\mathbb {D})$ stands for the linear space of bounded holomorphic functions on $\mathbb {D}$. It follows that $A(\mathbb {D}) \subset H^\infty (\mathbb {D}) \subset N$. It is also known that every function in N has finite radial limit almost everywhere (a.e., from now on) on $\mathbb {T}$.

In 1929, Bloch and Nevanlinna [28] posed the problem of whether the derivative of a function of bounded characteristic is also of bounded characteristic; in terms of operator theory, this is equivalent to ask whether $D(N) \subset N$, where D is the derivative operator on $H(\mathbb {D})$ ($Df:= f'$). Since then, a number of counterexamples have been constructed by several mathematicians, as Campbell, Duren, Frostman, Hayman, Lohwater, Piranian, Rudin, and Wickes (see [12, 14,15,16, 21, 23, 30, 31]).

A common approach is to construct a function f whose derivative has finite radial limit almost nowhere along the unit circle, which implies (see, e.g., [32, Chapter 17]) $f' \not \in N$. It is even possible for such a function to have a continuous extension on the unit circle [30], that is, $f \in A(\mathbb {D})$. In 1972, Hahn [20] used Baire category methods to conclude the existence of a dense subset of $A(\mathbb {D})$ all of whose members f satisfy

$$\begin{aligned} \displaystyle \limsup _{r \rightarrow 1^-} |f'(re^{i\theta })| = +\infty \ \hbox {for all} \ \theta \in [0, 2\pi ]. \end{aligned}$$

(1.1)

Finally, Galanos [17], also employing the Baire category theorem, has been able to prove the desired generic existence, that is, the set of $f \in A(\mathbb {D})$ with $f' \not \in N$ is residual in $A(\mathbb {D})$ (see also the related paper [34]). In fact, Galanos established a stronger assertion, from which the last result is the special case $A=0,B=2\pi$: there is a residual subset ${\mathcal {R}}$ of $A(\mathbb {D})$ all of whose functions f satisfy $\sup _{0<r<1} \int _{A}^{B} \log ^+ |f'(re^{i \theta })| \,\textrm{d}\theta = +\infty$ for all $A,B \in \mathbb {R}$ with $A < B$ [17, Theorem 3.1]. Recall that, in a Baire topological space (in particular, in a completely metrizable topological space), a subset A is said to be residual provided that it contains a dense $G_\delta$-subset; such a subset A is “very large” in the topological sense.

The aim of the present paper is twofold. On the one hand, we strengthen the results by Hahn and Galanos by proving that the set of functions in the disc algebra satisfying (1.1) is in fact residual with respect to the natural topology (see Sect. 3). On the other hand, together with the mentioned topological genericity, it is shown in Sect. 4 that a slightly weaker version of the property (1.1) is enjoyed by the members of the disc algebra in an algebraically generic way, so that this set of functions contains, except for zero, large vector spaces and large algebras. Section 2 is devoted to furnish the necessary background on algebraic genericity and on a collection of spaces of holomorphic functions in the disc.

2 Lineability and Function Space Background

In the last two decades, there has been a crescent interest in the mathematical world about the search for large linear or algebraic structures inside families being not necessarily linear (for background on this line of research, called lineability, the reader is referred to the survey [11] and the monograph [2]; see also [33]). We gather in the next definition the basic concepts that are relevant to this note.

Definition 2.1

Assume that X is a vector space and $\alpha$ is a cardinal number. Then a subset $A \subset X$ is said to be:

• Lineable if there is an infinite-dimensional vector space M such that $M {\setminus } \{0\}\subset A$.

• $\alpha$-lineable if there exists a vector space M with dim$(M) = \alpha$ and $M {\setminus } \{0\} \subset A$.

If, in addition, X is a topological vector space, then the subset A is said to be:

• Spaceable in X whenever there is a closed infinite-dimensional vector subspace M of X such that $M {\setminus } \{0\} \subset A$.

• Dense-lineable in X whenever there is a dense vector subspace M of X satisfying $M {\setminus } \{0\} \subset A$.

• $\alpha$-dense-lineable in X whenever there is a dense vector subspace M of X with dim$(M) = \alpha$ and $M {\setminus } \{0\} \subset A$.

• Maximal dense-lineable in X if it is $\textrm{dim}(X)$-dense-lineable.

And, provided that X is a vector space contained in some (linear) algebra, then A is called:

• Algebrable if there is an algebra M so that $M {\setminus } \{0\} \subset A$ and M is infinitely generated, that is, the cardinality of any system of generators of M is infinite.

• Strongly $\alpha$-algebrable if there exists an $\alpha$-generated free algebra M with $M {\setminus } \{0\} \subset A$.

Recall that if X is contained in a commutative algebra, then a set $B \subset X$ is a generating set of some free algebra contained in A if and only if for any $N \in \mathbb {N}$, any nonzero polynomial P in N variables without constant term and any distinct $f_1, \dots ,f_N \in B$, we have $P(f_1, \dots ,f_N) \in A {\setminus } \{0\}$.

A number of implications are clear. For instance, if $\alpha$ is assumed to be infinite, we have: $\alpha$-lineable or spaceable or dense-lineable or algebrable $\Longrightarrow$ lineable; $\alpha$-dense-lineable $\Longrightarrow$ dense-lineable and $\alpha$-lineable; strongly $\alpha$-algebrable $\Longrightarrow$ algebrable and $\alpha$-lineable.

