Converse growth estimates for ODEs with slowly growing solutions

Abstract

Let $f_1,f_2$ be linearly independent solutions of $f''+Af=0$, where the coefficient A is an analytic function in the open unit disc ${\mathbb {D}}$ of the complex plane ${\mathbb {C}}$. It is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function $u=-\log \, (f_1/f_2)^{\#}$. For example, the case when $\sup _{z\in {\mathbb {D}}} |A(z)|(1-|z|^2)^2 < \infty$ and $f_1/f_2$ is normal, is characterized by the condition $\sup _{z\in {\mathbb {D}}} |\nabla u(z)|(1-|z|^2) < \infty$. Different types of Blaschke-oscillatory equations are also described in terms of harmonic majorants of u. Even if $f_1,f_2$ are bounded linearly independent solutions of $f''+Af=0$, it is possible that $\sup _{z\in {\mathbb {D}}} |A(z)|(1-|z|^2)^2 = \infty$ or $f_1/f_2$ is non-normal. These results relate to sharpness discussion of recent results in the literature, and are succeeded by a detailed analysis of differential equations with bounded solutions. Analogues for the Nevanlinna class are also considered, by taking advantage of Nevanlinna interpolating sequences. It is shown that, instead of considering solutions with prescribed zeros, it is possible to construct a bounded solution of $f''+Af=0$ in such a way that it solves an interpolation problem natural to bounded analytic functions, while $|A(z)|^2(1-|z|^2)^3\, dm(z)$ remains to be a Carleson measure.

1 Introduction

Let $\mathrm{Hol}({\mathbb {D}})$ be the collection of analytic functions in the open unit disc ${\mathbb {D}}$ of the complex plane ${\mathbb {C}}$. For $0\le \alpha < \infty$, let $L^\infty _\alpha$ denote the space of $f:{\mathbb {D}}\rightarrow {\mathbb {C}}$ for which $\Vert f\Vert _{L^\infty _\alpha } = \sup _{z\in {\mathbb {D}}} |f(z)| (1-|z|^2)^{\alpha }<\infty$, and write $H^\infty _\alpha =L^\infty _\alpha \cap \mathrm{Hol}({\mathbb {D}})$ and $H^\infty = H^\infty _0$ for short. We are interested in the relation between the growth of the coefficient $A\in \mathrm{Hol}({\mathbb {D}})$ and the oscillation and growth of solutions of

$$\begin{aligned} f''+Af=0. \end{aligned}$$

(1)

By [43, Theorems 3–4], the following conditions are equivalent:

(i):

$A\in H^\infty _2$;

(ii):

zero-sequences of all non-trivial solutions ($f\not \equiv 0$) of (1) are separated with respect to the hyperbolic metric.

We refer to [3] for a far reaching generalization concerning the connection between the growth of the coefficient $A\in \mathrm{Hol}({\mathbb {D}})$ and the minimal separation of zeros of non-trivial solutions of (1). It has been unclear whether

(iii):

all solutions of (1) belong to the Korenblum space $\bigcup _{0< \alpha < \infty } H^\infty _\alpha$,

is equivalent to the conditions above. Recall that, if $f_1,f_2$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, then the Wronskian determinant $W(f_1,f_2) = f_1 f_2'-f_1'f_2$ reduces to a non-zero complex constant, and consequently, any solution of (1) can be written as a linear combination of $f_1,f_2$.

In view of results in the literature, the condition (iii) is a natural candidate for a description of the growth of solutions of (1) under (i). Pommerenke used a classical comparison theorem [40, Example 1] to prove that $\mathrm{(i)} \Rightarrow \mathrm{(iii)}$. This implication has been rediscovered with different methods: growth estimates [20, Theorem 4.3(2)], [23, Theorem 3.1]; successive approximations [9, Theorem I]; and straight-forward integration [16, Theorem 2], [24, Corollary 4(a)]. We point out that, even if $\Vert A\Vert _{H^\infty _2}>0$ is arbitrarily small, some solutions of (1) may be unbounded. Any coefficient condition $A\in H^\infty _\alpha$ for $0<\alpha <2$ implies boundedness of all solutions of (1) by [20, Theorem 4.3(1)]. For more involved growth estimates in the case of slowly growing solutions, see [12, 14].

The difficulty in the converse assertion $\mathrm{(iii)} \Rightarrow \mathrm{(i)}$ lies in the fact that the assumption concerns all solutions. The existence of one non-trivial slowly growing solution is not sufficient, as $f(z)=\exp (-(1+z)/(1-z))$ is a bounded solution of (1) for $A(z)=-4z/(1-z)^4$, $z\in {\mathbb {D}}$. Two classical methods to attack problems of this type are the Bank-Laine approach and arguments based on the Schwarzian derivative. In the former case, let $E=f_1f_2$ denote the product of two linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$. By [27, pp. 76–77],

$$\begin{aligned} 4 A = \left( \frac{E'}{E} \right) ^2 - \left( \frac{W(f_1,f_2)}{E} \right) ^2 - 2 \, \frac{E''}{E}. \end{aligned}$$

The Bank-Laine representation is usually used in conjunction with estimates that appear in Wiman-Valiron and Nevanlinna theories. The latter method is based on [27, Theorem 6.1]: if $f_1,f_2$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, then $w=f_1/f_2$ is a locally univalent meromorphic function in ${\mathbb {D}}$ such that the Schwarzian derivative

$$\begin{aligned} S_w = \left( \frac{w''}{w'} \right) ' - \frac{1}{2} \left( \frac{w''}{w'} \right) ^2 \end{aligned}$$

is not only analytic in ${\mathbb {D}}$ but also satisfies $S_w=2A$. Both approaches represent the coefficient function A in terms of the linearly independent solutions $f_1,f_2$, and are indispensable tools in the case of fast growing solutions (and also in oscillation theory). However, if all solutions are slowly growing functions in ${\mathbb {D}}$, then neither of these techniques seem to be sufficiently delicate to produce sharp growth estimates for the coefficient A.

2 Results

Many of the following results are converse growth estimates as they measure the growth of the coefficient in terms of solutions. We begin with studying equations with bounded solutions. The preliminary results in Sect. 2.1 not only set the stage for forthcoming findings but also provide a sharpness discussion for [12, 44]. The significant part of this article is devoted to the study of the subharmonic auxiliary function $u=-\log \, (f_1/f_2)^{\#}$ where $f_1,f_2$ are linearly independent solutions of (1). This approach leads to several new characterizations which are, in essence, based on identities obtained in Sect. 2.2. Our intention is to compare properties of u to the coefficient A, to the quotient $f_1/f_2$ and to any non-trivial solution of (1). Results concerning equations with bounded solutions have natural counterparts in the setting of the Nevanlinna class, which are considered in Sect. 2.4. These results depend on recent advances concerning Nevanlinna interpolating sequences. Finally, in Sect. 2.6, we show that fixed points can be prescribed for a solution of (1) in such a way that all solutions remain bounded.

2.1 Bounded solutions

The following result indicates that the implication $\mathrm{(iii)} \Rightarrow \mathrm{(i)}$, mentioned in the Introduction, fails to be true.

Theorem 1

Consider the differential equation (1) in ${\mathbb {D}}$.

• (i) There exists $A\in \mathrm{Hol}({\mathbb {D}}) {\setminus } H^\infty _2$ such that all solutions of (1) are bounded.

• (ii) Let $0<p<\infty$. There exists $A\in \mathrm{Hol}({\mathbb {D}}) {\setminus } H^\infty _2$ such that all solutions of (1) belong to $H^\infty _p$ while one of the solutions is non-normal.

The class of normal functions consists of those meromorphic functions for which $\sup _{z\in {\mathbb {D}}} \, w^{\#}(z) (1-|z|^2) < \infty$, where $w^\# = |w'|/(1+|w|^2)$ is the spherical derivative. A function w meromorphic in ${\mathbb {D}}$ is normal if and only if

$$\begin{aligned} \{ w \circ \varphi : \varphi \text { conformal automorphism of } {\mathbb {D}}\} \end{aligned}$$

is a normal family in ${\mathbb {D}}$ in the sense of Montel [30]. We consider the normality of solutions of (1) as well as the normality of the quotient of two linearly independent solutions. If $A\in H^\infty _2$, then normal solutions of (1) are described by [17, Proposition 7], and the case when the quotient is normal will be characterized in Sect. 2.5. Note that the coefficient condition $A\in H^\infty _2$ allows non-normal solutions by [10, Theorem 3] and [11, Theorem 1]; and even the normality of all solutions is not sufficient for $A\in H^\infty _2$ by Theorem 1(i) above.

If $f_1, f_2 \in H^\infty$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, then $A\in H^\infty _3$ by a result of Steinmetz [44, p. 130]. Theorem 1(i) shows that this result cannot be improved to $A\in H^\infty _2$. The intermediate conclusion $A\in H^\infty _{\alpha }$ for $\alpha =5/2$ has been obtained in [12, Theorem 6] under the weaker assumption $f_1,f_2\in {\mathcal {B}}$, while the question of finding the best possible $\alpha$ remains open. Here ${\mathcal {B}}$ is the Bloch space, which contains $f\in \mathrm{Hol}({\mathbb {D}})$ for which $\Vert f\Vert _{{\mathcal {B}}} = \Vert f'\Vert _{H^\infty _1}<\infty$. The desired conclusion $A\in H^\infty _2$ has been obtained in [12, Theorem 7] under the additional assumption $\inf _{z\in {\mathbb {D}}}( |f_1(z)| + |f_2(z)| )>0$. We proceed to state two generalizations in this respect. Theorem 15 in Sect. 4 shows that it is not necessary to take the infimum over the whole unit disc while Theorem 2 below implies that we may take the infimum of a function which is significantly larger than $|f_1|+|f_2|$. The latter generalization is based on having specific information about the structure of the ideal $I_{H^\infty }(f_1,f_2)$ generated by the solutions $f_1,f_2 \in H^\infty$.

A positive Borel measure $\mu$ on ${\mathbb {D}}$ is called a Carleson measure, if for fixed $0<p<\infty$ there exists $C=C(p)$ with $0<C<\infty$ such that

$$\begin{aligned} \int _{{\mathbb {D}}} |f(z)|^p \, d\mu (z) \le C \, \lim _{r\rightarrow 1^-} \frac{1}{2\pi } \int _0^{2\pi } |f(re^{i\theta })|^p \, d\theta = C \, \Vert f\Vert _{H^p}^p, \quad f\in \mathrm{Hol}({\mathbb {D}}). \end{aligned}$$

Here $H^p$ is the standard Hardy space. By [8, Lemma 3.3, p. 231], such measures $\mu$ are characterized by $\sup _{a\in {\mathbb {D}}} \int _{\mathbb {D}}|\varphi _a'(z)| \, d\mu (z) < \infty$, where $\varphi _a(z)=(\zeta -z)/(1-{\overline{a}}z)$ is a conformal automorphism of ${\mathbb {D}}$ which coincides with its own inverse. Since $|A|^2$ is subharmonic for $A\in \mathrm{Hol}({\mathbb {D}})$, we deduce $A\in H^\infty _2$ whenever $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure. This Carleson measure condition appears several times in the literature: in connection to solutions of (1) with uniformly separated zeros [11, 15] and in relation to solutions in Hardy spaces [14, 17].

Theorem 2

If $f_1,f_2\in H^\infty$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$ such that

$$\begin{aligned} \inf _{a\in {\mathbb {D}}} \, \sum _{k=0}^n \left( \big | (f_1 \circ \varphi _a)^{(k)}(0)\big | + \big | (f_2 \circ \varphi _a)^{(k)}(0)\big | \right) > 0 \end{aligned}$$

(2)

for some $n\in {\mathbb {N}}\cup \{0\}$, then $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure.

Let $f_1,f_2\in H^\infty$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$. In [44], Steinmetz proved $(f_1/f_2)^\#\in L^\infty _2$ and asked whether this can be improved to $(f_1/f_2)^\#\in L^\infty _1$? It turns out that Steinmetz’s result is best possible up to a multiplicative constant. Recall that the sequence $\{z_n\}\subset {\mathbb {D}}$ is said to be uniformly separated, if it is separated in the hyperbolic metric and $\sum _n (1-|z_n|)\delta _{z_n}$ is a Carleson measure. Here $\delta _{z_n}$ is the Dirac measure with point mass at $z_n\in {\mathbb {D}}$.

Theorem 3

Let $\varLambda \subset {\mathbb {D}}$ be uniformly separated. Then, there exists $A\in \mathrm{Hol}({\mathbb {D}})$ such that $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure and (1) admits two linearly independent solutions $f_1,f_2\in H^\infty$ such that $\inf _{z_n\in \varLambda } \, (f_1/f_2)^\#(z_n) (1-|z_n|^2)^2 >0$.

In [7, Theorem 1.1], Fournier, Kraus and Roth obtain sharp estimates for $w^\#(0)$, where w is a meromorphic function in ${\mathbb {D}}$ with spherical derivative uniformly bounded away from zero.

Instead of considering prescribed zeros of solutions—which is the approach in Theorem 3, among many other results—we may also consider solutions which satisfy an interpolation problem natural for bounded analytic functions. Such result has been the objective of recent research. Our solution to this problem is based on combining classical interpolation results by Earl and Øyma.

Theorem 4

Let $\{z_n\}\subset {\mathbb {D}}$ be uniformly separated and $\{w_n\}\subset {\mathbb {C}}$ bounded. Then, there exists $A\in \mathrm{Hol}({\mathbb {D}})$ such that $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure (1) admits a solution $f\in H^\infty$ which satisfies $f(z_n)=w_n$ for all n, while all solutions of (1) are bounded.

In Sect. 5 we consider oscillation of solutions of such differential equations whose solutions are bounded, and concentrate on the zeros and critical points.

2.2 Identities

We take a short side-track to consider properties of the differential equation (1) assuming that the coefficient A is merely analytic in ${\mathbb {D}}$. Suppose for a moment that f is a zero-free solution of (1). In this case $\log f \in \mathrm{Hol}({\mathbb {D}})$ and

$$\begin{aligned} A = - f''/f = - ( \log f)'' - \big ( (\log f)' \big )^2. \end{aligned}$$

(3)

Our next objective is to obtain a similar representation which takes account on both linearly independent solutions and allows them to have zeros in ${\mathbb {D}}$. Let

$$\begin{aligned} \partial f = \frac{1}{2} \left( \frac{\partial f}{\partial x} - i \, \frac{\partial f}{\partial y} \right) , \quad {\overline{\partial }} f = \frac{1}{2} \left( \frac{\partial f}{\partial x} + i \, \frac{\partial f}{\partial y} \right) , \end{aligned}$$

denote the complex partial derivatives of f. Note that $\partial f$ and ${\overline{\partial }} f$ exist as long as $\partial f / \partial x$ and $\partial f / \partial y$ exist, and then the gradient $\nabla f = (\partial f / \partial x, \partial f / \partial y)$ satisfies $|\nabla f|^2 = 2 ( |\partial f|^2 + |{\overline{\partial }} f|^2 )$. If f has continuous second-order derivatives (denoted by $f\in C^2$), then the Laplacian $\varDelta f$ can be written in the form $\varDelta f = 4 \, {\overline{\partial }} \partial f = 4 \, \partial {\overline{\partial }} f$.

We have been unable to find a reference for the following result, which is known for experts in another form. We will present a short proof of Theorem 5 for the convenience of the reader.

Theorem 5

Let $f_1,f_2$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, and define $u = -\log \, (f_1/f_2)^{\#}$. Then,

• (i) $\varDelta u = 4 \, e^{-2u}$;

• (ii) $\varDelta u + | \nabla u |^2 = e^{-u} \varDelta e^u$;

• (iii) $A = - \partial ^2 u - (\partial u)^2$.

Let $f_1,f_2$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$. The function $u = -\log \, (f_1/f_2)^{\#}$ has several interesting properties, which make up the bulk of this paper. The underlying reason for the relevance of u is its connection to regular conformal metrics of constant curvature. Actually, u is closely related to the general solution of Liouville’s equation in the case of ${\mathbb {D}}$. This point of view is elaborated further in Remark 1, Sect. 6. For the classical representation of regular conformal metrics of constant curvature in terms of analytic functions, see [31]. Nevertheless, we choose to proceed without the notation of conformal metrics.

