Generalized weighted composition operators on Bloch-type spaces

Abstract

In this paper, we give three different characterizations for the boundedness and compactness of generalized weighted composition operators on Bloch-type spaces, especially we characterize them in terms of the sequence of Bloch-type norms of the generalized weighted composition operator applied to the functions $I^{j}(z)=z^{j}$.

1 Introduction

Let $\mathbb{D}$ be an open unit disk in the complex plane ℂ and $H(\mathbb{D})$ be the space of analytic functions on $\mathbb {D}$. For $0<\alpha <\infty$, the Bloch-type space (or α-Bloch space) $\mathcal{B}^{\alpha}$ is the space that consists of all analytic functions f on $\mathbb{D}$ such that

$$B_{\alpha }(f)=\sup_{z \in\mathbb{D}}\bigl(1-|z|^{2} \bigr)^{\alpha}\bigl|f'(z)\bigr|< \infty. $$

$\mathcal{B}^{\alpha}$ becomes a Banach space under the norm $\|f\|_{\mathcal{B}^{\alpha}}=|f(0)|+B_{\alpha }(f)$. When $\alpha=1$, $\mathcal{B}^{1}=\mathcal{B}$ is the well-known Bloch space. See [1, 2] for more information on Bloch-type spaces.

Throughout this paper, φ denotes a nonconstant analytic self-map of $\mathbb{D}$. The composition operator $C_{\varphi}$ induced by φ is defined by $C_{\varphi}f = f \circ\varphi$ for $f \in H(\mathbb{D})$. For a fixed $u \in H(\mathbb{D})$, define a linear operator $uC_{\varphi}$ as follows:

$$uC_{\varphi}f =u ( f\circ\varphi) ,\quad f \in H(\mathbb{D}). $$

The operator $uC_{\varphi}$ is called the weighted composition operator. The weighted composition operator is a generalization of the composition operator and the multiplication operator defined by $M_{u}f=uf$.

A basic problem concerning composition operators on various Banach function spaces is to relate the operator theoretic properties of $C_{\varphi}$ to the function theoretic properties of the symbol φ, which attracted a lot of attention recently; the reader can refer to [3].

The differentiation operator D is defined by $Df=f'$, $f\in H(\mathbb{D})$. For a nonnegative integer n, we define

$$\bigl(D^{0} f\bigr) (z)=f(z),\qquad \bigl(D^{n} f\bigr) (z)=f^{(n)}(z),\quad n\ge1, f \in H(\mathbb{D}). $$

Let φ be an analytic self-map of $\mathbb{D}$, $u \in H(\mathbb {D})$, and let n be a nonnegative integer. Define the linear operator $D^{n}_{\varphi, u}$, called the generalized weighted composition operator, by (see [4–6])

$$\begin{aligned} \bigl(D^{n}_{\varphi, u} f\bigr) (z) =u(z)\cdot\bigl(D^{n} f\bigr) \bigl(\varphi(z)\bigr) ,\quad f \in H(\mathbb{D}), z\in\mathbb{D}. \end{aligned}$$

When $n=0$ and $u(z)=1$, $D^{n}_{\varphi,u}$ is the composition operator $C_{\varphi }$. If $n=0$, then $D^{n}_{\varphi,u}$ is the weighted composition operator $uC_{\varphi }$. If $n=1$, $u(z)=\varphi'(z)$, then $D^{n}_{\varphi, u}= DC_{\varphi}$, which was studied in [7–10]. For $u(z)=1$, $D^{n}_{\varphi, u}= C_{\varphi}D^{n}$, which was studied in [7, 11–14]. For the study of the generalized weighted composition operator on various function spaces, see, for example, [4–6, 15–19].

It is well known that the composition operator is bounded on the Bloch space by the Schwarz-Pick lemma. Composition operators and weighted composition operators on Bloch-type spaces were studied, for example, in [20–28]. The product-type operators on or into Bloch-type spaces have been studied in many papers recently, see [7–11, 13, 14, 18, 29–36] for example. In [27], Wulan et al. obtained a characterization for the compactness of the composition operators acting on the Bloch space as follows.

Theorem A

Let φ be an analytic self-map of $\mathbb{D}$. Then $C_{\varphi}: \mathcal{B}\rightarrow \mathcal{B}$ is compact if and only if

$$\lim_{j\rightarrow\infty}\bigl\| \varphi^{j} \bigr\| _{\mathcal{B}}=0. $$

In [14], Wu and Wulan obtained two characterizations for the compactness of the product of differentiation and composition operators acting on the Bloch space as follows.

Theorem B

Let φ be an analytic self-map of $\mathbb{D}$, $n\in \mathbb {N}$. Then the following statements are equivalent.

