Solutions of complex differential equation having zeros on pre-given sequences

TL;DR

This paper investigates solutions to a complex differential equation with zeros on prescribed sequences, revealing new conditions under which solutions can have zeros not satisfying classical criteria and connecting zero-sequences with Carleson measures.

Contribution

It demonstrates that solutions can have zeros outside the Blaschke condition and characterizes zero-sequences via Carleson measures and uniform separation, answering open questions.

Findings

Existence of solutions with zeros not satisfying the Blaschke condition.

Characterization of zero-sequences via Carleson measures and uniform separation.

Improvement of classical results on non-normal functions and local univalence.

Abstract

Behavior of solutions of $f''+Af=0$ is discussed under the assumption that $A$ is analytic in $\mathbb{D}$ and $\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty$, where $\mathbb{D}$ is the unit disc of the complex plane. As a main result it is shown that such differential equation may admit a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature. It is also proved that $\Lambda\subset\mathbb{D}$ is the zero-sequence of a non-trivial solution of $f''+Af=0$ where $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure if and only if $\Lambda$ is uniformly separated. As an application an old result, according to which there exists a non-normal function which is uniformly locally univalent, is improved.

Equations

$$f^{\prime \prime }+Af=0$$

$$\underset{k}{\inf }\prod _{n\ne k}\left|\frac{z_n-z_k}{1-{\overline{z}}_n{z}_k}\right|>0,$$

$$D^{-}(\Lambda )$$

$$D^{+}(\Lambda )$$

$$D^{+}(\Lambda)\leq(2\pi+1)\!\left(1-\frac{2\lVert A\rVert_{H^{\infty}_{2}}^{1/2}}{\lVert A\rVert_{H^{\infty}_{2}}+1}\right)^{1/2}\!\!\Bigg{(}1-\left(1-\frac{2\lVert A\rVert_{H^{\infty}_{2}}^{1/2}}{\lVert A\rVert_{H^{\infty}_{2}}+1}\right)^{1/2}\,\Bigg{)}^{-2}.$$

$$(1-|z{|}^2{)}^{\beta }|g(z)|\simeq \varrho (z,{\Lambda }^{\star }),z\in \mathbb{D},$$

$$\sup_{I\subset\partial\mathbb{D}}\,\frac{\mu(Q(I))}{|I|^{p}}<\infty,\quad Q(I)=\big{\{}re^{i\theta}:1-|I|\leq r<1,\,e^{i\theta}\in I\big{\}},$$

$$\underset{a\in \mathbb{D}}{\sup }{\int }_{\mathbb{D}}{\left(\frac{1-|a{|}^2}{|1-\overline{a}z{|}^2}\right)}^{p}d\mu (z)<\infty .$$

$$\sum _{z_n\in \Lambda }(1-|z_n|{)}^p{\delta }_{z_n}$$

$$d{\mu }_{A,p}(z)=|A(z){|}^2(1-|z{|}^2{)}^{2+p}dm(z)$$

$${\int }_0^{2\pi }\log \mathrm{dist}(e^{i\theta },\Lambda )d\theta >-\infty ,$$

$$S_w={\left(\frac{w^{\prime \prime }}{w^{\prime }}\right)}^{\prime }-\frac{1}{2}{\left(\frac{w^{\prime \prime }}{w^{\prime }}\right)}^2$$

$$\Vert g{\Vert }_{H_{\alpha }^{\infty }}=\underset{z\in \mathbb{D}}{\sup }(1-|z{|}^2{)}^{\alpha }|g(z)|<\infty .$$

$$|g(z)|\le \frac{C\Vert g{\Vert }_{H_{\alpha }^{\infty }}\varrho (z,z_0)}{(1-|z_0{|}^2{)}^{\alpha }},z\in \Delta (z_0,\delta ).$$

$$|h(z)|\le \frac{|g(z)|}{\varrho (z,z_0)}\le \frac{\Vert g{\Vert }_{H_{\alpha }^{\infty }}}{\delta (1-|z{|}^2{)}^{\alpha }}\le \frac{C^{\star }\Vert g{\Vert }_{H_{\alpha }^{\infty }}}{\delta (1-|z_0{|}^2{)}^{\alpha }},z\in \partial \Delta (z_0,\delta ).$$

$$A=-\frac{f^{\prime \prime }}{f}=-\frac{g^{\prime \prime }+2g^{\prime }{h}^{\prime }}{g}-(h^{\prime }{)}^2-h^{\prime \prime },$$

$$h^{\prime }(z_n)=-\frac{1}{2}\frac{g^{\prime \prime }(z_n)}{g^{\prime }(z_n)},z_n\in {\Lambda }^{\star }.$$

$$\underset{z_n\in {\Lambda }^{\star }}{\inf }(1-|z_n{|}^2{)}^{\beta +1}|g^{\prime }(z_n)|>0$$

$$\underset{z_n\in {\Lambda }^{\star }}{\sup }(1-|z_n{|}^2)\left|\frac{g^{\prime \prime }(z_n)}{g^{\prime }(z_n)}\right|<\infty .$$