Theorem 5(i) implies $\varDelta u = 4 \, ( (f_1/f_2)^{\#})^2 \ge 0$. Therefore u is subharmonic, and $r \mapsto (1/(2\pi )) \int _0^{2\pi } u(re^{i\theta }) \, d\theta$ is a non-decreasing and convex function of $\log r$. Theorem 5(iii) is a counterpart of (3). As $W(f_1,f_2)$ is a non-zero complex constant, $\partial u = (f_1'{\overline{f}}_1+f_2' {\overline{f}}_2)/(|f_1|^2+|f_2|^2)$ is finite-valued throughout ${\mathbb {D}}$.

2.3 Blaschke-oscillatory equations

The differential equation (1) is said to be Blaschke-oscillatory, if $A\in \mathrm{Hol}({\mathbb {D}})$ and the zero-sequence $\{z_n\}$ of any non-trivial solution of (1) satisfies the Blaschke condition $\sum _n (1-|z_n|)<\infty$. Such differential equations are characterized by the fact that the quotient of any two linearly independent solutions belongs to the Nevanlinna class [22, Lemma 3]. The Nevanlinna class ${\mathcal {N}}$ consists of those meromorphic functions w in ${\mathbb {D}}$ such that $\int _{{\mathbb {D}}} w^\#(z)^2 (1-|z|^2) \, dm(z) <\infty$; see Sect. 7. A meromorphic function w is said to be of uniformly bounded characteristic, that is $w\in \mathrm {UBC}$, if $w^\#(z)^2 (1-|z|^2) \, dm(z)$ is a Carleson measure. We refer to [38, Theorem 3] for more details.

Let $u\not \equiv -\infty$ be a subharmonic function in ${\mathbb {D}}$. A harmonic function h is said to be a harmonic majorant for u if $u \le h$ in ${\mathbb {D}}$. The least harmonic majorant ${\hat{u}}$ is a harmonic majorant which is point-wise smaller than any other harmonic majorant for u. If $f\in \mathrm{Hol}({\mathbb {D}})$, then it is well-known that $f\in {\mathcal {N}}$ if and only if $\log ^+ |f|$ admits a harmonic majorant, while $f\in H^p$ if and only if $|f|^p$ has a harmonic majorant.

Theorem 6

Let $f_1,f_2$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, and define $u = -\log \, (f_1/f_2)^{\#}$. Then,

• (i) $f_1/f_2\in {\mathcal {N}}$ if and only if u has a harmonic majorant;

• (ii) $f_1/f_2\in {\mathcal {N}}$ and is normal if and only if $u_a(z)= u(a+(1-|a|)z)-u(a)$, $a\in {\mathbb {D}}$, have harmonic majorants with $\sup _{a\in {\mathbb {D}}} \widehat{u_a}(0) < \infty$;

• (iii) $f_1/f_2 \in \mathrm {UBC}$ if and only if $\sup _{a\in {\mathbb {D}}} ( {\hat{u}}(a) - u(a) ) < \infty$.

Moreover,

• (iv) all solutions of (1) belong to ${\mathcal {N}}$ if and only if u has a positive harmonic majorant;

• (v) all solutions of (1) belong to $H^p$, for $0<p<\infty$, if and only if $\exp (\frac{p}{2} \, u)$ has a harmonic majorant;

• (vi) all solutions of (1) belong to $H^\infty$ if and only if $\exp (u) \in L^\infty$.

Recall that the following conditions are equivalent for any subharmonic function u in the unit disc (see [8, p. 66] for more details): (a) u has a positive harmonic majorant; (b) the subharmonic function $u^+ = \max \{ u, 0\}$ has a harmonic majorant; (c) u is majorized by a Poisson integral of a finite measure on $\partial {\mathbb {D}}$. In Theorem 6, it is possible that u admits a harmonic majorant which takes negative values, since there are Blaschke-oscillatory equations (1) whose non-trivial solutions lie outside ${\mathcal {N}}$ [22, Section 4.3]. Although the items (iv)–(vi) are immediate, their assertions raise an interesting observation. Since we may describe the behavior of all solutions of (1) in terms of $f_1/f_2$, no essential information is reduced in this quotient. In Remark 2, Sect. 7, we illustrate that the growth of solutions of Blaschke-oscillatory equations is severely restricted.

There are normal functions which do not belong to ${\mathcal {N}}$. Classical example of such a function is the elliptic modular function [30, p. 57]. If $f_1,f_2$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, then $f_1/f_2\in {\mathcal {N}}$ provided that $f_1/f_2$ is normal and the set where $|f_1|^2+|f_2|^2$ takes small values, is not too large.

Proposition 1

Let $f_1,f_2$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$. The differential equation (1) is Blaschke-oscillatory if $f_1/f_2$ is normal and there exists $0<\delta <\infty$ such that $\int _{\{z\in {\mathbb {D}}\; \! : \; \! |f_1(z)|^2+|f_2(z)|^2< \delta \}} dm(z)/(1-|z|^2) < \infty$.

2.4 Nevanlinna interpolating sequences

By recent advances concerning free interpolation in ${\mathcal {N}}$ [18, 19, 32], there is an astounding resemblance between uniformly separated sequences and Nevanlinna interpolating sequences. Therefore the following results can be interpreted as Nevanlinna analogues of ones that are either presented in Sect. 2.1 or already appear in the literature.

A sequence $\varLambda \subset {\mathbb {D}}$ is called (free) interpolating for ${\mathcal {N}}$ if the trace of ${\mathcal {N}}$ on $\varLambda$ is ideal [18, p. 3]. That is, for any $g\in {\mathcal {N}}$ and for any bounded sequence $\{w_n\}\in {\mathbb {C}}$, there exists $f\in {\mathcal {N}}$ such that $f(z_n) = w_n \, g(z_n)$ for all $z_n\in \varLambda$. The collection of (free) interpolating sequences for ${\mathcal {N}}$ is denoted by ${{\,\mathrm{Int}\,}}{\mathcal {N}}$. Note that $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$ if and only if the trace ${\mathcal {N}}\mid \varLambda$ contains all bounded sequences [18, Remark 1.1], and in particular, all sequences in ${{\,\mathrm{Int}\,}}{\mathcal {N}}$ satisfy the Blaschke condition.

Let $\mathrm{Har^+}({\mathbb {D}})$ denote the space of positive harmonic functions in ${\mathbb {D}}$. By [18, Theorem 1.2], $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$ if and only if there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that

$$\begin{aligned} \prod _{z_k\in \varLambda {\setminus } \{z_n\}} \left| \frac{z_k-z_n}{1-{\overline{z}}_kz_n}\right| \ge e^{-h(z_n)}, \quad z_n\in \varLambda . \end{aligned}$$

(4)

The reader is invited to compare (4) to the classical description (20) of uniformly separated sequences, which are precisely the interpolating sequences for $H^\infty$.

Theorem 7

Let $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$. Then, there exist $h \in \mathrm{Har^+}({\mathbb {D}})$ and $A\in \mathrm{Hol}({\mathbb {D}})$ such that $|A(z)|(1-|z|^2)^2 \le e^{h(z)}$, $z\in {\mathbb {D}}$, and (1) admits a non-trivial solution whose zero-sequence is $\varLambda$.

By [18, Corollary 1.9], Theorem 7 allows us the prescribe any separated Blaschke sequence to be a zero-sequence of a non-trivial solution of (1). Theorem 7 should be compared to [11, Theorem 1], according to which any separated sequence of sufficiently small upper uniform density can appear as a subset of the zero-sequence of a non-trivial solution of (1) under the coefficient condition $A\in H^\infty _2$. The coefficient condition in Theorem 7 is of different nature as it controls the growth in an average sense. On one hand, the restriction $|A(z)|(1-|z|^2)^2 \le e^{h(z)}$, $z\in {\mathbb {D}}$ and $h\in \mathrm{Har^+}({\mathbb {D}})$, passes through functions such as $A(z) = (e/(1-z))^k$ for any $0<k<\infty$. On the other hand, it implies that there exists $0<C<\infty$ such that

$$\begin{aligned} \int _0^{2\pi } \log ^+ |A(re^{i\theta })| \, d\theta \le 2 \log ^+ \frac{1}{1-r} + C, \quad r\rightarrow 1^-, \end{aligned}$$

(5)

which is an estimate that cannot be improved even if $A\in H^\infty _2$. Estimate (5) reveals that such coefficient A lies close to ${\mathcal {N}}$ as it is non-admissible.

The following result is an analogue of [10, Theorem 5], and is related to the classical 0, 1-interpolation result due to Carleson [2, Theorem 2]. The Nevanlinna counterpart of Carleson’s result is presented in Sect. 9.

Theorem 8

Assume that $\alpha ,\beta \in {\mathbb {C}}{\setminus }\{0\}$ are distinct values. Let $\{z_n\},\{\zeta _n\}$ be any Blaschke sequences, and let $B_{\{z_n\}}$ and $B_{\{\zeta _n\}}$ be the corresponding Blaschke products. If there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that

$$\begin{aligned} \big |B_{\{z_n\}}(z)\big | + \big |B_{\{\zeta _n\}}(z)\big | \ge e^{-h(z)}, \quad z\in {\mathbb {D}}, \end{aligned}$$

(6)

then there exists $A \in \mathrm{Hol}({\mathbb {D}})$ and $H\in \mathrm{Har^+}({\mathbb {D}})$ such that $|A(z)|(1-|z|^2)^2 \le e^{H(z)}$, $z\in {\mathbb {D}}$, and (1) admits a solution f with $f(z_n)=\alpha$ and $f(\zeta _n)=\beta$ for all n.

We turn to study differential equations with solutions in ${\mathcal {N}}$. It turns out that Steinmetz’s approach from [44, Theorem, p. 129] applies with obvious changes.

Theorem 9

If $f_1,f_2\in {\mathcal {N}}$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, then there exists $H\in \mathrm{Har^+}({\mathbb {D}})$ such that $|A(z)|(1-|z|^2)^3 \le e^{H(z)}$ and $(f_1/f_2)^{\#}(z)(1-|z|^2)^2 \le e^{H(z)}$, $z\in {\mathbb {D}}$.

We may also ask when the stronger estimate $|A(z)|(1-|z|^2)^2 \le e^{H(z)}$, $z\in {\mathbb {D}}$, is obtained? The following result is analogous to Theorem 2; generalization of the assumption (7) to higher derivatives is left to the interested reader.

Theorem 10

If $f_1,f_2\in {\mathcal {N}}$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$ such that

$$\begin{aligned} \sum _{j=1,2} \Big ( |f_j(z)| + |f_j'(z)|(1-|z|^2) \Big ) \ge e^{-h(z)}, \quad z\in {\mathbb {D}}, \end{aligned}$$

(7)

for $h\in \mathrm{Har^+}({\mathbb {D}})$, then there exists $H\in \mathrm{Har^+}({\mathbb {D}})$ such that $|A(z)|(1-|z|^2)^2 \le e^{H(z)}$, $z\in {\mathbb {D}}$.

The sequence $\varLambda \subset {\mathbb {D}}$ is called h-separated, if there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that the pseudo-hyperbolic discs $\varDelta _p(z_n,e^{-h(z_n)})$, $z_n\in \varLambda$, are pairwise disjoint. Recall that the pseudo-hyperbolic disc of radius $0<\delta <1$, centered at $z\in {\mathbb {D}}$, is given by $\varDelta _p(z,\delta ) = \{ w \in {\mathbb {D}}: \varrho _p(z,w) < \delta \}$ where $\varrho _p(z,w) = |w-z|/|1-{\overline{w}}z|$ is the pseudo-hyperbolic distance between $z,w\in {\mathbb {D}}$. The following result corresponds to Schwarz’s findings [43, Theorems 3-4] in the case $A\in H^\infty _2$.

Proposition 2

Suppose that there exist $A\in \mathrm{Hol}({\mathbb {D}})$ and $H\in \mathrm{Har^+}({\mathbb {D}})$ such that $|A(z)|(1-|z|^2)^2 \le e^{H(z)}$, $z\in {\mathbb {D}}$. Then, there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that the zero-sequence of any non-trivial solution of (1) is h-separated.

Conversely, suppose that $A\in \mathrm{Hol}({\mathbb {D}})$ and there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that the zero-sequence of any non-trivial solution of (1) is h-separated. Then, there exists $H\in \mathrm{Har^+}({\mathbb {D}})$ such that $|A(z)|(1-|z|^2)^2 \le e^{H(z)}$, $z\in {\mathbb {D}}$.

2.5 Point-wise growth restrictions

A function $\omega : {\mathbb {D}}\rightarrow (0,\infty )$ is said to be a weight if it is bounded and continuous. The weight $\omega$ is radial if $\omega (z) = \omega (|z|)$ for all $z\in {\mathbb {D}}$, and is called regular if it is radial and for each $0\le s <1$ there exists a constant $C=C(s,\omega )$ with $1\le C<\infty$ such that

$$\begin{aligned} C^{-1} \, \omega (t)\le \omega (r) \le C \, \omega (t), \quad 0\le r \le t \le r+s(1-r) < 1. \end{aligned}$$

(8)

For a general reference for regular weights, see [39, Chapter 1]. For a weight $\omega$, let $L^\infty _\omega$ denote the growth space which consists of functions $f:{\mathbb {D}}\rightarrow {\mathbb {C}}$ for which $\Vert f\Vert _{L^\infty _\omega } = \sup _{z\in {\mathbb {D}}} |f(z)| \, \omega (z) <\infty$, and denote $H^\infty _\omega =L^\infty _\omega \cap \mathrm{Hol}({\mathbb {D}})$.

Theorem 11

Let $f_1,f_2$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, and define $u = -\log \, (f_1/f_2)^{\#}$. Suppose that $\omega$ is a regular weight which satisfies $\sup _{z\in {\mathbb {D}}}\, \omega (z)/(1-|z|) < \infty$. Then, $|\nabla u | \in L^\infty _\omega$ if and only if $A\in H^\infty _{\omega ^2}$ and $(f_1/f_2)^\#\in L^\infty _\omega$.

The following result follows directly from Theorem 11 with $\omega (z)=1-|z|^2$, $z\in {\mathbb {D}}$. This corollary concerns those differential equations (1) which have both desired properties mentioned in Sect. 2.1: $A\in H^\infty _2$ and $(f_1/f_2)^\# \in L^\infty _1$.

Corollary 1

Let $f_1,f_2$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, and define $u = -\log \, (f_1/f_2)^{\#}$. Then, $|\nabla u|\in L^\infty _1$ if and only if $A\in H^\infty _2$ and $f_1/f_2$ is normal.

Corollary 1 can also be deduced by combining several results in the literature. The first part follows from [1, Theorem 6], while the second part can be concluded from [48, Theorem 1] and [48, Corollary to Theorem 2]. Note that $f_1/f_2$ is uniformly locally univalent provided that $A\in H^\infty _2$, which can be seen by applying Nehari’s univalency criterion [34, Theorem I] locally.

Corollary 2

Let $f_1,f_2$ be linearly independent solutions of (1) for $A\in H^\infty _{\omega ^2}$, and define $u = -\log \, (f_1/f_2)^{\#}$. Suppose that $\omega$ is a regular weight which satisfies $\sup _{z\in {\mathbb {D}}}\, \omega (z)/(1-|z|) < \infty$. Then, the following statements are equivalent:

• (i) $|\nabla u |\in L^\infty _\omega$;

• (ii) $(f_1/f_2)^\#\in L^\infty _\omega$;

• (iii) $(|f_1'|+|f_2'|)/(|f_1|+|f_2|)\in L^\infty _\omega$;

• (iv) $\varDelta u \in L^\infty _{\omega ^2}$.

If $\omega (z)=1-|z|^2$, $z\in {\mathbb {D}}$, then Corollary 2 provides a complete description of those differential equations (1) for $A\in H^\infty _2$, where the quotient of two linearly independent solutions is normal. Such characterizations are important in oscillation theory. Since normal functions are Lipschitz-continuous, as mappings from ${\mathbb {D}}$ equipped with the hyperbolic metric to the Riemann sphere equipped with the chordal metric, the normality of $f_1/f_2$ implies that its the zeros and poles (which correspond to the zeros of $f_1$ and $f_2$, respectively) are separated in the hyperbolic metric. Finally, we point out that Corollary 2(iii) does not extend to higher derivatives, since there are differential equations (1) with $A\in \mathrm{Hol}({\mathbb {D}})$ and $|A| = (|f_1''|+|f_2''|)/(|f_1|+|f_2|)\in L^\infty _2$ such that the quotient $f_1/f_2$ of linearly independent solutions $f_1,f_2$ is non-normal; see [28] and Theorem 3.