• (a) $C_{\varphi}D^{n}:\mathcal{B}\rightarrow \mathcal{B}$ is compact.

• (b) $\lim_{j\rightarrow\infty}\|C_{\varphi}D^{n} I^{j} \|_{\mathcal{B}}=0$, where $I^{j}(z)=z^{j}$.

• (c) $\lim_{|a|\rightarrow1}\|C_{\varphi}D^{n}\sigma_{a}(z)\|_{\mathcal{B}}=0$, where $\sigma_{a}(z)=(a-z)/(1-\overline{a}z)$ is the Möbius map on $\mathbb{D}$.

Motivated by Theorems A and B, in this work we show that $D^{n}_{\varphi,u}:\mathcal {B}^{\alpha }\to\mathcal{B}^{\beta}$ is bounded (respectively, compact) if and only if the sequence $(j^{\alpha -1}\|D^{n}_{\varphi,u}I^{j}\|_{\mathcal{B}^{\beta}})_{j=n}^{\infty}$ is bounded (respectively, convergent to 0 as $j\to\infty$), where $I^{j}(z)=z^{j}$. Moreover, we use two families of functions to characterize the boundedness and compactness of the operator $D^{n}_{\varphi, u}$.

Throughout the paper, we denote by C a positive constant which may differ from one occurrence to the next. In addition, we say that $A\preceq B$ if there exists a constant C such that $A\leq CB$. The symbol $A\approx B$ means that $A \preceq B \preceq A$.

2 Main results and proofs

In this section, we give our main results and proofs. First we characterize the boundedness of the operator $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha }\to\mathcal{B}^{\beta}$.

Theorem 1

Let n be a positive integer, $0<\alpha , \beta<\infty$, $u \in H(\mathbb{D})$ and φ be an analytic self-map of $\mathbb{D}$. Then the following statements are equivalent.

• (a) The operator $D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to \mathcal{B}^{\beta}$ is bounded.

• (b) $\sup_{j\geq n} j^{ \alpha-1}\|D^{n}_{\varphi, u} I^{j}(z)\|_{\mathcal{B}^{\beta}}<\infty$, where $I^{j}(z)=z^{j}$.

• (c) $u\in\mathcal{B}^{\beta}$, $\sup_{z\in\mathbb {D}}(1-|z|^{2})^{\beta}|u(z) | |\varphi'(z)|<\infty$ and

  $$\sup_{a\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}f_{a} \bigr\| _{\mathcal{B}^{\beta}} < \infty,\qquad \sup_{a\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}h_{a}\bigr\| _{\mathcal{B}^{\beta}} <\infty, $$

  where

  $$f_{a}(z)=\frac{1-|a|^{2}}{(1-\overline{a} z)^{\alpha}} \quad\textit{and} \quad h_{a}(z)= \frac{(1-|a|^{2})^{2}}{(1-\overline{a} z)^{\alpha+1}},\quad z\in \mathbb {D}. $$
• (d)
  $$\sup_{z\in\mathbb{D} } \frac{(1-|z |^{2})^{\beta}|u(z)|| \varphi' (z) | }{(1-|\varphi(z)|^{2})^{\alpha +n}} < \infty \quad\textit{and}\quad \sup _{z\in\mathbb{D} } \frac{(1-|z|^{2})^{\beta}|u'(z)|}{(1-|\varphi(z)|^{2})^{\alpha +n-1}}<\infty . $$

Proof

(a) ⇒ (b) This implication is obvious, since for $j\in\mathbb{N}$, the function $j^{ \alpha-1} I^{j}$ is bounded in $\mathcal{B}^{\alpha }$ and $j^{ \alpha-1}\|I^{j}\|_{\mathcal{B}^{\alpha }} \approx1$.

(b) ⇒ (c) Assume that (b) holds and let $Q=\sup_{j\ge n}j^{ \alpha-1}\|D^{n}_{\varphi,u}I^{j}\|_{\mathcal{B}^{\beta}}$. For any $a\in \mathbb {D}$, it is easy to see that $f_{a}$ and $h_{a}$ have bounded norms in $\mathcal{B}^{\alpha}$. It is clear that

$$\begin{aligned}& f_{a}(z)=\bigl(1-|a|^{2}\bigr)\sum _{j=0}^{\infty}\frac{\Gamma(j+\alpha)}{j!\Gamma (\alpha)} \overline{a}^{j}z^{j}, \\& h_{a}(z)=\bigl(1-|a|^{2}\bigr)^{2}\sum _{j=0}^{\infty}\frac{\Gamma(j+1+\alpha )}{j!\Gamma (\alpha+1)}\overline{a}^{j}z^{j}. \end{aligned}$$