$$\sup\bigg{\{}(1-|z|^{2})^{2}|A(z)|\,:\,z\in\mathbb{D}\setminus\bigcup_{z_{n}\in\Lambda^{\star}}\Delta(z_{n},\delta)\bigg{\}}<\infty.$$

$$\underset{r\to 1^{-}}{\mathrm{limsup}}\frac{|h(r\xi )|}{\Psi (r)}\le \Vert h^{\prime }{\Vert }_{H_1^{\infty }},\Psi (r)=\sqrt{\log \frac{1}{1-r}\log \log \log \frac{1}{1-r}},$$

$$|h(z_{n})|\leq\big{|}h(|z_{n}|\xi)\big{|}+\frac{\lVert h^{\prime}\rVert_{H^{\infty}_{1}}}{2}\,\log\frac{1+\varrho(z_{n},|z_{n}|\xi)}{1-\varrho(z_{n},|z_{n}|\xi)},\quad z_{n}\in\Lambda^{\prime},$$

$$(1-|z_n{|}^2)|f^{\prime }(z_n)|$$

$$\displaystyle\gtrsim\frac{1}{(1-|z_{n}|^{2})^{\beta}}\,\exp\!\Big{(}-\big{(}\lVert h^{\prime}\rVert_{H^{\infty}_{1}}+1\big{)}\,\Psi(|z_{n}|)\Big{)}$$

$$\underset{z_n\in \Lambda }{\inf }(1-|z_n{|}^2)|B^{\prime }(z_n)|=\underset{z_n\in \Lambda }{\inf }\prod _{z_k\ne z_n}\left|\frac{z_k-z_n}{1-{\overline{z}}_k{z}_n}\right|>0$$

$$\underset{z_n\in \Lambda }{\sup }\frac{\left|B^{\prime \prime }(z_n)\right|}{{\left|B^{\prime }(z_n)\right|}^2}<\infty .$$

$$k(z_n)=-\frac{B^{\prime \prime }(z_n)}{2{\left(B^{\prime }(z_n)\right)}^2},z_n\in \Lambda .$$

$$A=-\frac{f^{\prime \prime }}{f}=-\frac{B^{\prime \prime }+2B^{\prime }{h}^{\prime }}{B}-(h^{\prime }{)}^2-h^{\prime \prime }.$$

$$|A(z)|=\frac{|f^{\prime\prime}(z)|}{|f(z)|}\lesssim|f^{\prime\prime}(z)|\,\big{|}e^{-h(z)}\big{|}\lesssim|f^{\prime\prime}(z)|,\quad z\in\Omega.$$

$$\underset{a\in \mathbb{D}}{\sup }{\int }_{\Omega }{\left(\frac{1-|a{|}^2}{|1-\overline{a}z{|}^2}\right)}^{p}d{\mu }_{A,p}(z)$$

$$\lesssim \underset{a\in \mathbb{D}}{\sup }{\int }_{\mathbb{D}}|f^{\prime \prime }(z){|}^2(1-|z{|}^2{)}^{2+p}{\left(\frac{1-|a{|}^2}{|1-\overline{a}z{|}^2}\right)}^{p}dm(z)<\infty$$

$$I=\underset{a\in \mathbb{D}}{\sup }\sum _{z_n\in \Lambda }{\int }_{\Delta (z_n,\delta )}{\left|\frac{B^{\prime \prime }(z)+2B^{\prime }(z)h^{\prime }(z)}{B(z)}\right|}^2\frac{(1-|z{|}^2{)}^{2+p}(1-|a{|}^2{)}^p}{|1-\overline{a}z{|}^{2p}}dm(z)$$

Full text

It is also proved that $\Lambda\subset\mathbb{D}$ is the zero-sequence of a non-trivial solution of $f^{\prime\prime}+Af=0$ where $|A(z)|^{2}(1-|z|^{2})^{3}\,dm(z)$ is a Carleson measure if and only if $\Lambda$ is uniformly separated. As an application an old result, according to which there exists a non-normal function which is uniformly locally univalent, is improved.

The author is supported in part by the Academy of Finland #286877 and the Finnish Academy of Science and Letters (Vilho, Yrjö and Kalle Väisälä Foundation).

1. Introduction

Let $\mathcal{H}(\mathbb{D})$ be the collection of analytic functions in the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. This research concerns zero-sequences of non-trivial solutions ($f\not\equiv 0$) of

[TABLE]

under the assumption $A\in H^{\infty}_{2}$, which means that $A\in\mathcal{H}(\mathbb{D})$ and $\lVert A\rVert_{H^{\infty}_{2}}=\sup_{z\in\mathbb{D}}\,(1-|z|^{2})^{2}|A(z)|<\infty$. In particular, we are interested in the following question:

• (Q)

Is it true that the zero-sequence $\{z_{n}\}$ of any non-trivial solution of (1) satisfies the Blaschke condition $\sum_{n}(1-|z_{n}|)<\infty$ if $A\in H^{\infty}_{2}$?