2.6 Prescribed fixed points

The point $z_0\in {\mathbb {D}}$ is said to be a fixed point of $f\in \mathrm{Hol}({\mathbb {D}})$ if $f(z_0)=z_0$. There are a lot of known results according to which zeros and critical points (i.e., zeros of the derivative) can be prescribed for solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$. See [11, 13, 21, 22] among many others. For example, the proof of Theorem 3 depends on such an argument. It turns out that fixed points can be prescribed for a solution of (1) under the coefficient condition $A\in \mathrm{Hol}({\mathbb {D}})$ in such a way that all solutions of the differential equation remain bounded. Such differential equations were studied in detail in Sect. 2.1.

Theorem 12

Let $\varLambda \subset {\mathbb {D}}$ be a Blaschke sequence, and let $0<\varepsilon <1$. Then, there exists a coefficient $A\in \mathrm{Hol}({\mathbb {D}})$ such that $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure; the differential equation (1) admits a solution f, which satisfies $\Vert f\Vert _{H^\infty }< 1 + \varepsilon$ and has fixed points $\{0\} \cup \varLambda$; all solutions of (1) are bounded.

If we assume that prescribed fixed points are uniformly separated, then we can go further and dictate their type. In this paper, we make distinction between three different types: the fixed point $z_0\in {\mathbb {D}}$ of $f\in \mathrm{Hol}({\mathbb {D}})$ is said to be attractive if $|f'(z_0)|<1$, neutral if $|f'(z_0)|=1$, and repulsive if $|f'(z_0)|>1$.

Theorem 13

Let $\varLambda \subset {\mathbb {D}}{\setminus }\{0 \}$ be uniformly separated. Then, there exists a coefficient $A\in \mathrm{Hol}({\mathbb {D}})$ such that $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure; the differential equation (1) admits a bounded solution for which every point in $\varLambda$ is a fixed point of prescribed type; all solutions of (1) are bounded.

Theorem 13 has a natural counterpart in the setting of Nevanlinna interpolating sequences. Note that Theorem 12 is valid for sequences $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$ as it is.

Theorem 14

Let $\varLambda \subset {\mathbb {D}}{\setminus }\{0 \}$ and $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$. Then, there exists a coefficient $A\in \mathrm{Hol}({\mathbb {D}})$ and $H\in \mathrm{Har^+}({\mathbb {D}})$ such that $|A(z)|^2(1-|z|^2)^2\le e^{H(z)}$, $z\in {\mathbb {D}}$, and (1) admits a solution for which every point in $\varLambda$ is a fixed point of prescribed type.

3 Proof of Theorem 1

The following argument is based on concrete construction.

Proof

(of Theorem 1) (i) Let $0<p<1/2$, and

$$\begin{aligned} f_1(z)=\exp \bigg ( i \cdot \frac{p}{2\pi } \bigg ( \log \frac{2i}{1-z} \bigg )^2 \, \bigg ), \quad z\in {\mathbb {D}}. \end{aligned}$$

Note that the function $z\mapsto 2i/(1-z)$ maps ${\mathbb {D}}$ onto $\{z\in {\mathbb {C}}: {{\,\mathrm{Im}\,}}z > 1\}$. Since

$$\begin{aligned} {{\,\mathrm{Re}\,}}\bigg ( i \cdot \frac{p}{2\pi } \bigg ( \log \frac{2i}{1-z} \bigg )^2 \, \bigg ) = - \frac{p}{\pi } \, \log \frac{2}{|1-z|} \, \arg \frac{2i}{1-z}, \quad z\in {\mathbb {D}}, \end{aligned}$$

we deduce $2^{-p} (1-|z|)^p \le |f_1(z)| \le 1$ for $z\in {\mathbb {D}}$. Since $f_1$ is zero-free, we conclude $A=-f_1''/f_1 \in \mathrm{Hol}({\mathbb {D}})$. Moreover, $A\not \in H^\infty _2$ because

$$\begin{aligned} A(z) = p \, \frac{p\big ( \log \frac{2i}{1-z} \big )^2 - i \pi \log \frac{2i}{1-z}-i\pi }{\pi ^2 (1-z)^2}, \quad z\in {\mathbb {D}}. \end{aligned}$$

It remains to show that all solutions of (1) are bounded. Note that

$$\begin{aligned} f_2(z) = f_1(z) \, \int _0^z \frac{1}{f_1(\zeta )^2} \, d\zeta , \quad z\in {\mathbb {D}}, \end{aligned}$$

(9)

is a bounded solution of (1), and $f_2$ is linearly independent to $f_1$. Here we integrate along the straight line segment. This completes the proof of (i), since every solution of (1) is a linear combination of $f_1,f_2$.

(ii) Let $0<p<1/2$, and

$$\begin{aligned} f_1(z)=\exp \bigg ( i \cdot \frac{p}{\pi } \bigg ( \log \frac{1+z}{1-z} \bigg )^2 \, \bigg ), \quad z\in {\mathbb {D}}. \end{aligned}$$

Similar function has been utilized in [29, pp. 142–143]. We point out that $f_1$ has asymptotic values 0 and $\infty$ at $z=1$, and hence $f_1$ is not normal. This fact alone implies that the zero-free function $f_1$ cannot be a solution of (1) for $A\in H^\infty _2$; see [17, Proposition 7]. As in the part (i), we deduce

$$\begin{aligned} \left( \frac{1-|z|}{1+|z|} \right) ^p \le |f_1(z)| \le \left( \frac{1+|z|}{1-|z|} \right) ^p, \quad z\in {\mathbb {D}}. \end{aligned}$$

(10)

Since $f_1$ is zero-free, we conclude $A=-f_1''/f_1 \in \mathrm{Hol}({\mathbb {D}})$. Moreover, $A\not \in H^\infty _2$ as

$$\begin{aligned} A(z) = 8p \, \frac{2p\big ( \log \frac{1+z}{1-z} \big )^2 - i \pi z \log \frac{1+z}{1-z}-i\pi }{\pi ^2 (1-z^2)^2}, \quad z\in {\mathbb {D}}. \end{aligned}$$

It remains to show that all solutions of (1) belong to $H^\infty _p$. On one hand, it is clear that $f_1\in H^\infty _p$ by (10). On the other hand, (9) is a solution of (1) which is linearly independent to $f_1$. Since $z \mapsto \int _0^z d\zeta /f_1(\zeta )^2$ is bounded in ${\mathbb {D}}$, we have $f_2\in H^\infty _p$. This completes the proof of Theorem 1. $\square$

4 Proof of Theorem 2

We offer two different proofs for Theorem 2. We begin by considering a more general result which implies Theorem 2 as a corollary. The following lemma indicates that any analytic function, which satisfies $H^\infty _\alpha$-type estimate outside a small exceptional set, actually belongs to $H^\infty _\alpha$.

Lemma 1

Let $f\in \mathrm{Hol}({\mathbb {D}})$ and $0\le \alpha <\infty$. Then $f\in H^\infty _\alpha$, if there exist pairwise disjoint discs $\varDelta _p(z_n,\delta )$, $z_n\in {\mathbb {D}}$ and $0<\delta <1$, such that

$$\begin{aligned} \sup \bigg \{ |f(z)| (1-|z|^2)^\alpha : z \in {\mathbb {D}}{\setminus } \bigcup _n \, \varDelta _p(z_n,\delta ) \bigg \} < \infty . \end{aligned}$$

(11)

Proof

Let $z\in \varDelta _p(z_n,\delta )$ for some n, and let S be the supremum in (11). By the maximum modulus principle, there exists $\zeta \in \partial \varDelta _p(z_n,\delta )$ such that $|f(\zeta )| = \max \big \{ |f(\xi )| : \xi \in \overline{\varDelta _p(z_n,\delta )}\big \}$. By the standard estimates, there exists a constant $C=C(\delta )$ with $0<C<\infty$ such that

$$\begin{aligned} |f(z)| (1-|z|^2)^\alpha \le |f(\zeta )| (1-|z|^2)^\alpha \le C^\alpha |f(\zeta )| (1-|\zeta |^2)^\alpha \le C^\alpha S. \end{aligned}$$

The assertion $f\in H^\infty _\alpha$ follows. $\square$

Recall that the space $\mathrm BMOA$ consists of those functions in $H^2$ whose boundary values have bounded mean oscillation on $\partial {\mathbb {D}}$, or equivalently, of those functions $f\in \mathrm{Hol}({\mathbb {D}})$ for which $|f'(z)|^2(1-|z|^2)\, dm(z)$ is a Carleson measure. We write $\Vert f\Vert _{\mathrm{BMOA}}^2 = \sup _{a\in {\mathbb {D}}} \, \Vert f_a\Vert _{H^2}^2$ where $f_a(z)=f(\varphi _a(z)) - f(a)$ for $a,z\in {\mathbb {D}}$.

Theorem 15

If $f_1,f_2\in {\mathcal {B}}$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, and there exist pairwise disjoint discs $\varDelta _p(z_n,\delta )$, $z_n\in {\mathbb {D}}$ and $0<\delta <1$, with

$$\begin{aligned} \inf \bigg \{ |f_1(z)| + |f_2(z)| : z \in {\mathbb {D}}{\setminus } \bigcup _n \, \varDelta _p(z_n,\delta ) \bigg \} > 0, \end{aligned}$$

(12)

then $A\in H^\infty _2$. If $f_1,f_2\in \mathrm BMOA$ and the sequence $\{z_n\}\subset {\mathbb {D}}$ in (12) is uniformly separated, then $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure.

The first part of Theorem 15 improves [12, Theorem 7] by Example 1(ii) below. When comparing Theorem 15 to Theorem 2 note that in the former result it is not required that $f_1,f_2\in H^\infty$.

Proof

(of Theorem 15) Let $f_1,f_2\in {\mathcal {B}}$ be linearly independent solutions of (1) and suppose that (12) holds. Denote $\varOmega = \bigcup _{n} \varDelta _p(z_n,\delta )$. Since

$$\begin{aligned} |A| = \frac{|f_1|+|f_2|}{|f_1|+|f_2|} \, |A| = \frac{|f_1''|+|f_2''|}{|f_1|+|f_2|}, \end{aligned}$$

(13)

we deduce

$$\begin{aligned} \sup _{z\in {\mathbb {D}}{\setminus }\varOmega } |A(z)| (1-|z|^2)^2 \le \frac{\Vert f_1''\Vert _{H^\infty _2} + \Vert f_2''\Vert _{H^\infty _2}}{\inf _{z\in {\mathbb {D}}{\setminus }\varOmega } \big (|f_1(z)| + |f_2(z)|\big )}. \end{aligned}$$

Since $A\in \mathrm{Hol}({\mathbb {D}})$, we conclude $A\in H^\infty _2$ by Lemma 1. This completes the proof of the first part of Theorem 15.

If $f_1,f_2\in \mathrm BMOA$ and $\{z_n \}\subset {\mathbb {D}}$ in (12) is uniformly separated, then we write

$$\begin{aligned} \sup _{a\in {\mathbb {D}}} \int _{{\mathbb {D}}} |A(z)|^2(1-|z|^2)^3 \, \frac{1-|a|^2}{|1-{\overline{a}}z|^2} \, dm(z) = I_1 + I_2, \end{aligned}$$

where $I_1,I_2$ are defined as below. By (13) and [42, Theorem 4.2.1], we deduce

$$\begin{aligned} I_1&= \, \sup _{a\in {\mathbb {D}}} \int _{{\mathbb {D}}{\setminus } \varOmega } |A(z)|^2(1-|z|^2)^3 \, \frac{1-|a|^2}{|1-{\overline{a}}z|^2} \, dm(z) \nonumber \\&\, \lesssim \sup _{a\in {\mathbb {D}}} \int _{{\mathbb {D}}} \big ( |f_1''(z)|^2 + |f_2''(z)|^2 \big ) (1-|z|^2)^3 \, \frac{1-|a|^2}{|1-{\overline{a}}z|^2} \, dm(z) < \infty . \end{aligned}$$

(14)

Actually, (14) is bounded above by a constant multiple of $\Vert f_1\Vert _{\mathrm{BMOA}}^2 + \Vert f_2\Vert _{\mathrm{BMOA}}^2$. Since $A\in H^\infty _2$ by the first part of the proof, standard estimates yield

$$\begin{aligned} I_2&= \sup _{a\in {\mathbb {D}}} \, \sum _n \, \int _{\varDelta _p(z_n,\delta )} |A(z)|^2(1-|z|^2)^3 \, \frac{1-|a|^2}{|1-{\overline{a}}z|^2} \, dm(z) \nonumber \\&\lesssim \Vert A\Vert ^2_{H^\infty _2} \, \sup _{a\in {\mathbb {D}}} \, \sum _n \frac{(1-|a|^2)(1-|z_n|^2)}{|1-{\overline{a}}z_n|^2} < \infty . \end{aligned}$$

(15)

The sum in (15) is finite by the uniform separation of $\{z_n\}$. This completes the proof of Theorem 15. $\square$

If $\{z_n\}\subset {\mathbb {D}}$ is a Blaschke sequence, then the Blaschke product

$$\begin{aligned} B(z) = B_{\{z_n\}}(z) = \prod _n \frac{|z_n|}{z_n} \, \frac{z_n-z}{1-{\overline{z}}_n z}, \quad z\in {\mathbb {D}}, \end{aligned}$$

is a bounded analytic function which vanishes precisely on $\{z_n\}$. Let $f_1,f_2\in H^\infty$. By [45, Theorem 3], the ideal

$$\begin{aligned} J_{H^\infty }(f_1,f_2) = \Big \{ f\in H^\infty : \exists \, c = c(f)>0\text { such that }|f| \le c \big (|f_1| + |f_2|\big ) \!\Big \} \end{aligned}$$

contains a Blaschke product whose zeros form a finite union of uniformly separated sequences if and only if (2) holds. If B is such a Blaschke product, then there exists a constant $0<\delta <1$ and a subsequence $\{z_n'\}$ of zeros of B such that the discs $\varDelta _p(z_n',\delta )$, $n\in {\mathbb {N}}$, are pairwise disjoint and

$$\begin{aligned} \inf \bigg \{ |B(z)| : z \in {\mathbb {D}}{\setminus } \bigcup _n \, \varDelta _p(z_n',\delta ) \bigg \} > 0. \end{aligned}$$

(16)

This follows from [26, Lemmas 1 and 3]; see also [35, Lemma 1]. Therefore Theorem 15 gives an immediate proof for Theorem 2. We also present another proof which, in addition, provides a concrete representation for the coefficient A.

Proof

(of Theorem 2) By (2) and [45, Theorem 3], the ideal $I_{H^\infty }(f_1,f_2)$ contains a Blaschke product B whose zeros form a finite union of uniformly separated sequences. This is equivalent to the fact that there exist functions $g_1,g_2\in H^\infty$ such that $f_1 g_1 + f_2 g_2 = B$. Differentiate this identity twice, and then apply (1) to $f_1''$ and $f_2''$, to obtain

$$\begin{aligned} A = \frac{2(f_1'g_1'+f_2'g_2') + f_1g_1''+ f_2g_2''-B''}{B}. \end{aligned}$$

(17)

As in the proof of Theorem 15, by taking account on (16), we conclude that $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure. $\square$

One of the objectives in Sect. 2.1 was to generalize a result according to which $A\in H^\infty _2$ if $f_1,f_2\in {\mathcal {B}}$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$ such that $\inf _{z\in {\mathbb {D}}} (|f_1(z)| + |f_2(z)|) >0$. The Cauchy-Schwarz inequality gives

$$\begin{aligned} |W(f_1,f_2)|^2 \le ( |f_1|^2+|f_2|^2 ) \, ( |f_1'|^2+|f_2'|^2 ). \end{aligned}$$

(18)

Since $f_1',f_2'\in H^\infty _1$, we deduce $|f_1(z)|+|f_2(z)| \gtrsim 1-|z|^2$, $z\in {\mathbb {D}}$, without using any additional assumptions.

Example 1

Let $f_1,f_2\in H^\infty$ be linearly independent solutions of (1) for $A\in H^\infty _2$. This example concerns different situations that may happen.

• (i) There are a lot of examples in which $\inf _{z\in {\mathbb {D}}} ( |f_1(z)| + |f_2(z)| ) >0$. See the discussion after the proof of [11, Theorem 2], for example.

• (ii) The proof of Theorem 3 below produces an example, where the condition (12) holds; take $\{z_n\}$ as in Theorem 3 and note that $|f_1|+|f_2|\ge |f_1|$, where $f_1$ has the desired property. At the same time, $|f_1(z_n)|+|f_2(z_n)| \asymp 1-|z_n|^2$ as $n\rightarrow \infty$. Not only $\inf _{z\in {\mathbb {D}}} ( |f_1(z)| + |f_2(z)| ) >0$ fails to be true but also it breaks down in the worst possible way.

