1 Introduction

Let \({\mathbb{D}}=\{z\in {\mathbb{C}}:|z|<1\}\) be the open unit disk in the complex plane \({\mathbb{C}}\) and \(H({\mathbb{D}})\) the class of all analytic functions on \({\mathbb{D}}\). Let φ be an analytic self-map of \({\mathbb{D}}\) and \(\psi\in H({\mathbb{D}})\). The weighted composition operator \(W_{\varphi ,\psi}\) on \(H({\mathbb{D}})\) is defined by

$$W_{\varphi ,\psi}f(z)=\psi(z)f\bigl(\varphi (z)\bigr),\quad z\in {\mathbb{D}}. $$

If \(\psi\equiv1\), it becomes the composition operator, usually denoted by \(C_{\varphi }\). If \(\varphi (z)=z\), it becomes the multiplication operator, usually denoted by \(M_{\psi}\). Hence, since \(W_{\varphi ,\psi}=M_{\psi}C_{\varphi }\), it is a product-type operator. A standard problem is to provide function theoretic characterizations when φ and ψ induce a bounded or compact weighted composition operator (see, e.g., [1–5] and the references therein).

A systematic study of other product-type operators started by Stević et al. since the publication of papers [6] and [7]. Before that there were a few papers in the topic, e.g., [8]. The differentiation operator on \(H({\mathbb{D}})\) is defined by

$$D f(z)=f'(z),\quad z\in {\mathbb{D}}. $$

The next two product-type operators \(D C_{\varphi }\) and \(C_{\varphi }D\), attracted some attention first (see, e.g., [9–12] and the references therein). The publication of [7] attracted some attention in product-type operators involving integral-type ones (see, e.g., [13–17] and the references therein). Since that time there has been a great interest in various product-type operators on spaces of holomorphic functions. For example, the six product-type operators from Bergman spaces to Bloch type spaces

$$\begin{aligned} M_{\psi}C_{\varphi }D, \qquad M_{\psi}DC_{\varphi }, \qquad C_{\varphi }M_{\psi}D,\qquad C_{\varphi }D M_{\psi}, \qquad DC_{\varphi }M_{\psi},\qquad DM_{\psi}C_{\varphi } \end{aligned}$$
(1)

were studied by Sharma in [18]. The next product-type operators \(W_{\varphi ,\psi}D\) and \(DW_{\varphi ,\psi}\), which were considered in [19] and [20], are included in (1) as the first and sixth operators, respectively. For some other product-type operators, see, e.g., [14, 21–29] and the references therein.

In order to treat operators in (1) in a unified manner, Stević and Sharma introduced the following so-called Stević-Sharma operator:

$$\begin{aligned} T_{\psi_{1},\psi_{2},\varphi }f(z)=\psi_{1}f\bigl(\varphi (z)\bigr)+ \psi_{2}(z)f'\bigl(\varphi (z)\bigr),\quad f\in H( {\mathbb{D}}). \end{aligned}$$
(2)

For example, in [30] and [31] the operator was studied on the weighted Bergman space.

By using Stević-Sharma operator all six possible products of composition, multiplication, and differentiation operators can be obtained. More specifically we have

$$\begin{aligned}& M_{\psi}C_{\varphi }D=T_{0,\psi, \varphi },\qquad M_{\psi}D C_{\varphi }=T_{0,\psi \varphi ',\varphi },\qquad C_{\varphi }M_{\psi}D=T_{0,\psi\circ \varphi ,\varphi }, \\& C_{\varphi }DM_{\psi}=T_{\psi'\circ \varphi ,\psi\circ \varphi ,\varphi },\qquad DM_{\psi}C_{\varphi }=T_{\psi',\psi \varphi ',\varphi },\qquad DC_{\varphi }M_{\psi}=T_{\varphi '\psi'\circ \varphi ,\varphi '\psi\circ \varphi ,\varphi }. \end{aligned}$$

Furthermore, by using this operator all possible difference operators of product-type operators in (1) can also be obtained. For example

