The Bloch–Nevanlinna Problem Under Lineability

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.


Introduction, Notation, Preliminaries and Aim of This Paper
Throughout this paper, we shall use the following (mostly standard) notation. If D is the open unit disc of the complex plane C, then H(D) will stand for the vector space of all holomorphic functions D → 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(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 T denotes the unit circle {z ∈ C : |z| = 1}, then A(D) will represent the disk algebra, that is, the vector space of all holomorphic functions in D admitting continuous extension on the closed unit disc D = D ∪ T. In other words, A(D) = H(D) ∩ C(D). It is well known that, under the supremum norm f ∞ := sup |z|<1 |f (z)|, the set A(D) is a (separable) Banach space. log + |f (re iθ )| dθ (0 < r < 1).
where log + t = max{0, log t} for t > 0. The Nevanlinna class N is defined as the class of holomorphic functions f in D having bounded characteristic, that is, satisfying The class N is a linear subspace, and even a linear algebra. It is known that f ∈ N if and only if f = g/h for some pair of functions g, h ∈ H ∞ (D), where h is zero-free and H ∞ (D) stands for the linear space of bounded holomorphic functions on D. It follows that A(D) ⊂ H ∞ (D) ⊂ N . It is also known that every function in N has finite radial limit almost everywhere (a.e., from now on) on 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 ) ⊂ N , where D is the derivative operator on H(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 ∈ N . It is even possible for such a function to have a continuous extension on the unit circle [30], that is, f ∈ A(D). In 1972, Hahn [20] used Baire category methods to conclude the existence of a dense subset of A(D) all of whose members f satisfy lim sup (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 ∈ A(D) with f ∈ N is residual in A(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 = 2π: there is a residual subset R of A(D) all of whose functions f satisfy sup 0<r<1 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 δ -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.

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 α is a cardinal number. Then a subset A ⊂ X is said to be: If, in addition, X is a topological vector space, then the subset A is said to be: 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 \{0} ⊂ A and M is infinitely generated, that is, the cardinality of any system of generators of M is infinite.
Recall that if X is contained in a commutative algebra, then a set B ⊂ X is a generating set of some free algebra contained in A if and only if for any N ∈ N, any nonzero polynomial P in N variables without constant term and any distinct f 1 , . . . , f N ∈ B, we have P (f 1 , . . . , f N ) ∈ A\{0}.
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: 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 E we denote the family of exponential-like functions on C, that is, the functions of the form

Theorem 2.3. Let Ω be a nonempty set and let F be a family of functions
Now, we turn our attention to several vector subspaces of H(D), apart from the disc algebra A(D). By H ∞ (D) we denote, as usual, the linear space of all bounded holomorphic functions on D. It becomes a (non-separable) Banach space under the norm · ∞ . Moreover, the symbol A s (D) will represent the family of all power series f (z) = ∞ k=0 a k z k with f s := ∞ k=0 |a k | < +∞. In the related literature, this family is known as the Wiener algebra. The Weierstrass M test yields A s (D) ⊂ A(D), while a standard argument proves that (A s (D), · s ) is a Banach space whose topology is finer than that inherited from A(D). Other interesting subspaces are (see [19] for a study of them, related to lineability): n=0 a n z n . Both spaces are Banach spaces under the norm f b = sup z∈T |S n f (z)|, and satisfy all inclusions being strict, continuous and dense. Moreover, the set of polynomials is dense in each of the spaces A s (D), A uc (D), A(D). We remark that,