• (iii) Let $f_1(z)=(1-z^2)^{1/2}$ and $f_2(z)=(1-z^2)^{1/2} \log ( (1+z)/(1-z) )$, $z\in {\mathbb {D}}$. These functions are linearly independent solutions of (1) for the coefficient $A(z)=1/(1-z^2)^2$, $z\in {\mathbb {D}}$, which evidently satisfies $A\in H^\infty _2$. Since both solutions have radial limit zero along the positive real axis, the condition (12) cannot hold for any pairwise disjoint pseudo-hyperbolic discs.

5 Proofs of Theorems 3 and 4

The first part of the proof of Theorem 3 follows directly from that of [11, Corollary 3]. The new contribution lies in the fact that the differential equation in question admits only bounded solutions.

Proof

(of Theorem 3) Let $B=B_\varLambda$ be the Blaschke product corresponding to the uniformly separated sequence $\varLambda =\{z_n\}$. By (20) and Cauchy’s integral formula, $\sup _{z_n\in \varLambda } |B''(z_n)|/|B'(z_n)|^2 < \infty$. Let $f_1=Be^{Bk}$, where $k\in H^\infty$ is a solution of the interpolation problem

$$\begin{aligned} k(z_n) = - \frac{B''(z_n)}{2 \, B'(z_n)^2}, \quad z_n\in \varLambda . \end{aligned}$$

As in the proof of Theorem 15, the coefficient $A=-f_1''/f_1 \in \mathrm{Hol}({\mathbb {D}})$ induces a Carleson measure $|A(z)|^2(1-|z|^2)^3\, dm(z)$. Now, $f_1$ is a solution of (1) which has precisely the prescribed zeros $\varLambda$.

Since $\varLambda$ is uniformly separated, there exists a constant $0<\delta <1$ such that $\varOmega = \bigcup _{z_n\in \varLambda } \varDelta _p(z_n,\delta )$ is a union of pairwise disjoint pseudo-hyperbolic discs. Fix any $\alpha \in {\mathbb {D}}{\setminus } \varOmega$, and define the meromorphic function $f_2$ by

$$\begin{aligned} f_2(z) = f_1(z)\, \int _\alpha ^z \frac{1}{f_1(\zeta )^2} \, d\zeta , \quad z\in {\mathbb {D}}. \end{aligned}$$

(19)

Choose the path of integration by the following rules. If $z\in {\mathbb {D}}{\setminus } \varOmega$, then the whole path lies in ${\mathbb {D}}{\setminus } \varOmega$. If $z\in \varDelta _p(z_n,\delta )$ for some $z_n\in \varLambda$, then the path stays in $({\mathbb {D}}{\setminus } \varOmega ) \cup \varDelta _p(z_n,\delta )$. Then, each point $z\in {\mathbb {D}}$ can be reached by a path which satisfies these properties and is also of uniformly bounded Euclidean length. The following argument is standard. In a sufficiently small pseudo-hyperbolic neighborhood of $\alpha$, $f_2$ represents an analytic function such that $f_1f_2'-f_1'f_2$ is identically one. As a solution of (1) function $f_2$ admits an analytic continuation to ${\mathbb {D}}$, and this continuation agrees with the representation (19).

There exists a constant $\mu =\mu (\varLambda )$ such that $|B(\zeta )|\ge \mu >0$ for $\zeta \in {\mathbb {D}}{\setminus } \varOmega$; see [4, Theorem 1] for example. We deduce

$$\begin{aligned} |f_2(z)| \le |B(z)| e^{|B(z)| |k(z)|} \int _\alpha ^z \frac{|d\zeta |}{|B(\zeta )|^2 e^{-2 |B(\zeta )| |k(\zeta )|}} \le \frac{e^{3 \,\! \Vert k\Vert _{H^\infty }}}{\mu ^2} \int _{\alpha }^z |d\zeta |, \end{aligned}$$

for $z\in {\mathbb {D}}{\setminus } \varOmega$. Lemma 1 implies that $f_2\in H^\infty$. Since $W(f_1,f_2)=1$, we obtain

$$\begin{aligned} (f_1/f_2)^\#(z_n) \, (1-|z_n|^2)^2&= \frac{1}{|f_2(z_n)|^2} \, (1-|z_n|^2)^2 = |f_1'(z_n)|^2 (1-|z_n|^2)^2\\&= |B'(z_n)|^2 (1-|z_n|^2)^2, \quad z_n\in \varLambda . \end{aligned}$$

This completes the proof as $\varLambda$ is uniformly separated. $\square$

The proof of Theorem 4 depends on a supporting result, which is considered next. Suppose that $f\in \mathrm{Hol}({\mathbb {D}})$, $f:{\mathbb {D}}\rightarrow {\mathbb {D}}$, $f(0)=0$ and $|f'(0)|\ge \delta$ for some $0<\delta \le 1$. By Cauchy’s integral formula and Schwarz’s lemma,

$$\begin{aligned} |f'(0)|-|f'(z)| \le \frac{12\, |z|}{(1-|z|)^2}, \quad z\in {\mathbb {D}}. \end{aligned}$$

If $0<\eta <1$ satisfies $12\eta /(1-\eta )^2< \delta /2$, then $|f'(z)|\ge \delta /2$ for all $|z|< \eta$. The following lemma is a conformally invariant version of this property.

Lemma 2

Suppose that $f\in \mathrm{Hol}({\mathbb {D}})$ and $f:{\mathbb {D}}\rightarrow {\mathbb {D}}$. Assume that there exists a sequence $\varLambda \subset {\mathbb {D}}$ such that $\inf _{z_n\in \varLambda } \, |f'(z_n)| (1-|z_n|^2) \ge \delta >0$. If $0<\eta <1$ satisfies $12\eta /(1-\eta )^2< \delta /2$, then there exist a constant $\nu =\nu (\delta )$ such that

$$\begin{aligned} |f'(z)| (1-|z|^2) \ge \nu >0, \quad z\in \bigcup _{z_n\in \varLambda } \varDelta _p(z_n,\eta ). \end{aligned}$$

Proof

Let $z_n\in \varLambda$ be fixed, and define $g_{z_n} = \varphi _{f(z_n)} \circ f \circ \varphi _{z_n}$. Now $g_{z_n}:{\mathbb {D}}\rightarrow {\mathbb {D}}$ is analytic, $g_{z_n}(0) = 0$, and

$$\begin{aligned} |g_{z_n}'(0)| = \big |\varphi _{f(z_n)}' \big ( f( z_n) \big ) \big | \, | f'(z_n)| (1-|z_n|^2 ) \ge | f'(z_n)| (1-|z_n|^2 ) \ge \delta . \end{aligned}$$

The property above implies

$$\begin{aligned} |g_{z_n}'(z)| = \big |\varphi _{f(z_n)}' \big ( f( \varphi _{z_n}(z)) \big ) \big | \cdot | f'( \varphi _{z_n}(z))| \cdot | \varphi _{z_n}'(z) | \ge \delta /2, \quad |z|< \eta . \end{aligned}$$

If we denote $w=\varphi _{z_n}(z)$, then $|z|<\eta$ if and only if $w\in \varDelta _p(z_n,\eta )$. Consequently,

$$\begin{aligned} | f'(w)| (1-|w|^2) \ge \frac{\delta }{2} \cdot \frac{1-|\varphi _{z_n}(w)|^2}{1-|\varphi _{f(z_n)}\big (f(w)\big )|^2} \big ( 1-|f(w)|^2 \big ), \quad w\in \varDelta (z_n,\eta ). \end{aligned}$$

Since $\varrho _p(w, z_n) < \eta$, we have $\varrho _p(f(w), f(z_n)) < \eta$ by Schwarz’s lemma. Therefore there exists a constant $\delta ^\star =\delta ^\star (\delta ,\eta )>0$ such that

$$\begin{aligned} | f'(w)| (1-|w|^2) \ge \delta ^\star \big ( 1-|f(z_n)|^2 \big ) \ge \delta ^\star | f'(z_n)| (1-|z_n|^2) \ge \delta ^\star \delta , \quad w\in \varDelta (z_n,\eta ), \end{aligned}$$

by the Schwarz–Pick lemma. The claim follows for $\nu =\delta ^\star \delta$. $\square$

Proof

(of Theorem 4) The proof is divided into two steps. The first step takes advantage of two results concerning interpolation in $H^\infty$.

Construction of auxiliary functions. Let $B=B_\varLambda$ be the Blaschke product corresponding to the uniformly separated sequence $\varLambda =\{z_n\}$, and let $\{w_n\}$ be the bounded target sequence for the desired interpolation. Consequently,

$$\begin{aligned} \inf _{z_n\in \varLambda } \, |B'(z_n)| (1-|z_n|^2) = \inf _{z_n\in \varLambda } \, \prod _{z_k\in \varLambda {\setminus } \{z_n\}} \left| \frac{z_k-z_n}{1-{\overline{z}}_kz_n}\right| =\delta >0. \end{aligned}$$

(20)

Let $0<\eta <1$ satisfy $12\eta /(1-\eta )^2< \delta /2$. Then, in particular, $\eta <\delta /3$. Earl’s interpolation theorem [6, Theorem 2], applied with $\eta$ instead of $\delta$, shows that

$$\begin{aligned} \big \{ h \in H^\infty : h(z_n)=w_n\text { for all }z_n\in \varLambda \big \} \end{aligned}$$

(21)

can be solved by a constant multiple of a Blaschke product. More precisely, there exist $C=C(\varLambda , \{w_n\}, \eta ) \in {\mathbb {C}}$ and a Blaschke product $I=I(\varLambda , \{w_n\}, \eta )$ such that

• (i) $h=C I$ solves the interpolation problem (21);

• (ii) the zeros $\varLambda ^\star = \{\zeta _n\}$ of $I=I_{\{\zeta _n\}}$ satisfy $\zeta _n\in \varDelta _p(z_n,\eta )$ for all n.

The standard estimates show that

$$\begin{aligned} \inf _{\zeta _n\in \varLambda ^\star } \, |I'(\zeta _n)| (1-|\zeta _n|^2) = \inf _{\zeta _n\in \varLambda ^\star } \, \prod _{\zeta _k\in \varLambda ^\star {\setminus } \{\zeta _n\}} \left| \frac{\zeta _k-\zeta _n}{1-{\overline{\zeta }}_k \zeta _n}\right| \ge \frac{\delta }{3} >0, \end{aligned}$$

and therefore $\{\zeta _n\}$ is also uniformly separated.

By applying Lemma 2 to the Blaschke product B, there exists another constant $\nu$ such that $| B'(\zeta _n) | (1-|\zeta _n|^2) \ge \nu >0$ for all $\zeta _n\in \varLambda ^\star$. According to Øyma’s interpolation theorem [37, Theorem 1], there exists $g\in H^\infty$ such that

$$\begin{aligned} g(\zeta _n) = - \frac{I''(\zeta _n)}{2 \, I'(\zeta _n) \, B'(\zeta _n)}, \quad g'(\zeta _n)=0, \quad \zeta _n\in \varLambda ^\star . \end{aligned}$$

(22)

Note that the target sequence for g is bounded by the obtained estimates.

Construction of the differential equation. Let $f=C I \, e^{Bg} \in \mathrm{Hol}({\mathbb {D}})$, where $C\in {\mathbb {C}}$ and I, B, g are functions as in the above construction. Clearly, $f(z_n)=w_n$ for all $z_n\in \varLambda$. The zeros of f are precisely the points in $\varLambda ^\star$, and they are pairwise pseudo-hyperbolically close to the corresponding points in $\varLambda$ by (ii). Since

$$\begin{aligned} f''(\zeta _n) = C e^{B(\zeta _n) g(\zeta _n)} \Big ( I''(\zeta _n) + 2 \, I'(\zeta _n) B'(\zeta _n) g(\zeta _n) \Big ) =0, \quad \zeta _n\in \varLambda ^\star , \end{aligned}$$

the function

$$\begin{aligned} A = -\frac{f''}{f} = - \frac{I''+2\, I' (Bg)'}{I} - \big ( (Bg)' \big )^2 - (Bg)'', \end{aligned}$$

is analytic in ${\mathbb {D}}$. More precisely, the points in $\varLambda ^\star$ are removable singularities for the coefficient A as g solves the interpolation problem (22). As in the proof of Theorem 15, we conclude that $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure. The fact that all solutions of (1) are bounded follows as in the proof of Theorem 3. This completes the proof of Theorem 4. $\square$

5.1 Separation of zeros and critical points

Let $A\in H^\infty _2$, and let f be a non-trivial solution of (1). By [43, Theorem 3], the zeros of f are separated in the hyperbolic metric by a constant depending only on $\Vert A\Vert _{H^\infty _2}$, and by [10, Corollary 2], the hyperbolic distance between any zero and any critical point of f is uniformly bounded away from zero in a similar fashion. Moreover, [10, Example 1] shows that critical points of f need not to obey any kind of separation. The situation becomes more difficult if we consider similar questions between zeros and critical points of linearly independent solutions. See [10, Section 4] for related discussion.

The following result concerns differential equations with bounded solutions. The proof is based on an auxiliary estimate [5, Lemma 7, p. 209]: If $f\in H^\infty _\alpha$ for $0\le \alpha < \infty$, then there exists a constant $C=C(\alpha )$ with $0<C<\infty$ such that

$$\begin{aligned} \big | |f(z_1)|(1-|z_1|^2)^\alpha - |f(z_2)|(1-|z_2|^2)^\alpha \big | \le C \, \varrho _p(z_1,z_2) \Vert f\Vert _{H^\infty _\alpha } \end{aligned}$$

(23)

for all points $z_1,z_2\in {\mathbb {D}}$ with $\varrho _p(z_1,z_2)\le 1/2$. The sharpness discussion of Proposition 3 below is omitted.

Proposition 3

Suppose that $A\in \mathrm{Hol}({\mathbb {D}})$ and all solutions of (1) are bounded.

• (i) It is possible that for each $0<\delta <1$ there exists a solution of (1), depending on $\delta$, which has two distinct zeros $z_1,z_2\in {\mathbb {D}}$ such that $\varrho _p(z_1,z_2) <\delta$.

• (ii) Critical points of non-trivial solutions are not separated in any way.

Let $f_1,f_2\in H^\infty$ be linearly independent solutions of (1).

• (iii) If $z_1\in {\mathbb {D}}$ is a zero and $z_2\in {\mathbb {D}}$ is a critical point of $f_1$, then there exists a constant $0<C<\infty$ such that

  $$\begin{aligned} \varrho _p(z_1,z_2) \ge C \, \frac{|W(f_1,f_2)|}{\Vert f_1\Vert _{H^\infty } \Vert f_2\Vert _{H^\infty }} \, \max \!\big \{1-|z_1|, 1-|z_2|\big \}. \end{aligned}$$

  (24)

• (iv) If $z_1\in {\mathbb {D}}$ is a zero of $f_1$, and $z_2\in {\mathbb {D}}$ is a zero of $f_2$, then (24) holds.

• (v) If $z_1\in {\mathbb {D}}$ is a critical point of $f_1$, and $z_2\in {\mathbb {D}}$ is that of $f_2$, then (24) holds.

Proof

(i) Let the coefficient $A\in \mathrm{Hol}({\mathbb {D}}) {\setminus } H^\infty _2$ be as in Theorem 1(i). If the pseudo-hyperbolic distance between any distinct zeros of any non-trivial solution of (1) is uniformly bounded away from zero, then $A\in H^\infty _2$ by [43, Theorem 4]. This is a contradiction, and therefore (i) holds in this particular case.

(ii) The assertion follows from [10, Example 1], since in this example all solutions of (1) are bounded; use (9) to obtain a bounded linearly independent solution.