By Stirling’s formula, we have $\frac{\Gamma(j+\alpha)}{j!\Gamma(\alpha)}\approx j^{\alpha-1}$ as $j\rightarrow\infty$. Using linearity we get

$$\begin{aligned}& \bigl\| D^{n}_{\varphi,u}f_{a}\bigr\| _{\mathcal{B}^{\beta}} \le C\bigl(1-|a|^{2}\bigr) \sum_{j=0}^{\infty}|a|^{j} j^{\alpha-1}\bigl\| D^{n}_{\varphi,u}I^{j}\bigr\| _{\mathcal{B}^{\beta}} \preceq Q\quad\mbox{and }\\& \bigl\| D^{n}_{\varphi,u}h_{a}\bigr\| _{\mathcal{B}^{\beta}} \le C\bigl(1-|a|^{2} \bigr)^{2}\sum_{j=0}^{\infty}(j+1)|a|^{j} j^{\alpha-1}\bigl\| D^{n}_{\varphi,u}I^{j}\bigr\| _{\mathcal{B}^{\beta}} \preceq Q. \end{aligned}$$

Therefore, by the arbitrariness of $a\in \mathbb {D}$,

$$\sup_{a\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}f_{a} \bigr\| _{\mathcal{B}^{\beta}} < \infty,\qquad \sup_{a\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}h_{a}\bigr\| _{\mathcal{B}^{\beta}} <\infty. $$

In addition, applying the operator $D^{n}_{\varphi, u}$ to $I^{j}$ with $j=n, n+1$, we obtain

$$\begin{aligned}& \bigl(D^{n}_{\varphi,u}I^{n}\bigr)'(z)=u'(z)n! \quad\mbox{and}\\& \bigl(D^{n}_{\varphi,u}I^{n+1}\bigr)'(z)=u'(z) (n+1)! \varphi (z)+u(z) (n+1)!\varphi '(z), \end{aligned}$$

while for $j< n$, $(D^{n}_{\varphi,u}I^{j})'(z)=0$. Thus, using the boundedness of the function φ, we have $u\in\mathcal{B}^{\beta}$ and $\sup_{z\in \mathbb{D}}(1-|z|^{2})^{\beta}|u(z) | |\varphi'(z)|<\infty$.

(c) ⇒ (d) Assume that (c) holds. Let

$$C_{1}:=\sup_{a\in \mathbb {D}} \bigl\| D^{n}_{\varphi,u}f_{a} \bigr\| _{\mathcal{B}^{\beta}},\qquad C_{2}:= \sup_{a\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}h_{a}\bigr\| _{\mathcal{B}^{\beta}} . $$

For $w\in\mathbb{D}$, set

$$g_{w}(z)=\frac{1-|w|^{2}}{(1-\overline{w} z)^{\alpha} } - \frac{\alpha}{\alpha+n}\frac{(1-|w|^{2})^{2}}{(1-\overline{w} z)^{\alpha+1 }} ,\quad w \in \mathbb {D}. $$

It is easy to check that $g_{w}\in\mathcal{B}^{\alpha }$, $\|g_{w}\|_{\mathcal{B}^{\alpha }} <\infty$ for every $w\in\mathbb{D}$. Moreover,

$$\begin{aligned} \sup_{w\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}g_{w} \bigr\| _{\mathcal{B}^{\beta}} \leq& \sup_{w\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}f_{w}\bigr\| _{\mathcal{B}^{\beta}}+ \frac{\alpha}{\alpha+n} \sup_{w\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}h_{w} \bigr\| _{\mathcal{B}^{\beta}}\\ \leq& C_{1}+\frac{\alpha }{\alpha +n}C_{2} < \infty. \end{aligned}$$

In addition,

$$g^{(n)}_{\varphi(\lambda)}\bigl(\varphi(\lambda)\bigr)=0, \qquad \bigl|g^{(n+1)}_{\varphi(\lambda)}\bigl(\varphi(\lambda)\bigr)\bigr|=\alpha (\alpha +1)\cdot \cdot \cdot (\alpha +n-1) \frac{|\varphi(\lambda)|^{n+1}}{(1-|\varphi(\lambda)|^{2})^{\alpha +n}}. $$

It follows that

$$\begin{aligned} C_{1}+\frac{\alpha }{\alpha +n}C_{2} >& \bigl\| D^{n}_{\varphi ,u}g_{\varphi (\lambda)}\bigr\| _{\mathcal{B}^{\beta}} \\ \geq& \alpha (\alpha +1)\cdot\cdot\cdot(\alpha +n-1) \frac{(1-|\lambda|^{2})^{\beta}|u(\lambda)||\varphi'(\lambda)| |\varphi(\lambda)|^{n+1}}{(1-|\varphi(\lambda)|^{2})^{\alpha +n}} \end{aligned}$$

