Skip to main content

On a product-type operator from Zygmund-type spaces to Bloch-Orlicz spaces

Abstract

The boundedness and compactness of a product-type operator from Zygmund-type spaces to Bloch-Orlicz spaces are investigated in this paper.

1 Introduction

Let \(\mathcal{D}\) denote the unit disk in the complex plane \(\mathcal {C}\), and let \(\mathcal{H}(\mathcal{D})\) be the space of all holomorphic functions on \(\mathcal{D}\) with the topology of uniform convergence on compacts of \(\mathcal{D}\).

For \(0<\alpha<\infty\), the α-Bloch space, denoted by \(\mathcal{B}^{\alpha}\), consists of all functions \(f\in\mathcal {H}(\mathcal{D})\) such that

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

By \(\mathcal{Z}^{\alpha}\) we denote the Zygmund-type space consisting of those functions \(f\in\mathcal{H}(\mathcal{D})\) satisfying

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

\(\mathcal{B}^{\alpha}\) and \(\mathcal{Z}^{\alpha}\) are Banach spaces under the norms

$$\begin{aligned}& \|f\|_{\mathcal{B}^{\alpha}}=\bigl\vert f(0)\bigr\vert +\sup_{z\in\mathcal {D}} \bigl(1-|z|^{2}\bigr)^{\alpha}\bigl\vert f'(z)\bigr\vert , \\& \|f\|_{\mathcal{Z}^{\alpha}}=\bigl\vert f(0)\bigr\vert +\bigl\vert f'(0)\bigr\vert +\sup_{z\in\mathcal {D}}\bigl(1-|z|^{2} \bigr)^{\alpha}\bigl\vert f''(z)\bigr\vert , \end{aligned}$$

respectively. For some results on the Zygmund-type spaces on various domains in the complex plane and \(\mathcal{C}^{n}\) and operators on them, see, for example, [1–18]. The α-Bloch space is introduced and studied by numerous authors. For the general theory of α-Bloch or Bloch-type spaces and operators of them, see, e.g., [4, 19–41]. Recently, many authors studied different classes of Bloch-type spaces, where the typical weight function, \(\omega(z)=1-|z|^{2}\), \(z\in\mathcal{D}\), is replaced by a bounded continuous positive function μ defined on \(\mathcal{D}\). More precisely, a function \(f\in\mathcal{H}(\mathcal{D})\) is called a μ-Bloch function, denoted by \(f\in\mathcal{B}^{\mu}\), if \(\|f\|_{\mu}=\sup_{z\in\mathcal{D}}\mu(z)|f'(z)|<\infty\). If \(\mu(z)=\omega(z)^{\alpha}\), \(\alpha>0\), \(\mathcal{B}^{\mu}\) is just the α-Bloch space \(\mathcal{B}^{\alpha}\). It is readily seen that \(\mathcal{B}^{\mu}\) is a Banach space with the norm \(\|f\|_{\mathcal{B}^{\mu}}=|f(0)|+\|f\|_{\mu}\).

Recently, Ramos Fernández in [42] used Young’s functions to define the Bloch-Orlicz space. More precisely, let \(\varphi: [0,+\infty)\rightarrow[0,+\infty)\) be a strictly increasing convex function such that \(\varphi(0)=0\) and note that from these conditions it follows that \(\lim_{t\rightarrow+\infty}\varphi(t)=+\infty\). The Bloch-Orlicz space associated with the function φ, denoted by \(\mathcal{B}^{\varphi}\), is the class of all analytic functions f in \(\mathcal{D}\) such that

$$\sup_{z\in\mathcal{D}}\bigl(1-|z|^{2}\bigr)\varphi\bigl(\lambda \bigl\vert f'(z)\bigr\vert \bigr)< \infty $$

for some \(\lambda>0\) depending on f. Also, since φ is convex, it is not hard to see that Minkowski’s functional

$$\|f\|_{\varphi}=\inf \biggl\{ k>0:S_{\varphi} \biggl(\frac{f'}{k} \biggr)\leq1 \biggr\} $$

defines a seminorm for \(\mathcal{B}^{\varphi}\), which, in this case, is known as Luxemburg’s seminorm, where

$$S_{\varphi}(f)=\sup_{z\in\mathcal{D}}\bigl(1-|z|^{2}\bigr) \varphi\bigl(\bigl\vert f(z)\bigr\vert \bigr). $$

Moreover, it can be shown that \(\mathcal{B}^{\varphi}\) is a Banach space with the norm \(\|f\|_{\mathcal{B}^{\varphi}}=|f(0)|+\|f\|_{\varphi}\). We also have that the Bloch-Orlicz space is isometrically equal to a particular μ-Bloch space, where \(\mu(z)=\frac{1}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}\) with \(z\in \mathcal{D}\). Thus, for any \(f\in\mathcal{B}^{\varphi}\), we have

$$\|f\|_{\mathcal{B}^{\varphi}}=\bigl\vert f(0)\bigr\vert +\sup_{z\in\mathcal{D}} \mu(z)\bigl\vert f'(z)\bigr\vert . $$

When φ is the identity map on \([0,+\infty)\), \(\mathcal {B}^{\varphi}\) is the so-called Bloch space ℬ.

Let \(u\in\mathcal{H}(\mathcal{D})\) and Ï• be an analytic self-map of \(\mathcal{D}\). The differentiation operator D, the multiplication operator \(M_{u}\) and the composition operator \(C_{\phi}\) are defined by

$$(Df) (z)=f'(z),\qquad (M_{u}f) (z)=u(z)f(z),\qquad (C_{\phi}f) (z)=f\bigl(\phi(z)\bigr), \quad f\in\mathcal{H}(\mathcal{D}). $$

There is a considerable interest in studying the above mentioned operators as well as their products (see, e.g., [1–38, 41–56] and the related references therein).

A product-type operator \(DM_{u}C_{\phi}\) is defined as follows:

$$(DM_{u}C_{\phi}f) (z)=u'(z)f\bigl(\phi(z) \bigr)+u(z)\phi'(z)f'\bigl(\phi(z)\bigr),\quad u,f\in \mathcal{H}(\mathcal{D}). $$

For \(0<\alpha<\infty\) and \(\frac{1}{2}<|a|<1\), we define the test functions (see [1])

$$\begin{aligned}& f_{a}(z)=\frac{1}{\overline{a}^{2}} \biggl[\frac{(1-|a|^{2})^{2}}{ (1-\overline{a}z)^{\alpha}}-\frac{1-|a|^{2}}{(1-\overline {a}z)^{\alpha-1}} \biggr], \\& h_{a}(z)=\frac{1}{\overline{a}}\int^{z}_{0} \frac {1-|a|^{2}}{(1-\overline{a}\lambda)^{\alpha}}\, d\lambda,\quad z\in\mathcal{D}. \end{aligned}$$

It is easy to show that \(f_{a},h_{a}\in\mathcal{Z}^{\alpha}\) and \(f_{a}(a)=0\),

$$f'_{a}(a)=h'_{a}(a)= \frac{1}{\overline {a}}\bigl(1-|a|^{2}\bigr)^{1-\alpha},\qquad f''_{a}(a)=\frac{2\alpha}{(1-|a|^{2})^{\alpha}}, \qquad h''_{a}(a)=\frac {\alpha}{(1-|a|^{2})^{\alpha}}. $$

Esmaeili and Lindström in [1] investigated weighted composition operators between Zygmund-type spaces. Ramos Fernández in [42] studied the boundedness and compactness of composition operators on Bloch-Orlicz spaces. Li and Stević in [5] investigated products of Volterra-type operator and composition operator from \(H^{\infty}\) and Bloch spaces to Zygmund spaces, and they in [8] studied products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Liu and Yu in [25] characterized the boundedness and compactness of products of composition, multiplication and radial derivative operators from logarithmic Bloch spaces to weighted-type spaces on the unit ball. Sharma in [27] studied the boundedness and compactness of products of composition multiplication and differentiation between Bergman and Bloch-type spaces. In [52], Stević investigated the properties of weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Stević in [13] studied weighted radial operators from the mixed-norm space to the nth weighted-type space on the unit ball. Stević et al. in [54] characterized the boundedness and compactness of products of multiplication composition and differentiation operators on weighted Bergman spaces. Zhu in [18] studied extended Cesàro operators from mixed-norm spaces to Zygmund-type spaces.

Motivated by the above papers, in this paper, we investigate the boundedness and compactness of the product-type operator \(DM_{u}C_{\phi}\) from Zygmund-type spaces to the Bloch-Orlicz space. The paper is organized as follows. In Section 2, we give some necessary and sufficient conditions for the boundedness of the operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\). In Section 3, we give some necessary and sufficient conditions for the compactness of the operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\).

Throughout this paper,

$$\mu(z)=\frac{1}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}, $$

and we use letter C to denote a positive constant whose value may change at each occurrence.

2 The boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha }\rightarrow\mathcal{B}^{\varphi}\)

The following lemma was essentially proved in [3] and [11] (see also [1]).

Lemma 1

For \(f\in\mathcal{Z}^{\alpha}\) and \(\alpha>0\). Then:

  1. (i)

    For \(0<\alpha<1\), \(|f'(z)|\leq\frac{2}{1-\alpha}\|f\|_{\mathcal{Z}^{\alpha}}\) and \(|f(z)|\leq\frac{2}{1-\alpha}\|f\|_{\mathcal{Z}^{\alpha}}\).

  2. (ii)

    For \(\alpha=1\), \(|f'(z)|\leq\log\frac{e}{1-|z|^{2}}\|f\|_{\mathcal{Z}}\) and \(|f(z)|\leq\|f\|_{\mathcal{Z}}\).

  3. (iii)

    For \(\alpha>1\), \(|f'(z)|\leq\frac{2}{\alpha-1}\frac{\|f\|_{\mathcal{Z}^{\alpha }}}{(1-|z|^{2})^{\alpha-1}}\). For \(\alpha=2\), \(|f'(z)|\leq\frac{e}{1-|z|^{2}}\|f\|_{\mathcal{Z}^{2}}\).