(iii) Let $f_1\in H^\infty$ be the non-trivial solution of (1) with $f_1(z_1)=0=f_1'(z_2)$. If $\varrho _p(z_1,z_2) > 1/2$, then there is nothing to prove. Otherwise, let $f_2\in H^\infty$ be a solution of (1), which is linearly independent to $f_1$. Since $f_1(z_2)f_2'(z_2) = W(f_1,f_2)$, there exists a constant $0<C_1<\infty$ such that

$$\begin{aligned} \varrho _p(z_1,z_2) \ge \frac{|f_1(z_2)|}{C_1 \, \Vert f_1\Vert _{H^\infty }} = \frac{|W(f_1,f_2)|}{C_1 \, \Vert f_1\Vert _{H^\infty } |f_2'(z_2)|} \ge \frac{|W(f_1,f_2)|(1-|z_2|^2)}{C_1 \, \Vert f_1\Vert _{H^\infty } \Vert f_2\Vert _{H^\infty }} \end{aligned}$$

by (23); note that $\Vert f_2'\Vert _{H^\infty _1}\le \Vert f_2\Vert _{H^\infty }$ by standard estimates. Analogously, since $- f_1'(z_1)f_2(z_1) = W(f_1,f_2)$, there exists another constant $0<C_2<\infty$ such that

$$\begin{aligned} \varrho _p(z_1,z_2) \ge \frac{ |f_1'(z_1)|(1-|z_1|^2)}{C_2 \, \Vert f_1'\Vert _{H^\infty _1}} = \frac{|W(f_1,f_2)|(1-|z_1|^2)}{C_2 \, \Vert f_1'\Vert _{H^\infty _1} |f_2(z_1)|} \ge \frac{|W(f_1,f_2)|(1-|z_1|^2)}{C_2 \, \Vert f_1\Vert _{H^\infty } \Vert f_2\Vert _{H^\infty }}. \end{aligned}$$

Statements (iv) and (v) are proved similarly. In the case of (iv) apply (23) to $f_1,f_2\in H^\infty$, and in the case of (v) apply (23) to $f_1',f_2'\in H^\infty _1$. $\square$

6 Proof of Theorem 5

After the proof of Theorem 5, we consider its relation to conformal metrics of constant curvature. We also discuss an application concerning Carleson measures induced by bounded solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$.

Proof

(of Theorem 5) It is clear that u is sufficiently smooth to be in the class $C^2$.

(i) Since $(f_1/f_2)'=-W(f_1,f_2)/f_2^2$, we deduce

$$\begin{aligned} \left( \frac{f_1}{f_2} \right) ^\# = \frac{|W(f_1,f_2)|}{|f_1|^2+|f_2|^2}, \qquad \partial u = \frac{f_1' \overline{f_1} + f_2' \overline{f_2}}{|f_1|^2+|f_2|^2}. \end{aligned}$$

We compute

$$\begin{aligned} \varDelta u = 4 \, {\overline{\partial }}(\partial u) = 4 \, \frac{|f_1f_2'-f_1'f_2|^2}{(|f_1|^2+|f_2|^2)^2} = 4 \, e^{-2u}. \end{aligned}$$

(ii) As above, we obtain

$$\begin{aligned} \frac{1}{4} \, \varDelta u&= \partial \big ( {\overline{\partial }} u \big ) = \frac{|f_1'|^2+|f_2'|^2}{|f_1|^2+|f_2|^2} - \frac{f_1\overline{f_1'} + f_2 \overline{f_2'}}{|f_1|^2+|f_2|^2} \cdot \frac{f_1'\overline{f_1} + f_2' \overline{f_2}}{|f_1|^2+|f_2|^2}\\&= \frac{|f_1'|^2+|f_2'|^2}{|f_1|^2+|f_2|^2} - \big ( {\overline{\partial }}u \big ) \cdot \big ( \partial u \big ). \end{aligned}$$

Since u is real-valued, $\varDelta u = (\varDelta e^u)/(e^u) - 4 \, |\partial u|^2 = (\varDelta e^u)/(e^u) - |\nabla u|^2$.

(iii) We deduce

$$\begin{aligned} \partial ^2 u = \, \frac{f_1'' {\overline{f}}_1 + f_2'' {\overline{f}}_2}{|f_1|^2+|f_2|^2} - (\partial u)^2 = - \, \frac{A |f_1|^2 + A |f_2|^2}{|f_1|^2+|f_2|^2} - (\partial u)^2 = - A - (\partial u)^2, \end{aligned}$$

which completes the proof. $\square$

Remark 1

Let $f_1$ and $f_2$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$. As in the proof of Theorem 5(i), we deduce that $v= -u = \log \, (f_1/f_2)^{\#}$ is a solution of the Liouville equation $\varDelta v = -4 \, e^{2 v}$. Recall that $\lambda (z)|dz|$ is said to be a conformal metric on ${\mathbb {D}}$ if the conformal density $\lambda : {\mathbb {D}}\rightarrow {\mathbb {R}}$ is strictly positive and continuous. If $\lambda \in C^2$, then $\lambda (z)|dz|$ is called a regular conformal metric on ${\mathbb {D}}$. The (Gauss) curvature $\kappa : {\mathbb {D}}\rightarrow {\mathbb {R}}$ of the regular conformal metric $\lambda (z)|dz|$ is given by $\kappa = - \varDelta (\log \lambda )/\lambda ^2$. In conclusion, $(f_1/f_2)^{\#}(z)|dz|$ defines a regular conformal metric of constant curvature 4 on ${\mathbb {D}}$.

As an application of Theorem 5, we return to consider differential equations with bounded solutions. Theorem 3 shows that, even if $f_1,f_2\in H^\infty$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, it may happen that $f_1/f_2$ is non-normal and $((f_1/f_2)^\#)^2 \log (1/|z|) \, dm(z)$ is not a Carleson measure. The following result and Theorem 5(ii) imply that this Carleson measure condition becomes true if the exponent 2 is replaced by any smaller value.

Theorem 16

Let $f_1,f_2\in H^\infty$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$. Then, $(|f_1'|^2+|f_2'|^2)(|f_1|^2 + |f_2|^2)^{\varepsilon -1} \log (1/|z|) \, dm(z)$ is a Carleson measure for any $0<\varepsilon <\infty$.

Let $\varOmega \subset {\mathbb {R}}^2$ be a domain with smooth boundary, and let $u_1,u_2$ be $C^2$-functions on ${\overline{\varOmega }}$. The classical Green theorem asserts

$$\begin{aligned} \int _{\partial \varOmega } \left( u_1 \, \frac{\partial u_2}{\partial n} - u_2 \, \frac{\partial u_1}{\partial n} \right) ds = \int _{\varOmega } \big ( u_1 \, \varDelta u_2 - u_2 \, \varDelta u_1 \big ) \, dxdy, \end{aligned}$$

(25)

where $\partial /\partial n$ denotes differentiation in the direction of outward pointing normal and ds is the arc length on $\partial \varOmega$. The following argument is based on a modification of Uchiyama’s lemma. We refer to [36, p. 290] and [46, Lemma 2.1] for the original statement. Suppose that $f\in \mathrm{Hol}({\mathbb {D}})$ and $\varphi \in C^2$ is a subharmonic function in ${\mathbb {D}}$. By the theorems of Green and Fubini, we deduce

$$\begin{aligned}&\frac{1}{2\pi } \int _0^{2\pi } e^{\varphi (re^{i\theta })} |f(re^{i\theta })|^2 \, d\theta - e^{\varphi (0)} |f(0)|^2\\&\qquad = \frac{1}{2\pi } \int _{D(0,r)} \varDelta ( e^{\varphi } |f|^2)(z) \log \frac{r}{|z|} \, dm(z) \end{aligned}$$

for any $0<r<1$. Since

$$\begin{aligned} |\zeta _1+\zeta _2|^2 \ge \big (|\zeta _1|-|\zeta _2|\big )^2 \ge \frac{1}{2} \, |\zeta _1|^2 - |\zeta _2|^2, \quad \zeta _1,\zeta _2\in {\mathbb {C}}, \end{aligned}$$

we obtain

$$\begin{aligned} \varDelta ( e^\varphi |f|^2 ) = e^\varphi (\varDelta \varphi ) |f|^2 + 4 \, e^\varphi | (\partial \varphi ) f + f' |^2 \ge e^\varphi \Big ( \varDelta \varphi + 2 \, |\partial \varphi |^2 \Big ) |f|^2 - 4 \, e^\varphi |f'|^2, \end{aligned}$$

and finally

$$\begin{aligned} \begin{aligned}&\frac{1}{2\pi } \int _{D(0,r)} |f(z)|^2 e^{\varphi (z)} \Big ( \varDelta \varphi (z) + 2 \, |\partial \varphi (z)|^2 \Big ) \log \frac{r}{|z|} \, dm(z)\\&\qquad \le \frac{1}{2\pi } \int _0^{2\pi } e^{\varphi (re^{i\theta })} |f(re^{i\theta })|^2 \, d\theta + \frac{2}{\pi } \int _{D(0,r)} e^{\varphi (z)} |f'(z)|^2 \log \frac{r}{|z|} \, dm(z) \end{aligned} \end{aligned}$$

(26)

for any $0<r<1$.

Proof

(of Theorem 16) Let $f_1,f_2\in H^\infty$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$. Without loss of generality, we may assume that $W(f_1,f_2)=1$. We conclude that $\varphi = \varepsilon \, u = \varepsilon \, \log ( |f_1|^2+|f_2|^2 )$ is bounded from above and subharmonic in ${\mathbb {D}}$, as $\varDelta \varphi = 4 \, \varepsilon \, ( (f_1/f_2)^\# )^2\ge 0$ by Theorem 5(i). By the Littlewood-Paley formula [8, Lemma 3.1], we obtain

$$\begin{aligned} \Vert f\Vert _{H^2}^2 = |f(0)|^2 + \frac{2}{\pi } \int _{{\mathbb {D}}} |f'(z)|^2 \log \frac{1}{|z|} \, dm(z), \quad f\in \mathrm{Hol}({\mathbb {D}}), \end{aligned}$$

and therefore a standard convergence argument applied to (26) reveals that

$$\begin{aligned} \begin{aligned}&\frac{1}{2\pi } \!\int _{{\mathbb {D}}} |f(z)|^2 e^{\varphi (z)} \Big ( \varDelta \varphi (z) + 2 \, |\partial \varphi (z)|^2 \Big )\log \frac{1}{|z|} dm(z)\\&\qquad \le 2 \big ( \Vert f_1\Vert _{H^\infty }^2 + \Vert f_2\Vert _{H^\infty }^2 \big )^\varepsilon \Vert f\Vert _{H^2}^2 \end{aligned} \end{aligned}$$

for any $f\in H^2$. This proves that $e^{\varphi (z)} ( \varDelta \varphi (z) + 2 \, |\partial \varphi (z)|^2 ) \log (1/|z|) \, dm(z)$ is a Carleson measure, and therefore by Theorem 5(ii), we deduce

$$\begin{aligned} \begin{aligned} e^\varphi \big ( \varDelta \varphi + 2 \, |\partial \varphi |^2 \big )&= \big ( |f_1|^2+|f_2|^2 \big )^\varepsilon \, \left( \varepsilon \, \varDelta u + \frac{\varepsilon ^2}{2} \, |\nabla u|^2 \right) \\&\ge \min \!\big \{ \varepsilon , \varepsilon ^2/2 \big \} \big ( |f_1|^2+|f_2|^2 \big )^\varepsilon \, \frac{\varDelta e^u}{e^u}. \end{aligned} \end{aligned}$$

This completes the proof of Theorem 16. $\square$

7 Proofs of Theorem 6 and Proposition 1

Recall that the meromorphic function g in the unit disc belongs to the Nevanlinna class ${\mathcal {N}}$ if and only if the Ahlfors-Shimizu characteristic

$$\begin{aligned} T_0(r,g) = \frac{1}{\pi } \, \int _0^r \bigg ( \int _{D(0,t)} g^{\#}(z)^2\, dm(z) \bigg ) \frac{dt}{t} = \frac{1}{\pi } \, \int _{D(0,r)} g^{\#}(z)^2\, \log \frac{r}{|z|} \, dm(z) \end{aligned}$$

is uniformly bounded for $0<r<1$. The equivalence of the representations above follows from Fubini’s theorem.

Let $u\not \equiv -\infty$ be subharmonic in ${\mathbb {D}}$. Function u admits a harmonic majorant in ${\mathbb {D}}$ if and only if $\lim _{r\rightarrow 1^-} \int _0^{2\pi } u(re^{i\theta }) \, d\theta < \infty$, and in this case, the least harmonic majorant for u is

$$\begin{aligned} {\hat{u}}(z) = \lim _{r\rightarrow 1^-} \, \frac{1}{2\pi } \int _0^{2\pi } u(re^{i\theta }) \, \frac{r^2-|z|^2}{|re^{i\theta }-z|^2}\, d\theta < \infty , \quad z\in {\mathbb {D}}. \end{aligned}$$

See [41, Theorem 3.3] for more details. In the proof of Theorem 6 we take advantage of the following well-known fact: If $u\in C^2$ is subharmonic and $\varphi$ is analytic, then $u\circ \varphi$ is subharmonic with $\varDelta (u\circ \varphi ) = ((\varDelta u) \circ \varphi ) \, |\varphi '|^2$.

Proof

(of Theorem 6) (i) By Green’s theorem (25) with $u_1 = 1$, $u_2 = u$, we obtain

$$\begin{aligned} \frac{d}{dt} \int _0^{2\pi } u(te^{i\theta }) \, d\theta = \frac{4}{t} \, \int _{D(0,t)} \big ( (f_1/f_2)^{\#}(z) \big )^2 \, dm(z), \quad 0<t<1, \end{aligned}$$

as $\varDelta u = 4 \, ( (f_1/f_2)^{\#})^2$ by Theorem 5(i). By integrating from 0 to r, we conclude $1/(2\pi ) \int _0^{2\pi } u(re^{i\theta }) \, d\theta = u(0) + 2 \, T_0\big ( r,f_1/f_2 \big )$ for any $0<r<1$. Consequently, u admits a harmonic majorant if and only if $f_1/f_2\in {\mathcal {N}}$.

(ii) Let $a\in {\mathbb {D}}$. By Green’s theorem and Theorem 5(i),

$$\begin{aligned} \begin{aligned}&\frac{1}{2\pi } \int _0^{2\pi } u\big (a+(1-|a|) \,re^{i\theta }\big ) \, d\theta - u(a)\\&\qquad = \frac{2}{\pi } \, \int _0^r \bigg ( \int _{D(a,t(1-|a|))} (f_1/f_2)^{\#}(z)^2\, dm(z) \bigg ) \frac{dt}{t}, \quad 0<r<1. \end{aligned} \end{aligned}$$

By letting $r\rightarrow 1^-$, we deduce

$$\begin{aligned} \sup _{a\in {\mathbb {D}}} \, \widehat{u_a}(0) = \sup _{a\in {\mathbb {D}}}\, \frac{2}{\pi } \, \int _0^1 \bigg ( \int _{D(a,t(1-|a|))} (f_1/f_2)^{\#}(z)^2\, dm(z) \bigg ) \frac{dt}{t}. \end{aligned}$$

(27)

This completes the proof of (ii), as $f_1/f_2$ is a normal function in the Nevanlinna class if and only if the right-hand side of (27) is finite [38, Theorem 1].

(iii) The assertion is in some sense a meromorphic counterpart of [47, Theorem 5.1]. Fix $a\in {\mathbb {D}}$, and take r to be sufficiently large to satisfy $|a|<r<1$. Define $\psi (z) = r \, \varphi _{a/r}(z/r)$, $z\in {\mathbb {D}}$. By Green’s theorem,

$$\begin{aligned} \frac{1}{2\pi } \int _0^{2\pi } u(re^{i\theta }) \, \frac{r^2 - |a|^2}{|re^{i\theta }-a|^2}\, d\theta - u(a)&= \frac{1}{2\pi } \int _0^{2\pi } (u \circ \psi )(re^{it}) \, dt - (u\circ \psi )(0)\\&=\frac{1}{2\pi } \int _{D(0,r)} \varDelta u(z) \log \frac{1}{| \varphi _{a/r}(z/r) |} \, dm(z). \end{aligned}$$

By using standard estimates and letting $r\rightarrow 1^-$, we conclude that ${\hat{u}}(a) - u(a) \asymp \int _{{\mathbb {D}}} \varDelta u(z) ( 1 - |\varphi _{a}(z)|^2 ) \, dm(z)$, where the comparison constants are independent of $a\in {\mathbb {D}}$. Theorem 5(i) implies

$$\begin{aligned} \sup _{a\in {\mathbb {D}}} \big ( {\hat{u}}(a) - u(a) \big ) \asymp \, \sup _{a\in {\mathbb {D}}} \,\int _{{\mathbb {D}}} \big ( (f_1/f_2)^\#(z) \big )^2 (1-|z|^2) \, \frac{1-|a|^2}{|1-{\overline{a}}z|^2} \, dm(z). \end{aligned}$$

The part (iii) follows as $f_1/f_2\in \mathrm {UBC}$ if and only if $( (f_1/f_2)^\#(z))^2 (1-|z|^2)\, dm(z)$ is a Carleson measure [38, Theorem 3].