Topological Genericity of the Bloch-Nevanlinna Functions
Inspired by condition (1.1), we consider any mapping S : H(D) → H(D) and any nonempty subset X ⊂ H(D), and define the following sets: The classes BN X are relevant for the Bloch-Nevanlinna problem as soon as X ⊂ N . It is plain that A X,S ⊂ A X,S,ef ⊂ A X,S,ae . In the Bloch-Nevanlinna problem, the mapping S is the derivative operator D : In this case, all sets above are contained in BN X. Under the terminology introduced above, the results by Hahn [20] and Galanos [17] Recall that an F-space is a complete metrizable topological vector space. For instance, H(D), when endowed with the topology of uniform convergence in compacta, is an F-space. Every Banach space is an F-space, and every Fspace 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. (a) The topology τ on X is stronger than the one inherited from H(D).
Then 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\A X,S is of first category in X, that is, that E is a countable union of nowhere dense subsets. For each n ∈ N we define: It is evident that E ⊂ n∈N E n . It suffices to show that each E n is closed and has empty interior. Fix n ∈ N, as well as a sequence , 1). Now, the compactness of [0, 2π] implies the existence of a subsequence (α k ) of (θ k ) converging to some α 0 ∈ [0, 2π]. We have that Let us fix r ∈ [1 − 1 n , 1) and ε > 0. Since Sf is continuous in D, there is . Since S is a continuous selfmapping of H(D), we get that Sf k → Sf uniformly on the circle |z| = r. This entails the existence of a k 2 ∈ N such that Thus, we have for every k ≥ k 0 := max{k 1 , k 2 } that Keeping in mind the extreme terms of the above inequality and letting ε → 0, we get which tells us that f ∈ E n . This shows that E n is closed. Now, we fix f ∈ X and a τ -neighbourhood V of f . Our aim is to show that V ∩ (X\E n ) = ∅. 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 ⊂ U 0 . From (b), there is g ∈ D such that g ∈ f + U . Then there exists K ∈ (0, +∞) such that |(Sg)(z)| ≤ K for all z ∈ D. From (c), there is a function h ∈ U such that sup{|(Sh)(re iθ )| : r ∈ [1 − 1 n , 1)} > n + 1 + K for all θ ∈ [0, 2π]. Therefore, for every θ ∈ [0, 2π], there is r 0 = r 0 (θ) ∈ [1 − 1 n , 1) such that |(Sh)(r 0 e iθ )| > n + 1 + K. Let us define ϕ := g + h. Then, on the one hand, ϕ ∈ f + U + U ⊂ f + U 0 = V and, on the other hand, we have that where we have used the linearity of S. We have obtained |(Sϕ)(r 0 e iθ )| > n for all θ ∈ [0, 2π]. In other words, ϕ ∈ X\E n . Thus, V ∩ (X\E n ) = ∅, as required. We have seen that every E n is closed and has empty interior, which finishes the proof.
Recall that every polynomial P (z) = a 0 +a 1 z +· · ·+a N z n with complex coefficients generates a continuous linear differential operator P (D) = a 0 I + a 1 D + · · · + a N D N , where I denotes the identity operator and D is the differentiation operator. we have that 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 (D) ⊂ A uc (D) ⊂ A(D) are continuous, it is sufficient to fix M > 0, γ ∈ [0, 1) and an A s (D)-neighbourhood U of zero, and to exhibit a function f ∈ U satisfying that, for every θ ∈ [0, 2π], there is r = r(θ) ∈ [γ, 1) such that where m ∈ N will be determined later. It is obvious that f ∈ A s (D) and that f s = ε 2 , and so f ∈ U . By assumption, we can write P (D) = a 0 I + a 1 D + · · · + a N D N for certain N ∈ N and a 0 , a 1 , . . . , a N ∈ C with a N = 0. Choose m ∈ N satisfying m > N and m > 1 1−γ . Then and k ∈ {0, 1, . . . , N} together with the triangle inequality, we get where Q is a polynomial (not depending on r) with real coefficients and degree not greater than N − 1. Letting r := 1 − 1 m , we obtain r ∈ [γ, 1) and, for all θ ∈ [0, 2π], that To summarize, we have that, thanks to (3.1), with this selection of m, the radius r = 1 − 1 m satisfies r ∈ [γ, 1) and |(P (D)f )(re iθ )| > M for all θ ∈ [0, 2π], as required. +∞ a.e. on [0, 2π]. Note that this property is stronger than lim sup r→1 − |f 0 (re iθ )| = +∞ a.e. on [0, 2π]. In fact, it is enough that (n k ) satisfies n k > k 2 · n 2 k−1 and [30]). Note that f 0 ∈ A A(D),D,ae . Since the set P of polynomials is dense in A(D) and, evidently, P + A A(D),D,ae ⊂ A A(D),D,ae , we have that f 0 + P provides an explicit example of a set that is dense in A(D) and is contained in A A(D),D,ae . Note that, in addition, f 0 ∈ A s (D), and so the same example holds by replacing A(D) with A s (D) or A uc (D). 3. Starting from the function f 0 and the set 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