Question (Q) relates to so-called Blaschke-oscillatory equations, and appears in [12, pp. 61–62]. Note that the characterization [12, Lemma 3] of Blaschke-oscillatory equations does not provide an immediate answer to (Q). Moreover, it is known that all non-trivial solutions of (1) may lie outside the Nevanlinna class $N$, even if $A\in H^{\infty}_{2}$ [12, pp. 57–58].

We will make repeated use of [15, Theorem 6.1], which connects non-trivial solutions of (1) to a locally univalent meromorphic function in $\mathbb{D}$. If $\lVert A\rVert_{H^{\infty}_{2}}\leq 1$ then all non-trivial solutions of (1) vanish at most once in $\mathbb{D}$ by [19, Theorem I], and if $\lVert A\rVert_{H^{\infty}_{2}}>1$ then non-trivial solutions may have infinitely many zeros [13]. The condition $A\in H^{\infty}_{2}$ is equivalent to the fact that zero-sequences of non-trivial solutions of (1) are separated with respect to the pseudo-hyperbolic metric [25, Theorems 3–4], by a constant depending on $\lVert A\rVert_{H^{\infty}_{2}}$, and hence zero-sequences almost satisfy the Blaschke condition [3, p. 162]. Many sufficient coefficient conditions implying an affirmative answer to (Q) are known. For coefficient conditions placing all solutions of (1) to the Nevanlinna class, see [11, 21], and for coefficient conditions placing all solutions to some Hardy space, see [8, 10, 21, 24]. For a more direct approach to zero-sequences of solutions of (1), see [9].

2. Results

As the main result, we prove that there is $A\in H^{\infty}_{2}$ such that (1) admits a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This answers (Q) in the negative. Actually, we show that a non-trivial solution may vanish on any pre-given sequence of sufficiently small density.

We also obtain a complete description of zero-sequences of solutions of (1) in the case that $A$ satisfies a condition stronger than $A\in H^{\infty}_{2}$. In Section 2.2, we consider an application concerning normal meromorphic functions.

2.1. Zero-sequences of solutions

The sequence $\Lambda=\{z_{n}\}$ of points in $\mathbb{D}$ is said to be uniformly separated if

[TABLE]

while $\Lambda$ is called separated if there exists a constant $\delta=\delta(\Lambda)>0$ such that $\varrho(z_{n},z_{k})=|z_{n}-z_{k}|/|1-\overline{z}_{n}z_{k}|>\delta$ for any $n\neq k$. Unless otherwise stated, separation is understood with respect to the pseudo-hyperbolic metric.

Let $\Delta(\zeta,r)=\{z\in\mathbb{D}:\varrho(z,\zeta)<r\}$ be the (open) pseudo-hyperbolic disc of radius $0<r<1$, centered at $\zeta\in\mathbb{D}$, and let $n(\Lambda,\zeta,r)$ be the counting function for those points in $\Lambda$ which lie in $\Delta(\zeta,r)$. The lower and upper uniform densities of $\Lambda$ are

[TABLE]

respectively. For a comprehensive treatment of these densities, and their connection to interpolation and sampling, see [3, Chapters 6–7] and [28, Chapter 3]. See also the seminal papers [26, 27] by Seip.

Theorem 1**.**

If $\Lambda\subset\mathbb{D}$ is a separated sequence for which $D^{+}(\Lambda)<1$, then there exists $A=A(\Lambda)\in H^{\infty}_{2}$ such that (1) admits a non-trivial solution which vanishes at all points on $\Lambda$.

Conversely, if $A\in H^{\infty}_{2}$ and $f$ is a non-trivial solution of (1) whose zero-sequence is $\Lambda\subset\mathbb{D}$, then $\Lambda$ is separated, and contains at most one point if $\lVert A\rVert_{H^{\infty}_{2}}\leq 1$, while otherwise

[TABLE]

Let $\lesssim$ denote a one-sided estimate up to a constant and write $\simeq$ for a two-sided estimate up to constants. In the first part of Theorem 1 we construct a solution $f$ of (1) which vanishes on $\Lambda$ (but has other zeros also). The idea is to find an auxiliary function $g\in\mathcal{H}(\mathbb{D})$ such that

[TABLE]

for some constant $0<\beta<\infty$, and for some separated sequence $\Lambda^{\star}\supset\Lambda$ which satisfies $D^{+}(\Lambda^{\star})<1$. Such function $g$ exists in the literature. Then, $f=ge^{h}$ is a solution of (1) for $A\in H^{\infty}_{2}$ if $h$ belongs to the Bloch space and solves a certain interpolation problem. Note that the additional assumption $D^{+}(\Lambda)<1$ in Theorem 1 holds for any sequence $\Lambda\subset\mathbb{D}$ which is separated by a constant sufficiently close to one [3, Lemma 9, p. 190]. The second part of Theorem 1 follows immediately by combining [3, Lemma 9, p. 190], [19, Theorem I] and [25, Theorem 3]. It implies that $D^{+}(\Lambda)\to 0^{+}$ at the linear rate as $\lVert A\rVert_{H^{\infty}_{2}}\to 1^{+}$.