In Sect. 4, we shall use the following two criteria for dense lineability and algebrability. The first of them was established in a rather general way in [1, Section 7.3] (see also [2, 7, 8, 10] for related results), and reads as follows:

Theorem 2.2

Assume that X is a metrizable separable topological vector space, that $\alpha$ is an infinite cardinal number, that A is $\alpha$-lineable, and that there is a dense-lineable subset B of X with $A \cap B = \varnothing$ and $A + B \subset A$. Then A is $\alpha$-dense-lineable.

The algebrability criterion can be found in [1, Section 7.5] (see also [9]). It contains a complex version of a very useful method developed by Balcerzak et al. in [4, Proposition 7] (see also [5, Theorem 1.5 and Section 6] and [6]). By ${\mathcal {E}}$ we denote the family of exponential-like functions on $\mathbb {C}$, that is, the functions of the form

$$\begin{aligned} \varphi = \sum _{j=1}^m a_j e_{b_j} \end{aligned}$$

for some $m \in \mathbb {N}$, some $a_1, \dots , a_m \in \mathbb {C}{\setminus } \{0\}$ and some distinct $b_1, \dots ,b_m \in \mathbb {C}{\setminus } \{0\}$, where $e_\alpha (z):= e^{\alpha z}$ ($\alpha \in \mathbb {C}$).

Theorem 2.3

Let $\Omega$ be a nonempty set and let ${{\mathcal {F}}}$ be a family of functions from $\Omega$ into $\mathbb {C}$. Assume that there exists a function $f: \Omega \rightarrow \mathbb {C}$ such that $f(\Omega )$ is uncountable and $\varphi \circ f \in {{\mathcal {F}}}$ for every $\varphi \in {{\mathcal {E}}}$. Then ${{\mathcal {F}}}$ is strongly ${\mathfrak {c}}$-algebrable.

Now, we turn our attention to several vector subspaces of $H(\mathbb {D})$, apart from the disc algebra $A(\mathbb {D})$. By $H^\infty (\mathbb {D})$ we denote, as usual, the linear space of all bounded holomorphic functions on $\mathbb {D}$. It becomes a (non-separable) Banach space under the norm $\Vert \cdot \Vert _\infty$. Moreover, the symbol $A_s(\mathbb {D})$ will represent the family of all power series $f(z) = \sum _{k=0}^\infty a_k z^k$ with $\Vert f\Vert _s:= \sum _{k=0}^\infty |a_k| < +\infty$. In the related literature, this family is known as the Wiener algebra. The Weierstrass M test yields $A_s(\mathbb {D}) \subset A(\mathbb {D})$, while a standard argument proves that $(A_s(\mathbb {D}),\Vert \cdot \Vert _s)$ is a Banach space whose topology is finer than that inherited from $A(\mathbb {D})$. Other interesting subspaces are (see [19] for a study of them, related to lineability):

$$\begin{aligned} A_{uc}(\mathbb {D}):= \{f \in H(\mathbb {D}): (S_nf) \ \hbox {converges uniformly on} \ T \} \end{aligned}$$

and

$$\begin{aligned} A_{ub}(\mathbb {D}):= \{f \in H(\mathbb {D}): (S_nf) \ \hbox {is uniformly bounded on} \ T \}, \end{aligned}$$

where $S_nf (z) = \sum _{k=0}^n a_k z^k$ if $f(z) = \sum _{n=0}^\infty a_n z^n$. Both spaces are Banach spaces under the norm $\Vert f\Vert _b = \sup _{z \in \mathbb {T}} |S_nf(z)|$, and satisfy

$$\begin{aligned} A_s(\mathbb {D}) \subset A_{uc}(\mathbb {D}) \subset A_{ub}(\mathbb {D}) \subset A(\mathbb {D}), \end{aligned}$$

all inclusions being strict, continuous and dense. Moreover, the set of polynomials is dense in each of the spaces $A_s(\mathbb {D}), A_{uc}(\mathbb {D}), A(\mathbb {D})$. We remark that, under $\Vert \cdot \Vert _b$, the space $A_{uc}(\mathbb {D})$ is separable but $A_{ub}(\mathbb {D})$ is not.

3 Topological Genericity of the Bloch–Nevanlinna Functions

Inspired by condition (1.1), we consider any mapping $S: H(\mathbb {D}) \rightarrow H(\mathbb {D})$ and any nonempty subset $X \subset H(\mathbb {D})$, and define the following sets:

• $BNX:= \{f \in X: f' \not \in N \}$.

• ${\mathcal {A}}_{X,S}:= \{f \in X: \limsup _{r \rightarrow 1^-} |(Sf)(re^{i \theta })| = +\infty$ for all $\theta \in [0,2\pi ]\}$.

• ${\mathcal {A}}_{X,S,ae}:= \{f \in X: \limsup _{r \rightarrow 1^-} |(Sf)(re^{i \theta })| = +\infty$ a.e. on $[0,2\pi ]\}$.

• ${\mathcal {A}}_{X,S,ef}:= \{f \in X: \limsup _{r \rightarrow 1^-} |(Sf)(re^{i \theta })| = +\infty$ for all $\theta$ except for a finite

  subset $F(f) \subset [0,2\pi ]\}$.

The classes BNX are relevant for the Bloch–Nevanlinna problem as soon as $X \subset N$. It is plain that ${\mathcal {A}}_{X,S} \subset {\mathcal {A}}_{X,S,ef} \subset {\mathcal {A}}_{X,S,ae}$. In the Bloch–Nevanlinna problem, the mapping S is the derivative operator $D: f \in H(\mathbb {D}) \mapsto f' \in H(\mathbb {D})$. In this case, all sets above are contained in BNX. Under the terminology introduced above, the results by Hahn [20] and Galanos [17] (case $A = 0, \, B = 2\pi$; see Sect. 1) assert, respectively, that ${\mathcal {A}}_{A_s(\mathbb {D}),D}$ is residual in $A_s(\mathbb {D})$ and $BNA(\mathbb {D})$ is residual in $A(\mathbb {D})$.