$$\begin{aligned}& M_{\psi_{1}}C_{\varphi }D-M_{\psi_{2}}DC_{\varphi }=T_{0,\psi_{1}-\psi_{2}\varphi ',\varphi },\qquad M_{\psi_{1}}C_{\varphi }D-C_{\varphi }M_{\psi_{2}}D=T_{0,\psi_{1}-\psi_{2}\circ \varphi ,\varphi }, \\& M_{\psi_{1}}C_{\varphi }D-C_{\varphi }DM_{\psi_{2}}=T_{-\psi'_{2}\circ \varphi ,\psi _{1}-\psi_{2}\circ \varphi ,\varphi },\qquad M_{\psi_{1}}C_{\varphi }D-DM_{\psi_{2}}C_{\varphi }=T_{-\psi'_{2},\psi_{1}-\psi_{2}\varphi ',\varphi }, \\& M_{\psi_{1}}C_{\varphi }D-DC_{\varphi }M_{\psi_{2}}=T_{-\varphi '\psi_{2}'\circ \varphi ,\psi _{1}-\varphi '\psi_{2}\circ \varphi ,\varphi },\qquad M_{\psi_{1}}D C_{\varphi }-C_{\varphi }M_{\psi_{2}}D=T_{0,\psi_{1}\varphi '-\psi_{2}\circ \varphi ,\varphi }, \\& M_{\psi_{1}}D C_{\varphi }-C_{\varphi }DM_{\psi_{2}}=T_{-\psi_{2}'\circ \varphi ,\psi _{1}\varphi '-\psi_{2}\circ \varphi ,\varphi },\qquad M_{\psi_{1}}D C_{\varphi }-DM_{\psi_{2}}C_{\varphi }=T_{-\psi_{2}',(\psi_{1}-\psi _{2})\varphi ',\varphi }, \\& M_{\psi_{1}}D C_{\varphi }-DC_{\varphi }M_{\psi_{2}}=T_{-\varphi '\psi_{2}'\circ \varphi ,\psi _{1}\varphi '-\varphi '\psi_{2}\circ \varphi ,\varphi },\\& C_{\varphi }M_{\psi_{1}}D-C_{\varphi }DM_{\psi_{2}}=T_{-\psi_{2}'\circ \varphi ,(\psi _{1}-\psi_{2})\circ \varphi ,\varphi }, \\& C_{\varphi }M_{\psi_{1}}D-DM_{\psi_{2}}C_{\varphi }=T_{-\psi_{2}',\psi_{1}\varphi -\psi _{2}\varphi ',\varphi },\\& C_{\varphi }M_{\psi_{1}}D-DC_{\varphi }M_{\psi_{2}}=T_{-\varphi '\psi_{2}'\circ \varphi ,\psi_{1}\circ \varphi -\varphi '\psi_{2}\circ \varphi ,\varphi }, \\& C_{\varphi }DM_{\psi_{1}}-DM_{\psi_{2}} C_{\varphi }=T_{\psi_{1}'\circ \varphi -\psi _{2}',\psi_{1}\circ \varphi -\psi_{2}\varphi ,\varphi },\\& C_{\varphi }DM_{\psi_{1}}-DC_{\varphi }M_{\psi_{2}}=T_{\psi_{1}'\circ \varphi -\varphi '\psi _{2}\circ \varphi ,\psi_{1}\circ \varphi -\varphi '\psi_{2}\circ \varphi ,\varphi }, \\& DM_{\psi_{1}}C_{\varphi }-DC_{\varphi }M_{\psi_{2}}=T_{\psi_{1}'-\varphi '\psi_{2}\circ \varphi ,\psi_{1}\varphi '-\varphi '\psi_{2}\circ \varphi ,\varphi }, \end{aligned}$$

etc., where \(\psi_{1},\psi_{2}\in H({\mathbb{D}})\). In this paper we characterize the boundedness and compactness of the Stević-Sharma operator from the Zygmund space to the Bloch-Orlicz space. As the applications of our main results, readers can obtain some characterizations for the boundedness and compactness for all six product-type operators in (1), as well as above mentioned differences operators from the Zygmund space to the Bloch-Orlicz space.

Now we present the needed spaces and some facts. For \(\alpha >0\), the weighted Zygmund space \(\mathcal{Z}_{\alpha }\) consists of all \(f\in H({\mathbb{D}})\) such that

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

It is a Banach space with the norm

$$\|f\|_{\mathcal{Z}_{\alpha }}=\bigl|f(0)\bigr|+\bigl|f'(0)\bigr|+\sup_{z\in {\mathbb{D}}} \bigl(1-|z|^{2}\bigr)^{\alpha}\bigl|f''(z)\bigr|. $$

When \(\alpha =1\), this space is the Zygmund space and is denoted by \(\mathcal{Z}\) [32]. From Zygmund’s theorem (see Theorem 5.3 in [33]), we know that \(f\in \mathcal{Z}\) if and only if f is continuous on \(\overline{{\mathbb{D}}}\) and

$$\sup_{h>0,\theta\in {\mathbb{R}}}\frac{|f(e^{i(\theta+h)})+f(e^{i(\theta -h)})-2f(e^{i\theta})|}{h}< \infty. $$

For some results on Zygmund-type spaces and some concrete operators on them, see, for example, [15, 23, 32] and the references therein.