The following construction is due to Seip [26, pp. 214–215]. For $a>1$ and $b>0$, let $\Lambda=\Lambda(a,b)$ be the image of $\{a^{j}(bk+i)\}_{j,k\in\mathbb{Z}}$ under the Cayley transform (conformal map from the upper half-plane onto $\mathbb{D}$). The set $\Lambda\subset\mathbb{D}$ satisfies $D^{-}(\Lambda)=D^{+}(\Lambda)=2\pi/(b\log a)$ by [27, p. 23]. In particular, there exists $g\in\mathcal{H}(\mathbb{D})$ which satisfies (3) for $\Lambda^{\star}=\Lambda$ and $\beta=2\pi/(b\log a)$. Now $\Lambda$ is a separated sequence which behaves (essentially) as badly as possible in terms of the Blaschke condition; recall that the lower uniform density of any separated Blaschke sequence is zero by (2). As a straightforward consequence of Theorem 1, we obtain:

Corollary 2**.**

Let $\Lambda=\Lambda(a,b)\subset\mathbb{D}$ be as above, and let $2\pi/(b\log a)<1$. Then, there exists $A=A(a,b)\in H^{\infty}_{2}$ such that (1) admits a non-trivial solution whose zero-sequence is $\Lambda$, and hence does not satisfy the Blaschke condition.

Let $0<p<\infty$. A positive Borel measure $\mu$ on $\mathbb{D}$ is called a (bounded) $p$-Carleson measure provided that

[TABLE]

where the supremum is taken over all subarcs $I\subset\partial\mathbb{D}$ and $|I|$ denotes the length of $I$ (normalized so that $|\partial\mathbb{D}|=1$). These measures can be described in conformally invariant terms [1, Lemma 2.1]. In fact, the positive Borel measure $\mu$ is a $p$-Carleson measure if and only if

[TABLE]

For $p=1$ the condition (4) characterizes the classical Carleson measures, which were invented to study interpolation by bounded analytic functions. See [5] for a general reference.

There are two types of measures which play a role in this study. First, let $\delta_{z}$ be the Dirac mass at the point $z\in\mathbb{D}$. We consider separated sequences $\Lambda\subset\mathbb{D}$ for which

[TABLE]

is a $p$-Carleson measure. Such sequences are uniformly separated for any $0<p\leq 1$. Second, we consider functions $A\in\mathcal{H}(\mathbb{D})$ for which

[TABLE]

is a $p$-Carleson measure. Here $dm(z)$ is the Lebesgue area measure on $\mathbb{D}$. We write $\mu_{A}=\mu_{A,1}$ for short. Such functions satisfy $A\in H^{\infty}_{2}$ by the subharmonicity of $|A|^{2}$. The effect of the parameter $p$ is more evident when second primitives of $A\in\mathcal{H}(\mathbb{D})$ are considered [23, Theorem 3.2]: (6) is a $p$-Carleson measure if and only if the second primitive of $A$ belongs to $Q_{p}$. The space $Q_{p}$ consists of those $f\in\mathcal{H}(\mathbb{D})$ for which $|f^{\prime}(z)|^{2}(1-|z|^{2})^{p}\,dm(z)$ is a $p$-Carleson measure. For $1<p<\infty$, $Q_{p}$ is the Bloch space while $Q_{p_{1}}\subsetneq Q_{p_{2}}\subsetneq Q_{1}=\rm BMOA$ for any $0<p_{1}<p_{2}<1$. See [1, 4, 20] and the references therein, for more details.

Theorem 3**.**

Let $0<p\leq 1$. If $\Lambda\subset\mathbb{D}$ is a separated sequence such that (5) is a $p$-Carleson measure, then there exists $A=A(\Lambda)\in\mathcal{H}(\mathbb{D})$ such that (6) is a $p$-Carleson measure and (1) admits a non-trivial solution whose zero-sequence is $\Lambda$.

Theorems 1 and 3 improve [7, Corollary 7], which states that any uniformly separated sequence can appear as the zero-sequence of a non-trivial solution of (1) where $A\in H^{\infty}_{2}$, i.e., the second primitive of $A$ is in the Bloch space. Theorem 1 shows that we can prescribe zero-sequences of strictly positive uniform density under the same coefficient condition while Theorem 3 implies that any sufficiently separated sequence can be prescribed such that the second primitive of $A$ belongs to $Q_{p}$ for fixed $0<p\leq 1$. When prescribing infinite zero-sequences to non-trivial solutions of (1), we cannot expect that the coefficient $A$ is even close to be bounded. The breaking point lies inside $H^{\infty}_{2}$: if $A\in\mathcal{H}(\mathbb{D})$ and there exists $0<R<1$ such that $(1-|z|^{2})^{2}|A(z)|\leq 1$ for $R<|z|<1$, then all non-trivial solutions of (1) have at most finitely many zeros in $\mathbb{D}$ [25, Theorem 1].

By Theorem 3, we obtain a complete description of zero-sequences of non-trivial solutions of (1) in the case that $d\mu_{A}(z)=|A(z)|^{2}(1-|z|^{2})^{3}\,dm(z)$ is a Carleson measure.