Recall that an F-space is a complete metrizable topological vector space. For instance, $H(\mathbb {D})$, when endowed with the topology of uniform convergence in compacta, is an F-space. Every Banach space is an F-space, and every F-space is a Baire space.

We shall use the next general theorem to extend the mentioned results by Hahn and Galanos, which are contained in Theorem 3.2.

Theorem 3.1

Assume that X is a vector subspace of $H(\mathbb {D})$ endowed with a topology $\tau$ under which X becomes an F-space. Assume also that $S: H(\mathbb {D}) \rightarrow H(\mathbb {D})$ is a continuous linear operator. Assume that the following conditions are satisfied:

• (a) The topology $\tau$ on X is stronger than the one inherited from $H(\mathbb {D})$.

• (b) There is a $\tau$-dense subset ${\mathcal {D}}$ of X such that $S({\mathcal {D}} ) \subset H^\infty (\mathbb {D})$.

• (c) Given $M > 0$, a $\tau$-neighbourhood U of zero, and $\gamma \in [0,1)$, there exists $f \in U$ such that $\sup \{ |(Sf)(re^{i \theta })|: r \in [\gamma ,1) \} > M$ for all $\theta \in [0,2\pi ]$.

Then ${\mathcal {A}}_{X,S}$ is residual in X.

Proof

We follow the approach of [20]. Being X a complete metric space, the Baire category theorem tells us that it is enough to prove that $E:= X {\setminus } {\mathcal {A}}_{X,S}$ is of first category in X, that is, that E is a countable union of nowhere dense subsets. For each $n \in \mathbb {N}$ we define:

$$\begin{aligned} E_n\!:= \!\big \{f \!\in \!X: \exists \theta \!=\! \theta (f) \in [0,2\pi ] \hbox { such that } |(Sf)(r e^{i \theta })|\! \le \!n \ \forall r \in [1-{1 \over n},1) \big \}. \end{aligned}$$

It is evident that $E \subset \bigcup _{n \in \mathbb {N}} E_n$. It suffices to show that each $E_n$ is closed and has empty interior.

Fix $n \in \mathbb {N}$, as well as a sequence $\{f_k\}_{k \ge 1} \subset E_n$ with $f_k \rightarrow f \in X$ as $k \rightarrow \infty$. Since $f_n \in E_k$, there exists $\theta _k \in [0,2\pi ]$ such that $|(Sf_k)(re^{i \theta _k})| \le n$ for all $r \in [1-{1 \over n},1)$. Now, the compactness of $[0,2 \pi ]$ implies the existence of a subsequence $(\alpha _k)$ of $(\theta _k)$ converging to some $\alpha _0 \in [0,2\pi ]$. We have that

$$\begin{aligned}{} & {} |(Sf)(re^{i\alpha _0})| \le |(Sf)(re^{i\alpha _0}) - (Sf)(re^{i\alpha _k})| + |(Sf)(re^{i\alpha _k}) - (Sf_k)(re^{i\alpha _k})| \\{} & {} \quad + |(Sf_k)(re^{i\alpha _k})|. \end{aligned}$$

Let us fix $r \in [1-{1 \over n},1)$ and $\varepsilon > 0$. Since Sf is continuous in $\mathbb {D}$, there is $k_1 \in \mathbb {N}$ such that

$$\begin{aligned} |(Sf)(re^{i\alpha _0}) - (Sf)(re^{i\alpha _k})| < {\varepsilon \over 2} \ \ \hbox {for all } \ k \ge k_1. \end{aligned}$$

Now, the condition (a) implies that $f_k \rightarrow f$ ($k \rightarrow \infty$) in the topology of $H(\mathbb {D})$. Since S is a continuous selfmapping of $H(\mathbb {D})$, we get that $Sf_k \rightarrow Sf$ uniformly on the circle $|z| = r$. This entails the existence of a $k_2 \in \mathbb {N}$ such that

$$\begin{aligned} |(Sf)(re^{i\alpha _k}) - (Sf_k)(re^{i\alpha _k})| < {\varepsilon \over 2} \ \ \hbox {for all } \ k \ge k_2. \end{aligned}$$

Thus, we have for every $k \ge k_0:= \max \{k_1,k_2\}$ that

$$\begin{aligned} |(Sf)(re^{i\alpha _0})| < {\varepsilon \over 2} + {\varepsilon \over 2} + |(Sf_k)(re^{i\alpha _k})| \le \varepsilon + n. \end{aligned}$$

Keeping in mind the extreme terms of the above inequality and letting $\varepsilon \rightarrow 0$, we get

$$\begin{aligned} |(Sf)(re^{i\alpha _0})| \le n \ \hbox {for all} \ r \in [1-{1 \over n},1), \end{aligned}$$

which tells us that $f \in E_n$. This shows that $E_n$ is closed.