The proofs of (iv)-(vi) are straight-forward and hence omitted. Note that the function $e^u = (|f_1|^2+|f_2|^2)/|W(f_1,f_2)|$ is subharmonic in ${\mathbb {D}}$. $\square$

It is well-known that non-trivial solutions of a Blaschke-oscillatory equation (1), $A\in \mathrm{Hol}({\mathbb {D}})$, may lie outside the Nevanlinna class ${\mathcal {N}}$ [22, Section 4.3]. In the following remark, we deduce an estimate according to which the Nevanlinna characteristic of solutions of Blaschke-oscillatory equations cannot grow arbitrarily fast.

Remark 2

Let $f_1$ be a non-trivial solution of a Blaschke-oscillatory equation (1) for $A\in \mathrm{Hol}({\mathbb {D}})$. Let $f_2$ be another solution of (1), which is linearly independent to $f_1$. Note that $f_2/f_1 \in {\mathcal {N}}$ by [22, Lemma 3], and $(f_2/f_1)'=W(f_1,f_2)/f_1^2$ by straight-forward computation. Kennedy’s estimate [25, Theorem 1] implies

$$\begin{aligned} S = \int _0^1 (1-r) e^{2 \, T(r,(f_2/f_1)')} \, dr < \infty . \end{aligned}$$

(28)

Nevanlinna’s first theorem shows that (28) remains to be true, if $T(r,(f_2/f_1)')$ is replaced by $2 \, T(r,f_1)$. This places a severe restriction for the growth of $T(r,f_1)$ as $r\rightarrow 1^-$. It implies $T(r,f_1) \le (1/2) \log (\sqrt{2S}/(1-r))$ for all $0<r<1$. Therefore all solutions of (1) are non-admissible [22, p. 53].

Proof

(of Proposition 1) Recall that $(f_1/f_2)^\#= |W(f_1,f_2)|/(|f_1|^2+|f_2|^2)$. Now

$$\begin{aligned}&\int _{{\mathbb {D}}} \big ( ( f_1/f_2 )^\# \big )^2 (1-|z|^2)\, dm(z)\\&\qquad \le \frac{|W(f_1,f_2)|^2}{\delta ^2} \int _{\{z\in {\mathbb {D}}\; \! : \; \! |f_1(z)|^2+|f_2(z)|^2 \ge \delta \}} (1-|z|^2)\, dm(z)\\&\qquad \qquad + \left( \, \sup _{z\in {\mathbb {D}}} \, ( f_1/f_2 )^\#(z)^2 (1-|z|^2)^2 \right) \int _{\{z\in {\mathbb {D}}\; \! : \; \! |f_1(z)|^2+|f_2(z)|^2 < \delta \}} \frac{dm(z)}{1-|z|^2}. \end{aligned}$$

Therefore $f_1/f_2$ belongs to the Nevanlinna class by the assumption. $\square$

We briefly consider two applications of Proposition 1. Suppose that $f_1,f_2$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$ and assume that (12) holds for some Blaschke sequence $\{z_n\}\subset {\mathbb {D}}$ and $0<\delta <1$. Denote this infimum by $0<s<\infty$. We deduce

$$\begin{aligned} \int _{\{z\in {\mathbb {D}}\; \! : \; \! |f_1(z)|^2+|f_2(z)|^2< s^2/2\}} \frac{dm(z)}{1-|z|^2}&\le \sum _n \int _{\varDelta _p(z_n,\delta )} \frac{dm(z)}{1-|z|^2}\\&\asymp \sum _n (1-|z_n|) < \infty , \end{aligned}$$

where the pseudo-hyperbolic discs $\varDelta _p(z_n,\delta )$ are not necessarily pairwise disjoint. In such a case the normality of $f_1/f_2$ implies that $f_1/f_2\in {\mathcal {N}}$ by Proposition 1.

The same conclusion is obtained if $f_1/f_2$ is normal and $|f_1|+|f_2|$ is uniformly bounded from below for all points in ${\mathbb {D}}$ which lie outside a horodisc (that is, a disc internally tangent to $\partial {\mathbb {D}}$). The details are omitted.

8 Proof of Theorem 7

We begin with a lemma, which is needed in the proof of Theorem 7. This auxiliary result is based on the well-known Harnack inequalities: if $h\in \mathrm{Har^+}({\mathbb {D}})$, then

$$\begin{aligned} \frac{1-\varrho _p(z,w)}{1+\varrho _p(z,w)} \le \frac{h(z)}{h(w)} \le \frac{1+\varrho _p(z,w)}{1-\varrho _p(z,w)}, \quad z,w\in {\mathbb {D}}. \end{aligned}$$

Let $f\in \mathrm{Hol}({\mathbb {D}})$ and recall that $f\in {\mathcal {N}}$ if and only if there exists $h \in \mathrm{Har^+}({\mathbb {D}})$ such that $\log ^+ |f| \le h$, which is equivalent to the fact $|f| \le e^h$. There is no reason to expect that any order derivative of f would belong to ${\mathcal {N}}$. However, for every $k\in {\mathbb {N}}$, there exists a constant $C=C(k)$ with $0<C<\infty$ such that

$$\begin{aligned} |f^{(k)}(z)| (1-|z|^2)^k \le e^{C\, h(z)}, \quad z\in {\mathbb {D}}, \end{aligned}$$

(29)

by Cauchy’s integral formula and Harnack’s inequality. See [19, Lemma 2.1].

Lemma 3

Suppose that $f\in \mathrm{Hol}({\mathbb {D}})$ and it satisfies $|f(z)| (1-|z|^2)^k \le e^{h(z)}$, $z\in {\mathbb {D}}$, for $k\in {\mathbb {N}}\cup \{0\}$ and $h\in \mathrm{Har^+}({\mathbb {D}})$. If f vanishes on a sequence $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$, then there exists $H\in \mathrm{Har^+}({\mathbb {D}})$ such that $|f(z)| (1-|z|^2)^k \le \varrho _p(\varLambda , z) \, e^{H(z)}$, $z\in {\mathbb {D}}$.

Proof

Consider a dyadic partition of ${\mathbb {D}}$ into Whitney squares of the type

$$\begin{aligned} Q =Q_I = \big \{ z\in {\mathbb {D}}: 1-|I|/(2\pi ) \le |z| < 1, \, \arg {z}\in I \big \} \end{aligned}$$

where $\ell (Q)=|I|$ is the arc-length of the interval $I\subset \partial {\mathbb {D}}$. The top part of Q is $T(Q) = \{ z\in Q : 1-\ell (Q)/(2\pi ) \le |z| \le 1-\ell (Q)/(4\pi ) \}$.

Let Q be any Whitney square in the dyadic partition. Let $\varOmega _1\subset {\mathbb {D}}$ such that

$$\begin{aligned} T(Q) \subset \varOmega _1, \quad \varrho _p\big ( \partial \varOmega _1, \partial T(Q)\big ) = {{\,\mathrm{diam}\,}}_p\!\big ( T(Q) \big ), \end{aligned}$$

and let $\varOmega _2$ be another set such that $\varOmega _1 \subset \varOmega _2 \subset {\mathbb {D}}$ and $\varrho _p( \partial \varOmega _2, \partial \varOmega _1 ) = 4 {{\,\mathrm{diam}\,}}_p\varOmega _1$. Here ${{\,\mathrm{diam}\,}}_p$ denotes the pseudo-hyperbolic diameter. Define $g\in {\mathcal {H}}({\mathbb {D}})$ by

$$\begin{aligned} g(z) = f(z) \Bigg ( \, \prod _{z_k \in \varLambda \, \cap \, \varOmega _1} \frac{z_k - z}{1-{\overline{z}}_k z} \Bigg )^{-1}, \quad z\in {\mathbb {D}}. \end{aligned}$$

We may assume that $\varLambda \cap \varOmega _1$ is not empty, for otherwise the assertion follows for all $z\in T(Q)$ by trivial reasons. Fix any $z_n \in \varLambda \, \cap \, \varOmega _1$. We deduce

$$\begin{aligned} |g(\zeta )| \le \frac{(1-|\zeta |^2)^{-k} e^{h(\zeta )}}{\varrho _p(z_n,\zeta )} \Bigg ( \, \prod _{z_k \in \varLambda \, \cap \, \varOmega _1 \, : \, z_k\ne z_n} \left| \frac{z_k - \zeta }{1-{\overline{z}}_k \zeta } \right| \Bigg )^{-1}, \quad \zeta \in \partial \varOmega _2. \end{aligned}$$

Since $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$, [18, Theorem 1.2] implies that there exists $h_1\in \mathrm{Har^+}({\mathbb {D}})$ with

$$\begin{aligned} |g(\zeta )| \lesssim (1-|\zeta |^2)^{-k} e^{h(\zeta )+h_1(z_n)} \lesssim (1-|z_n|^2)^{-k} e^{(Ch+h_1)(z_n)}, \quad \zeta \in \partial \varOmega _2, \end{aligned}$$

where $0<C<\infty$ is a universal constant by Harnack’s inequalities. The maximum modulus principle extends this estimate for all $z\in \varOmega _2$, and therefore

$$\begin{aligned} |f(z)| \le |g(z)| \prod _{z_k \in \varLambda \, \cap \, \varOmega _1} \left| \frac{z_k - z}{1-{\overline{z}}_k z} \right| \lesssim (1-|z_n|^2)^{-k} e^{(Ch+h_1)(z_n)} \, \varrho _p(\varLambda ,z), \end{aligned}$$

for $z\in T(Q)$. By Harnack’s inequalities, there exists $H\in \mathrm{Har^+}({\mathbb {D}})$ such that the assertion holds for all $z\in T(Q)$. Since the argument is independent of the Whitney square Q, the proof is complete. $\square$

Proof

(of Theorem 7) Let $B=B_\varLambda$ be the Blaschke product with zeros $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$ and let $f=Be^{Bg}$, where $g\in \mathrm{Hol}({\mathbb {D}})$ is a solution of the interpolation problem

$$\begin{aligned} g(z_n) = w_n, \quad w_n = - \frac{B''(z_n)}{2 \big (B'(z_n)\big )^2}, \quad z_n\in \varLambda . \end{aligned}$$

(30)

As $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$, [18, Theorem 1.2] implies that there exists $h_1\in \mathrm{Har^+}({\mathbb {D}})$ with

$$\begin{aligned} |B'(z_n)| (1-|z_n|^2) = \prod _{z_k \in \varLambda \, : \, z_k\ne z_n} \left| \frac{z_k - z_n}{1-{\overline{z}}_k z_n} \right| \ge e^{-h_1(z_n)}, \quad z_n\in \varLambda . \end{aligned}$$

(31)

Since there exists a constant $0<C<\infty$ such that

$$\begin{aligned} \log ^+ |w_n| = \log ^+ \left| \frac{B''(z_n)}{2 \big (B'(z_n)\big )^2} \right| \le C + 2\, h_1(z_n), \quad z_n\in {\mathbb {D}}, \end{aligned}$$

[18, Theorem 1.2] ensures that $\{w_n\} \in {\mathcal {N}}\mid \varLambda$. Therefore we may assume $g\in {\mathcal {N}}$.

By straight-forward computation, f is a solution of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, where

$$\begin{aligned} A = - \frac{f''}{f} = - \frac{B'' + 2 B' (B'g+Bg')}{B} - {\big ( (Bg)' \big )}^2-(Bg)''. \end{aligned}$$

(32)

The interpolation property (30) guarantees that every point $z_n\in \varLambda$ is a removable singularity for A. It remains to show that there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that $|A(z)|(1-|z|^2)^2 \le e^{h(z)}$, $z\in {\mathbb {D}}$. Since $Bg \in {\mathcal {N}}$, (29) implies that the two right-most terms in (32) are of the desired type. Since $B'' + 2 B' (B'g+Bg')$ vanishes on the sequence $\varLambda$, Lemma 3 shows that there exists $h_2\in \mathrm{Har^+}({\mathbb {D}})$ such that

$$\begin{aligned} \big |B''(z) + 2 B'(z) \big (B'(z)g(z)+B(z)g'(z)\big )\big | (1-|z|^2)^{2} \le \varrho _p(\varLambda , z) \, e^{h_2(z)}, \quad z\in {\mathbb {D}}. \end{aligned}$$

And finally, by [19, Theorem 1.2], there exists $h_3\in \mathrm{Har^+}({\mathbb {D}})$ such that $|B(z)| \ge \varrho _p(\varLambda , z) e^{-h_3(z)}$, $z\in {\mathbb {D}}$. We deduce Theorem 7 by combining the estimates. $\square$

9 Proof of Theorem 8

The following result is an analogue of Carleson’s [2, Theorem 2], which characterizes those cases in which the classical 0, 1-interpolation is possible. The proof of Proposition 4 is based on the Nevanlinna corona theorem by Mortini [33, Satz 4]: Given $f_1,f_2\in {\mathcal {N}}$, the Bézout equation $f_1g_2+f_2g_2 = 1$ can be solved with functions $g_1,g_2\in {\mathcal {N}}$ if and only if there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that $|f_1(z)|+|f_2(z)|\ge e^{-h(z)}$, $z\in {\mathbb {D}}$.

Proposition 4

Let $\{z_n\},\{\zeta _n\}$ be Blaschke sequences. Then, there exists $f\in {\mathcal {N}}$ such that $f(z_n)=0$ and $f(\zeta _n)=1$ for all n if and only if there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that (6) holds.

Proof

Assume that there exists $f\in {\mathcal {N}}$ such that $f(z_n)=0$ and $f(\zeta _n)=1$ for all n. By the classical factorization theorem, there exist functions $g_1,g_2\in {\mathcal {N}}$ such that $f=B_{\{z_n\}} g_1 = 1+B_{\{\zeta _n\}}g_2$. Here $B_{\{z_n\}}$ and $B_{\{\zeta _n\}}$ are Blaschke products with zeros $\{z_n\}$ and $\{\zeta _n\}$, respectively. As $g_1,g_2\in {\mathcal {N}}$, there exist $h_1,h_2\in \mathrm{Har^+}({\mathbb {D}})$ such that $|g_1| \le e^{h_1}$ and $|g_2| \le e^{h_2}$. We deduce

$$\begin{aligned} 1 = \big | B_{\{z_n\}} g_1 - B_{\{\zeta _n\}}g_2 \big | \le e^{h_1+h_2} \big ( | B_{\{z_n\}} | + | B_{\{\zeta _n\}} |\big ), \end{aligned}$$

which proves the first part of the assertion.

Assume that there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that (6) holds. By the Nevanlinna corona theorem, there exist $g_1,g_2\in {\mathcal {N}}$ such that $B_{\{ z_n \}} g_1 + B_{\{\zeta _n\}} g_2=1$. Then, the function $f=B_{\{ z_n \}} g_1 \in {\mathcal {N}}$ satisfies the desired 0, 1-interpolation. $\square$

Proof

(of Theorem 8) By Proposition 4, there exists $g\in {\mathcal {N}}$ such that $g(z_n)=0$ and $g(\zeta _n)=1$ for all n. Now $f(z)=\exp (\log \alpha + g(z) \log (\beta /\alpha ))$, $z\in {\mathbb {D}}$, satisfies the desired interpolation property, and is a zero-free solution of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$,

$$\begin{aligned} A(z) = - \frac{f''(z)}{f(z)} = - \left( g'(z) \log \frac{\beta }{\alpha } \right) ^2 - g''(z) \log \frac{\beta }{\alpha }, \quad z\in {\mathbb {D}}. \end{aligned}$$

By (29), there exists $H\in \mathrm{Har^+}({\mathbb {D}})$ such that $|A(z)|(1-|z|^2)^2 \le e^{H(z)}$, $z\in {\mathbb {D}}$. $\square$

10 Proofs of Theorems 9 and 10, and Proposition 2

The following proof proceeds along the same lines as that in [44, p. 129].

Proof

(of Theorem 9) If $f_1,f_2$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, then

$$\begin{aligned} W(f_1,f_2)\, A = f_1'f_2''-f_1''f_2', \qquad (f_1/f_2)^\# = |W(f_1,f_2)|/(|f_1|^2+|f_2|^2). \end{aligned}$$

Since $W(f_1,f_2)$ is a non-zero complex constant, the estimate (29) and the fact $f_1,f_2\in {\mathcal {N}}$ imply that there exists $h_1\in \mathrm{Har^+}({\mathbb {D}})$ such that $|A(z)|(1-|z|^2)^3 \le e^{h_1(z)}$, $z\in {\mathbb {D}}$. Moreover, the Cauchy-Schwarz inequality (18) and the estimate (29) show that there exists $h_2\in \mathrm{Har^+}({\mathbb {D}})$ such that $(f_1/f_2)^{\#}(z)(1-|z|^2)^2 \le e^{h_2(z)}$, $z\in {\mathbb {D}}$. The claim follows by choosing $H=h_1+h_2 \in \mathrm{Har^+}({\mathbb {D}})$. $\square$

The proof of Theorem 10 is analogous to that proof of Theorem 2, which is presented in the end of Sect. 4.