Recently, the Bloch-Orlicz space was introduced in [4] by Ramos Fernández. More precisely, let Ψ be a strictly increasing convex function such that \(\Psi(0)=0\). From these conditions it follows that \(\lim_{t\to+\infty}\Psi(t)=+\infty\). The Bloch-Orlicz space associated with the function Ψ, denoted by \(\mathcal{B}^{\Psi}\), is the class of all \(f\in H({\mathbb{D}})\) such that

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

for some \(\lambda>0\) depending on f. The Minkowski functional

$$\|f\|_{\Psi}=\inf \biggl\{ k>0:S_{\Psi}\biggl(\frac{f'}{k} \biggr)\leq 1 \biggr\} $$

defines a seminorm for \(\mathcal{B}^{\Psi}\), where

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

Moreover, \(\mathcal{B}^{\Psi}\) is a Banach space with the norm

$$\|f\|_{\mathcal{B}^{\Psi}}=\bigl|f(0)\bigr|+\|f\|_{\Psi}. $$

In fact, Ramos Fernández in [4] proved that \(\mathcal{B}^{\Psi}\) is isometrically equal to \(\mu_{\Psi}\)-Bloch space, where

$$\mu_{\Psi}(z)=\frac{1}{\Psi^{-1}(\frac{1}{1-|z|^{2}})},\quad z\in {\mathbb{D}}. $$

Thus, for \(f\in \mathcal{B}^{\Psi}\) it follows that

$$\|f\|_{\mathcal{B}^{\Psi}}=\bigl|f(0)\bigr|+\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z)\bigl|f'(z)\bigr|. $$

This equivalent norm is useful to us for the study of operator \(T_{\psi_{1},\psi_{2},\varphi }\) from the Zygmund space to the Bloch-Orlicz space. It is obvious to see that if \(\Psi(t)=t^{p}\) with \(p>0\), then the space \(\mathcal{B}^{\Psi}\) coincides with the weighted Bloch space \(\mathcal{B}^{\alpha }\), where \(\alpha =1/p\). Also, if \(\Psi(t)=t\log(1+t)\), then \(\mathcal{B}^{\Psi}\) coincides with the Log-Bloch space (see [34]). For the generalization of the Log-Bloch spaces, see, for example, [35, 36].

Let X and Y be Banach spaces. It is said that a linear operator \(L:X\to Y \) is bounded if there exists a positive constant K such that

$$\|Lf\|_{Y}\leq K\|f\|_{X} $$

for all \(f\in X\). The operator \(L:X\rightarrow Y\) is said to be compact if it maps bounded sets into relatively compact sets. It is well known that the norm of operator \(L:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is defined by

$$\|L\|_{\mathcal{Z}\to \mathcal{B}^{\Psi}}=\sup_{\|f\|_{\mathcal{Z}}\leq1}\|Lf\|_{\mathcal{B}^{\Psi}} $$

and written by \(\|L\|\).

Throughout this paper, a positive constant C may differ from one occurrence to the other. The notation \(a\lesssim b\) means that there exists a positive constant C such that \(a\leq Cb\). When \(a\lesssim b\) and \(b\lesssim a\), we write \(a\simeq b\).

2 Main results and proofs

In order to characterize the compactness, we need the following result, which is proved in a standard way [5]. So, the proof is omitted.

Lemma 1

Let φ be an analytic self-map of \({\mathbb{D}}\) and \(\psi_{1},\psi_{2}\in H({\mathbb{D}})\). Then the bounded operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is compact if and only if for every bounded sequence \(\{f_{j}\}_{j\in {\mathbb{N}}}\) in \(\mathcal{Z}\) such that \(f_{j}\to0\) uniformly on every compact subset of \({\mathbb{D}}\) as \(j\to\infty\), it follows that

$$\lim_{j\to\infty}\|T_{\psi_{1},\psi_{2},\varphi }f_{j} \|_{\mathcal{B}^{\Psi}}=0. $$

We state the following useful result whose first estimate was essentially proved in [37], while the second essentially follows from the point evaluation estimate for the Bloch functions (see, e.g., [38]). See also [2].

Lemma 2

For each \(f\in \mathcal{Z}\) and \(z\in {\mathbb{D}}\), it follows that

$$\bigl|f(z)\bigr|\leq\|f\|_{\mathcal{Z}}\quad\textit{and}\quad \bigl|f'(z)\bigr|\leq\log \frac {e}{1-|z|^{2}}\|f\|_{\mathcal{Z}}. $$

The following lemma was proved in [37], Lemma 2.5.