  4. (iv)

    For \(1<\alpha<2\), \(|f(z)|\leq\frac{2}{(\alpha-1)(2-\alpha)}\|f\|_{\mathcal {Z}^{\alpha}}\).

  5. (v)

    For \(\alpha=2\), \(|f(z)|\leq2\log\frac{e}{1-|z|^{2}}\|f\|_{\mathcal{Z}^{2}}\).

  6. (vi)

    For \(\alpha>2\), \(|f(z)|\leq\frac{2}{(\alpha-1)(\alpha-2)}\frac{\|f\|_{\mathcal {Z}^{\alpha}}}{(1-|z|^{2})^{\alpha-2}}\).

Lemma 2

If \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded and \(0<\alpha<\infty\), then the following conditions hold:

$$\begin{aligned}& k_{1}=\sup_{z\in\mathcal{D}}\mu(z)\bigl\vert u''(z)\bigr\vert < \infty, \end{aligned}$$
(1)
$$\begin{aligned}& k_{2}=\sup_{z\in\mathcal{D}}\mu(z)\bigl\vert 2u'(z)\phi'(z)+u(z)\phi ''(z) \bigr\vert < \infty, \end{aligned}$$
(2)
$$\begin{aligned}& k_{3}=\sup_{z\in\mathcal{D}}\mu(z)\bigl\vert u(z)\bigr\vert \bigl\vert \phi'(z)\bigr\vert ^{2}< \infty. \end{aligned}$$
(3)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded. Taking the function \(f(z)=1\in\mathcal{Z}^{\alpha}\) and using the obvious fact that \(\|f\|_{\mathcal{Z}^{\alpha}}=1\), we have that

$$S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi}f)'(z)}{C\|f\|_{\mathcal {Z}^{\alpha}}} \biggr) =S_{\varphi} \biggl( \frac{u''(z)}{C} \biggr)=\sup_{z\in \mathcal{D}}\bigl(1-|z|^{2} \bigr)\varphi \biggl(\frac{|u''(z)|}{C} \biggr)\leq1, $$

from which it follows that (1) holds. Taking the function \(f(z)=z\in\mathcal{Z}^{\alpha}\) and using the fact that \(\|f\|_{\mathcal{Z}^{\alpha}}=1\), we obtain

$$\begin{aligned}& S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi}f)'(z)}{C\|f\|_{\mathcal {Z}^{\alpha}}} \biggr) \\& \quad = S_{\varphi} \biggl(\frac{u''(z)\phi(z)+2u'(z)\phi'(z)+u(z)\phi ''(z)}{C} \biggr) \\& \quad = \sup_{z\in \mathcal{D}}\bigl(1-|z|^{2}\bigr)\varphi \biggl(\frac{|u''(z)\phi (z)+2u'(z)\phi'(z)+u(z)\phi''(z)|}{C} \biggr)\leq1. \end{aligned}$$

Hence

$$\sup_{z\in\mathcal{D}}\mu(z)\bigl\vert 2u'(z) \phi'(z)+u(z)\phi ''(z)+u''(z) \phi(z)\bigr\vert < \infty. $$

From this, (1) and by the boundedness of \(\phi(z)\), condition (2) easily follows. Now taking the function \(f(z)=z^{2}\in\mathcal{Z}^{\alpha}\) and using the fact that \(\|f\|_{\mathcal{Z}^{\alpha}}=2\), we get

$$ S_{\varphi} \biggl(\frac{u''(z)(\phi(z))^{2}+2\phi (z)(2u'(z)\phi'(z)+u(z)\phi''(z))+2u(z) \phi'(z)^{2}}{2C} \biggr)\leq1. $$

Hence

$$\sup_{z\in\mathcal{D}}\mu(z)\bigl\vert u''(z) \bigl(\phi(z)\bigr)^{2}+2\phi (z) \bigl(2u'(z) \phi'(z)+u(z)\phi''(z)\bigr)+2u(z) \phi'(z)^{2}\bigr\vert < \infty. $$

From this, (1), (2) and the boundedness of \(\phi(z)\), we obtain (3). □

Now, we are ready to characterize the boundedness of the product-type operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\). For this purpose we need to break the problem into five different cases: \(0<\alpha<1\), \(\alpha=1\), \(1<\alpha<2\), \(\alpha=2\) and \(\alpha>2\).

Theorem 3

Let \(u\in\mathcal{H}(\mathcal{D})\), Ï• be an analytic self-map of \(\mathcal{D}\) and \(0<\alpha<1\). Then \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded if and only if \(k_{1}<\infty\), \(k_{2}<\infty\) and

$$ k_{4}=\sup_{z\in\mathcal{D}}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{(1-|\phi(z)|^{2})^{\alpha}}< \infty. $$
(4)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded, by Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). Now we will prove (4). Let

$$g_{\phi(\omega)}(z)=f_{\phi(\omega)}(z)-h_{\phi(\omega )}(z)+h_{\phi(\omega)} \bigl(\phi(\omega)\bigr) $$

for all \(z\in\mathcal{D}\) and \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega)|<1\), then \(g_{\phi(\omega)}\in\mathcal{Z}^{\alpha}\), and

$$g_{\phi(\omega)}\bigl(\phi(\omega)\bigr)=g'_{\phi(\omega)}\bigl( \phi(\omega)\bigr)=0,\qquad g''_{\phi(\omega)}\bigl( \phi(\omega)\bigr)=\frac{\alpha}{(1-|\phi(\omega )|^{2})^{\alpha}}. $$

By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}g_{\phi(\omega)}\|_{\mathcal{B}^{\varphi}}\leq C\), then

$$ 1 \geq S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi }g_{\phi(\omega)})'(z)}{C} \biggr)\geq \sup _{\frac{1}{2}< |\phi(\omega)|<1}\bigl(1-|\omega|^{2}\bigr)\varphi \biggl( \frac{\alpha|u(\omega)| |\phi'(\omega)|^{2}}{C(1-|\phi(\omega)|^{2})^{\alpha}} \biggr). $$

It follows that

$$ \sup_{\frac{1}{2}< |\phi(\omega)|<1} \frac{\mu(\omega)|u(\omega)||\phi'(\omega)|^{2}}{ (1-|\phi(\omega)|^{2})^{\alpha}}<\infty. $$
(5)

By \(k_{3}<\infty\), we see that

$$ \sup_{|\phi(\omega)|\leq\frac{1}{2}}\frac{\mu (\omega)|u(\omega)||\phi'(\omega)|^{2}}{ (1-|\phi(\omega)|^{2})^{\alpha}}\leq C \sup _{|\phi(\omega)|\leq\frac{1}{2}}\mu(\omega)\bigl\vert u(\omega)\bigr\vert \bigl\vert \phi '(\omega)\bigr\vert ^{2}< \infty. $$
(6)

From (5) and (6), we obtain (4).

Suppose that \(k_{1}, k_{2}, k_{4}<\infty\). For each \(f\in\mathcal{Z}^{\alpha}\setminus\{0\}\), by Lemma 1(i) we have

$$\begin{aligned}& S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi }f)'(z)}{C\|f\|_{\mathcal{Z}^{\alpha}}} \biggr) \\& \quad \leq \sup_{z\in\mathcal{D}}\bigl(1-|z|^{2}\bigr)\varphi \biggl[\frac{(k_{1}|f(\phi(z))| +k_{2}|f'(\phi(z))|+k_{4}(1-|\phi(z)|^{2})^{\alpha}|f''(\phi (z))|)}{C\mu(z)\|f\|_{\mathcal{Z}^{\alpha}}} \biggr] \\& \quad \leq \sup_{z\in\mathcal{D}}\bigl(1-|z|^{2}\bigr)\varphi \biggl[\frac{k_{1}\frac{2}{1-\alpha}+ k_{2}\frac{2}{1-\alpha}+k_{4}}{C\mu(z)} \biggr] \leq1, \end{aligned}$$

where C is a constant such that \(C\geq k_{1}\frac{2}{1-\alpha}+k_{2}\frac{2}{1-\alpha}+k_{4}\). Here we use the fact that

$$ \sup_{z\in\mathcal{D}}\bigl(1-\bigl\vert \phi(z)\bigr\vert ^{2}\bigr)^{\alpha }\bigl\vert f'' \bigl(\phi(z)\bigr)\bigr\vert \leq\|f\|_{\mathcal{Z}^{\alpha}}. $$

Now, we can conclude that there exists a constant C such that \(\|DM_{u}C_{\phi}f\|_{\mathcal{B}^{\varphi}}\leq C\|f\|_{\mathcal {Z}^{\alpha}}\) for all \(f\in\mathcal{Z}^{\alpha}\), so the product-type operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded. □

Theorem 4

Let \(u\in\mathcal{H}(\mathcal{D})\) and Ï• be an analytic self-map of \(\mathcal{D}\). Then \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\) is bounded if and only if \(k_{1}<\infty\),

$$\begin{aligned}& k_{5}=\sup_{z\in\mathcal{D}}\mu(z)\bigl\vert 2u'(z)\phi'(z)+u(z)\phi ''(z) \bigr\vert \log\frac{e}{1-|\phi(z)|^{2}}< \infty, \end{aligned}$$
(7)
$$\begin{aligned}& k_{6}=\sup_{z\in\mathcal{D}}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{1-|\phi(z)|^{2}}< \infty. \end{aligned}$$
(8)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow \mathcal{B}^{\varphi}\) is bounded, by Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). Let

$$\begin{aligned}& r(z)=(z-1) \biggl[ \biggl(1+\log\frac{e}{1-z} \biggr)^{2}+1 \biggr], \\& s_{a}(z)=\frac{r(\bar{a}z)}{\bar{a}} \biggl(\log\frac {e}{1-|a|^{2}} \biggr)^{-1} -\int^{z}_{0}\log \frac{e}{1-\bar{a}\lambda}\, d\lambda-c_{1}+c_{2}, \end{aligned}$$

where

$$ c_{1}=\frac{r(|a|^{2})}{\bar{a}} \biggl(\log\frac {e}{1-|a|^{2}} \biggr)^{-1},\qquad c_{2}=\int^{a}_{0} \log\frac{e}{1-\bar{a}\lambda}\, d\lambda $$

for any \(a\in\mathcal{D}\) such that \(\frac{1}{2}<|a|<1\). Then we have

$$ \bigl\vert s''_{a}(z)\bigr\vert = \frac{2}{1-|z|} \biggl(C+\log\frac{e}{1-|a|} \biggr) \biggl(\log \frac{e}{1-|a|^{2}} \biggr)^{-1} +\frac{1}{1-|z|}\leq\frac{C}{1-|z|} $$

for \(\frac{1}{2}<|a|<1\) and \(\sup_{\frac{1}{2}<|a|<1}\|s_{a}\|_{\mathcal{Z}}<\infty\).