Algebraic Genericity of the Bloch-Nevanlinna Functions
If we relax slightly the condition that for all θ ∈ [0, 2π] it holds that lim sup r→1 − |f (re iθ )| = +∞, 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(D), S = D. Recall also that E denotes the family of exponential-like functions, so that E ∪ {0} = span{e α : α ∈ C\{0}}. A number of auxiliary assertions will be needed in our proofs. After j derivations, we get a 1 b j 1 e b1 + · · · + a p b j p e bp = 0. By evaluating this function at 0 and letting j ∈ {0, 1, . . . , p − 1}, we obtain the p × p linear system a 1 b j 1 + · · · + a p b j p = 0 (j = 0, 1, . . . ), whose matrix is a Vardermonde one having (b 1 , · · · , 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 = · · · = a p = 0, which yields the desired independence. (b) Assume, by way of contradiction, that ϕ = p n=1 a n e bn ∈ E (with the b n 's pairwise different and a 1 , . . . , a p , b 1 , . . . , b p ∈ C\{0}) is identically constant. Then 0 = ϕ = p n=1 a n b n e bn . It follows from (a) that a n b n = 0 for all n ∈ {1, . . . , p}, which is absurd. Proof. The conclusion is evident if ϕ is constant, so that we may assume that ϕ is not constant. Then ϕ is not identically zero. It is evident, on the one hand, that ϕ · f and ϕ · f belong to A(D). Then there is a constant K ∈ (0, +∞) such that |(ϕ · f )(z)| ≤ K for all z ∈ D. On the other hand, the analytic continuation principle prevents ϕ to have infinitely many zeros on the compact set T. In other words, there exists a finite set F ⊂ [0, 2π] such that |ϕ(e iθ )| > 0 for all θ ∈ [0, 2π]\F . Then we obtain for every θ in the last set that |ϕ(re iθ )| → |ϕ(e iθ )| > 0 as r → 1 − . Finally, using the assumption about the limits, we get as r → 1 − . Thus, |(ϕ · f ) (re iθ )| shares the same limit property whenever θ ∈ F , and this is the conclusion of the lemma. Proof. It is evident, on the one hand, that ϕ • f and ϕ • f belong to A(D). On the other hand, the function ϕ 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 ϕ • f cannot be identically zero. From the fact ϕ •f ∈ A(D) we infer that it is nonzero a.e. on T (see [22,Chapter 6]). In other words, there is a null measure set Z ⊂ [0, 2π] such that (ϕ • f )(e iθ ) = 0 for all θ ∈ [0, 2π]\Z, and so |(ϕ • f )(re iθ )| −→ |(ϕ • f )(e iθ )| > 0 as r → 1 − . Consequently, provided that θ belongs to the last set, it follows from the assumptions that To conclude, we take X := A(D), α := c, A := A A(D),D,ef , B := 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 ∩ B = ∅ and A + B ⊂ A. On the other hand, it has been proved that A is α-lineable. Finally, since P is itself a dense vector subspace of A(D), we get that B is dense-lineable. An application of Theorem 2.2 yields that A is α-denselineable, and this is exactly what had to be proved.
The next theorem tells us that our family A A(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 T supporting infinite radial limits. Finally, we shall prove that the family A A(D),D,ae enjoys a property that is close to spaceability. This entails that BN A(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 (C) of entire functions associated to a continuous function v : C → (0, +∞) (see, e.g., [25] and the references contained in it), that is defined as H v (C) := {f ∈ H (C) : sup z∈C v (z) · |f (z)| < +∞}. It is known that H v (C) is a Banach space when it is endowed with the norm f v := sup z∈C v (z) · |f (z)| and that convergence in · v implies uniform convergence on compacta in C.