Lemma 3

Let \(\{f_{j}\}_{j\in {\mathbb{N}}}\) be a bounded sequence in \(\mathcal{Z}\) which uniformly converges to zero on compact subsets of \({\mathbb{D}}\) as \(j\to\infty\). Then

$$\lim_{j\to\infty}\sup_{z\in {\mathbb{D}}}\bigl|f_{j}(z)\bigr|=0. $$

For \(w\in {\mathbb{D}}\) and \(1/2<|w|<1\), we define the function

$$f_{w}(z)= \biggl(z-\frac{1}{\overline{w}} \biggr) \biggl[ \biggl(1+\log \frac {e}{1-\overline{w}z} \biggr)^{2}+1 \biggr]. $$

By using this function, the test functions in the Zygmund space can be obtained as follows:

$$\begin{aligned}& g_{w}(z)=f_{w}(z) \biggl(\log\frac{e}{1-|w|^{2}} \biggr)^{-1}, \\& h_{w}(z)=f_{w}(z) \biggl(\log\frac{e}{1-|w|^{2}} \biggr)^{-1} -\int_{0}^{z}\log \frac{e}{1-\overline{w}\lambda}\,d\lambda. \end{aligned}$$

From [9] we have the next result on the functions \(g_{w}\) and \(h_{w}\).

Lemma 4

Let \(w\in {\mathbb{D}}\) and \(1/2<|w|<1\). Then

$$g_{w}'(w)=\log\frac{e}{1-|w|^{2}},\qquad g''_{w}(w)= \frac{2\overline {w}}{1-|w|^{2}}, \qquad h''_{w}(w)= \frac{\overline{w}}{1-|w|^{2}}. $$

Moreover,

$$\sup_{1/2< |w|< 1}\|g_{w}\|_{\mathcal{Z}}\lesssim1, \qquad\sup _{1/2< |w|< 1}\| h_{w}\|_{\mathcal{Z}}\lesssim1. $$

Now we characterize the boundedness of the operator \(T_{\psi_{1},\psi _{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\).

Theorem 1

Let φ be an analytic self-map of \({\mathbb{D}}\) and \(\psi_{1},\psi_{2}\in H({\mathbb{D}})\). Then the following statements are equivalent.

  1. (i)

    The operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded.

  2. (ii)

    The functions \(\psi_{1}\), \(\psi_{2}\), and φ satisfy the following conditions:

    $$\begin{aligned}& M_{1}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z)\bigl| \psi_{1}'(z)\bigr|< \infty,\\& M_{2}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl| \psi_{1}(z)\varphi '(z)+\psi _{2}'(z) \bigr| \log\frac{e}{1-|\varphi (z)|^{2}}< \infty, \end{aligned}$$

    and

    $$M_{3}:=\sup_{z\in {\mathbb{D}}}\frac{\mu_{\Psi}(z)|\psi_{2}(z)||\varphi '(z)|}{1-|\varphi (z)|^{2}}< \infty. $$

Moreover, if the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is nonzero and bounded, then

$$\|T_{\psi_{1},\psi_{2},\varphi } \|\simeq1+M_{1}+M_{2}+M_{3}. $$

Proof

(i) ⇒ (ii). Suppose that \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded. For a fixed \(w\in {\mathbb{D}}\) and \(|\varphi (w)|>1/2\), let \(f(z)=h_{\varphi (w)}(z)-c_{1}+c_{2}\), where

$$c_{1}=g_{\varphi (w)}\bigl(\varphi (w)\bigr)=f_{\varphi (w)}\bigl(\varphi (w) \bigr) \biggl(\log\frac {e}{1-|\varphi (w)|^{2}} \biggr)^{-1},\qquad c_{2}=\int _{0}^{\varphi (w)}\log\frac{e}{1-\overline{\varphi (w)}\lambda }\,d\lambda. $$

Then by Lemma 4

$$f\bigl(\varphi (w)\bigr)=f'\bigl(\varphi (w)\bigr)=0,\qquad f'' \bigl(\varphi (w)\bigr)=h''_{\varphi (w)}\bigl(\varphi (w) \bigr)=\frac{\overline{\varphi (w)}}{1-|\varphi (w)|^{2}}. $$