Corollary 4**.**

A sequence $\Lambda\subset\mathbb{D}$ is the zero-sequence of a non-trivial solution of (1) where $\mu_{A}$ is a Carleson measure if and only if $\Lambda$ is uniformly separated.

In Corollary 4, all zero-sequences are uniformly separated by [9, Corollary 3]. The converse assertion follows from Theorem 3 by taking $p=1$. The following observation concerns the case $A\in H^{\infty}_{2,0}$, which means that $A\in\mathcal{H}(\mathbb{D})$ and $\lim_{|z|\to 1^{-}}\,(1-|z|^{2})^{2}|A(z)|=0$. Sequence $\Lambda\subset\mathbb{D}$ is the zero-sequence of a non-trivial solution of (1) where $A\in H^{\infty}_{2,0}$ if and only if $\Lambda$ is a finite sequence of distinct points in $\mathbb{D}$. The fact that all zero-sequences are finite follows from [25, Theorem 1], while the converse assertion is proved by constructing a non-trivial solution of (1), which has finitely many prescribed zeros [12, Section 10].

Conformally invariant collections of zero-sequences

Let $X$ be a space of analytic functions, and let $\mathcal{Z}(X)$ be the collection of sequences $\Lambda\subset\mathbb{D}$ for which there exists $A=A(\Lambda)\in X$ such that (1) admits a non-trivial solution whose zero-sequence is (precisely) $\Lambda$. Some parts of the following result are known in another form, see the proof for references.

Proposition 5**.**

The following statements hold:

1. (a)

$\mathcal{Z}(H^{\infty}_{2})$* is conformally invariant.* 2. (b)

If $\Lambda$ is separated and $D^{+}(\Lambda)<1$, then $\Lambda\subset\Lambda^{\prime}$ for some $\Lambda^{\prime}\in\mathcal{Z}(H^{\infty}_{2})$. Conversely, if $\Lambda\in\mathcal{Z}(H^{\infty}_{2})$, then $\Lambda$ is separated and $D^{+}(\Lambda)<\infty$. 3. (c)

$\mathcal{Z}(H^{\infty}_{2})$* contains non-Blaschke sequences. However, if $\Lambda\in\mathcal{Z}(H^{\infty}_{2})$ and*

[TABLE]

where $\operatorname{dist}$ is the Euclidean distance, then $\Lambda$ is a Blaschke sequence.

Let $K$ be the space which consists of the second derivatives of $\rm BMOA$ functions. Consequently, $\mathcal{Z}(K)$ is the collection of zero-sequences of non-trivial solutions of (1) induced by those coefficients $A\in\mathcal{H}(\mathbb{D})$ for which $d\mu_{A}(z)=|A(z)|^{2}(1-|z|^{2})^{3}\,dm(z)$ is a Carleson measure. By Corollary 4, $\Lambda\in\mathcal{Z}(K)$ if and only if $\Lambda$ is uniformly separated. The following observations follow from known properties of uniformly separated sequences:

1. (a)

$\mathcal{Z}(K)$ is conformally invariant. 2. (b)

If $\Lambda\subset\Lambda^{\prime}$ for some $\Lambda^{\prime}\in\mathcal{Z}(K)$, then $\Lambda\in\mathcal{Z}(K)$. 3. (c)

If $\Lambda_{1},\Lambda_{2}\in\mathcal{Z}(K)$, and $\Lambda_{1}\cup\Lambda_{2}$ is separated, then $\Lambda_{1}\cup\Lambda_{2}\in\mathcal{Z}(K)$. 4. (d)

If $\Lambda=\{z_{n}\}\in\mathcal{Z}(K)$, and $\Lambda^{\prime}=\{z_{n}^{\prime}\}\subset\mathbb{D}$ is a sequence such that $\sup_{n}\varrho(z_{n},z_{n}^{\prime})$ is sufficiently small, then $\Lambda^{\prime}\in\mathcal{Z}(K)$. 5. (e)

$\mathcal{Z}(K)$ contains only Blaschke sequences. However, there are separated Blaschke sequences which are not in $\mathcal{Z}(K)$.

By subharmonicity $K\subset H^{\infty}_{2}$, and hence $\mathcal{Z}(K)\subset\mathcal{Z}(H^{\infty}_{2})$. It is curious that $\mathcal{Z}(H^{\infty}_{2,0})\subset\mathcal{Z}(K)$ even though there exist functions $A\in H^{\infty}_{2,0}$ for which $\mu_{A}$ is not a Carleson measure; typical examples of such functions are given in terms of lacunary series. Note that $\mathcal{Z}(H^{\infty}_{2,0})$ is the collection of finite sequences of distinct points in $\mathbb{D}$.