Now let \(a=\phi(\omega)\), \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega)|<1\), then

$$ s_{\phi(\omega)}\bigl(\phi(\omega)\bigr)=s'_{\phi(\omega)}\bigl( \phi(\omega)\bigr)=0,\qquad s''_{\phi(\omega)}\bigl( \phi(\omega)\bigr)=\frac{\overline{\phi(\omega )}}{1-|\phi(\omega)|^{2}}. $$

By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}s_{\phi(\omega)}\|_{\mathcal{B}^{\varphi }}\leq C\), then

$$ 1 \geq S_{\varphi } \biggl(\frac{(DM_{u}C_{\phi}s_{\phi(\omega)})'(z)}{C} \biggr)\geq \sup _{\frac{1}{2}< |\phi(\omega)|<1}\bigl(1-|\omega|^{2}\bigr)\varphi \biggl( \frac{|u(\omega)||\phi'(\omega)|^{2}|\phi(\omega )|}{C(1-|\phi(\omega)|^{2})} \biggr). $$

From this it follows that

$$ \frac{1}{2}\sup_{\frac{1}{2}< |\phi(\omega)|<1} \frac{\mu(\omega)|u(\omega)||\phi'(\omega)|^{2}}{1-|\phi(\omega )|^{2}} \leq \sup_{\frac{1}{2}<|\phi(\omega)|<1} \frac{\mu(\omega)|u(\omega)||\phi'(\omega)|^{2}|\phi(\omega )|}{1-|\phi(\omega)|^{2}}<\infty. $$
(9)

By \(k_{3}<\infty\) we see that

$$ \sup_{|\phi(\omega)|\leq\frac{1}{2}}\frac{\mu (\omega)|u(\omega)||\phi'(\omega)|^{2}}{ 1-|\phi(\omega)|^{2}}\leq \frac{4}{3} \sup_{|\phi(\omega)|\leq\frac{1}{2}}\mu(\omega)\bigl\vert u(\omega) \bigr\vert \bigl\vert \phi '(\omega)\bigr\vert ^{2} < \infty. $$
(10)

From (9) and (10) we obtain \(k_{6}<\infty\).

Let

$$ t_{\phi(\omega)}(z)=\frac{r(\overline{\phi(\omega)}z)}{ \overline{\phi(\omega)}} \biggl(\log\frac{e}{1-|\phi(\omega )|^{2}} \biggr)^{-1}-c_{1} $$

for \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega )|<1\), then, as above, we can get that \(t_{\phi(\omega)}\in\mathcal{Z}\) and

$$ t_{\phi(\omega)}\bigl(\phi(\omega)\bigr)=0, \qquad t'_{\phi (\omega)} \bigl(\phi(\omega)\bigr)=\log\frac{e}{1-|\phi(\omega)|^{2}},\qquad t''_{\phi(\omega)} \bigl(\phi(\omega)\bigr)=\frac{2\overline{\phi(\omega )}}{1-|\phi(\omega)|^{2}}. $$

By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}t_{\phi(\omega)}\|_{\mathcal{B}^{\varphi }}\leq C\), then

$$\begin{aligned} 1 \geq& S_{\varphi } \biggl(\frac{(DM_{u}C_{\phi}t_{\phi(\omega)})'(z)}{C} \biggr) \\ \geq& \sup _{\frac{1}{2}< |\phi(\omega)|<1}\bigl(1-|\omega|^{2}\bigr)\varphi \biggl( \frac{|(DM_{u}C_{\phi}t_{\phi(\omega)})'(\omega)|}{C} \biggr) \\ \geq&\sup_{\frac{1}{2}<|\phi(\omega )|<1}\bigl(1-|\omega|^{2}\bigr) \\ &{}\cdot\varphi \biggl(\frac{ |(2u'(\omega)\phi '(\omega)+u(\omega) \phi''(\omega))\log\frac{e}{1-|\phi(\omega)|^{2}}+u(\omega)(\phi '(\omega))^{2}\frac{2\overline{\phi(\omega)}}{1-|\phi(\omega)|^{2}} |}{C} \biggr). \end{aligned}$$

From this and by \(k_{6}<\infty\), we get

$$\begin{aligned}& \sup_{\frac{1}{2}< |\phi (\omega)|<1} \mu(\omega)\bigl\vert 2u'(\omega)\phi'(\omega)+u(\omega)\phi''( \omega )\bigr\vert \log\frac{e}{1-|\phi(\omega)|^{2}} \\& \quad \leq C+2Ck_{6}<\infty. \end{aligned}$$
(11)

By \(k_{2}<\infty\) we see that

$$\begin{aligned}& \sup_{|\phi(\omega)|\leq\frac {1}{2}}\mu(\omega)\bigl\vert 2u'(\omega)\phi'(\omega)+u(\omega)\phi ''(\omega) \bigr\vert \log\frac{e}{1-|\phi(\omega)|^{2}} \\& \quad \leq C \sup_{|\phi(\omega)|\leq\frac{1}{2}}\mu(\omega)\bigl\vert 2u'(\omega)\phi '(\omega)+u(\omega) \phi''(\omega) \bigr\vert < \infty. \end{aligned}$$
(12)

From (11) and (12) we obtain (7).

Suppose that \(k_{1}, k_{5}, k_{6}<\infty\). Then, by Lemma 1(ii) and similar to the proof of Theorem 3, we get that \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal{B}^{\varphi}\) is bounded. □

Theorem 5

Let \(u\in\mathcal{H}(\mathcal{D})\), Ï• be an analytic self-map of \(\mathcal{D}\) and \(1<\alpha<2\). Then \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded if and only if \(k_{1}<\infty\),

$$\begin{aligned}& k_{7}=\sup_{z\in\mathcal{D}}\frac{\mu(z)|2u'(z)\phi '(z)+u(z)\phi''(z)|}{(1-|\phi(z)|^{2})^{\alpha-1}}< \infty, \end{aligned}$$
(13)
$$\begin{aligned}& k_{8}=\sup_{z\in\mathcal{D}}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{(1-|\phi(z)|^{2})^{\alpha}}< \infty. \end{aligned}$$
(14)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded, by Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). Inequality (14) can be proved as in Theorem 3. Using the test function \(f_{\phi(\omega)}(z)\) in Section 1, where \(z\in\mathcal{D}\), \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega)|<1\), then we have that \(f_{\phi(\omega)}\in\mathcal{Z}^{\alpha}\), and

$$\begin{aligned}& f_{\phi(\omega)}\bigl(\phi(\omega)\bigr)=0, \\& f'_{\phi(\omega)} \bigl(\phi(\omega)\bigr)=\frac{1}{\overline{\phi(\omega )}(1-|\phi(\omega)|^{2})^{\alpha-1}}, \\& f''_{\phi(\omega)} \bigl(\phi(\omega)\bigr)=\frac{2\alpha}{(1-|\phi(\omega )|^{2})^{\alpha}}. \end{aligned}$$

By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}f_{\phi(\omega)}\|_{\mathcal{B}^{\varphi}}\leq C\), then

$$\begin{aligned} 1 \geq& S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi }f_{\phi(\omega)})'(z)}{C} \biggr) \\ \geq& \sup_{\frac{1}{2}< |\phi(\omega)|<1}\bigl(1-|\omega|^{2}\bigr)\varphi \biggl(\frac{|(DM_{u}C_{\phi} f_{\phi(\omega)})'(\omega)|}{C} \biggr) \\ \geq&\sup_{\frac{1}{2}<|\phi(\omega)|<1}\bigl(1-|\omega |^{2}\bigr)\varphi \biggl(\frac{ | \frac{2u'(\omega)\phi'(\omega)+u(\omega) \phi''(\omega)}{\overline{\phi(\omega)}(1-|\phi(\omega )|^{2})^{\alpha-1}} +\frac{2\alpha u(\omega)\phi'(\omega)^{2}}{(1-|\phi(\omega)|^{2})^{\alpha}} |}{C} \biggr). \end{aligned}$$

From this and by \(k_{8}<\infty\), we get

$$ \sup_{\frac{1}{2}< |\phi(\omega)|<1} \frac{\mu(\omega)|2u'(\omega)\phi'(\omega)+u(\omega)\phi ''(\omega)|}{(1-|\phi(\omega)|^{2})^{\alpha-1}} \leq C+2C\alpha k_{8}<\infty. $$

Then, according to the former proof with \(k_{2}<\infty\), we can get (13).