Now, we fix $f \in X$ and a $\tau$-neighbourhood V of f. Our aim is to show that $V \cap (X {\setminus } E_n) \ne \varnothing$. We can write $V = f + U_0$ for a certain neighbourhood $U_0$ of zero. Choose a neighbourhood U of zero in X such that $U + U \subset U_0$. From (b), there is $g \in {\mathcal {D}}$ such that $g \in f + U$. Then there exists $K \in (0,+\infty )$ such that $|(Sg)(z)| \le K$ for all $z \in \mathbb {D}$. From (c), there is a function $h \in U$ such that $\sup \{ |(Sh)(re^{i \theta })|: r \in [1 - {1 \over n},1) \} > n + 1 + K$ for all $\theta \in [0,2\pi ]$. Therefore, for every $\theta \in [0,2\pi ]$, there is $r_0 = r_0(\theta ) \in [1-{1 \over n},1)$ such that $|(Sh)(r_0 e^{i \theta })| > n + 1 + K$. Let us define $\varphi := g + h$. Then, on the one hand, $\varphi \in f + U + U \subset f + U_0 = V$ and, on the other hand, we have that

$$\begin{aligned} |(S \varphi )(r_0 e^{i \theta })| \ge |(S h)(r_0 e^{i \theta })| - |(S g)(r_0 e^{i \theta })| \ge (n+1+K) - K = n+1, \end{aligned}$$

where we have used the linearity of S. We have obtained $|(S \varphi )(r_0 e^{i \theta })| > n$ for all $\theta \in [0,2\pi ]$. In other words, $\varphi \in X {\setminus } E_n$. Thus, $V \cap (X {\setminus } E_n) \ne \varnothing$, as required. We have seen that every $E_n$ is closed and has empty interior, which finishes the proof. $\square$

Recall that every polynomial $P(z) = a_0 + a_1z + \cdots + a_Nz^n$ with complex coefficients generates a continuous linear differential operator $P(D) = a_0 I + a_1 D + \cdots + a_ND^N$, where I denotes the identity operator and D is the differentiation operator.

Theorem 3.2

If P is a nonconstant polynomial and X is one of the spaces $A(\mathbb {D})$, $A_s(\mathbb {D})$, $A_{uc}(\mathbb {D})$, endowed with their respective topologies, then the set ${\mathcal {A}}_{X,P(D)}$ is residual in X. In particular, BNX is residual in X.

Proof

Recall that, on the one hand, each space $X \in \{A(\mathbb {D}), A_s(\mathbb {D}), A_{uc}(\mathbb {D})\}$ satisfies condition (a) in Theorem 3.1 and that, on the other hand, the operator $S:= P(D)$ is linear and continuous on $H(\mathbb {D})$. Letting ${\mathcal {D}}:= \{$polynomials$\}$, we have that ${\mathcal {D}}$ is dense in X and, since every P(D)Q is a polynomial if Q is, we obtain that condition (b) is also satisfied. Therefore, our only task is to show that condition (c) in Theorem 3.1 is fulfilled.

With this aim, and taking into account that the inclusions $A_s(\mathbb {D}) \subset A_{uc}(\mathbb {D}) \subset A(\mathbb {D})$ are continuous, it is sufficient to fix $M > 0$, $\gamma \in [0,1)$ and an $A_s(\mathbb {D})$-neighbourhood U of zero, and to exhibit a function $f \in U$ satisfying that, for every $\theta \in [0,2\pi ]$, there is $r = r(\theta ) \in [\gamma ,1)$ such that $|(P(D)f)(re^{i \theta })| > M$. Choose $\varepsilon > 0$ such that $\{f \in A_s(\mathbb {D}): \Vert f\Vert _s < \varepsilon \} \subset U$. Let f be the monomial $f(z) = {\varepsilon \over 2} \cdot z^m$, where $m \in \mathbb {N}$ will be determined later. It is obvious that $f \in A_s (\mathbb {D})$ and that $\Vert f\Vert _s = {\varepsilon \over 2}$, and so $f \in U$. By assumption, we can write $P(D) = a_0 I + a_1 D + \cdots + a_ND^N$ for certain $N \in \mathbb {N}$ and $a_0, a_1, \dots ,a_N \in \mathbb {C}$ with $a_N \ne 0$. Choose $m \in \mathbb {N}$ satisfying $m > N$ and $m > {1 \over 1 - \gamma }$. Then

$$\begin{aligned} (P(D)f)(z) = {\varepsilon \over 2} \cdot \sum _{k=0}^N a_k \cdot m(m-1)(m-2) \cdots (m-k+1) \cdot z^{m-k}, \end{aligned}$$

where $m(m-1) \cdots (m-k+1)$ is understood as 1 if $k=0$.

Using $|re^{i\theta }|^{m-k} = r^{m-k} < 1$ for $(r,\theta ) \in [0,1) \times [0,2\pi ]$ and $k \in \{0,1, \dots ,N\}$ together with the triangle inequality, we get

$$\begin{aligned}{} & {} |(P(D)f)(re^{i \theta })| \ge {\varepsilon \over 2} \cdot |a_N|(m - n)^N r^{m-N} + Q(m) \ \hbox {for all} \ (r,\theta ) \in [0,1) \\{} & {} \quad \times [0,2\pi ], \end{aligned}$$

where Q is a polynomial (not depending on r) with real coefficients and degree not greater than $N-1$. Letting $r:= 1-{1 \over m}$, we obtain $r \in [\gamma ,1)$ and, for all $\theta \in [0,2\pi ]$, that

$$\begin{aligned}{} & {} |(P(D)f)(re^{i \theta })| \ge {\varepsilon \over 2} \cdot |a_N|(m - n)^N \left( 1 - {1 \over m}\right) ^{m-N} + Q(m) \longrightarrow \\{} & {} \quad +\infty \ \hbox { as} \ m \rightarrow \infty ,\nonumber \end{aligned}$$

(3.1)

because ${\varepsilon \over 2} \cdot |a_N|(1 - {1 \over m})^{m-N} \longrightarrow {\varepsilon |a_N| \over 2 e} > 0$. Therefore there is $m \in \mathbb {N}$ with $m > N$ and $m > {1 \over 1 - \gamma }$ such that ${\varepsilon \over 2} \cdot |a_N|(m - n)^N (1 - {1 \over m})^{m-N} + Q(m) > M$.