(2.1)

for any $\lambda\in \mathbb {D}$. For any fixed $r\in (0,1)$, from (2.1) we have

$$\begin{aligned} \sup_{|\varphi(\lambda)|>r} \frac{(1-|\lambda|^{2})^{\beta}|u(\lambda)||\varphi'(\lambda)| }{(1-|\varphi(\lambda)|^{2})^{\alpha +n}} \leq& \sup_{|\varphi(\lambda)|>r} \frac{1}{r^{n+1}} \frac{(1-|\lambda|^{2})^{\beta}|u(\lambda)||\varphi'(\lambda)| |\varphi(\lambda)|^{n+1}}{(1-|\varphi(\lambda)|^{2})^{\alpha +n}} \\ \leq& \frac{ C_{1}+\frac{\alpha }{\alpha +n}C_{2} }{r^{n+1}\alpha (\alpha +1)\cdot\cdot\cdot (\alpha +n-1)} < \infty. \end{aligned}$$

(2.2)

From the assumption that $\sup_{z\in\mathbb{D}}(1-|z|^{2})^{\beta}|u(z) | |\varphi'(z)|<\infty$, we get

$$\begin{aligned} \sup_{|\varphi(\lambda)|\leq r}\frac{(1-|\lambda|^{2})^{\beta}|u(\lambda)||\varphi'(\lambda)| }{(1-|\varphi(\lambda)|^{2})^{\alpha +n}} \leq \frac{ \sup_{|\varphi(\lambda)|\leq r} (1-|\lambda|^{2})^{\beta}|u(\lambda)||\varphi'(\lambda )|}{(1-r^{2})^{\alpha +n}} < \infty. \end{aligned}$$

(2.3)

Therefore, (2.2) and (2.3) yield the first inequality of (d).

Next, note that

$$\begin{aligned} &C_{1}\ge\bigl\| D^{n}_{\varphi,u}f_{\varphi(\lambda)} \bigr\| _{\mathcal {B}^{\beta}} \\ &\quad \geq \alpha (\alpha +1)\cdot\cdot\cdot(\alpha +n-1) \frac{(1-|\lambda|^{2})^{\beta}|u'(\lambda)||\varphi(\lambda )|^{n}}{(1-|\varphi (\lambda)|^{2})^{\alpha +n-1}} \\ &\qquad{}-\alpha (\alpha +1)\cdot\cdot\cdot(\alpha +n)\frac{ (1-|\lambda|^{2})^{\beta}|u(\lambda)||\varphi'(\lambda)| |\varphi(\lambda)|^{n+1}}{(1-|\varphi(\lambda)|^{2})^{\alpha +n}} \end{aligned}$$

for any $\lambda\in \mathbb {D}$. From (2.1) we get

$$\begin{aligned} &\frac{(1-|\lambda|^{2})^{\beta}|u'(\lambda)||\varphi(\lambda )|^{n}}{(1-|\varphi(\lambda)|^{2})^{\alpha +n-1}}\\ &\quad \leq \frac{ \|D^{n}_{\varphi,u}f_{\varphi(\lambda)}\|_{\mathcal{B}^{\beta}}}{ \alpha (\alpha +1)\cdot\cdot\cdot(\alpha +n-1)} + \frac{(\alpha +n)(1-|\lambda|^{2})^{\beta}|u(\lambda)||\varphi'(\lambda)| |\varphi(\lambda)|^{n+1}}{(1-|\varphi(\lambda)|^{2})^{\alpha +n}}\\ &\quad \leq \frac{ C_{1}}{\alpha (\alpha +1)\cdot\cdot\cdot(\alpha +n-1)} + \frac{(\alpha +n) C_{1}+\alpha C_{2}}{\alpha (\alpha +1)\cdot\cdot\cdot(\alpha +n-1)}\\ &\quad \leq \frac{ (\alpha +n+1)C_{1}+\alpha C_{2}}{\alpha (\alpha +1)\cdot\cdot\cdot(\alpha +n-1)} . \end{aligned}$$

By arbitrary $\lambda\in\mathbb{D}$, we get

$$\begin{aligned} \sup_{\lambda\in\mathbb{D} }\frac{(1-|\lambda|^{2})^{\beta}|u'(\lambda)||\varphi(\lambda )|^{n}}{(1-|\varphi (\lambda)|^{2})^{\alpha +n-1} } < \infty. \end{aligned}$$

(2.4)

Combining (2.4) with the fact that $u \in\mathcal{B}^{\beta}$, similarly to the former proof, we get the second inequality of (d).