Suppose that \(k_{1}, k_{7}, k_{8}<\infty\). Then, by Lemma 1(iii) and (iv) and similar to the proof of Theorem 3, we get that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded. □

Theorem 6

Let \(u\in\mathcal{H}(\mathcal{D})\) and Ï• be an analytic self-map of \(\mathcal{D}\). Then \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow \mathcal{B}^{\varphi}\) is bounded if and only if

$$\begin{aligned}& k_{9}=\sup_{z\in\mathcal{D}}\mu(z)\bigl\vert u''(z)\bigr\vert \log\frac{e}{1-|\phi (z)|^{2}}< \infty, \end{aligned}$$
(15)
$$\begin{aligned}& k_{10}=\sup_{z\in\mathcal{D}}\frac{\mu(z)|2u'(z)\phi '(z)+u(z)\phi''(z)|}{1-|\phi(z)|^{2}}< \infty, \end{aligned}$$
(16)
$$\begin{aligned}& k_{11}=\sup_{z\in\mathcal{D}}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{(1-|\phi(z)|^{2})^{2}}< \infty. \end{aligned}$$
(17)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal{B}^{\varphi}\) is bounded, from Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). By repeating the arguments in the proof of Theorem 3 and Theorem 5, (16) and (17) can be proved similarly. Hence we only need to show \(k_{9}<\infty\). For every \(z\in\mathcal{D}\) and \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega)|<1\), let \(p_{\phi(\omega)}(z)=\log\frac{e}{1-\overline{\phi(\omega)}z}\). Clearly \(p_{\phi(\omega)}\in\mathcal{Z}^{2}\), and \(p_{\phi(\omega)}(\phi(\omega))=\log\frac{e}{1-|\phi(\omega)|^{2}}\),

$$\begin{aligned}& p'_{\phi(\omega)}\bigl(\phi(\omega)\bigr)=\frac{\overline{\phi(\omega )}}{1-|\phi(\omega)|^{2}}, \\& p''_{\phi(\omega)}\bigl(\phi(\omega)\bigr)= \frac{\overline{\phi(\omega)}^{2}}{ (1-|\phi(\omega)|^{2})^{2}}. \end{aligned}$$

By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal{B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}p_{\phi(\omega)}\|_{\mathcal{B}^{\varphi}}\leq C\), then

$$\begin{aligned} 1 \geq& S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi }p_{\phi(\omega)})'(z)}{C} \biggr) \\ \geq& \sup_{\frac{1}{2}< |\phi(\omega)|<1}\bigl(1-|\omega|^{2}\bigr)\varphi \biggl(\frac{|(DM_{u}C_{\phi}p_{\phi(\omega)})'(\omega)|}{C} \biggr) \\ \geq&\sup_{\frac{1}{2}<|\phi(\omega)|<1}\bigl(1-|\omega|^{2}\bigr) \\ &{}\cdot\varphi \biggl(\frac{ |u''(\omega)\log\frac {e}{1-|\phi(\omega)|^{2}} +\frac {(2u'(\omega)\phi'(\omega)+u(\omega)\phi''(\omega))\overline{\phi(\omega)}}{1-|\phi(\omega)|^{2}} + \frac{u(\omega)(\phi'(\omega))^{2}\overline{\phi(\omega )}^{2}}{(1-|\phi(\omega)|^{2})^{2}}|}{C}\biggr). \end{aligned}$$

By \(k_{10}, k_{11}<\infty\) we get

$$ \sup_{\frac{1}{2}< |\phi(\omega)|<1}\mu(\omega)\bigl\vert u''(\omega)\bigr\vert \log \frac{e}{1-|\phi(\omega)|^{2}} \leq C+Ck_{10}+Ck_{11}<\infty. $$
(18)

By \(k_{1}<\infty\) we see that

$$ \sup_{|\phi(\omega)|\leq\frac{1}{2}}\mu(\omega)\bigl\vert u''(\omega )\bigr\vert \log\frac{e}{1-|\phi(\omega)|^{2}} \leq C \sup_{|\phi(\omega)|\leq\frac{1}{2}}\mu(\omega)\bigl\vert u''( \omega)\bigr\vert < \infty. $$
(19)

From (18) and (19) we obtain (15).

Suppose that \(k_{9}, k_{10}, k_{11}<\infty\). Then, by Lemma 1(iii) and (v) and similar to the proof of Theorem 3, we get that \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal{B}^{\varphi}\) is bounded. □

Theorem 7

Let \(u\in\mathcal{H}(\mathcal{D})\), Ï• be an analytic self-map of \(\mathcal{D}\) and \(\alpha>2\). Then \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded if and only if

$$\begin{aligned}& k_{12}=\sup_{z\in\mathcal{D}}\frac{\mu(z)|u''(z)|}{(1-|\phi (z)|^{2})^{\alpha-2}}< \infty, \end{aligned}$$
(20)
$$\begin{aligned}& k_{13}=\sup_{z\in\mathcal{D}}\frac{\mu(z)|2u'(z)\phi '(z)+u(z)\phi''(z)|}{(1-|\phi(z)|^{2})^{\alpha-1}}< \infty, \end{aligned}$$
(21)
$$\begin{aligned}& k_{14}=\sup_{z\in\mathcal{D}}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{(1-|\phi(z)|^{2})^{\alpha}}< \infty. \end{aligned}$$
(22)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded, by Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). With the same argument as in Theorem 5 one can show that (21) and (22) hold.

Now we prove that \(k_{12}<\infty\). For every \(a, z\in\mathcal{D}\), define \(q_{a}(z)=\frac{(1-|a|^{2})^{2}}{(1-\bar{a}z)^{\alpha}}\). Then \(\sup_{z\in\mathcal{D}}(1-|z|^{2})^{\alpha}|q''_{a}(z)|\leq4\alpha \cdot2^{\alpha}\cdot(\alpha+1)\), which shows that \(q_{a}\in\mathcal{Z}^{\alpha}\). Now we let \(a=\phi (\omega)\) for every \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi (\omega)|<1\), and we have

$$\begin{aligned}& q_{\phi(\omega)}\bigl(\phi(\omega)\bigr)=\frac{1}{(1-|\phi(\omega )|^{2})^{\alpha-2}}, \\& q'_{\phi(\omega)}\bigl(\phi(\omega)\bigr)=\frac{\alpha\overline{\phi (\omega)}}{(1-|\phi(\omega)|^{2})^{\alpha-1}},\qquad q''_{\phi(\omega)}\bigl(\phi(\omega)\bigr)= \frac{\alpha(\alpha+1)\overline {\phi(\omega)}^{2}}{(1-|\phi(\omega)|^{2})^{\alpha}}. \end{aligned}$$

By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}q_{\phi(\omega)}\|_{\mathcal{B}^{\varphi}}\leq C\), then

$$\begin{aligned} 1 \geq& S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi}q_{\phi(\omega )})'(z)}{C} \biggr) \\ \geq& \sup_{\frac{1}{2}< |\phi(\omega)|<1}\bigl(1-|\omega|^{2}\bigr) \\ &{}\cdot\varphi \biggl(\frac{ |\frac{u''(\omega )}{(1-|\phi(\omega)|^{2})^{\alpha-2}} +\frac{\alpha\overline{\phi(\omega)}(2u'(\omega)\phi'(\omega )+u(\omega)\phi''(\omega))}{(1-|\phi(\omega)|^{2})^{\alpha-1}} +\frac{\alpha(\alpha+1)\overline{\phi(\omega)}^{2}u(\omega )\phi'(\omega)^{2}}{(1-|\phi(\omega)|^{2})^{\alpha}} |}{C} \biggr). \end{aligned}$$

Then we have

$$\begin{aligned}& \sup_{\frac{1}{2}< |\phi(\omega)|<1}\frac{\mu(\omega )|u''(\omega)|}{(1-|\phi(\omega)|^{2})^{\alpha-2}} \\& \quad \leq C+\sup_{\frac{1}{2}<|\phi(\omega)|<1}\alpha \mu(\omega)\biggl\vert \frac{2u'(\omega)\phi'(\omega)+u(\omega)\phi ''(\omega)}{\overline{\phi(\omega)}(1-|\phi(\omega)|^{2})^{\alpha-1}} +\frac{\alpha+1}{2\alpha}\frac{2\alpha u(\omega)\phi'(\omega)^{2}}{(1-|\phi(\omega)|^{2})^{\alpha}}\biggr\vert \\& \quad \leq C +\sup_{\frac{1}{2}<|\phi(\omega)|<1} \alpha \mu(\omega) \biggl\{ \biggl\vert \frac{2u'(\omega)\phi'(\omega)+u(\omega )\phi''(\omega)}{\overline{\phi(\omega)} (1-|\phi(\omega)|^{2})^{\alpha-1}}\biggr\vert +\frac{\alpha+1}{2\alpha}\biggl\vert \frac{2\alpha u(\omega)\phi'(\omega)^{2}}{(1-|\phi(\omega)|^{2})^{\alpha}}\biggr\vert \biggr\} . \end{aligned}$$
(23)

Since \(\frac{\alpha+1}{2\alpha}<1\), then by (21), (22), (23) and according to the former proof with \(k_{1}<\infty\) for \(|\phi(\omega)|\leq\frac{1}{2}\), then \(k_{12}<\infty\). Suppose that \(k_{12}, k_{13}, k_{14}<\infty\). Then, by Lemma 1(iii) and (vi) and similar to the proof of Theorem 3, we get that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded. □

3 The compactness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha }\rightarrow\mathcal{B}^{\varphi}\)

In order to prove the compactness of the product-type operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\), we need the following lemmas. The proof of the following lemma is similar to that of Proposition 3.11 in [43]. The details are omitted.

Lemma 8

Let \(u\in\mathcal{H}(\mathcal{D})\), Ï• be an analytic self-map of \(\mathcal{D}\) and \(0<\alpha<\infty\). Then \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact if and only if \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded and for any bounded sequence \(\{f_{n}\}_{n\in\mathcal{N}}\) in \(\mathcal{Z}^{\alpha}\) which converges to zero uniformly on compact subsets of \(\mathcal{D}\) as \(n\rightarrow\infty\), we have \(\|DM_{u}C_{\phi}f_{n}\|_{\mathcal{B}^{\varphi}}\rightarrow0\) as \(n\rightarrow\infty\).