To summarize, we have that, thanks to (3.1), with this selection of m, the radius $r = 1 - {1 \over m}$ satisfies $r \in [\gamma ,1)$ and $|(P(D)f)(re^{i \theta })| > M$ for all $\theta \in [0,2\pi ]$, as required. $\square$

Remarks 3.3

• 1. Letting $P(z) = z$ and, respectively, $X = A_s(\mathbb {D}), A(\mathbb {D})$, the main theorems by Hahn [20] and Galanos [17] (here with $A = 0,B=2\pi$) are special cases of the last theorem. Note also that Theorem 2 in [20] asserts that ${\mathcal {A}}_{A(\mathbb {D}),D}$ is dense in $A(\mathbb {D})$, due to the denseness of $A_s (\mathbb {D})$ in $A(\mathbb {D})$; but the residuality of ${\mathcal {A}}_{A(\mathbb {D}),D}$ in $A(\mathbb {D})$ cannot be extracted from this fact, because the topology of $A(\mathbb {D})$ is coarser than the one of $A_s(\mathbb {D})$.

• 2. In 1955, Rudin [30] constructed a sequence $\{n_1< n_2< n_3 < \cdots \} \subset \mathbb {N}$ such that the function $f_0(z) = {\sum _{k=1}^{\infty } {1 \over k^2} z^{n_k}}$ satisfies $\lim _{r \rightarrow 1^-} |f_0'(re^{i \theta })| = + \infty$ a.e. on $[0,2\pi ]$. Note that this property is stronger than $\limsup _{r \rightarrow 1^-} |f_0'(re^{i \theta })| = + \infty$ a.e. on $[0,2\pi ]$. In fact, it is enough that $(n_k)$ satisfies $n_k > k^2 \cdot n^2_{k-1}$ and $\sum _{j=1}^{k-1} n_j(1 - (1 - n_k^{-1/2})^{n_j} )$ for all $k \in \mathbb {N}$ (see [30]). Note that $f_0 \in {\mathcal {A}}_{A(\mathbb {D}),D,ae}$. Since the set ${\mathcal {P}}$ of polynomials is dense in $A(\mathbb {D})$ and, evidently, ${\mathcal {P}} + {\mathcal {A}}_{A(\mathbb {D}),D,ae} \subset {\mathcal {A}}_{A(\mathbb {D}),D,ae}$, we have that $f_0 + {\mathcal {P}}$ provides an explicit example of a set that is dense in $A(\mathbb {D})$ and is contained in ${\mathcal {A}}_{A(\mathbb {D}),D,ae}$. Note that, in addition, $f_0 \in A_s(\mathbb {D})$, and so the same example holds by replacing $A(\mathbb {D})$ with $A_s (\mathbb {D})$ or $A_{uc} (\mathbb {D})$.

• 3. Starting from the function $f_0$ and the set ${\mathcal {P}}$ in the preceding remark, it is possible to give a quick proof of Galanos’ result (see Sect. 1) in its full extension. Observe that $f_0 + {\mathcal {P}} \subset {\mathcal {R}}$, where ${\mathcal {R}}$ is the set of all $f \in A(\mathbb {D})$ such that $\int _{A}^{B} \log ^+ |f'(r e^{i \theta })| \textrm{d}\theta = + \infty$ for all $A,B \in \mathbb {R}$ with $A < B$. Since $f_0 + {\mathcal {P}}$ is dense in $A(\mathbb {D})$, we derive the denseness of ${\mathcal {R}}$. Let us define, for each triple $A,B,r \in \mathbb {R}$ with $A < B$ and $r > 0$, the mapping $\Phi _{A,B,r}: f \in A(\mathbb {D}) \mapsto \int _{A}^{B} \log ^+ |f'(r e^{i \theta })| \,\textrm{d}\theta \in \mathbb {R}$. An easy argument involving the continuity of the derivative operator $f \in H(\mathbb {D}) \mapsto f' \in H(\mathbb {D})$ yields the continuity of every $\Phi _{A,B,r}$. Hence, for every $\alpha > 0$, the set $S_{A,B,\alpha }:= \bigcup _{0< r < 1} \Phi _{A,B,r}^{-1} ((\alpha ,+\infty ))$ is open in $A(\mathbb {D})$. Since ${\mathcal {R}} = \bigcap _{A,B \in \mathbb {Q}, A < B, n \in \mathbb {N}} S_{A,B,n}$, we obtain that ${\mathcal {R}}$ is a (dense) $G_\delta$-subset of $A(\mathbb {D})$ and, consequently, it is residual in $A(\mathbb {D})$.

4 Algebraic Genericity of the Bloch–Nevanlinna Functions

If we relax slightly the condition that for all $\theta \in [0,2 \pi ]$ it holds that $\limsup _{r \rightarrow 1^-} |f'(re^{i \theta })| = +\infty$, then we can obtain algebraic genericity for the family of solutions of the Bloch–Nevanlinna conjecture. Under the terminology of Sect. 2, here we restrict ourselves to the case $X = A(\mathbb {D})$, $S = D$. Recall also that ${\mathcal {E}}$ denotes the family of exponential-like functions, so that ${\mathcal {E}} \cup \{0\} = \textrm{span} \{e_\alpha : \alpha \in \mathbb {C}{\setminus } \{0\} \}$. A number of auxiliary assertions will be needed in our proofs.