By using the boundedness of \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi }\) to the function f, we have

$$\begin{aligned} M_{3}(w)&:=\frac{\mu_{\Psi}(w)|\varphi (w)||\psi_{2}(w)||\varphi '(w)|}{1-|\varphi (w)|^{2}} =\mu_{\Psi}(w) \bigl| (T_{\psi_{1},\psi_{2},\varphi }f )'(w) \bigr| \leq C \|T_{\psi_{1},\psi_{2},\varphi } \|, \end{aligned}$$
(3)

from which we get

$$\begin{aligned} \sup_{|\varphi (z)|>1/2}M_{3}(z)\leq C\|T_{\psi_{1},\psi_{2},\varphi } \|. \end{aligned}$$
(4)

From (4) it follows that

$$\begin{aligned} \sup_{|\varphi (z)|>1/2}&\frac{\mu_{\Psi}(z)|\psi_{2}(z)||\varphi '(z)|}{1-|\varphi (z)|^{2}} \leq2\sup_{|\varphi (z)|>1/2}M_{3}(z) \leq C\|T_{\psi_{1},\psi_{2},\varphi }\| . \end{aligned}$$
(5)

Let \(h_{0}(z)\equiv1\in \mathcal{Z}\). Then by the boundedness of \(T_{\psi _{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\), we obtain

$$\begin{aligned} M_{1}=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z)\bigl| \psi_{1}'(z)\bigr|\leq\|T_{\psi_{1},\psi _{2},\varphi }h_{0}\|\leq C\|T_{\psi_{1},\psi_{2},\varphi }\|. \end{aligned}$$
(6)

Considering \(h_{1}(z)=z\in \mathcal{Z}\), by the boundedness of \(T_{\psi_{1},\psi _{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) we have

$$\begin{aligned} \sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl|\psi_{1}'(z) \varphi (z)+\psi_{1}(z)\varphi '(z)+\psi_{2}'(z) \bigr| \leq C\|T_{\psi_{1},\psi_{2},\varphi }\|. \end{aligned}$$
(7)

From (6), (7), the boundedness of φ, and the triangle inequality, we obtain

$$\begin{aligned} L_{1}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl| \psi_{1}(z)\varphi '(z)+\psi _{2}'(z) \bigr| \leq C\|T_{\psi_{1},\psi_{2},\varphi }\|. \end{aligned}$$
(8)

Considering \(h_{2}(z)=z^{2}\in \mathcal{Z}\), we have

$$\begin{aligned} \sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl|\psi_{1}'(z) \bigl(\varphi (z)\bigr)^{2}+2 \bigl(\psi _{1}(z) \varphi '(z)+\psi_{2}'(z) \bigr)\varphi (z)+2 \psi_{2}(z)\varphi '(z) \bigr| \leq C\|T_{\psi_{1},\psi_{2},\varphi } \|. \end{aligned}$$
(9)

From (6), (8), (9), the boundedness of \(\varphi ^{2}\), and the triangle inequality, we get

$$\begin{aligned} L_{2}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z)\bigl| \psi_{2}(z)\bigr|\bigl|\varphi '(z)\bigr| \leq C\|T_{\psi_{1},\psi_{2},\varphi } \|. \end{aligned}$$
(10)

Then from (10) we have

$$\begin{aligned} \sup_{|\varphi (z)|\leq1/2}\frac{\mu_{\Psi}(z)|\psi_{2}(z)||\varphi '(z)|}{1-|\varphi (z)|^{2}} \leq C\|T_{\psi_{1},\psi_{2},\varphi } \|. \end{aligned}$$
(11)

From (5) and (11) we finally have \(M_{3}<\infty\).

Now we prove that \(M_{2}<\infty\). For a fixed \(w\in {\mathbb{D}}\) and \(|\varphi (w)|>1/2\), let \(g(z)=g_{\varphi (w)}(z)-c_{1}\). Then

$$g\bigl(\varphi (w)\bigr)=0,\qquad g'\bigl(\varphi (w)\bigr)=\log\frac{e}{1-|\varphi (w)|^{2}},\qquad g''\bigl(\varphi (w)\bigr)=\frac{2\overline{\varphi (w)}}{1-|\varphi (w)|^{2}}. $$

By using the boundedness of \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\), we have

$$\begin{aligned} &\mu_{\Psi}(w) \biggl| \bigl(\psi_{1}(w)\varphi '(w)+ \psi_{2}'(w) \bigr)\log \frac{e}{1-|\varphi (w)|^{2}} +2 \frac{\overline{\varphi (w)}\psi_{2}(w)\varphi '(w)}{1-|\varphi (w)|^{2}}\biggr| \\ &\quad=\mu_{\Psi}(w) \bigl| (T_{\psi_{1},\psi_{2},\varphi }g )'(w) \bigr|\leq C \|T_{\psi_{1},\psi_{2},\varphi }\|. \end{aligned}$$
(12)