Proof

(of Theorem 10) By (7) and [19, Theorem 1], the ideal $I_{{\mathcal {N}}}(f_1,f_2)$ contains a Blaschke product B whose zero-sequence belongs to ${{\,\mathrm{Int}\,}}{\mathcal {N}}$. This is equivalent to the fact that there exist functions $g_1,g_2\in {\mathcal {N}}$ such that $f_1 g_1 + f_2 g_2 = B$. Differentiate $f_1 g_1 + f_2 g_2 = B$ twice, and apply (1) to $f_1''$ and $f_2''$ to obtain (17). Note that $A\in \mathrm{Hol}({\mathbb {D}})$ by assumption. As in the proof of Theorem 7, we conclude that there exists $H\in \mathrm{Har^+}({\mathbb {D}})$ such that $\sup _{z\in {\mathbb {D}}} |A(z)|(1-|z|^2)^2 \le e^{H(z)}$, $z\in {\mathbb {D}}$. $\square$

Proof

(of Proposition 2) Proposition 2 follows directly from [3, Theorem 15] if $\psi : {\mathbb {D}}\rightarrow (0,1/2)$ given by $\psi (z) = e^{-H(z)/2}e^{-1}$, $z\in {\mathbb {D}}$ and $H\in \mathrm{Har^+}({\mathbb {D}})$, satisfies $\sup _{a,z\in {\mathbb {D}}} \, \psi (a)/\psi \big ( \varphi _a(\psi (a)z) \big ) < \infty$. Now

$$\begin{aligned}&\sup _{a,z\in {\mathbb {D}}} \,\exp \!\left( \frac{H(a)}{2} \left( \frac{H\big ( \varphi _a(e^{-H(a)/2}e^{-1}z)\big )}{H(\varphi _a(0))} - 1 \right) \right) \\&\qquad \le \sup _{a,z\in {\mathbb {D}}} \,\exp \!\left( \frac{H(a)}{2} \left( \frac{1+\varrho _p\big (0, e^{-H(a)/2}e^{-1}z\big )}{1-\varrho _p\big (0,e^{-H(a)/2}e^{-1}z\big )} - 1 \right) \right) \end{aligned}$$

by Harnack’s inequalities. This is bounded by

$$\begin{aligned} \sup _{0\le x< \infty } \exp \!\left( \frac{x}{2} \left( \frac{1+e^{-x/2}e^{-1}}{1-e^{-x/2}e^{-1}} - 1 \right) \right) < \frac{3}{2}, \end{aligned}$$

which implies the assertion. $\square$

10.1 Separation of zeros and critical points

We proceed to state an analogue of Proposition 3. If $f\in \mathrm{Hol}({\mathbb {D}})$ and

$$\begin{aligned} \Vert f\Vert = \sup _{z\in {\mathbb {D}}} \, |f(z)| (1-|z|^2)^\alpha e^{-h(z)} < \infty \end{aligned}$$

(33)

for $0\le \alpha < \infty$ and $h\in \mathrm{Har^+}({\mathbb {D}})$, then there exists $C=C(\alpha )>0$ such that

$$\begin{aligned} \begin{aligned} \Big | |f(z_1)|(1-|z_1|^2)^\alpha e^{-h(z_1)} - |f(z_2)|(1-|z_2|^2)^\alpha e^{-h(z_2)} \Big | \le C \, \varrho _p(z_1,z_2) \, \Vert f\Vert , \end{aligned} \end{aligned}$$

for all points $z_1,z_2\in {\mathbb {D}}$ with $\varrho _p(z_1,z_2)\le 1/2$. This estimate follows immediately from (23): If $f\in \mathrm{Hol}({\mathbb {D}})$ satisfies (33) for $0\le \alpha < \infty$ and $h\in \mathrm{Har^+}({\mathbb {D}})$, then (23) can be applied to $f e^{-h - i h^\star } \in H^\infty _\alpha$, where $h^\star$ is a harmonic conjugate of h.

Proposition 5

Let $f_1,f_2\in {\mathcal {N}}$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$.

(i):

If $z_1\in {\mathbb {D}}$ is a zero and $z_2\in {\mathbb {D}}$ is a critical point of $f_1$, then there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that

$$\begin{aligned} \varrho _p(z_1,z_2) \gtrsim \max \Big \{ (1-|z_1|)e^{-h(z_1)}, (1-|z_2|)e^{-h(z_2)} \Big \}. \end{aligned}$$

(34)

(ii):

If $z_1\in {\mathbb {D}}$ is a zero of $f_1$, and $z_2\in {\mathbb {D}}$ is a zero of $f_2$, then there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that (34) holds.

(iii):

If $z_1\in {\mathbb {D}}$ is a critical point of $f_1$, and $z_2\in {\mathbb {D}}$ is a critical point of $f_2$, then there exists $h\in \mathrm{Har^+}({\mathbb {D}})$ such that (34) holds.

The proof of Proposition 5 is omitted.

11 Proofs of Theorem 11 and Corollary 2

The proof of Theorem 11 is based on a smoothness property, which is considered first. Let $\omega$ be a radial weight on ${\mathbb {D}}$. Then,

$$\begin{aligned} \varrho _\omega (z_1,z_2) = \int _{\langle z_1, z_2 \rangle } \frac{|dz|}{\omega (z)}, \quad z_1,z_2\in {\mathbb {D}}, \end{aligned}$$

defines a distance function. Here, we integrate along the hyperbolic segment $\langle z_1,z_2 \rangle$ between the points $z_1,z_2\in {\mathbb {D}}$, where the hyperbolic segment is a closed subset of the corresponding hyperbolic geodesic. For $\omega (z)=1-|z|^2$, $z\in {\mathbb {D}}$, the function $\varrho _\omega$ reduces to the standard hyperbolic distance $\varrho _h$:

$$\begin{aligned} \varrho _h(z_1,z_2) = \frac{1}{2} \log \frac{1+\varrho _p(z_1,z_2)}{1-\varrho _p(z_1,z_2)}, \quad \varrho _p(z_1,z_2)=\left| \frac{z_2-z_1}{1-{\overline{z}}_2z_1}\right| , \quad z_1,z_2\in {\mathbb {D}}. \end{aligned}$$

Lemma 4

Let $f_1,f_2$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, and define $u = -\log \, (f_1/f_2)^{\#}$. Let $\omega$ be a radial weight. If

$$\begin{aligned} \sup _{z\in {\mathbb {D}}} \, |\nabla u(z)| \, \omega (z) \le \varLambda < \infty , \end{aligned}$$

(35)

then

$$\begin{aligned} e^{-\varLambda \varrho _\omega (z_1,z_2)} \le \frac{|f_1(z_1)|^2+|f_2(z_1)|^2}{|f_1(z_2)|^2+|f_2(z_2)|^2} \le e^{ \varLambda \varrho _\omega (z_1,z_2)}, \quad z_1,z_2\in {\mathbb {D}}. \end{aligned}$$

(36)

Conversely, if (36) holds for some constant $0<\varLambda <\infty$, then (35) is satisfied.

Proof

Assume that (35) holds. Let $z_1,z_2\in {\mathbb {D}}$ be distinct points, and let $\gamma =\gamma (t)$, $0\le t \le 1$, be a parametrization of $\langle z_1,z_2 \rangle$. Schwarz’s inequality and (35) imply

$$\begin{aligned} \left| \log \frac{|f_1(z_1)|^2+|f_2(z_1)|^2}{|f_1(z_2)|^2+|f_2(z_2)|^2} \right|&= \big | u(z_1) - u(z_2) \big | \le \left| \int _0^1 \nabla u (\gamma (t)) \cdot \gamma '(t) \, dt \right| \\&\le \int _0^1 | \nabla u (\gamma (t)) | \, |\gamma '(t) | \, dt \le \varLambda \, \varrho _\omega (z_1,z_2). \end{aligned}$$

From this estimate we deduce (36).

Assume that (36) holds for some constant $0<\varLambda <\infty$. Fix $z_2\in {\mathbb {D}}$. Since

$$\begin{aligned} \lim _{z_1\rightarrow z_2} \frac{|z_1-z_2|}{\varrho _h(z_1,z_2)} = \lim _{z_1\rightarrow z_2} \frac{\varrho _p(z_1,z_2)}{\frac{1}{2} \log \frac{1+\varrho _p(z_1,z_2)}{1-\varrho _p(z_1,z_2)}} \cdot |1-{\overline{z}}_1z_2| = 1-|z_2|^2, \end{aligned}$$

and

$$\begin{aligned} \frac{|z_1-z_2|}{\varrho _h(z_1,z_2)} \cdot \frac{1}{\max _{z\in \langle z_1,z_2 \rangle } \frac{1-|z|^2}{\omega (z)}} \le \frac{|z_1-z_2|}{\varrho _\omega (z_1,z_2)} \le \frac{|z_1-z_2|}{\varrho _h(z_1,z_2)} \cdot \frac{1}{\min _{z\in \langle z_1,z_2 \rangle } \frac{1-|z|^2}{\omega (z)}} \end{aligned}$$

for any $z_1\in {\mathbb {D}}$, we conclude that $\lim _{z_1\rightarrow z_2} |z_1-z_2|/\varrho _\omega (z_1,z_2) = \omega (z_2)$ by the continuity of $\omega$. Therefore,

$$\begin{aligned} \left| \nabla u(z_2) \right| \omega (z_2) = \lim _{z_1\rightarrow z_2} \left| \frac{u(z_1) - u(z_2)}{z_1-z_2} \right| \frac{|z_1-z_2|}{\varrho _\omega (z_1,z_2)} \le \lim _{z_1\rightarrow z_2} \frac{\varLambda \, \varrho _\omega (z_1,z_2)}{\varrho _\omega (z_1,z_2)} = \varLambda . \end{aligned}$$

This completes the proof of Lemma 4. $\square$

The following lemma is important for our cause due to the representation (13).

Lemma 5

Let $f_1,f_2$ be linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, and define $u = -\log \, (f_1/f_2)^{\#}$. Suppose that $\omega$ is a regular weight which satisfies $\sup _{z\in {\mathbb {D}}}\, \omega (z)/(1-|z|) < \infty$. If $|\nabla u| \in L^\infty _\omega$, then

$$\begin{aligned} \frac{|f_1^{(j)}|+|f_2^{(j)}|}{|f_1| + |f_2|} \in L^\infty _{\omega ^j}, \quad j\in {\mathbb {N}}. \end{aligned}$$

Proof

By the assumption, there exists a positive constant c such that the discs ${\mathfrak {D}}(z)= D(z,c \,\omega (z))$ satisfy ${\mathfrak {D}}(z) \subset D(z,(1-|z|)/2)$, $z\in {\mathbb {D}}$. Let $\zeta \in \partial {\mathfrak {D}}(z)$. Since $\langle z, \zeta \rangle \subset D(z,(1-|z|)/2)$, a straight-forward argument based on (8) reveals

$$\begin{aligned} \varrho _\omega (z,\zeta ) \lesssim \frac{1-|z|^2}{\omega (z)} \, \varrho _h(z,\zeta ) \lesssim \frac{1-|z|^2}{\omega (z)} \, \varrho _p(z,\zeta ) \lesssim \frac{|z-\zeta |}{\omega (z)} = c. \end{aligned}$$

Therefore $\sup _{z\in {\mathbb {D}}} \max _{\zeta \in \partial {\mathfrak {D}}(z)} \varrho _\omega (z,\zeta ) < \infty$.

By Cauchy’s integral formula,

$$\begin{aligned} \begin{aligned} |f_1^{(j)}(z)|+|f_2^{(j)}(z)|&\le 2 \, \max \big \{ |f_1^{(j)}(z)|, |f_2^{(j)}(z)| \big \} \\&\le \left( \, \max _{\zeta \in \partial {\mathfrak {D}}(z)} \big ( |f_1(\zeta )| + |f_2(\zeta )| \big ) \right) \, \frac{2 j!}{c^j \, \omega (z)^j}, \quad z\in {\mathbb {D}}. \end{aligned} \end{aligned}$$

(37)

Now (37) and Lemma 4 imply

$$\begin{aligned} \frac{|f_1^{(j)}(z)|+|f_2^{(j)}(z)|}{|f_1(z)| + |f_2(z)|}&\le \frac{2 j! \sqrt{2}}{c^j \, \omega (z)^j} \!\left( \max _{\zeta \in \partial {\mathfrak {D}}(z)} \frac{|f_1(\zeta )|^2 + |f_2(\zeta )|^2}{|f_1(z)|^2 + |f_2(z)|^2} \right) ^{1/2}\\&\le \frac{2 j! \sqrt{2}}{c^j \, \omega (z)^j} \, \exp \left( \frac{\Vert |\nabla u| \Vert _{L^\infty _\omega }}{2} \max _{\zeta \in \partial {\mathfrak {D}}(z)} \, \varrho _\omega (z,\zeta ) \right) \lesssim \frac{1}{\omega (z)^j} \end{aligned}$$

for $z\in {\mathbb {D}}$. The assertion of Lemma 5 follows. $\square$

Finally, proceed to prove Theorem 11. We take advantage of Yamashita’s [48, Corollary to Theorem 2, p. 161], which uses the following notation. For a meromorphic function f and $z\in {\mathbb {D}}$, let $\rho (z,f)$ be the maximum of $0<r\le 1$ such that f is univalent in $\varDelta _p(z,r)$, and let $\rho _a(z,f)$ be the maximum of $0<r\le 1$ such that $f(w)\ne -1/\overline{f(z)}$ for all $w\in \varDelta _p(z,r)$. Note that $-1/\overline{f(z)}$ is the antipodal point of f(z) in the Riemann sphere.

Proof

(of Theorem 11) First, assume that $|\nabla u|\in L^\infty _\omega$. By the representation (13) and Lemma 5, we conclude that $A\in H^\infty _{\omega ^2}$. By Theorem 5(i) and (ii),

$$\begin{aligned} 4 \, \Big ( \big ( f_1/f_2\big )^\# \Big )^2 = \varDelta u \le \frac{\varDelta e^u}{e^u} = \frac{|f_1'|^2+|f_2'|^2}{|f_1|^2+|f_2|^2} \le 2 \left( \frac{|f_1'|+|f_2'|}{|f_1|+|f_2|} \right) ^2, \end{aligned}$$

and therefore $(f_1/f_2)^\# \in L^\infty _\omega$ by Lemma 5.

Second, let $A \in H^\infty _{\omega ^2}$ and $(f_1/f_2)^\# \in L^\infty _\omega$. Since $f_1/f_2$ is meromorphic in ${\mathbb {D}}$ and has zero-free spherical derivative, Yamashita’s [48, Corollary to Theorem 2, p. 161] implies

$$\begin{aligned} (1-|z|^2) \left| \frac{-{\overline{z}}}{1-|z|^2} - \partial u(z) \right| \le \frac{2}{\min \{ \rho (z,f_1/f_2), \rho _a(z,f_1/f_2) \}}, \quad z\in {\mathbb {D}}. \end{aligned}$$

We deduce

$$\begin{aligned} | \nabla u(z) | \le \frac{2}{1-|z|^2} \left( 1 + \frac{2}{\min \big \{ \rho (z,f_1/f_2), \rho _a(z,f_1/f_2) \big \}} \right) , \quad z\in {\mathbb {D}}. \end{aligned}$$

Denote $h=f_1/f_2$. It suffices to show that both $\rho (z,h)$ and $\rho _a(z,h)$ are bounded from below by a constant multiple of $\omega (z)/(1-|z|^2)$ as $|z|\rightarrow 1^-$.