Lemma 4.1

• (a) The system of functions $\{e_\alpha : \alpha \in \mathbb {C}{\setminus } \{0\} \}$ is linearly independent. In particular, $\textrm{dim} ({\mathcal {E}} \cup \{0\}) = {\mathfrak {c}}$.

• (b) No function in ${\mathcal {E}}$ can be identically constant.

Proof

• (a) Assume that $b_1, \dots , b_p$ are nonzero, mutually different, complex numbers, and that there are scalars $a_1, \dots ,a_p$ such that

  $$\begin{aligned} a_1 e_{b_1} + \cdots + a_p e_{b_p} = 0. \end{aligned}$$

  After j derivations, we get $a_1b_1^je_{b_1} + \cdots + a_pb_p^je_{b_p} = 0$. By evaluating this function at 0 and letting $j \in \{0,1, \dots , p-1\}$, we obtain the $p \times p$ linear system

  $$\begin{aligned} a_1b_1^j + \cdots + a_pb_p^j = 0 \quad (j = 0,1, \dots ), \end{aligned}$$

  whose matrix is a Vardermonde one having $(b_1, \cdots ,b_p)$ as its second row. It is well known that its determinant is nonzero if and only if the elements of this row are pairwise different, which is the case. Since any matrix with nonzero determinant determines an injective linear operator, we derive that $a_1 = a_2 = \cdots = a_p = 0$, which yields the desired independence.

• (b) Assume, by way of contradiction, that $\varphi = \sum _{n=1}^p a_n e_{b_n} \in {\mathcal {E}}$ (with the $b_n$’s pairwise different and $a_1, \dots ,a_p, b_1, \dots ,b_p \in \mathbb {C}{\setminus } \{0\}$) is identically constant. Then $0 = \varphi ' = \sum _{n=1}^p a_n b_n e_{b_n}$. It follows from (a) that $a_n b_n = 0$ for all $n \in \{1, \dots ,p\}$, which is absurd. $\square$

Lemma 4.2

Assume that $f \in A(\mathbb {D})$ and that $\varphi$ is a non-identically zero holomorphic function on some open set containing ${\overline{\mathbb {D}}}$. Suppose also that $\lim _{r \rightarrow 1^-} |f'(re^{i \theta })| = + \infty$ for all $\theta \in [0,2\pi ]$. Then $\varphi \cdot f \in {\mathcal {A}}_{A(\mathbb {D}),D,ef}$.

Proof

The conclusion is evident if $\varphi$ is constant, so that we may assume that $\varphi$ is not constant. Then $\varphi$ is not identically zero. It is evident, on the one hand, that $\varphi \cdot f$ and $\varphi ' \cdot f$ belong to $A(\mathbb {D})$. Then there is a constant $K \in (0,+\infty )$ such that $|(\varphi ' \cdot f) (z)| \le K$ for all $z \in {\overline{\mathbb {D}}}$. On the other hand, the analytic continuation principle prevents $\varphi$ to have infinitely many zeros on the compact set $\mathbb {T}$. In other words, there exists a finite set $F \subset [0,2\pi ]$ such that $|\varphi (e^{i \theta })| > 0$ for all $\theta \in [0,2\pi ] {\setminus } F$. Then we obtain for every $\theta$ in the last set that $|\varphi (re^{i \theta })| \rightarrow |\varphi (e^{i \theta })| > 0$ as $r \rightarrow 1^-$. Finally, using the assumption about the limits, we get

$$\begin{aligned} |(\varphi \cdot f)'(re^{i \theta })|\ge & {} |(\varphi \cdot f')(re^{i \theta })| - |(\varphi ' \cdot f)(re^{i \theta })| \ge |\varphi (re^{i \theta })||f'(re^{i \theta })| \\{} & {} \quad - K \longrightarrow \infty \end{aligned}$$

as $r \rightarrow 1^-$. Thus, $|(\varphi \cdot f)'(re^{i \theta })|$ shares the same limit property whenever $\theta \not \in F$, and this is the conclusion of the lemma. $\square$

Lemma 4.3

Assume that $f \in A(\mathbb {D})$ and that $\varphi$ is a nonconstant entire function. Suppose also that $\lim _{r \rightarrow 1^-} |f'(re^{i \theta })| = + \infty$ for all $\theta \in [0,2\pi ]$. Then $\varphi \circ f \in {\mathcal {A}}_{A(\mathbb {D}),D,ae}$.

Proof

It is evident, on the one hand, that $\varphi \circ f$ and $\varphi ' \circ f$ belong to $A(\mathbb {D})$. On the other hand, the function $\varphi '$ is not identically zero. Clearly, f cannot be constant, so it is open, that is, it sends open sets to open set. Then an application of the analytic continuation principle yields that $\varphi ' \circ f$ cannot be identically zero. From the fact $\varphi ' \circ f \in A(\mathbb {D})$ we infer that it is nonzero a.e. on $\mathbb {T}$ (see [22, Chapter 6]). In other words, there is a null measure set $Z \subset [0,2\pi ]$ such that $(\varphi ' \circ f)(e^{i \theta } ) \ne 0$ for all $\theta \in [0,2\pi ] {\setminus } Z$, and so $|(\varphi ' \circ f)(re^{i \theta } )| \longrightarrow |(\varphi ' \circ f)(e^{i \theta } )| > 0$ as $r \rightarrow 1^-$. Consequently, provided that $\theta$ belongs to the last set, it follows from the assumptions that

$$\begin{aligned} \lim _{r \rightarrow 1^-} |(\varphi \circ f)'(re^{i \theta })| = \lim _{r \rightarrow 1^-} |f'(re^{i \theta })||(\varphi ' \circ f)(re^{i \theta } )| = +\infty \end{aligned}$$

which finishes the proof. $\square$

Theorem 4.4

The set ${\mathcal {A}}_{A(\mathbb {D}),D,ef}$ is maximal dense-lineable in $A(\mathbb {D})$. Hence ${\mathcal {A}}_{A(\mathbb {D}),D,ae}$ and $BNA(\mathbb {D})$ are maximal dense-lineable in $A(\mathbb {D})$ as well.