2.2. Normal functions

A function $w$ meromorphic in the unit disc $\mathbb{D}$ is said to be normal if $\sup_{z\in\mathbb{D}}\,(1-|z|^{2})\,w^{\#}(z)<\infty$, where $w^{\#}=|w^{\prime}|/(1+|w|^{2})$ is the spherical derivative of $w$. Actually, a meromorphic function $w$ is normal if and only if \{w\circ\varphi:\text{\varphi$conformal automorphism of$\mathbb{D}}\} is a normal family in $\mathbb{D}$ (in the sense of Montel). For more details, see [18].

In [16], Lappan gives an answer to a question of Hayman by showing that there exists a non-normal $w\in\mathcal{H}(\mathbb{D})$ whose Schwarzian derivative

[TABLE]

belongs to $H^{\infty}_{2}$. In a subsequent paper [17, Theorem 5] a concrete function having these properties is presented. Recall that, for $w$ meromorphic in $\mathbb{D}$, $S_{w}\in H^{\infty}_{2}$ if and only if $w$ is uniformly locally univalent; see for example [6, Lemma B] and the references therein. As a consequence of the proof of Theorem 3, we obtain:

Corollary 6**.**

For any $0<p\leq 1$, there exists a non-normal meromorphic function $w$ in $\mathbb{D}$ such that $S_{w}\in\mathcal{H}(\mathbb{D})$ and $|S_{w}(z)|^{2}(1-|z|^{2})^{2+p}\,dm(z)$ is a $p$-Carleson measure.

In Corollary 6, $S_{w}\in H^{\infty}_{2}$ by the subharmonicity of $|S_{w}|^{2}$. By construction, the function $w$ has prescribed separated poles $\Lambda$ such that (5) is a $p$-Carleson measure, $w$ belongs to the Nevanlinna class of meromorphic functions and emerges as a primitive of $1/f^{2}$ where $f\in Q_{p}\cap H^{\infty}$ is a solution of (1) for $A=S_{w}/2$.

3. Proofs of the results

The growth space $H^{\infty}_{\alpha}$, for $0\leq\alpha<\infty$, consists of those $g\in\mathcal{H}(\mathbb{D})$ for which

[TABLE]

In particular, $H^{\infty}=H^{\infty}_{0}$. Before the proof of Theorem 1, we consider an auxiliary result which resembles the classical Schwarz lemma. See also [3, Lemma 7, p. 209]. The proof of Lemma 7 is presented for convenience of the reader.

Lemma 7**.**

Let $g\in H^{\infty}_{\alpha}$ where $0\leq\alpha<\infty$, and let $0<\delta<1$. If $g(z_{0})=0$ for some $z_{0}\in\mathbb{D}$, then there exists a positive constant $C=C(\alpha,\delta)$ such that

[TABLE]

Proof.

Consider the function $h(z)=g(z)/((z_{0}-z)/(1-\overline{z}_{0}z))$, and note that $h\in\mathcal{H}(\mathbb{D})$. There exists a positive constant $C^{\star}=C^{\star}(\alpha,\delta)$ such that

[TABLE]

By the maximum modulus principle this inequality holds for all $z\in\Delta(z_{0},\delta)$, which implies (8) for $C=C^{\star}/\delta$. ∎

Proof of Theorem 1.

Let $\Lambda$ be a separated sequence for which $D^{+}(\Lambda)<1$. By [3, Theorem 5, p. 220], there exist a separated sequence $\Lambda^{\star}\supset\Lambda$ and $g\in\mathcal{H}(\mathbb{D})$ such that (3) holds for $\beta=(D^{+}(\Lambda)+1)/2<1$. According to [3, Lemma 19, p. 235], we have $D^{-}(\Lambda^{\star})=D^{+}(\Lambda^{\star})=\beta$. Let $f=ge^{h}$ where $h\in\mathcal{H}(\mathbb{D})$ is defined later. The function $f$ is a solution of (1) with $A\in\mathcal{H}(\mathbb{D})$,

[TABLE]

provided that $h\in\mathcal{H}(\mathbb{D})$ and $h^{\prime}$ satisfies the interpolation property

[TABLE]

In particular, because of (9), $A$ has a removable singularity at each zero of $g$. Since $g^{\prime\prime}\in H^{\infty}_{\beta+2}$ by Cauchy’s integral formula, and

[TABLE]

by (3), we deduce

[TABLE]

Since $D^{+}(\Lambda^{\star})=\beta<1$, [27, Theorem 1.2] and (11) imply that there exists $h\in\mathcal{H}(\mathbb{D})$ such that $h^{\prime}\in H^{\infty}_{1}$ and the interpolation property (9) holds.