(d) ⇒ (a) For any $f \in\mathcal{B}^{\alpha }$, we have

$$\begin{aligned} &\bigl(1-|z |^{2}\bigr)^{\beta}\bigl| \bigl(D^{n}_{\varphi,u} f\bigr)'(z) \bigr| \\ &\quad=\bigl(1-|z |^{2}\bigr)^{\beta}\bigl| \bigl(f^{(n)}(\varphi)u \bigr)'(z) \bigr| \\ &\quad\leq \bigl(1-|z |^{2}\bigr)^{\beta}\bigl|u(z)\bigr|\bigl| \varphi' (z) \bigr| \bigl|f^{(n+1)}\bigl(\varphi(z)\bigr)\bigr|+ \bigl(1-|z |^{2}\bigr)^{\beta}\bigl| u' (z) \bigr| \bigl|f^{(n)} \bigl(\varphi(z)\bigr)\bigr| \\ &\quad \leq C\frac{(1-|z |^{2})^{\beta}|u(z)|| \varphi' (z) | }{(1-|\varphi(z)|^{2})^{\alpha +n}}\|f\|_{\mathcal{B}^{\alpha }} +C \frac{(1-|z|^{2})^{\beta}|u'(z)|}{(1-|\varphi(z)|^{2})^{\alpha +n-1}}\|f\| _{\mathcal {B}^{\alpha }}, \end{aligned}$$

(2.5)

where in the last inequality we used the fact that for $f \in \mathcal{B}^{\alpha }$ (see [2])

$$\sup_{z\in \mathbb {D}}\bigl(1-|z|^{2}\bigr)^{\alpha }\bigl|f'(z)\bigr| \asymp \bigl|f'(0)\bigr|+\cdots+\bigl|f^{(n)}(0)\bigr|+\sup _{z\in \mathbb {D}}\bigl(1-|z|^{2}\bigr)^{\alpha +n}\bigl|f^{(n+1)}(z)\bigr|. $$

Moreover

$$\bigl|\bigl(D^{n}_{\varphi,u} f\bigr) (0)\bigr|=\bigl|f^{(n)}\bigl( \varphi(0) \bigr)u(0) \bigr|\leq\frac{|u(0) |}{(1-|\varphi(0)|^{2})^{\alpha +n-1}}\|f\|_{\mathcal{B}^{\alpha }}. $$

From (d) we see that

$$\bigl\| D^{n}_{\varphi,u} f\bigr\| _{\mathcal{B}^{\beta}}=\bigl|\bigl(D^{n}_{\varphi,u} f\bigr) (0)\bigr|+ \sup_{z\in \mathbb {D}}\bigl(1-|z |^{2} \bigr)^{\beta}\bigl| \bigl(D^{n}_{\varphi,u} f\bigr)'(z) \bigr|< \infty. $$

Therefore the operator $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha }\rightarrow\mathcal{B}^{\beta}$ is bounded. The proof is complete. □

For the study of the compactness of $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\rightarrow\mathcal{B}^{\beta}$, we need the following lemma, which can be proved in a standard way; see, for example, Proposition 3.11 in [3].

Lemma 2

Let n be a positive integer, $0<\alpha , \beta<\infty$, $u \in H(\mathbb{D})$ and φ be an analytic self-map of $\mathbb{D}$. Then $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\rightarrow\mathcal{B}^{\beta}$ is compact if and only if $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\rightarrow\mathcal{B}^{\beta}$ is bounded and for any bounded sequence $(f_{j})_{j\in{ \mathbb {N}}}$ in $\mathcal{B}^{\alpha}$ which converges to zero uniformly on compact subsets of $\mathbb{D}$, $\|D^{n}_{\varphi,u} f_{j} \|_{\mathcal {B}^{\beta}}\to0$ as $j\to\infty$.

Theorem 3

Let n be a positive integer, $0<\alpha , \beta<\infty$, $u \in H(\mathbb{D})$ and φ be an analytic self-map of $\mathbb{D}$ such that $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\to\mathcal{B}^{\beta}$ is bounded. Then the following statements are equivalent.

• (a) $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\to\mathcal{B}^{\beta}$ is compact.

• (b) $\lim_{j\rightarrow\infty} j^{\alpha -1}\|D^{n}_{\varphi, u} I^{j} \|_{\mathcal{B}^{\beta}}=0$, where $I^{j}(z)=z^{j}$.

• (c) $\lim_{|\varphi (a)|\to1}\|D^{n}_{\varphi,u}f_{\varphi(a)}\|_{\mathcal{B}^{\beta}}=0$ and $\lim_{|\varphi (a)|\to1}\|D^{n}_{\varphi,u}h_{\varphi(a)}\|_{\mathcal{B}^{\beta}}=0$.