Let $\psi : {\mathbb {D}}\rightarrow (0,\infty )$ be the weight $\psi (z)=c \, \omega (z)/(1-|z|^2)$, where $0<c<1$ is a sufficiently small constant whose value is determined later. By the assumption, we may assume that $\psi : {\mathbb {D}}\rightarrow (0,1/2)$ and therefore $\varphi _a(\psi (a)z) \in \varDelta _p(a,1/2)$ for all $a,z\in {\mathbb {D}}$. By (8) and standard estimates,

$$\begin{aligned} \sup _{a,z\in {\mathbb {D}}} \, \frac{\psi (a)}{\psi \big (\varphi _a(\psi (a)z)\big )} = \sup _{a,z\in {\mathbb {D}}} \, \frac{\omega (a)}{\omega \big (\varphi _a(\psi (a)z)\big )} \cdot \frac{1-|\varphi _a(\psi (a)z)|^2}{1-|a|^2} <\infty . \end{aligned}$$

Function h is locally univalent and meromorphic, and its Schwarzian derivative satisfies $S_h= 2A$. Let $g_a(z) = (h \circ \varphi _a)(\psi (a) z)$ for $a,z\in {\mathbb {D}}$. By the chain rule,

$$\begin{aligned} |S_{g_a}(z)|&= \big | S_h\big ( \varphi _a(\psi (a)z) \big ) \big | \, \big | \varphi _a'\big ( \psi (a) z\big )\big |^2 \, \psi (a)^2\\&\le \frac{2 \, \Vert A\Vert _{H^\infty _{\omega ^2}}}{\omega \big ( \varphi _a(\psi (a)z) \big )^2} \, \frac{\big (1-| \varphi _a( \psi (a) z )|^2 \big )^2}{\big (1-|\psi (a)z|^2\big )^2}\, \frac{c^2 \omega (a)^2}{(1-|a|^2)^2}, \quad a,z\in {\mathbb {D}}. \end{aligned}$$

We deduce that $\Vert S_{g_a}\Vert _{H^\infty } \le \pi ^2/2$ for any $a\in {\mathbb {D}}$, provided that $0<c<1$ is sufficiently small. Therefore $g_a$ is univalent in the unit disc [34, Theorem II] for any $a\in {\mathbb {D}}$. This is equivalent to the fact that h is univalent in $\varDelta _p(a,\psi (a))$ for any $a\in {\mathbb {D}}$, and therefore $\rho (a,h)\ge \psi (a)$ for $a\in {\mathbb {D}}$.

It remains to estimate $\rho _a(z,h)$. Let $\sigma$ denote the spherical distance on the Riemann sphere. By the assumption $h^\# \in L^\infty _\omega$, we obtain

$$\begin{aligned} \sigma \big ( h(z), h(\zeta ) \big )&\le \int _{h(\langle z,\zeta \rangle )} \frac{|d\xi |}{1+|\xi |^2} = \int _{\langle z,\zeta \rangle } h^{\#}(\xi ) \, |d\xi | \\&\le \left( \, \sup _{\xi \in \langle z,\zeta \rangle } \frac{1-|\xi |^2}{\omega (\xi )} \right) \, \varrho _h(z,\zeta ) \end{aligned}$$

for any $z,\zeta \in {\mathbb {D}}$. If $\zeta \in \varDelta _p(z,\psi (z))$, which is a subset of $\varDelta _p(z,1/2)$, then

$$\begin{aligned} \sigma \big ( h(z), h(\zeta ) \big ) \le \left( \, \sup _{\xi \in \langle z,\zeta \rangle } \frac{1-|\xi |^2}{\omega (\xi )} \right) 2 \, \varrho _p(z,\zeta ) \lesssim \frac{1-|z|^2}{\omega (z)} \cdot \frac{c \, \omega (z)}{1-|z|^2} \end{aligned}$$

with an absolute comparison constant. Then, h(z) and $h(\zeta )$ cannot be antipodal points if $0<c<1$ is sufficiently small. Therefore $\rho _a(z,h)\ge \psi (z)$ for $z\in {\mathbb {D}}$, which completes the proof of Theorem 11. $\square$

Corollary 1 allows us to reach the desired conclusion $A\in H^\infty _2$ under the assumption $|\nabla u| \in L^\infty _1$. The following lemma shows that, in this sense, Corollary 1 improves [12, Theorem 7], according to which the same conclusion holds if the linearly independent solutions $f_1,f_2\in {\mathcal {B}}$ satisfy $\inf _{z\in {\mathbb {D}}}( |f_1(z)| + |f_2(z)| )>0$.

Lemma 6

The following assertions hold.

• (i) If $f_1,f_2\in {\mathcal {B}}$ are linearly independent solutions of (1) for $A\in \mathrm{Hol}({\mathbb {D}})$, and $\inf _{z\in {\mathbb {D}}}( |f_1(z)| + |f_2(z)| )>0$, then $|\nabla u| \in L^\infty _1$ for $u=-\log \, (f_1/f_2)^\#$.

• (ii) There exists $A\in H^\infty _2$ such that (1) admits linearly independent solutions $f_1,f_2$ such that $\inf _{z\in {\mathbb {D}}}( |f_1(z)| + |f_2(z)| )=0$ but $|\nabla u| \in L^\infty _1$.

• (iii) There exists $A\in \mathrm{Hol}({\mathbb {D}})$ such that (1) admits linearly independent solutions $f_1,f_2$ with $f_1/f_2$ bounded (and hence normal) but $|\nabla u| \not \in L^\infty _1$.

Proof

(i) Since $f_1,f_2\in {\mathcal {B}}$ satisfy $\inf _{z\in {\mathbb {D}}}( |f_1(z)| + |f_2(z)| )>0$, we deduce

$$\begin{aligned} \begin{aligned} |\nabla u(z)|&= 2 \left| \partial u(z) \right| = 2 \, \frac{\big | f_1'(z)\overline{f_1(z)}+f_2'(z) \overline{f_2(z)} \big |}{|f_1(z)|^2+|f_2(z)|^2} \\&\le \frac{2 \max \{ \Vert f_1\Vert _{{\mathcal {B}}}, \Vert f_2\Vert _{{\mathcal {B}}}\}}{1-|z|^2} \, \frac{|f_1(z)|+|f_2(z)|}{|f_1(z)|^2+|f_2(z)|^2} \lesssim \frac{1}{1-|z|^2}, \quad z\in {\mathbb {D}}. \end{aligned} \end{aligned}$$

(ii) Consider the analytic and univalent function $h(z)=-\log (1-z)$, $z\in {\mathbb {D}}$. Define $A=S_h/2$, where $S_h$ is the Schwarzian derivative of h. Then, $A(z)=4^{-1}(1-z)^{-2}$, $z\in {\mathbb {D}}$, and clearly $A\in H^\infty _2$. It is well-known that (1) admits two linearly independent solutions $f_1,f_2$ such that $h=f_1/f_2$. In this case

$$\begin{aligned} \frac{|W(f_1,f_2)|}{|f_1(z)|^2+|f_2(z)|^2} = h^{\#}(z) = \frac{1}{|1-z| \big ( 1 + | \log (1-z)|^2 \big )}, \quad z\in {\mathbb {D}}, \end{aligned}$$

is unbounded in ${\mathbb {D}}$, while $|\nabla u| \in L^\infty _1$ by Corollary 1 (h is normal as it is univalent).

Part (iii) follows by the proof of Theorem 1(ii). An application of Corollary 1 reveals that $|\nabla u| \not \in L^\infty _1$. $\square$

It is a natural question to ask how $|\nabla u |\in L^\infty _1$ compares to Theorem 15? On one hand, Lemma 6(ii) serves as an example where $|\nabla u| \in L^\infty _1$ but (12) fails for any pairwise disjoint pseudo-hyperbolic discs (consider the positive real axis). On the other hand, Example 1(ii) in Sect. 4 implies that there are cases in which (12) is satisfied but $|\nabla u |\not \in L^\infty _1$ ($f_1/f_2$ is not normal). In both of these examples, the coefficient function satisfies $A\in H^\infty _2$.

Proof

(of Corollary 2) The assertions (i) and (ii) are equivalent by Theorem 11. Note that (i) implies (iii) by Lemma 5, while (iii) implies (i), and also (ii), by Theorem 5(ii). Finally, (ii) is equivalent to (iv) according to Theorem 5(i). $\square$

The arguments in this section are build on the representation (13) for the coefficient A. Derivatives of the coefficient can be controlled by expressions of similar type. For example, by differentiating (1) we obtain $f'''+A'f +Af'=0$, and

$$\begin{aligned} |A|&= \frac{|f_1'|+|f_2'|}{|f_1'|+|f_2'|} \, |A| = \frac{|f_1'''+A'f_1| + |f_2'''+A'f_2|}{|f_1'|+|f_2'|}\\&\ge |A'| \, \frac{|f_1|+|f_2|}{|f_1'|+|f_2'|} - \frac{|f_1'''|+|f_2'''|}{|f_1'|+|f_2'|}. \end{aligned}$$

Therefore, by applying (13),

$$\begin{aligned} |A'| \le \frac{|f_1'|+|f_2'|}{|f_1|+|f_2|} \left( |A| + \frac{|f_1'''|+|f_2'''|}{|f_1'|+|f_2'|} \right) = \frac{|f_1'|+|f_2'|}{|f_1|+|f_2|} \cdot \frac{|f_1''|+|f_2''|}{|f_1|+|f_2|} + \frac{|f_1'''|+|f_2'''|}{|f_1|+|f_2|}. \end{aligned}$$

12 Proof of Theorem 12

It is natural to require that solution with prescribed fixed points is bounded in ${\mathbb {D}}$. Under this requirement, Theorem 12 is best possible. This is a consequence of the following auxiliary result.

Lemma 7

The following assertions hold.

• (i) The identity function is the only one in $\{ f\in H^\infty : \Vert f\Vert _{H^\infty } \le 1\}$ which has more than one fixed point.

• (ii) The identity function is the only one in ${\mathcal {N}}$ which has more fixed points than the Blaschke condition allows.

The proof of the lemma is straight-forward and hence omitted.

Proof

(of Theorem 12) Let $B=B_{\{z_n\}}$ be the Blaschke product with zeros $\{z_n\}$. Let $0<\varepsilon <1$, and define $f_1(z) = z + \varepsilon z^3 B(z)$, $z\in {\mathbb {D}}$. The fixed points of $f_1$ are precisely $\{0\} \cup \{z_n\}$. By the Schwarz lemma $| z^3\, B(z) | \le |z|$ for $z\in {\mathbb {D}}$, and therefore $(1-\varepsilon ) |z|\le |f_1(z)| \le (1+\varepsilon )|z|$ for any $z\in {\mathbb {D}}$.

Since $f_1$ has only one zero in ${\mathbb {D}}$ and $f_1''(0)=0$, we deduce $A=-f_1''/f_1 \in \mathrm{Hol}({\mathbb {D}})$. If $0<\delta <1$, then

$$\begin{aligned} \sup _{\delta<|z|<1} |A(z)| \le \frac{\varepsilon }{(1-\varepsilon )\delta } \, \sup _{\delta<|z|<1} \Big ( |B''(z)| + 6 \, |B'(z)| + 6 \, |B(z)| \Big ), \end{aligned}$$

and consequently, $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure. If $f_2$ is defined by (19) for fixed $\alpha \in {\mathbb {D}}{\setminus }\{0\}$, then $f_2\in H^\infty$ is a solution of (1) and is linearly independent to $f_1$. Consequently, all solutions of (1) are bounded. $\square$

13 Proofs of Theorems 13 and 14

Proof

(of Theorem 13) Let $\varLambda \subset {\mathbb {D}}{\setminus }\{0\}$ be a uniformly separated sequence. Then, the corresponding Blaschke product $B= B_\varLambda$ satisfies (20).

Let $h\in H^\infty$ be a function which satisfies $h(z_n)=\log {z_n}$ for $z_n\in \varLambda$. The existence of such h is guaranteed by Carleson’s interpolation theorem [2, Theorem 3]. Let $\{C_n\}$ be the sequence of real numbers defined as follows: Whenever $z_n\in \varLambda$ is prescribed to be an attractive fixed point define $C_n=1/2$, if neutral choose $C_n=1$, while otherwise take $C_n=2$. By (20), we obtain

$$\begin{aligned} \sup _{z_n\in \varLambda } \, \left| \frac{1}{B'(z_n)} \left( \frac{C_n}{z_n} - h'(z_n) \right) \right| \le \sup _{z_n\in \varLambda } \, \frac{1-|z_n|^2}{\delta } \left( \frac{2}{\inf _n |z_n|} + |h'(z_n)| \right) < \infty , \end{aligned}$$

and hence $\{w_n\}= \{(C_n/z_n - h'(z_n))/B'(z_n)\}$ is a bounded sequence. The aforementioned Carleson’s result guarantees that there exists $g\in H^\infty$ with $g(z_n)=w_n$ for $z_n\in \varLambda$. Define $f_1=\exp ( h+Bg)$, and note that $f_1$ is not only in $H^\infty$ but also is uniformly bounded away from zero. Moreover,

$$\begin{aligned} f_1(z_n)=z_n, \quad f_1'(z_n) = z_n \Big ( h'(z_n)+B'(z_n) g(z_n) \Big ) = C_n, \quad z_n\in \varLambda . \end{aligned}$$

The points $z_n\in \varLambda$ are fixed points of the prescribed type. The coefficient $A= -f_1''/f_1 \in \mathrm{Hol}({\mathbb {D}})$ satisfies $|A|\lesssim |f_1''|$ in ${\mathbb {D}}$, and therefore $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure. The fact that all solutions of (1) are bounded follows as in the proof of Theorem 12. $\square$

Note that the solution $f_1$ in Theorem 13, which has prescribed fixed points of pregiven type, may have fixed points which do not belong to $\varLambda$.

Remark 3

If $A\in \mathrm{Hol}({\mathbb {D}})$ and $z_0\in {\mathbb {D}}$, then (1) admits a unique solution f such that the initial conditions $f(z_0)=\alpha \in {\mathbb {C}}$ and $f'(z_0)=\beta \in {\mathbb {C}}$ are satisfied. Therefore fixed points of solutions of (1) are not always distinct from zeros or critical points. In the proof of Theorem 13, $\{C_n\}\subset {\mathbb {C}}$ can be any sequence with the property $\sup _n |C_n|(1-|z_n|^2)<\infty$. If we take $C_n=0$ for all n, then every point $z_n\in \varLambda$ is not only a fixed point but also a critical point of the solution $f_1$.

Proof

(of Theorem 14) Let $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$ be the sequence of non-zero points, and let $B=B_\varLambda$ be the corresponding Blaschke product. Since $\varLambda \in {{\,\mathrm{Int}\,}}{\mathcal {N}}$, [18, Theorem 1.2] implies that there exists $h_1\in \mathrm{Har^+}({\mathbb {D}})$ such that (31) holds.

Let $h\in {\mathcal {N}}$ be a function which satisfies $h(z_n)=\log {z_n}$ for $z_n\in \varLambda$. Since $\varLambda$ is Nevanlinna interpolating, the existence of such function h is guaranteed by [18, Theorem 1.2]. Let $\{C_n\}$ be the sequence of real numbers defined as in the proof of Theorem 13. As $h\in {\mathcal {N}}$, (29) implies that there exists a constant $0<C<\infty$ and $h_2\in \mathrm{Har^+}({\mathbb {D}})$ such that

$$\begin{aligned} \left| \frac{1}{B'(z_n)} \left( \frac{C_n}{z_n} - h'(z_n) \right) \right| \le \frac{1-|z_n|^2}{e^{-h_1(z_n)}} \left( \frac{2}{\inf _n |z_n|} + \frac{e^{C}e^{h_2(z_n)}}{1-|z_n|^2} \right) , \quad z_n\in \varLambda . \end{aligned}$$

Since $\{w_n\}= \{(C_n/z_n - h'(z_n))/B'(z_n)\} \in {\mathcal {N}}\, | \, \varLambda$ by [18, Theorem 1.2], there exists $g\in {\mathcal {N}}$ with $g(z_n)=w_n$ for $z_n\in \varLambda$. Define $f=\exp ( h+Bg)$, and note that

$$\begin{aligned} f(z_n)=z_n, \quad f'(z_n) = z_n \Big ( h'(z_n)+B'(z_n) g(z_n) \Big ) = C_n, \quad z_n\in \varLambda . \end{aligned}$$

The points $z_n\in \varLambda$ are fixed points of the prescribed type. Finally, the coefficient

$$\begin{aligned} A= -f''/f = - \big ( (h+Bg)' \big )^2 - (h+Bg)''\in \mathrm{Hol}({\mathbb {D}}) \end{aligned}$$

satisfies $|A(z)|(1-|z|^2)^2 \le e^{H(z)}$, $z\in {\mathbb {D}}$ and $H\in \mathrm{Har^+}({\mathbb {D}})$, by (29). $\square$