From (4), (12), and the triangle inequality, it follows that

$$\begin{aligned} \mu_{\Psi}(w) \bigl|\psi_{1}(w)\varphi '(w)+ \psi_{2}'(w) \bigr|\log\frac {e}{1-|\varphi (w)|^{2}} &\leq2M_{3}(w)+C \|T_{\psi_{1},\psi_{2},\varphi }\| \\ &\leq C\|T_{\psi_{1},\psi _{2},\varphi }\|, \end{aligned}$$
(13)

and then

$$\begin{aligned} \sup_{|\varphi (z)|>1/2}\mu_{\Psi}(z) \bigl|\psi_{1}(z) \varphi '(z)+\psi _{2}'(z) \bigr|\log\frac{e}{1-|\varphi (z)|^{2}} \leq C\|T_{\psi_{1},\psi_{2},\varphi }\|. \end{aligned}$$
(14)

From (8), we obtain

$$\begin{aligned} \sup_{|\varphi (z)|\leq1/2}\mu_{\Psi}(z) \bigl|\psi_{1}(z) \varphi '(z)+\psi _{2}'(z) \bigr|\log\frac{e}{1-|\varphi (z)|^{2}} \leq L_{1}\log\frac{4e}{3} \leq C\|T_{\psi_{1},\psi_{2},\varphi } \|. \end{aligned}$$
(15)

Hence, from (14) and (15) we have \(M_{2}<\infty\).

(ii) ⇒ (i). By Lemma 2, for all \(f\in \mathcal{Z}\) we have

$$\begin{aligned} &\mu_{\Psi}(z) \bigl|(T_{\psi_{1},\psi_{2},\varphi }f )'(z) \bigr| \\ &\quad=\mu_{\Psi}(z) \bigl|\psi_{1}'(z)f\bigl(\varphi (z) \bigr)+ \bigl(\psi_{1}(z)\varphi '(z)+\psi_{2}'(z) \bigr)f'\bigl(\varphi (z)\bigr)+\psi_{2}(z) \varphi '(z)f''\bigl(\varphi (z)\bigr) \bigr| \\ &\quad\leq\mu_{\Psi}(z) \bigl( \bigl|\psi_{1}'(z) \bigr| \bigl|f \bigl(\varphi (z)\bigr) \bigr|+ \bigl|\psi_{1}(z)\varphi '(z)+ \psi_{2}'(z) \bigr| \bigl|f'\bigl(\varphi (z)\bigr) \bigr| \\ &\qquad{} + \bigl| \psi_{2}(z) \bigr| \bigl|\varphi '(z) \bigr| \bigl|f'' \bigl(\varphi (z)\bigr) \bigr| \bigr) \\ &\quad\leq(M_{1}+M_{2}+M_{3})\|f\|_{\mathcal{Z}}. \end{aligned}$$
(16)

It is clear that

$$\begin{aligned} \bigl|T_{\psi_{1},\psi_{2},\varphi }f(0)\bigr|\leq C\|f\|_{\mathcal{Z}}. \end{aligned}$$
(17)

Hence from (16) and (17) it follows that \(T_{\psi _{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded.

Suppose that the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is nonzero and bounded. Then from the proof of (i) ⇒ (ii) it is not hard to see that

$$\begin{aligned} M_{1}+M_{2}+M_{3}\lesssim\|T_{\psi_{1},\psi_{2},\varphi } \|. \end{aligned}$$
(18)

Since the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is nonzero, we have \(\|T_{\psi_{1},\psi_{2},\varphi }\|>0\). From this we can find a positive constant C such that \(1\leq C\|T_{\psi_{1},\psi_{2},\varphi }\|\), which means that

$$\begin{aligned} 1\lesssim\|T_{\psi_{1},\psi_{2},\varphi }\|. \end{aligned}$$
(19)

Then combing (18) and (19) gives

$$\begin{aligned} 1+M_{1}+M_{2}+M_{3}\lesssim\|T_{\psi_{1},\psi_{2},\varphi } \|. \end{aligned}$$
(20)

It is clear from (16) and (17) that

$$\begin{aligned} \|T_{\psi_{1},\psi_{2},\varphi }\|\lesssim1+M_{1}+M_{2}+M_{3}. \end{aligned}$$
(21)

Hence from (20) and (21) the asymptotic expression of \(\| T_{\psi_{1},\psi_{2},\varphi }\|\) follows. The proof is finished. □

Next we characterize the compactness of operator \(T_{\psi_{1},\psi _{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\).