• (d)
  $$\lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z |^{2})^{\beta}|u(z)||\varphi'(z)|}{(1-|\varphi(z)|^{2})^{n+\alpha }}=0 \quad \textit{and}\quad \lim _{ |\varphi (z)|\rightarrow1}\frac{(1-|z|^{2})^{\beta}|u'(z)|}{(1-|\varphi(z)|^{2})^{n+\alpha -1}}=0. $$

Proof

(a) ⇒ (b) Assume that $D^{n}_{\varphi,u}:\mathcal {B}^{\alpha}\to\mathcal{B}^{\beta}$ is compact. Since the sequence $\{j^{\alpha -1}I^{j}\}$ is bounded in $\mathcal{B}^{\alpha}$ and converges to 0 uniformly on compact subsets, by Lemma 2 it follows that $j^{\alpha -1}\|D^{n}_{\varphi, u} I^{j}\| _{\mathcal{B}^{\beta}} \to0$ as $j\to\infty$.

(b) ⇒ (c) Suppose that (b) holds. Fix $\varepsilon >0$ and choose $N\in \mathbb {N}$ such that $j^{\alpha -1}\|D^{n}_{\varphi,u}I^{j}\|_{\mathcal {B}^{\beta}} <\varepsilon$ for all $j\ge N$. Let $z_{k} \in \mathbb{D}$ such that $|\varphi (z_{k})|\to1$ as $k\to\infty$. Arguing as in the proof of Theorem 1, we have

$$\begin{aligned} &\bigl\| D^{n}_{\varphi,u}f_{\varphi(z_{k})}\bigr\| _{\mathcal{B}^{\beta}} \\ &\quad\le C \bigl(1-\bigl|\varphi (z_{k})\bigr|^{2} \bigr)\sum_{j=0}^{\infty}\bigl|\varphi (z_{k})\bigr|^{j}j^{\alpha -1} \bigl\| D^{n}_{\varphi,u}I^{j}\bigr\| _{\mathcal {B}^{\beta}}\\ &\quad=C \bigl(1-\bigl|\varphi (z_{k})\bigr|^{2}\bigr) \Biggl(\sum _{j=0}^{N-1} \bigl|\varphi (z_{k})\bigr|^{j}j^{\alpha -1} \bigl\| D^{n}_{\varphi,u}I^{j}\bigr\| _{\mathcal{B}^{\beta}} + \sum_{j=N}^{\infty} \bigl|\varphi (z_{k})\bigr|^{j}j^{\alpha -1}\bigl\| D^{n}_{\varphi,u}I^{j} \bigr\| _{\mathcal{B}^{\beta}} \Biggr)\\ &\quad\le CQ\bigl(1-\bigl|\varphi (z_{k})\bigr|^{N}\bigr) + C\varepsilon , \end{aligned}$$

where $Q=\sup_{j\ge n}j^{\alpha -1}\|D^{n}_{\varphi,u}I^{j}\|_{\mathcal{B}^{\beta}}$. Since $|\varphi (z_{k})|\to1$ as $k\to\infty$, from the last inequality and the arbitrariness of ε, we get $\lim_{k\rightarrow\infty}\|D^{n}_{\varphi,u}f_{\varphi(z_{k})}\|_{\mathcal{B}^{\beta}} =0$, i.e., $\lim_{|\varphi (a)|\to 1}\|D^{n}_{\varphi,u}f_{\varphi(a)}\|_{\mathcal{B}^{\beta}} =0$.

Notice that

$$\sum_{j=0}^{N-1}(j+1)r^{j}= \frac{1-r^{N}-Nr^{N}(1-r)}{(1-r)^{2}},\quad 0\le r< 1, $$

arguing as in the proof of Theorem 1, we get

$$\begin{aligned} \bigl\| D^{n}_{\varphi,u}h_{\varphi (z_{k})}\bigr\| _{\mathcal{B}^{\beta}} \le& C\bigl(1-\bigl|\varphi (z_{k})\bigr|^{2} \bigr)^{2}\sum_{j=0}^{\infty}\bigl| \varphi (z_{k})\bigr|^{j}j^{\alpha }\bigl\| D^{n}_{\varphi,u}I^{j} \bigr\| _{\mathcal {B}^{\beta}}\\ \leq& C\bigl(1-\bigl|\varphi (z_{k})\bigr|^{2}\bigr)^{2}\sum _{j=0}^{N-1} (j+1)\bigl|\varphi (z_{k})\bigr|^{j}j^{\alpha -1} \bigl\| D^{n}_{\varphi,u}I^{j}\bigr\| _{\mathcal{B}^{\beta}}\\ &{} + C\bigl(1-\bigl|\varphi (z_{k})\bigr|^{2}\bigr)^{2}\sum _{j=N}^{\infty}(j+1)\bigl|\varphi (z_{k})\bigr|^{j}j^{\alpha -1} \bigl\| D^{n}_{\varphi,u}I^{j}\bigr\| _{\mathcal{B}^{\beta}}\\ \le& C (1-\bigl|\varphi (z_{k})\bigr|^{N}-N\bigl|\varphi (z_{k})\bigr|^{N} \bigl(1-\bigl|\varphi (z_{k})\bigr| \bigr)+ C\varepsilon . \end{aligned}$$