It remains to prove that $A\in H^{\infty}_{2}$. Let $0<\delta<1$ be a sufficiently small constant such that the pseudo-hyperbolic discs $\Delta(z_{n},\delta)$ are pairwise disjoint for $z_{n}\in\Lambda^{\star}$. On one hand, by (3) and $h^{\prime}\in H^{\infty}_{1}$, we obtain

[TABLE]

On the other hand, if $z\in\Delta(z_{n},\delta)$ for some $z_{n}\in\Lambda^{\star}$, then we apply Lemma 7 to $g^{\prime\prime}+2g^{\prime}h^{\prime}\in H^{\infty}_{\beta+2}$ (which vanishes at all points $z_{n}\in\Lambda^{\star}$ by the interpolation property) to deduce that $(1-|z|^{2})^{2}|A(z)|$ is uniformly bounded also for any $z\in\bigcup_{z_{n}\in\Lambda^{\star}}\Delta(z_{n},\delta)$. This completes the proof of Theorem 1. ∎

The proof of Theorem 1 produces a non-normal solution of (1) under the restriction $A\in H^{\infty}_{2}$. See [6, Theorem 3] for another example. To show that $f=ge^{h}$ in the proof of Theorem 1 is non-normal, we argue as follows. For any $\xi\in\partial\mathbb{D}$, there exists a subsequence $\Lambda^{\prime}=\Lambda^{\prime}(\xi)$ of $\Lambda^{\star}$ which converges non-tangentially to $\xi$. This follows from [3, Corollary, p. 188] as $D^{-}(\Lambda^{\star})>0$. The Makarov law of the iterated logarithm [22, Theorem 8.10] gives

[TABLE]

for almost every $\xi\in\partial\mathbb{D}$. Fix $\xi\in\partial\mathbb{D}$ such that (12) holds, and let $\Lambda^{\prime}=\Lambda^{\prime}(\xi)$ be the corresponding subsequence of $\Lambda^{\star}$. By [3, Proposition 1, p. 43],

[TABLE]

where $\sup\{\varrho(z_{n},|z_{n}|\xi):z_{n}\in\Lambda^{\prime}\}<1$. If $z_{n}\in\Lambda^{\prime}$ and $|z_{n}|$ is sufficiently close to one, then

[TABLE]

by (10), (12) and (13). It follows that $\sup_{z_{n}\in\Lambda^{\star}}\,(1-|z_{n}|^{2})\,|f^{\prime}(z_{n})|=\infty$, and hence $f$ is non-normal by [10, Proposition 7].

Proof of Theorem 3.

Let $0<p\leq 1$, and let $\Lambda\subset\mathbb{D}$ be any separated sequence such that (5) is a $p$-Carleson measure. If $B$ is a Blaschke product whose zero-sequence is $\Lambda$, then $B\in Q_{p}\cap H^{\infty}$ by [4, Theorem 2.2]; the case $p=1$ is of course trivial, since $Q_{1}\cap H^{\infty}=H^{\infty}$. Now

[TABLE]

by the uniform separation of $\Lambda$, and hence

[TABLE]

Define $f=Be^{h}$, where $h=Bk$ and $k\in\mathcal{H}(\mathbb{D})$ is a solution to the interpolation problem

[TABLE]

By [20, Theorem 1.3], where the condition (b) follows from the fact that (5) is a $p$-Carleson measure, or [2, Theorem 3] if $p=1$, we may assume that $k\in Q_{p}\cap H^{\infty}$. Then, as in the proof of Theorem 1, it follows that $f$ is a non-trivial solution of (1) where $A\in\mathcal{H}(\mathbb{D})$ and

[TABLE]

Since $Q_{p}\cap H^{\infty}$ is an algebra, we have $h\in Q_{p}\cap H^{\infty}$ and hence $f\in Q_{p}\cap H^{\infty}$.

It remains to prove that (6) is a $p$-Carleson measure. Let $0<\delta<1$ be a sufficiently small constant such that the pseudo-hyperbolic discs $\Delta(z_{n},\delta)$ are pairwise disjoint for $z_{n}\in\Lambda$. We proceed to verify (4) for $\mu=\mu_{A,p}$ in two parts. Denote $\Omega=\,\mathbb{D}\setminus\,\bigcup_{z_{n}\in\Lambda}\Delta(z_{n},\delta)$. Since $|B|$ is uniformly bounded away from zero on $\Omega$ and $h\in H^{\infty}$, we obtain

[TABLE]

Consequently,

[TABLE]

as $f\in Q_{p}$; see [23, Theorem 3.2]. Only the term $-(B^{\prime\prime}+2B^{\prime}h^{\prime})/B$ in (14) brings us additional trouble on $\mathbb{D}\setminus\Omega$, and hence it suffices to show that

[TABLE]

is finite. Since $|B(z)|\gtrsim\varrho(z,z_{n})$ for $z\in\Delta(z_{n},\delta)$, and $|1-\overline{a}z|\simeq|1-\overline{a}z_{n}|$ for $z\in\Delta(z_{n},\delta)$ and $a\in\mathbb{D}$ (with comparison constants independent of $a$),

[TABLE]

by Lemma 7 and the fact that (5) is a $p$-Carleson measure. This completes the proof of Theorem 3. ∎

We may apply the Corona theorem for the algebra $Q_{p}\cap H^{\infty}$ to sharpen a property in [8] (corresponding to the case $p=1$). Let $0<p<1$. Assume that $f_{1},f_{2}\in Q_{p}\cap H^{\infty}$ are linearly independent solutions of (1) such that

[TABLE]

By [20, Theorem 1.1] there exist $g_{1},g_{2}\in Q_{p}\cap H^{\infty}$ such that $f_{1}g_{1}+f_{2}g_{2}\equiv 1$. Differentiate twice and apply (1) to conclude $A=f_{1}g_{1}^{\prime\prime}+f_{2}g_{2}^{\prime\prime}+2(f_{1}^{\prime}g_{1}^{\prime}+f_{2}^{\prime}g_{2}^{\prime})$. Hence $d\mu_{A,p}(z)=|A(z)|^{2}(1-|z|^{2})^{2+p}\,dm(z)$ is a $p$-Carleson measure.