The following lemma was essentially proved in paper [11] in Lemma 2.5.

Lemma 9

Fix \(0<\alpha<2\) and let \(\{f_{n}\}_{n\in\mathcal{N}}\) be a bounded sequence in \(\mathcal{Z}^{\alpha}\) which converges to zero uniformly on compact subsets of \(\mathcal{D}\) as \(n\rightarrow\infty \). Then \(\lim_{n\rightarrow\infty}\sup_{z\in\mathcal{D}}|f_{n}(z)|=0\). Moreover, for \(0<\alpha<1\), if \(\{f_{n}\}_{n\in\mathcal{N}}\) is a bounded sequence in \(\mathcal{Z}^{\alpha}\) which converges to zero uniformly on compact subsets of \(\mathcal{D}\) as \(n\rightarrow\infty\), then \(\lim_{n\rightarrow\infty}\sup_{z\in\mathcal{D}}|f'_{n}(z)|=0\).

Theorem 10

Let \(u\in\mathcal{H}(\mathcal{D})\), Ï• be an analytic self-map of \(\mathcal{D}\) and \(0<\alpha<1\). Then \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact if and only if \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded,

$$ \lim_{|\phi(z)|\rightarrow1}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{(1-|\phi(z)|^{2})^{\alpha}}=0. $$
(24)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in\mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Taking the function

$$g_{n}(z)=\frac{1}{\overline{\phi(z_{n})}^{2}} \biggl[\frac{(1-|\phi (z_{n})|^{2})^{2}}{ (1-\overline{\phi(z_{n})}z)^{\alpha}}- \frac{1-|\phi (z_{n})|^{2}}{(1-\overline{\phi(z_{n})}z)^{\alpha-1}} \biggr] -\frac{1}{\overline{\phi(z_{n})}}\int^{z}_{0} \frac{1-|\phi (z_{n})|^{2}}{(1-\overline{\phi(z_{n})}\lambda)^{\alpha}}\, d\lambda. $$

Then \(\sup_{n\in\mathcal{N}}\|g_{n}\|_{\mathcal{Z}^{\alpha }}<\infty\), and \(g_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\). Since \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact, then \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}g_{n}\|_{\mathcal {B}^{\varphi}}=0\). Since \(\lim_{n\rightarrow\infty}|\phi(z_{n})|=1\), then \(\lim_{n\rightarrow\infty}\sup_{z\in\mathcal{D}}|g_{n}(z)|=0\). Moreover, we have

$$ g'_{n}\bigl(\phi(z_{n})\bigr)=0,\qquad g''_{n}\bigl(\phi(z_{n})\bigr)= \frac{\alpha}{(1-|\phi(z_{n})|^{2})^{\alpha}}. $$

Then

$$1\geq S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi}g_{n})'(z_{n})}{\| DM_{u}C_{\phi}g_{n}\|_{\mathcal{B}^{\varphi}}} \biggr)\geq \bigl(1-|z_{n}|^{2}\bigr)\varphi \biggl(\frac{ |u''(z_{n})g_{n}(\phi(z_{n}))+ \frac{\alpha u(z_{n})\phi'(z_{n})^{2}}{(1-|\phi(z_{n})|^{2})^{\alpha}} |}{ \|DM_{u}C_{\phi}g_{n}\|_{\mathcal{B}^{\varphi}}} \biggr). $$

Hence

$$ \biggl\vert \frac{\alpha \mu(z_{n})|u(z_{n})||\phi'(z_{n})|^{2}}{(1-|\phi (z_{n})|^{2})^{\alpha}} -\mu(z_{n})\bigl\vert u''(z_{n})\bigr\vert \bigl\vert g_{n}\bigl(\phi(z_{n})\bigr)\bigr\vert \biggr\vert \leq \|DM_{u}C_{\phi}g_{n}\|_{\mathcal{B}^{\varphi}}. $$

Therefore

$$ \lim_{|\phi(z_{n})|\rightarrow1}\frac{\mu(z_{n})|u(z_{n})||\phi '(z_{n})|^{2}}{(1-|\phi(z_{n})|^{2})^{\alpha}}= \lim_{n\rightarrow\infty} \frac{\alpha \mu(z_{n})|u(z_{n})||\phi'(z_{n})|^{2}}{(1-|\phi (z_{n})|^{2})^{\alpha}}=0. $$

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded and (24) holds. Then \(k_{1}, k_{2}, k_{3}<\infty\) by Lemma 2 and for every \(\epsilon>0\), there is \(\delta\in(0,1)\) such that

$$ \frac{\mu(z)|u(z)||\phi'(z)|^{2}}{(1-|\phi(z)|^{2})^{\alpha }}< \epsilon $$
(25)

whenever \(\delta<|\phi(z)|<1\). Assume that \(\{f_{n}\}_{n\in\mathcal {N}}\) is a sequence in \(\mathcal{Z}^{\alpha}\) such that \(\sup_{n\in\mathcal{N}} \|f_{n}\|_{\mathcal{Z}^{\alpha}}\leq L\), and \(f_{n}\) converges to 0 uniformly on compact subsets of \(\mathcal{D}\) as \(n\rightarrow \infty\). Let \(K=\{z\in\mathcal{D}:|\phi(z)|\leq\delta\}\). Then by \(k_{1}, k_{2}, k_{3}<\infty\) and (25) it follows that

$$\begin{aligned}& \sup_{z\in\mathcal{D}}\mu(z)\bigl\vert (DM_{u}C_{\phi }f_{n})'(z) \bigr\vert \\& \quad \leq \sup_{z\in\mathcal{D}}\mu(z)\bigl\vert u''(z) \bigr\vert \bigl\vert f_{n}\bigl(\phi(z)\bigr)\bigr\vert + \sup _{z\in\mathcal{D}}\mu(z)\bigl\vert 2u'(z) \phi'(z)+u(z)\phi ''(z)\bigr\vert \bigl\vert f'_{n}\bigl(\phi(z)\bigr)\bigr\vert \\& \qquad {}+\sup_{z\in K}\mu(z)\bigl\vert u(z)\bigr\vert \bigl\vert \phi'(z)\bigr\vert ^{2}\bigl\vert f''_{n}\bigl(\phi(z)\bigr)\bigr\vert + \sup _{z\in\mathcal{D}\setminus K}\mu(z)\bigl\vert u(z)\bigr\vert \bigl\vert \phi '(z)\bigr\vert ^{2}\bigl\vert f''_{n} \bigl(\phi(z)\bigr)\bigr\vert \\& \quad \leq k_{1}\sup_{z\in\mathcal{D}}\bigl\vert f_{n}\bigl(\phi(z)\bigr)\bigr\vert +k_{2}\sup _{z\in \mathcal{D}}\bigl\vert f'_{n}\bigl(\phi(z) \bigr)\bigr\vert +k_{3}\sup_{z\in K}\bigl\vert f''_{n}\bigl(\phi(z)\bigr)\bigr\vert \\& \qquad {} +\sup_{z\in\mathcal{D}\setminus K}\frac{\mu(z)\vert u(z)\vert \vert \phi'(z)\vert ^{2}(1-\vert \phi(z)\vert ^{2})^{\alpha }\vert f''_{n}(\phi(z))\vert }{ (1-\vert \phi(z)\vert ^{2})^{\alpha}} \\& \quad \leq k_{1}\sup_{\omega\in\mathcal{D}}\bigl\vert f_{n}(\omega)\bigr\vert +k_{2}\sup_{\omega \in\mathcal{D}} \bigl\vert f'_{n}(\omega)\bigr\vert + k_{3} \sup_{\vert \omega \vert \leq\delta}\bigl\vert f''_{n}( \omega)\bigr\vert +L\epsilon. \end{aligned}$$

Here we use the fact that \(\sup_{z\in\mathcal{D}}(1-|\phi(z)|^{2})^{\alpha}|f''_{n}(\phi (z))|\leq\|f_{n}\|_{\mathcal{Z}^{\alpha}}\leq L\). So we obtain

$$\begin{aligned} \begin{aligned}[b] &\|DM_{u}C_{\phi}f_{n} \|_{B^{\varphi}} \\ &\quad = \bigl\vert u'(0)f_{n}\bigl(\phi(0)\bigr)+u(0) \phi'(0)f'_{n}\bigl(\phi(0)\bigr)\bigr\vert + \sup_{z\in\mathcal{D}}\mu(z)\bigl\vert (DM_{u}C_{\phi}f_{n})'(z) \bigr\vert \\ &\quad \leq \bigl\vert u'(0)\bigr\vert \bigl\vert f_{n}\bigl(\phi(0)\bigr)\bigr\vert +\bigl\vert u(0)\bigr\vert \bigl\vert \phi'(0)\bigr\vert \bigl\vert f'_{n} \bigl(\phi (0)\bigr)\bigr\vert \\ &\qquad {} +k_{1}\sup_{\omega\in\mathcal{D}}\bigl\vert f_{n}(\omega)\bigr\vert +k_{2}\sup_{\omega\in\mathcal{D}} \bigl\vert f'_{n}(\omega)\bigr\vert + k_{3} \sup_{\vert \omega \vert \leq\delta}\bigl\vert f''_{n}( \omega)\bigr\vert +L\epsilon. \end{aligned} \end{aligned}$$
(26)