Theorem 2

Let φ be an analytic self-map of \({\mathbb{D}}\) and \(\psi_{1},\psi_{2}\in H({\mathbb{D}})\). Then the following statements are equivalent.

  1. (i)

    The operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is compact.

  2. (ii)

    The functions \(\psi_{1}\), \(\psi_{2}\), and φ satisfy the following conditions:

    $$\begin{aligned}& M_{1}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl| \psi'_{1}(z) \bigr|< \infty, \\& L_{1}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl| \psi_{1}(z)\varphi '(z)+\psi _{2}'(z) \bigr|< \infty, \\& L_{2}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z)\bigl| \psi_{2}(z)\bigr|\bigl|\varphi '(z)\bigr|< \infty , \\& \lim_{|\varphi (z)|\to1^{-}}\mu_{\Psi}(z) \bigl|\psi_{1}(z) \varphi '(z)+\psi _{2}'(z) \bigr|\log \frac{e}{1-|\varphi (z)|^{2}}=0, \end{aligned}$$

    and

    $$\lim_{|\varphi (z)|\to1^{-}}\frac{\mu_{\Psi}(z)|\psi_{2}(z)||\varphi '(z)|}{1-|\varphi (z)|^{2}}=0. $$

Proof

(i) ⇒ (ii). Suppose that (i) holds. Then it is clear that the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded. In the proof of Theorem 1, we have shown that \(M_{1}<\infty\), \(L_{1}<\infty\) and \(L_{2}<\infty\). Consider a sequence \(\{\varphi (z_{i})\}_{i\in {\mathbb{N}}}\) in \({\mathbb{D}}\) such that \(|\varphi (z_{i})|\to1^{-}\) as \(i\to\infty\). If such a sequence does not exist, then the last two conditions (ii) obviously hold. We may suppose, without loss of generality, that \(|\varphi (z_{i})|>1/2\) for all \(i\in {\mathbb{N}}\). Using this sequence, we define the function sequence

$$f_{i}(z)=f_{\varphi (z_{i})}(z) \biggl(\log\frac{e}{1-|\varphi (z_{i})|^{2}} \biggr)^{-1} - \biggl(\log\frac{e}{1-|\varphi (z_{i})|^{2}} \biggr)^{-2}\int _{0}^{z}\log ^{3}\frac{e}{1-\overline{\varphi (z_{i})}w}\,dw. $$

Then from a calculation we see that \(\sup_{i\in {\mathbb{N}}}\|f_{i}\|_{\mathcal{Z}}\leq C\) and \(f_{i}\to0\) uniformly on every compact subset of \({\mathbb{D}}\) as \(i\to\infty\). So by Lemma 1

$$\lim_{i\to\infty}\|T_{\psi_{1},\psi_{2},\varphi }f_{i} \|_{\mathcal{B}^{\Psi}}=0. $$

Moreover, we have

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

Hence we get

$$\begin{aligned} \biggl|\frac{\mu_{\Psi}(z_{i})|\psi_{2}(z_{i})||\varphi '(z_{i})||\varphi (z_{i})|}{1-|\varphi (z_{i})|^{2}} -\mu_{\Psi}(z_{i}) \bigl|\psi_{1}'(z_{i}) \bigr| \bigl|f_{i}\bigl(\varphi (z_{i})\bigr) \bigr| \biggr| \leq \|T_{\psi_{1},\psi_{2},\varphi }f_{i}\|_{\mathcal{B}^{\Psi}}. \end{aligned}$$

From this, Lemmas 1 and 3, and since \(M_{1}\) is finite, we obtain

$$\begin{aligned} \lim_{i\to\infty}\frac{\mu_{\Psi}(z_{i})|\psi_{2}(z_{i})||\varphi '(z_{i})|}{1-|\varphi (z_{i})|^{2}}=0. \end{aligned}$$
(22)

On the other hand, take the sequence \(g_{i}(z)=g_{\varphi (z_{i})}(z)-c_{i}\), \(i\in {\mathbb{N}}\), where \(c_{i}=g_{\varphi (z_{i})}(\varphi (z_{i}))\). Then \(\sup_{i\in {\mathbb{N}}}\|g_{i}\|_{\mathcal{Z}}\leq C\),

$$g_{i}\bigl(\varphi (z_{i})\bigr)=0,\qquad g_{i}' \bigl(\varphi (z_{i})\bigr)=\log\frac{e}{1-|\varphi (z_{i})|^{2}},\qquad g_{i}''(z_{i})= \frac{2\overline{\varphi (z_{i})}}{1-|\varphi (z_{i})|^{2}}. $$