To construct an example where (15) holds, we consider a method from [12, p. 58]. Let $0<p<1$, and choose $g\in Q_{p}\cap H^{\infty}$. Define $h\in\mathcal{H}(\mathbb{D})$ such that $h^{\prime}=e^{-2g}$, which implies that $h\in Q_{p}\cap H^{\infty}$ and $h^{\prime\prime}+2g^{\prime}h^{\prime}=0$. Then, functions $e^{g+h},e^{g-h}\in Q_{p}\cap H^{\infty}$ are zero-free linearly independent solutions of (1) where $A=-g^{\prime\prime}-(g^{\prime})^{2}-(h^{\prime})^{2}\in\mathcal{H}(\mathbb{D})$. Estimate (15) follows from the fact that both solutions are uniformly bounded away from zero. In this case it is easy to verify that $d\mu_{A,p}$ is a $p$-Carleson measure.

Proof of Proposition 5.

(a) Let $\Lambda\in\mathcal{Z}(H^{\infty}_{2})$ and let $\tau$ be a conformal automorphism of $\mathbb{D}$. We need to prove that $\tau(\Lambda)\in\mathcal{Z}(H^{\infty}_{2})$. By assumption, there exists $A=A(\Lambda)\in H^{\infty}_{2}$ such that (1) admits a non-trivial solution $f$ whose zero-sequence is $\Lambda$. Let $\varphi$ be the inverse of $\tau$. Consequently, $g=(f\circ\varphi)\,(\varphi^{\prime})^{-1/2}$ is a non-trivial solution of

[TABLE]

see [21, Lemma 1]. Since $g$ vanishes precisely on $\tau(\Lambda)$, and $\lVert B\rVert_{H^{\infty}_{2}}=\lVert A\rVert_{H^{\infty}_{2}}$ by standard estimates, we have $\tau(\Lambda)\in\mathcal{Z}(H^{\infty}_{2})$.

Note that (b) and the first part of (c) follow from Theorem 1, [25, Theorem 3] and Corollary 2. Hence, it suffices to prove the second part of (c). Suppose that $\Lambda\in\mathcal{Z}(H^{\infty}_{2})$ and (7) holds. Consequently, there exists a coefficient $A=A(\Lambda)\in H^{\infty}_{2}$ such that (1) admits a non-trivial solution $f$ whose zero-sequence is $\Lambda$. By [21, Example 1], $f\in H^{\infty}_{\alpha}$ for any sufficiently large $\alpha=\alpha(\lVert A\rVert_{H^{\infty}_{2}})<\infty$. According to (7) and [14, Theorem, p. 146], $\Lambda$ is a Blaschke set of $H^{\infty}_{\alpha}$, and hence $\Lambda$ is a Blaschke sequence. ∎

Proof of Corollary 6.

Let $\Lambda$ be a separated sequence having infinitely many points such that (5) is a $p$-Carleson measure, and let $f$ be the function in the proof of Theorem 3. In particular, $f=Be^{h}$ is a non-trivial solution of (1) where $A\in\mathcal{H}(\mathbb{D})$ and (6) is a $p$-Carleson measure. Here $B\in Q_{p}\cap H^{\infty}$ is the Blaschke product corresponding to $\Lambda$ and $h\in Q_{p}\cap H^{\infty}$.

Let $g$ be a solution of (1) which is linearly independent to $f$. We may assume that the Wronskian determinant satisfies $W(f,g)=fg^{\prime}-f^{\prime}g\equiv 1$. If we define $w=g/f$, then $w$ is a locally univalent meromorphic function in $\mathbb{D}$ such that $S_{w}=2A$ and $w^{\prime}=1/f^{2}$. Consequently, $S_{w}\in\mathcal{H}(\mathbb{D})$ and $|S_{w}(z)|^{2}(1-|z|^{2})^{2+p}\,dm(z)$ is a $p$-Carleson measure. It remains to show that $w$ is non-normal. Since $\Lambda$ is uniformly separated, there exists a constant $0<\delta<1$ such that $\delta<(1-|z_{n}|^{2})|B^{\prime}(z_{n})|\leq 1$ for all $z_{n}\in\Lambda$, and hence

[TABLE]

We conclude that

[TABLE]

which means that $w$ is non-normal. Finally, we point out that $w$ belongs to the Nevanlinna class of meromorphic functions by [9, Corollary 3] and [12, Lemma 3]. ∎

Acknowledgements

The author thanks A. Nicolau for helpful conversations, and gratefully acknowledges the hospitality of Departament de Matemàtiques, Universitat Autònoma de Barcelona.