Since \(f_{n}\) converges to 0 uniformly on compact subsets of \(\mathcal {D}\) as \(n\rightarrow\infty\), Cauchy’s estimation gives that \(f'_{n}\), \(f''_{n}\) also do as \(n\rightarrow\infty\). In particular, since \(\{\omega:|\omega|\leq\delta\}\) and \(\{\phi(0)\}\) are compact, it follows that

$$\lim_{n\rightarrow\infty}\bigl\{ \bigl\vert u'(0)\bigr\vert \bigl\vert f_{n}\bigl(\phi(0)\bigr)\bigr\vert +\bigl\vert u(0) \bigr\vert \bigl\vert \phi '(0)\bigr\vert \bigl\vert f'_{n}\bigl(\phi(0)\bigr)\bigr\vert \bigr\} =0\quad \mbox{and}\quad \lim_{n\rightarrow\infty}k_{3}\sup _{\vert \omega \vert \leq\delta }\bigl\vert f''_{n}( \omega)\bigr\vert =0. $$

Moreover, since \(0<\alpha<1\), by Lemma 9 we have

$$\lim_{n\rightarrow\infty}\sup_{\omega\in\mathcal {D}}\bigl\vert f_{n}(\omega)\bigr\vert =0,\qquad \lim_{n\rightarrow\infty}\sup _{\omega\in\mathcal {D}}\bigl\vert f'_{n}(\omega)\bigr\vert =0. $$

Hence, letting \(n\rightarrow\infty\) in (26), we get

$$\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}f_{n} \|_{B^{\varphi}}=0. $$

Employing Lemma 8 the implication follows. □

Theorem 11

Let \(u\in\mathcal{H}(\mathcal{D})\) and Ï• be an analytic self-map of \(\mathcal{D}\). Then \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\) is compact if and only if \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\) is bounded,

$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\mu(z)\bigl\vert 2u'(z) \phi'(z)+u(z)\phi ''(z)\bigr\vert \log \frac{e}{1-|\phi(z)|^{2}} =0, \end{aligned}$$
(27)
$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{1-|\phi(z)|^{2}}=0. \end{aligned}$$
(28)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow \mathcal{B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal{B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in \mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, we may suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Taking the function

$$s_{n}(z)=\frac{r(\overline{\phi(z_{n})}z)}{\overline{\phi(z_{n})}} \biggl(\log\frac{e}{1-|\phi(z_{n})|^{2}} \biggr)^{-1}- \biggl(\log\frac{e}{1-|\phi(z_{n})|^{2}} \biggr)^{-2} \int ^{z}_{0}\log^{3}\frac{e}{1-\overline{\phi(z_{n})}\lambda }\, d \lambda. $$

Then \(\sup_{n\in\mathcal{N}}\|s_{n}\|_{\mathcal{Z}}<\infty\) by the proof of Theorem 4, and \(s_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\) by a direct calculation. Consequently, \(\lim_{n\rightarrow\infty}\sup_{z\in\mathcal{D}}|s_{n}(z)|=0\) by Lemma 9. Since \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal{B}^{\varphi}\) is compact, then \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}s_{n}\|_{\mathcal {B}^{\varphi}}=0\). Moreover, we have

$$ s'_{n}\bigl(\phi(z_{n})\bigr)=0,\qquad s''_{n}\bigl(\phi(z_{n})\bigr)=- \frac{\overline {\phi(z_{n})}}{1-|\phi(z_{n})|^{2}}. $$

Then

$$\begin{aligned} 1 \geq& S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi}s_{n})'(z_{n})}{\| DM_{u}C_{\phi}s_{n}\|_{\mathcal{B}^{\varphi}}} \biggr) \\ \geq&\bigl(1-|z_{n}|^{2}\bigr)\varphi \biggl( \frac{ |u''(z_{n})s_{n}(\phi(z_{n}))+ \frac{-\overline{\phi(z_{n})}u(z_{n})\phi'(z_{n})^{2}}{1-|\phi (z_{n})|^{2}} |}{ \|DM_{u}C_{\phi}g_{n}\|_{\mathcal{B}^{\varphi}}} \biggr). \end{aligned}$$

It follows that

$$ \biggl\vert \frac{\mu(z_{n})|\phi(z_{n})||u(z_{n})||\phi '(z_{n})|^{2}}{1-|\phi(z_{n})|^{2}} -\mu(z_{n})\bigl\vert u''(z_{n})\bigr\vert \bigl\vert s_{n}\bigl(\phi(z_{n})\bigr)\bigr\vert \biggr\vert \leq \|DM_{u}C_{\phi}s_{n}\|_{\mathcal{B}^{\varphi}}. $$

Therefore

$$ \lim_{|\phi(z_{n})|\rightarrow1}\frac{\mu(z_{n})|u(z_{n})||\phi '(z_{n})|^{2}}{1-|\phi(z_{n})|^{2}}= \lim _{n\rightarrow\infty}\frac{\mu(z_{n})|\phi (z_{n})||u(z_{n})||\phi'(z_{n})|^{2}}{1-|\phi(z_{n})|^{2}}=0. $$
(29)

On the other hand, let

$$ t_{n}(z)=\frac{\overline{\phi(z_{n})}z-1}{\overline {\phi(z_{n})}} \biggl[ \biggl(1+\log\frac{e}{1-\overline{\phi(z_{n})}z} \biggr)^{2}+1 \biggr] \biggl(\log\frac{e}{1-|\phi(z_{n})|^{2}} \biggr)^{-1}-c_{n}, $$

where

$$ c_{n}=\frac{|\phi(z_{n})|^{2}-1}{\overline{\phi(z_{n})}} \biggl[ \biggl(1+\log\frac{e}{1-|\phi(z_{n})|^{2}} \biggr)^{2}+1 \biggr] \biggl(\log\frac{e}{1-|\phi(z_{n})|^{2}} \biggr)^{-1} $$

such that \(\lim_{n\rightarrow\infty}c_{n}=0\). By a direct calculation, we may easily prove that \(t_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\), and \(\sup_{n\in\mathcal{N}}\| t_{n}\|_{\mathcal{Z}}<\infty\) by the proof of Theorem 4. Since \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal{B}^{\varphi}\) is compact, then \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}t_{n}\|_{\mathcal {B}^{\varphi}}=0\). Moreover, we have

$$ t_{n}\bigl(\phi(z_{n})\bigr)=0,\qquad t'_{n} \bigl(\phi(z_{n})\bigr)=\log\frac{e}{1-|\phi(z_{n})|^{2}},\qquad t''_{n}\bigl(\phi(z_{n})\bigr)= \frac{2\overline{\phi(z_{n})}}{1-|\phi(z_{n})|^{2}}. $$

Then

$$\begin{aligned} 1 \geq& S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi}t_{n})'(z_{n})}{ \|DM_{u}C_{\phi}t_{n}\|_{\mathcal{B}^{\varphi}}} \biggr) \\ \geq& \bigl(1-|z_{n}|^{2}\bigr) \\ &{}\cdot\varphi \biggl(\frac{ |(2u'(z_{n})\phi'(z_{n})+u(z_{n})\phi ''(z_{n}))\log\frac{e}{1-|\phi(z_{n})|^{2}}+ \frac{2\overline{\phi(z_{n})}u(z_{n})\phi'(z_{n})^{2}}{1-|\phi (z_{n})|^{2}} |}{ \|DM_{u}C_{\phi}t_{n}\|_{\mathcal{B}^{\varphi}}} \biggr). \end{aligned}$$

It follows that

$$\begin{aligned}& \mu(z_{n})\bigl\vert 2u'(z_{n}) \phi'(z_{n})+u(z_{n})\phi ''(z_{n}) \bigr\vert \log\frac{e}{1-|\phi(z_{n})|^{2}} \\& \quad \leq\|DM_{u}C_{\phi}t_{n}\|_{\mathcal{B}^{\varphi}} + \frac{2\mu(z_{n})|\phi(z_{n})||u(z_{n})||\phi '(z_{n})|^{2}}{1-|\phi(z_{n})|^{2}}. \end{aligned}$$
(30)

Letting \(n\rightarrow\infty\) in (30) and combining with (29), we can get

$$ \lim_{|\phi(z_{n})|\rightarrow1}\mu(z_{n})\bigl\vert 2u'(z_{n})\phi '(z_{n})+u(z_{n}) \phi''(z_{n}) \bigr\vert \log \frac{e}{1-|\phi(z_{n})|^{2}} =0. $$
(31)

The implication follows from (29) and (31).

Conversely, by Lemma 1(ii), Lemma 2, Lemma 8 and Lemma 9, we can prove the converse implication similar to Theorem 10, so we omit the details. □

Theorem 12

Let \(u\in\mathcal{H}(\mathcal{D})\), Ï• be an analytic self-map of \(\mathcal{D}\) and \(1<\alpha<2\). Then \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact if and only if \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded,

$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\frac{\mu(z)|2u'(z)\phi '(z)+u(z)\phi''(z)|}{(1-|\phi(z)|^{2})^{\alpha-1}}=0, \end{aligned}$$
(32)
$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{(1-|\phi(z)|^{2})^{\alpha}}=0. \end{aligned}$$
(33)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in\mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, we may suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Then (33) can be proved as the method of (24) in Theorem 10, so we only need to show that (32) holds. Taking the function

$$ f_{n}(z)=\frac{1}{\overline{\phi(z_{n})}^{2}} \biggl[\frac{(1-|\phi (z_{n})|^{2})^{2}}{ (1-\overline{\phi(z_{n})}z)^{\alpha}}- \frac{1-|\phi(z_{n})|^{2}}{ (1-\overline{\phi(z_{n})}z)^{\alpha-1}} \biggr]. $$

Then \(\sup_{n\in\mathcal{N}}\|f_{n}\|_{\mathcal{Z}^{\alpha }}<\infty\), and \(f_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\). Since \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact, it gives \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}f_{n}\|_{\mathcal {B}^{\varphi}}=0\). Moreover, we have

$$f_{n}\bigl(\phi(z_{n})\bigr)=0,\qquad f'_{n} \bigl(\phi(z_{n})\bigr)=\frac{1}{\overline{\phi (z_{n})}(1-|\phi(z_{n})|^{2})^{\alpha-1}},\qquad f''_{n} \bigl(\phi(z_{n})\bigr)=\frac{2\alpha}{(1-|\phi(z_{n})|^{2})^{\alpha}}. $$