Therefore, $\lim_{k\to\infty}\|D^{n}_{\varphi,u}h_{\varphi (z_{k})}\|_{\mathcal{B}^{\beta}} \le C\varepsilon$. By the arbitrariness of ε, we obtain the desired result.

(c) ⇒ (d) To prove (d) we only need to show that if $(z_{k})_{k\in \mathbb {N}}$ is a sequence in $\mathbb{D}$ such that $|\varphi(z_{k})| \rightarrow1$ as $k\rightarrow\infty$, then

$$\lim_{k\to\infty}\frac{(1-|z_{k} |^{2})^{\beta}|u(z_{k})||\varphi'(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{\alpha +n}} =0, \qquad \lim_{k\to\infty} \frac{(1-|z_{k}|^{2})^{\beta}|u'(z_{k})|}{(1-|\varphi (z_{k})|^{2})^{\alpha +n-1}}=0. $$

Let $(z_{k})_{k\in \mathbb {N}}$ be such a sequence that $|\varphi(z_{k})| \rightarrow1$ as $k\rightarrow\infty$. Arguing as in the proof of Theorem 1, we obtain

$$\lim_{k\to\infty}\bigl\| D^{n}_{\varphi,u}g_{\varphi (z_{k})}\bigr\| _{\mathcal{B}^{\beta}} \le \lim _{k\to\infty}\bigl\| D^{n}_{\varphi,u}f_{\varphi (z_{k})}\bigr\| _{\mathcal{B}^{\beta}} + \frac {\alpha }{n+\alpha }\lim_{k\to\infty}\bigl\| D^{n}_{\varphi,u}h_{\varphi (z_{k})} \bigr\| _{\mathcal{B}^{\beta}} =0. $$

Hence $\lim_{k\to\infty}\|D^{n}_{\varphi,u}g_{\varphi (z_{k})}\|_{\mathcal{B}^{\beta}} = 0$. Similarly to the proof of Theorem 1, we have

$$\frac{n! (1-|z_{k}|^{2})^{\beta}|u(z_{k})||\varphi'(z_{k})||\varphi(z_{k})|^{n+1} }{(1-|\varphi(z_{k})|^{2})^{\alpha +n}}\leq\bigl\| D^{n}_{\varphi,u} g_{\varphi (z_{k})} \bigr\| _{{\mathcal{B}^{\beta}}}\rightarrow0 \quad\mbox{as } k\rightarrow\infty, $$

which implies

$$\begin{aligned} \lim_{k\to\infty}\frac{(1-|z_{k}|^{2})^{\beta}|u(z_{k})||\varphi'(z_{k})| }{(1-|\varphi(z_{k})|^{2})^{\alpha +n}} = \lim_{k\to\infty}\frac{(1-|z_{k}|^{2})^{\beta}|u(z_{k})||\varphi'(z_{k})||\varphi(z_{k})|^{n+1} }{(1-|\varphi(z_{k})|^{2})^{\alpha +n}}=0. \end{aligned}$$

(2.6)

In addition,

$$\begin{aligned} &\bigl\| D^{n}_{\varphi,u} f_{\varphi (z_{k})} \bigr\| _{{\mathcal {B}^{\beta}} }+ \frac{ (n+1)!(1-|z_{k}|^{2})^{\beta}|u(z_{k})||\varphi'(z_{k})| |\varphi(z_{k})|^{n+1}}{(1-|\varphi(z_{k})|^{2})^{\alpha +n}}\\ &\quad\geq \frac{ n!(1-|z_{k}|^{2})^{\beta}|u'(z_{k})||\varphi(z_{k})|^{n}}{(1-|\varphi (z_{k})|^{2})^{\alpha +n-1}}. \end{aligned}$$

From (2.6) and the assumption that $\|D^{n}_{\varphi,u} f_{\varphi (z_{k})} \|_{{\mathcal{B}^{\beta}} }\to0$ as $k\to\infty$, we have

$$\lim_{k\to\infty} \frac{(1-|z_{k}|^{2})^{\beta}|u'(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{n} } =\lim_{k\to\infty} \frac{(1-|z_{k}|^{2})^{\beta}|u'(z_{k})||\varphi(z_{k})|^{n}}{(1-|\varphi(z_{k})|^{2})^{\alpha +n-1}}=0, $$

as desired.