Proof

Since $A(\mathbb {D})$ is a separable completely metrizable infinite-dimensional topological vector space, an easy application of the Baire category theorem yields that $\textrm{dim} (A(\mathbb {D})) = {\mathfrak {c}}$, the cardinality of the continuum. Hence, our goal is to show that ${\mathcal {A}}_{A(\mathbb {D}),D,ef}$ is ${\mathfrak {c}}$-dense-lineable. We denote, as usual, $f \cdot S:= \{f \cdot \varphi : \varphi \in S \}$ if $f \in A(\mathbb {D}) \supset S$.

By [30] (see Remark 3.3.2) we can select a function $f \in A(\mathbb {D})$ satisfying

$$\begin{aligned} \lim _{r \rightarrow 1^-} |f'(re^{i\theta })| = + \infty \quad \text{ for } \text{ all } \theta \in [0, 2\pi ]. \end{aligned}$$

Consider the set $M:= {\mathcal {E}} \cup \{0\}$. According to Lemma 4.1, this set is a ${\mathfrak {c}}$-dimensional vector subspace of $A(\mathbb {D})$. Since f is not identically zero, an application of the analytic continuation principle gives that $f \cdot M$ is a ${\mathfrak {c}}$-dimensional vector subspace of $A(\mathbb {D})$. If $g \in (f \cdot M) {\setminus } \{0\}$ then there exists $\varphi \in {\mathcal {E}}$ with $g = f \cdot \varphi$. It follows from Lemma 4.2 that $g \in {\mathcal {A}}_{A(\mathbb {D}),D,ef}$. To summarize, we get $(f \cdot M) {\setminus } \{0\} \subset {\mathcal {A}}_{A(\mathbb {D}),D,ef}$, and so $A_{A(\mathbb {D}),D,ef}$ is ${\mathfrak {c}}$-lineable.

To conclude, we take $X:= A(\mathbb {D})$, $\alpha := {\mathfrak {c}}$, $A:= {\mathcal {A}}_{A(\mathbb {D}),D,ef}$, $B:= {\mathcal {P}}$ (the set of polynomials). On the one hand, from the facts that polynomials (hence their derivatives) are bounded on compact sets it is evident that $A \cap B = \varnothing$ and $A + B \subset A$. On the other hand, it has been proved that A is $\alpha$-lineable. Finally, since ${\mathcal {P}}$ is itself a dense vector subspace of $A(\mathbb {D})$, we get that B is dense-lineable. An application of Theorem 2.2 yields that A is $\alpha$-dense-lineable, and this is exactly what had to be proved. $\square$

The next theorem tells us that our family ${\mathcal {A}}_{A(\mathbb {D}),D,ae}$ can even contain large linear algebras. Note that our approach forces to relax again a little more the size of the subset of $\mathbb {T}$ supporting infinite radial limits.

Theorem 4.5

The set ${\mathcal {A}}_{A(\mathbb {D}),D,ae}$ is strongly ${\mathfrak {c}}$-algebrable. Hence $BNA(\mathbb {D})$ is strongly ${\mathfrak {c}}$-algebrable as well.

Proof

Let us choose, one more time, a function $f \in A(\mathbb {D})$ satisfying (see [30])

$$\begin{aligned} \lim _{r \rightarrow 1^-} |f'(re^{i\theta })| = + \infty \quad \text{ for } \text{ all } \theta \in [0, 2\pi ]. \end{aligned}$$

Consider again the family ${\mathcal {E}}$ of exponential functions, and fix $\varphi \in {\mathcal {E}}$. By Lemma 4.1, $\varphi$ is not identically constant. Of course, $\varphi$ is an entire function. According to Lemma 4.3, we get $\varphi \circ f \in {\mathcal {A}}_{A(\mathbb {D}),D,ae}$. Then the conclusion of the theorem follows from an application of Theorem 2.3 just by taking $\Omega := \mathbb {D}$ and ${\mathcal {F}}:= {\mathcal {A}}_{A(\mathbb {D}),D,ae}$. $\square$

Finally, we shall prove that the family ${\mathcal {A}}_{A(\mathbb {D}),D,ae}$ enjoys a property that is close to spaceability. This entails that $BNA(\mathbb {D})$ also enjoys such a property. Specifically, a large Banach space can be constructed so as to be contained, except for zero, in the mentioned family. Prior to stating our theorem, we recall the notion of weighted Banach space $H_v(\mathbb {C})$ of entire functions associated to a continuous function $v:\mathbb {C}\rightarrow \left( 0,+\infty \right)$ (see, e.g., [25] and the references contained in it), that is defined as $H_v(\mathbb {C}):= \left\{ f \in H\left( \mathbb {C}\right) : \sup _{z \in \mathbb {C}} v \left( z\right) \cdot \left| f \left( z\right) \right| < +\infty \right\}$. It is known that $H_v \left( \mathbb {C}\right)$ is a Banach space when it is endowed with the norm

$$\begin{aligned} \left\| f\right\| _v:=\sup _{z\in \mathbb {C}}v\left( z\right) \cdot \left| f\left( z\right) \right| \end{aligned}$$

and that convergence in $\left\| \cdot \right\| _v$ implies uniform convergence on compacta in $\mathbb {C}$.