Then

$$\begin{aligned} 1 \geq& S_{\varphi} \biggl(\frac{(DM_{u}C_{\phi}f_{n})'(z_{n})}{\| DM_{u}C_{\phi}f_{n}\|_{\mathcal{B}^{\varphi}}} \biggr) \\ \geq&\bigl(1-|z_{n}|^{2}\bigr)\varphi \biggl( \frac{ |\frac{2u'(z_{n})\phi '(z_{n})+u(z_{n})\phi''(z_{n})}{\overline{\phi(z_{n})}(1-|\phi(z_{n})|^{2})^{\alpha-1}}+ \frac{2\alpha u(z_{n})\phi'(z_{n})^{2}}{(1-|\phi(z_{n})|^{2})^{\alpha}} |}{ \|DM_{u}C_{\phi}f_{n}\|_{\mathcal{B}^{\varphi}}} \biggr). \end{aligned}$$

It follows that

$$\biggl\vert \frac{\mu(z_{n})|2u'(z_{n})\phi'(z_{n})+u(z_{n})\phi''(z_{n})|}{ |\phi(z_{n})|(1-|\phi(z_{n})|^{2})^{\alpha-1}} -\frac{2\alpha\mu(z_{n})|u(z_{n})||\phi'(z_{n})|^{2}}{(1-|\phi (z_{n})|^{2})^{\alpha}}\biggr\vert \leq \|DM_{u}C_{\phi}f_{n}\|_{\mathcal{B}^{\varphi}}. $$

Therefore

$$\lim_{|\phi(z_{n})|\rightarrow1}\frac{\mu(z_{n})|2u'(z_{n})\phi '(z_{n})+u(z_{n})\phi''(z_{n})|}{ (1-|\phi(z_{n})|^{2})^{\alpha-1}}=0. $$

By Lemma 1(iii), Lemma 2, Lemma 8 and Lemma 9, we can prove the converse implication similar to Theorem 10, so we omit the details. □

Theorem 13

Let \(u\in\mathcal{H}(\mathcal{D})\) and Ï• be an analytic self-map of \(\mathcal{D}\). Then \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow \mathcal{B}^{\varphi}\) is compact if and only if \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal {B}^{\varphi}\) is bounded,

$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\mu(z)\bigl\vert u''(z) \bigr\vert \log\frac{e}{1-|\phi (z)|^{2}}=0, \end{aligned}$$
(34)
$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\frac{\mu(z)|2u'(z)\phi '(z)+u(z)\phi''(z)|}{1-|\phi(z)|^{2}}=0, \end{aligned}$$
(35)
$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{(1-|\phi(z)|^{2})^{2}}=0. \end{aligned}$$
(36)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal{B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal {B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in\mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, we may suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Then, by repeating the arguments in the proof of Theorem 10 and Theorem 12, (35) and (36) can be proved similarly, so we only need to show that (34) holds. Taking the function

$$ p_{n}(z)= \biggl(1+ \biggl(\log\frac{e}{1-\overline{\phi (z_{n})}z} \biggr)^{2} \biggr) \biggl(\log\frac{e}{1-|\phi (z_{n})|^{2}} \biggr)^{-1}. $$
(37)

Then we have

$$\begin{aligned}& p'_{n}(z)=\frac{2\overline{\phi(z_{n})}}{1-\overline{\phi (z_{n})}z} \biggl(\log \frac{e}{ 1-\overline{\phi(z_{n})}z} \biggr) \biggl(\log\frac{e}{1-|\phi (z_{n})|^{2}} \biggr)^{-1}, \end{aligned}$$
(38)
$$\begin{aligned}& p''_{n}(z)=\frac{2\overline{\phi(z_{n})}^{2}}{(1-\overline{\phi (z_{n})}z)^{2}} \biggl(\log \frac{e}{ 1-\overline{\phi(z_{n})}z}+1 \biggr) \biggl(\log\frac{e}{1-|\phi (z_{n})|^{2}} \biggr)^{-1}. \end{aligned}$$
(39)

It is easy to show that \(\{p_{n}\}_{n\in\mathcal{N}}\) is a bounded sequence in \(\mathcal{Z}^{2}\), and \(p_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\). Since \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow \mathcal{B}^{\varphi}\) is compact, then \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}p_{n}\|_{\mathcal {B}^{\varphi}}=0\).

From (37), (38) and (39), we can get that

$$\begin{aligned}& \mu(z_{n})\bigl\vert u''(z_{n}) \bigr\vert \biggl[\log\frac{e}{1-|\phi(z_{n})|^{2}}+ \biggl(\log\frac{e}{1-|\phi(z_{n})|^{2}} \biggr)^{-1} \biggr] \\& \qquad {}- \frac{2\mu(z_{n})|\phi(z_{n})||2u'(z_{n})\phi'(z_{n})+u(z_{n})\phi ''(z_{n})|}{1-|\phi(z_{n})|^{2}} \\& \qquad {}-\frac{2\mu(z_{n})|\phi(z_{n})|^{2}|u(z_{n})||\phi '(z_{n})|^{2} [1+ (\log\frac{e}{1-|\phi(z_{n})|^{2}} )^{-1} ]}{ (1-|\phi(z_{n})|^{2})^{2}} \\& \quad \leq\|DM_{u}C_{\phi}p_{n}\|_{\mathcal{B}^{\varphi}}. \end{aligned}$$

Since \(\lim_{n\rightarrow\infty} (\log\frac{e}{1-|\phi (z_{n})|^{2}} )^{-1}=0\), and by (35) and (36), we can get (34).

By Lemma 1(iii) and (v), Lemma 2 and Lemma 8, we can prove the converse implication similar to Theorem 10, so we omit the details. □

Theorem 14

Let \(u\in\mathcal{H}(\mathcal{D})\), Ï• be an analytic self-map of \(\mathcal{D}\) and \(\alpha>2\). Then \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact if and only if \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded,

$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\frac{\mu(z)|u''(z)|}{(1-|\phi (z)|^{2})^{\alpha-2}}=0, \end{aligned}$$
(40)
$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\frac{\mu(z)|2u'(z)\phi '(z)+u(z)\phi''(z)|}{(1-|\phi(z)|^{2})^{\alpha-1}}=0, \end{aligned}$$
(41)
$$\begin{aligned}& \lim_{|\phi(z)|\rightarrow1}\frac{\mu(z)|u(z)||\phi '(z)|^{2}}{(1-|\phi(z)|^{2})^{\alpha}}=0. \end{aligned}$$
(42)

Proof

Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in\mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, we may suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Then, by repeating the arguments in the proof of Theorem 10 and Theorem 12, (41) and (42) can be proved similarly, so we only need to show that (40) holds.

Now let \(q_{n}(z)=\frac{(1-|\phi(z_{n})|^{2})^{2}}{(1-\overline{\phi (z_{n})}z)^{\alpha}}\), then \(\sup_{n\in\mathcal{N}}\|q_{n}\|_{\mathcal{Z}^{\alpha }}<\infty\), \(q_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\), and

$$\begin{aligned}& q_{n}\bigl(\phi(z_{n})\bigr)=\frac{1}{(1-|\phi (z_{n})|^{2})^{\alpha-2}}, \\& q'_{n}\bigl(\phi(z_{n})\bigr)= \frac{\alpha\overline{\phi(z_{n})}}{(1-|\phi (z_{n})|^{2})^{\alpha-1}},\qquad q''_{n}\bigl( \phi(z_{n})\bigr)=\frac{\alpha(\alpha +1)\overline{\phi(z_{n})}^{2}}{(1-|\phi(z_{n})|^{2})^{\alpha}}. \end{aligned}$$

Then we have

$$\begin{aligned}& \frac{\mu(z_{n})|u''(z_{n})|}{(1-|\phi (z_{n})|^{2})^{\alpha-2}} \\& \quad \leq\|DM_{u}C_{\phi}q_{n}\|_{\mathcal {B}^{\varphi}} \\& \qquad {}+\alpha\bigl\vert \phi(z_{n})\bigr\vert \mu(z_{n})\biggl\vert \frac{2u'(z_{n})\phi'(z_{n})+u(z_{n})\phi ''(z_{n})}{(1-|\phi(z_{n})|^{2})^{\alpha-1}}+ (\alpha+1)\overline{ \phi(z_{n})}\frac{u(z_{n})\phi '(z_{n})^{2}}{(1-|\phi(z_{n})|^{2})^{\alpha}}\biggr\vert \\& \quad \leq\|DM_{u}C_{\phi}q_{n}\|_{\mathcal {B}^{\varphi}}+ \alpha \mu(z_{n})\biggl\vert \frac{2u'(z_{n})\phi'(z_{n})+u(z_{n})\phi''(z_{n})}{ \overline{\phi(z_{n})}(1-|\phi(z_{n})|^{2})^{\alpha-1}} + \frac{\alpha+1}{2\alpha}\frac{2\alpha u(z_{n})\phi'(z_{n})^{2}}{ (1-|\phi(z_{n})|^{2})^{\alpha}}\biggr\vert \\& \quad \leq\|DM_{u}C_{\phi}q_{n}\|_{\mathcal {B}^{\varphi}} \\& \qquad {}+\alpha \mu(z_{n}) \biggl\{ \biggl\vert \frac{2u'(z_{n})\phi'(z_{n})+u(z_{n})\phi''(z_{n})}{ \overline{\phi(z_{n})}(1-|\phi(z_{n})|^{2})^{\alpha-1}} \biggr\vert + \frac{\alpha+1}{2\alpha}\biggl\vert \frac{2\alpha u(z_{n})\phi'(z_{n})^{2}}{ (1-|\phi(z_{n})|^{2})^{\alpha}}\biggr\vert \biggr\} . \end{aligned}$$
(43)

Since \(\frac{\alpha+1}{2\alpha}<1\), then by (41), (42) and letting \(n\rightarrow\infty\) in (43), we can get (40).