(d) ⇒ (a) Assume that $(f_{k})_{k\in \mathbb {N}}$ is a bounded sequence in $\mathcal{B}^{\alpha }$ converging to 0 uniformly on compact subsets of $\mathbb{D}$. By the assumption, for any $\varepsilon>0$, there exists $\delta\in(0,1)$ such that

$$\begin{aligned} \frac{(1-|z|^{2})^{\beta}|\varphi '(z)||u(z)|}{(1-|\varphi (z)|^{2})^{\alpha +n}}< \varepsilon \quad \mbox{and}\quad \frac{(1-|z|^{2})^{\beta}|u'(z)|}{(1-|\varphi(z)|^{2})^{\alpha +n-1}}< \varepsilon \end{aligned}$$

(2.7)

when $\delta<|\varphi(z)|<1$. Suppose that $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha }\to\mathcal{B}^{\beta}$ is bounded, by Theorem 1, we have

$$\begin{aligned} C_{3}=\sup_{z \in\mathbb {D}}\bigl(1-|z|^{2} \bigr)^{\beta}\bigl|u'(z)\bigr| < \infty \end{aligned}$$

(2.8)

and

$$\begin{aligned} C_{4}=\sup_{z \in\mathbb{D}}\bigl(1-|z |^{2} \bigr)^{\beta} \bigl|u(z)\bigr|\bigl|\varphi'(z)\bigr| < \infty. \end{aligned}$$

(2.9)

Let $K=\{ z\in\mathbb{D}:|\varphi(z)| \leq\delta\}$. Then by (2.8) and (2.9) we have that

$$\begin{aligned} & \sup_{z\in\mathbb{D}} \bigl(1-|z|^{2}\bigr)^{\beta}\bigl| \bigl(D^{n}_{\varphi ,u}f_{k}\bigr)'(z)\bigr|\\ &\quad\leq\sup_{z\in K}\bigl(1-|z|^{2}\bigr)^{\beta}\bigl|u(z)\bigr|\bigl| \varphi'(z)\bigr| \bigl|f_{k}^{(n+1)}\bigl(\varphi(z)\bigr)\bigr|+ \sup_{z\in K} \bigl(1-|z |^{2}\bigr)^{\beta}\bigl|u' (z) \bigr| \bigl|f_{k}^{(n)}\bigl(\varphi(z)\bigr)\bigr|\\ &\qquad{} +C\sup_{z\in\mathbb{D}\setminus K} \frac{(1-|z|^{2})^{\beta}|u(z)||\varphi'(z)|}{(1-|\varphi(z)|^{2})^{\alpha +n}} \|f_{k} \|_{\mathcal{B}^{\alpha }}+C\sup_{z\in\mathbb{D}\setminus K} \frac{(1-|z|^{2})^{\beta}|u'(z)|}{(1-|\varphi(z)|^{2})^{\alpha +n-1}} \|f_{k}\| _{\mathcal{B}^{\alpha }}\\ &\quad\leq C_{4} \sup_{z\in K} \bigl|f_{k}^{(n+1)} \bigl(\varphi(z)\bigr)\bigr|+C_{3}\sup_{z\in K} \bigl|f_{k}^{(n)}\bigl(\varphi(z)\bigr)\bigr| +C\varepsilon \|f_{k}\|_{{\mathcal{B}}^{\alpha }}, \end{aligned}$$

i.e., we get

$$\begin{aligned} \bigl\| D^{n}_{\varphi,u}f_{k}\bigr\| _{\mathcal{B}^{\beta}} =&C_{4} \sup_{|w| \leq\delta} \bigl|f_{k}^{(n+1)}(w)\bigr|+C_{3} \sup_{|w| \leq\delta} \bigl|f_{k}^{(n)}(w)\bigr| \\ &{} +C\varepsilon\|f_{k}\|_{{\mathcal{B}}^{\alpha }}+\bigl|u(0)\bigr|\bigl|f^{(n)}_{k} \bigl(\varphi (0)\bigr)\bigr|. \end{aligned}$$

(2.10)

Since $f_{k}$ converges to 0 uniformly on compact subsets of $\mathbb{D}$ as $k\to\infty$, Cauchy’s estimate gives that $f^{(n)}_{k} \to0$ as $k\to\infty$ on compact subsets of $\mathbb{D}$. Hence, letting $k\to\infty$ in (2.10) and using the fact that ε is an arbitrary positive number, we obtain $\|D^{n}_{\varphi,u} f_{k}\|_{\mathcal{B}^{\beta}}\rightarrow0$ as $k\to\infty$. Applying Lemma 2 the result follows. □