Theorem 4.6

There exists a Banach space $X \subset A(\mathbb {D})$ satisfying the following properties:

• (a) X is infinite-dimensional.

• (b) The norm topology of X is stronger than the one inherited from $A(\mathbb {D})$.

• (c) $X {\setminus } \{0\} \subset {\mathcal {A}}_{A(\mathbb {D}),D,ae}$.

Proof

Select again a function $f \in A(\mathbb {D})$ such that

$$\begin{aligned} \lim _{r \rightarrow 1^-} |f'(re^{i\theta })| = + \infty \quad \text{ for } \text{ all } \theta \in [0, 2\pi ], \end{aligned}$$

and consider the Banach space $H_v(\mathbb {C})$, where $v(z):= e^{-|z|}$. Let us define

$$\begin{aligned} X:= \big \{\varphi \circ f: \varphi \in H_v (\mathbb {C}), \varphi (0) = 0 \big \}. \end{aligned}$$

Plainly, X is a vector space and $X \subset A(\mathbb {D})$. If $g \in X$ and $\varphi _1 \circ f = g = \varphi _2 \circ f$, where $\varphi _1, \varphi _2 \in H_v (\mathbb {C})$, then $\varphi _1 = \varphi _2$ on the nonempty open set $f(\mathbb {D})$, so $\varphi _1 = \varphi _2$ on the whole $\mathbb {C}$ by the identity principle. Therefore, the mapping

$$\begin{aligned} g = \varphi \circ f \in X \mapsto \Vert g\Vert _X:= \Vert \varphi \Vert _v \end{aligned}$$

is well defined, and it is in fact a norm on X. Since convergence in $H_v(\mathbb {C})$ implies uniform convergence on compacta, it follows that $Y:= \{ \varphi \in H_v (\mathbb {C}): \varphi (0) = 0\}$ is a closed subspace of $H_v(\mathbb {C})$. Therefore, $\left( Y, \Vert \cdot \Vert _v \right)$ is a Banach space, which in turn implies that $\Vert \cdot \Vert _X$ is a complete norm on X, so as to make it a Banach space.

• (a) That dim$(X)= \infty$ follows from the facts that the monomials $\varphi _k (z) = z^k$ $(k \in \mathbb {N})$ are linearly independent, belong to $H_v (\mathbb {C})$ and satisfy $\varphi _k (0) = 0$. This conveys that the functions $f(z)^k$ $(k \in \mathbb {N})$ belong to X and are linearly independent. Indeed, if $\lambda _1 f(z) +\cdots +\lambda _p f^p(z) = 0$ on ${\overline{\mathbb {D}}}$, then $\lambda _1 z + \cdots + \lambda _p z^p = 0$ on the nonempty open set $f(\mathbb {D})$, so $\lambda _1 = \cdots = \lambda _p = 0$.

• (b) Consider the supremum norm $\Vert \cdot \Vert _\infty$ on $A(\mathbb {D})$. Let $K:= e^{\sup _{z \in f({\overline{\mathbb {D}}})} |z|}$. For each $g \in X$ there is $\varphi \in H_v (\mathbb {C})$ with $g = \varphi \circ f$ and then

  $$\begin{aligned} \Vert g\Vert _\infty= & {} \Vert \varphi \circ f \Vert _\infty = \sup _{z \in f({\overline{\mathbb {D}}})} |\varphi (z)| \\\le & {} \sup _{z \in f({\overline{\mathbb {D}}})} (K e^{-|z|} |\varphi (z)|) = K \cdot \sup _{z \in f({\overline{\mathbb {D}}})} e^{-|z|} |\varphi (z)| \\\le & {} K \cdot \sup _{z \in \mathbb {C}} e^{-|z|} |\varphi (z)| = K \cdot \Vert \varphi \Vert _v = K \cdot \Vert g\Vert _X. \end{aligned}$$

  Hence, the norm $\Vert \cdot \Vert _X$ is finer than $\Vert \cdot \Vert _\infty$, which proves (b).

• (c) Finally, suppose that $g \in X {\setminus } \{0\}$. Then there is $\varphi \in H_v(\mathbb {C})$ such that $\varphi \ne 0$, $\varphi (0) = 0$ and $g = \varphi \circ f$. Then $\varphi$ is a nonconstant (because $\varphi (0) = 0$ but $\varphi \ne 0$) entire function. Now, an application of Lemma 4.3 yields $g \in {\mathcal {A}}_{A(\mathbb {D}),D,ae}$. This completes the proof. $\square$

Remarks 4.7

• 1. According to their proofs, we have in fact obtained a little more in the theorems of this section. Namely, the set $\{f \in A(\mathbb {D}): \lim _{r \rightarrow 1^-} |f'(re^{i \theta })| = + \infty$ for all but finitely many $\theta \in [0,2\pi ] \}$ is maximal dense-lineable in $A(\mathbb {D})$, the set $\{f \in A(\mathbb {D}): \lim _{r \rightarrow 1^-} |f'(re^{i \theta })| = + \infty$ a.e. $\theta \in [0,2\pi ] \}$ is strongly ${\mathfrak {c}}$-algebrable, and the latter set contains, except for zero, an infinite-dimensional Banach space. The improvement is that “$\limsup$” can be replaced by a mere “$\lim$”.

• 2. It would be interesting to know whether or not the family ${\mathcal {A}}_{A(\mathbb {D}),D}$ (or some of its variants) is spaceable in $A(\mathbb {D})$.

• 3. In the recent papers [19, 24, 27], the algebraic genericity of several classes of holomorphic functions in the unit disc, different from the ones analyzed here, has been studied.