For the converse, by Lemma 1(iii) and (vi), Lemma 2 and Lemma 8, we can prove the converse implication similar to Theorem 10, so we omit the details. □

References

  1. Esmaeili, K, Lindström, M: Weighted composition operators between Zygmund type spaces and their essential norms. Integral Equ. Oper. Theory 75, 473-490 (2013)

    Article  MATH  Google Scholar 

  2. Li, S: Weighted composition operators from minimal Möbius invariant spaces to Zygmund spaces. Filomat 27(2), 267-275 (2013)

    Article  MathSciNet  Google Scholar 

  3. Li, S, Stević, S: Volterra type operators on Zygmund space. J. Inequal. Appl. 2007, Article ID 32124 (2007)

    Google Scholar 

  4. Li, S, Stević, S: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 338(2), 1282-1295 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Li, S, Stević, S: Products of Volterra type operator and composition operator from \(H^{\infty}\) and Bloch spaces to the Zygmund space. J. Math. Anal. Appl. 345, 40-52 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Li, S, Stević, S: Weighted composition operators from Zygmund spaces into Bloch spaces. Appl. Math. Comput. 206(2), 825-831 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Li, S, Stević, S: Integral-type operators from Bloch-type spaces to Zygmund-type spaces. Appl. Math. Comput. 215(2), 464-473 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Li, S, Stević, S: Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Appl. Math. Comput. 217(7), 3144-3154 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Li, S, Stević, S: On an integral-type operator from ω-Bloch spaces to μ-Zygmund spaces. Appl. Math. Comput. 215(12), 4385-4391 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Stević, S: On an integral operator from the Zygmund space to the Bloch-type space on the unit ball. Glasg. Math. J. 51, 275-287 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Stević, S: On an integral-type operator from Zygmund-type spaces to mixed-norm spaces on the unit ball. Abstr. Appl. Anal. 2010, Article ID 198608 (2010)

    Google Scholar 

  12. Stević, S: Weighted differentiation composition operators from the mixed-norm space to the nth weighted-type space on the unit disk. Abstr. Appl. Anal. 2010, Article ID 246287 (2010)

    Google Scholar 

  13. Stević, S: Weighted radial operator from the mixed-norm space to the nth weighted-type space on the unit ball. Appl. Math. Comput. 218(18), 9241-9247 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Stević, S, Sharma, AK: Composition operators from weighted Bergman-Privalov spaces to Zygmund type spaces on the unit disk. Ann. Pol. Math. 105(1), 77-86 (2012)

    Article  MATH  Google Scholar 

  15. Ye, S, Hu, Q: Weighted composition operators on the Zygmund spaces. Abstr. Appl. Anal. 2012, Article ID 462482 (2012)

    MathSciNet  Google Scholar 

  16. Zhu, X: Volterra type operators from logarithmic Bloch spaces to Zygmund type space. Int. J. Mod. Math. 3(3), 327-336 (2008)

    MATH  MathSciNet  Google Scholar 

  17. Zhu, X: Integral-type operators from iterated logarithmic Bloch spaces to Zygmund-type spaces. Appl. Math. Comput. 215(3), 1170-1175 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhu, X: Extended Cesàro operators from mixed norm spaces to Zygmund type spaces. Tamsui Oxf. J. Math. Sci. 26(4), 411-422 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Krantz, SG, Stević, S: On the iterated logarithmic Bloch space on the unit ball. Nonlinear Anal. 71, 1772-1795 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Li, S: Differences of generalized composition operators on the Bloch space. J. Math. Anal. Appl. 394(2), 706-711 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, S, Stević, S: Composition followed by differentiation between Bloch type spaces. J. Comput. Anal. Appl. 9(2), 195-205 (2007)

    MATH  MathSciNet  Google Scholar 

  22. Li, S, Stević, S: Weighted composition operators from Bergman-type spaces into Bloch spaces. Proc. Indian Acad. Sci. Math. Sci. 117(3), 371-385 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Li, S, Stević, S: Composition followed by differentiation from mixed-norm spaces to α-Bloch spaces. Sb. Math. 199(12), 1847-1857 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Li, S, Stević, S: Products of composition and integral type operators from \(H^{\infty}\) to the Bloch space. Complex Var. Elliptic Equ. 53(5), 463-474 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Liu, Y, Yu, Y: Products of composition, multiplication and radial derivative operators from logarithmic Bloch spaces to weighted-type spaces on the unit ball. J. Math. Anal. Appl. 423(1), 76-93 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  26. Madigan, K, Matheson, A: Compact composition operators on the Bloch space. Trans. Am. Math. Soc. 347(7), 2679-2687 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sharma, AK: Products of composition multiplication and differentiation between Bergman and Bloch type spaces. Turk. J. Math. 35, 275-291 (2011)

    MATH  Google Scholar 

  28. Stević, S: Norms of some operators from Bergman spaces to weighted and Bloch-type space. Util. Math. 76, 59-64 (2008)

    MATH  MathSciNet  Google Scholar 

  29. Stević, S: On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball. Discrete Dyn. Nat. Soc. 2008, Article ID 154263 (2008)

    Google Scholar 

  30. Stević, S: Integral-type operators from a mixed norm space to a Bloch-type space on the unit ball. Sib. Math. J. 50(6), 1098-1105 (2009)

    Article  MathSciNet  Google Scholar 

  31. Stević, S: Norm of weighted composition operators from α-Bloch spaces to weighted-type spaces. Appl. Math. Comput. 215, 818-820 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  32. Stević, S: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J. Math. Anal. Appl. 354, 426-434 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  33. Stević, S: Norm of an integral-type operator from Dirichlet to Bloch space on the unit disk. Util. Math. 83, 301-303 (2010)

    MATH  MathSciNet  Google Scholar 

  34. Stević, S: On an integral-type operator from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball. Appl. Math. Comput. 215(11), 3817-3823 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  35. Stević, S: On operator \(P_{\varphi}^{g}\) from the logarithmic Bloch-type space to the mixed-norm space on unit ball. Appl. Math. Comput. 215, 4248-4255 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  36. Stević, S: Characterizations of composition followed by differentiation between Bloch-type spaces. Appl. Math. Comput. 218(8), 4312-4316 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  37. Stević, S: On some integral-type operators between a general space and Bloch-type spaces. Appl. Math. Comput. 218(6), 2600-2618 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  38. Stević, S: Boundedness and compactness of an integral-type operator from Bloch-type spaces with normal weights to \(F(p,q,s)\) space. Appl. Math. Comput. 218(9), 5414-5421 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. Zhao, R: A characterization of Bloch-type spaces on the unit ball of \(C^{n}\). J. Math. Anal. Appl. 330(1), 291-297 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zhu, K: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 23(3), 1143-1177 (1993)

    Article  MATH  Google Scholar 

  41. Zhu, X: Generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces. J. Korean Math. Soc. 46(6), 1219-1232 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. Ramos Fernández, JC: Composition operators on Bloch-Orlicz type spaces. Appl. Math. Comput. 217(7), 3392-3402 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  43. Cowen, CC, MacCluer, BD: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Roton (1995)

    MATH  Google Scholar 

  44. Li, S, Stević, S: Weighted composition operators from \(H^{\infty}\) to the Bloch space on the polydisc. Abstr. Appl. Anal. 2007, Article ID 48478 (2007)

    Google Scholar 

  45. Li, S, Stević, S: Composition followed by differentiation between \(H^{\infty}\) and α-Bloch spaces. Houst. J. Math. 35(1), 327-340 (2009)

    MATH  Google Scholar 

  46. Sehba, B, Stević, S: On some product-type operators from Hardy-Orlicz and Bergman-Orlicz spaces to weighted-type spaces. Appl. Math. Comput. 233, 565-581 (2014)

    Article  MathSciNet  Google Scholar 

  47. Stević, S: Essential norms of weighted composition operators from the α-Bloch space to a weighted-type space on the unit ball. Abstr. Appl. Anal. 2008, Article ID 279691 (2008)

    Google Scholar 

  48. Stević, S: Norm and essential norm of composition followed by differentiation from α-Bloch spaces to \(H_{\mu}^{\infty}\). Appl. Math. Comput. 207, 225-229 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  49. Stević, S: On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces. Nonlinear Anal. 71(12), 6323-6342 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  50. Stević, S: Products of composition and differentiation operators on the weighted Bergman space. Bull. Belg. Math. Soc. Simon Stevin 16, 623-635 (2009)

    MATH  MathSciNet  Google Scholar 

  51. Stević, S: Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. Sib. Math. J. 50(4), 726-736 (2009)

    Article  MathSciNet  Google Scholar 

  52. Stević, S: Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput. 211(1), 222-233 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  53. Stević, S: Weighted differentiation composition operators from \(H^{\infty}\) and Bloch spaces to nth weighted-type spaces on the unit disk. Appl. Math. Comput. 216, 3634-3641 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  54. Stević, S, Sharma, AK, Bhat, A: Products of multiplication composition and differentiation operators on weighted Bergman spaces. Appl. Math. Comput. 217(20), 8115-8125 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  55. Stević, S, Ueki, SI: Integral-type operators acting between weighted-type spaces on the unit ball. Appl. Math. Comput. 215, 2464-2471 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  56. Zhu, X: Weighted composition operators from area Nevanlinna spaces into Bloch spaces. Appl. Math. Comput. 215, 4340-4346 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the NNSF of China (Nos. 11201127; 11271112) and IRTSTHN (14IRTSTHN023).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Haiying Li.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors completed the paper, read and approved the final manuscript.

Rights and permissions

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Guo, Z. On a product-type operator from Zygmund-type spaces to Bloch-Orlicz spaces. J Inequal Appl 2015, 132 (2015). https://doi.org/10.1186/s13660-015-0658-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13660-015-0658-8

MSC

Keywords