Hence we have

$$\mu_{\Psi}(z_{i}) \biggl| \bigl(\psi_{1}(z_{i}) \varphi '(z_{i})+\psi_{2}'(z_{i}) \bigr)\log\frac{e}{1-|\varphi (z_{i})|^{2}} +\frac{2\overline{\varphi (z_{i})}}{1-|\varphi (z_{i})|^{2}}\biggr|\leq\|T_{\psi _{1},\psi_{2},\varphi }g_{i} \|_{\mathcal{B}^{\Psi}}. $$

By the compactness \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\), Lemma 1 and (22), we get

$$\begin{aligned} \lim_{i\to\infty}\mu_{\Psi}(z_{i}) \bigl| \psi_{1}(z_{i})\varphi '(z_{i})+\psi _{2}'(z_{i}) \bigr|\log\frac{e}{1-|\varphi (z_{i})|^{2}}=0. \end{aligned}$$

(ii) ⇒ (i). We first prove that \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded. We observe that the conditions in (ii) imply that for every \(\varepsilon>0\), there is an \(\eta\in(0,1)\), such that for any \(z\in K=\{z\in {\mathbb{D}}:|\varphi (z)|>\eta\}\)

$$\begin{aligned} R_{1}(z):=\mu_{\Psi}(z) \bigl|\psi_{1}(z) \varphi '(z)+\psi_{2}'(z) \bigr|\log \frac{e}{1-|\varphi (z)|^{2}}< \varepsilon \end{aligned}$$
(23)

and

$$\begin{aligned} R_{2}(z):=\frac{\mu_{\Psi}(z)|\psi_{2}(z)||\varphi '(z)|}{1-|\varphi (z)|^{2}}< \varepsilon. \end{aligned}$$
(24)

From the fact \(L_{1}<\infty\) and (23), we obtain

$$M_{2}=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl| \psi_{1}(z)\varphi '(z)+\psi _{2}'(z) \bigr| \log\frac{e}{1-|\varphi (z)|^{2}}\leq\varepsilon+ L_{1}\log\frac{e}{1-\eta^{2}}. $$

From the fact \(L_{2}<\infty\) and (24), we also obtain

$$M_{3}=\sup_{z\in {\mathbb{D}}}\frac{\mu_{\Psi}(z)|\psi_{2}(z)||\varphi '(z)|}{1-|\varphi (z)|^{2}}\leq\varepsilon+ \frac{L_{2}}{1-\eta^{2}}. $$

Hence from Theorem 1 it follows that the operator \(T_{\psi_{1},\psi _{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded.

In order to prove that the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is compact, by Lemma 1 we just need to prove that, if \(\{f_{i}\}_{i\in {\mathbb{N}}}\) is a sequence in \(\mathcal{Z}\) such that \(\sup_{i\in {\mathbb{N}}}\|f_{i}\|_{\mathcal{Z}}\leq M\) and \(f_{i}\to0\) uniformly on any compact subset of \({\mathbb{D}}\) as \(i\to\infty\), then

$$\lim_{i\to\infty}\|T_{\psi_{1},\psi_{2},\varphi }f_{i} \|_{\mathcal{B}^{\Psi}}=0. $$

For such a chosen ε and η, by using (23), (24), and Lemma 2 we have

$$\begin{aligned} &\mu_{\Psi}(z) \bigl|(T_{\psi_{1},\psi_{2},\varphi }f_{i} )'(z) \bigr| \\ &\quad=\mu_{\Psi}(z) \bigl|\psi_{1}'(z)f_{i} \bigl(\varphi (z)\bigr)+ \bigl(\psi_{1}(z)\varphi '(z)+ \psi_{2}'(z) \bigr)f_{i}'\bigl( \varphi (z)\bigr)+\varphi '(z)\psi_{2}(z)f_{i}'' \bigl(\varphi (z)\bigr) \bigr| \\ &\quad\leq\mu_{\Psi}(z) \bigl( \bigl|\psi_{1}'(z) \bigr| \bigl|f_{i}\bigl(\varphi (z)\bigr) \bigr|+ \bigl|\psi_{1}(z) \varphi '(z)+\psi_{2}'(z) \bigr| \bigl|f_{i}' \bigl(\varphi (z)\bigr) \bigr| \\ &\qquad{}+ \bigl|\varphi '(z) \bigr| \bigl|\psi_{2}(z) \bigr| \bigl|f_{i}''\bigl(\varphi (z)\bigr) \bigr| \bigr) \\ &\quad\leq M_{1}\sup_{z\in {\mathbb{D}}}\bigl|f_{i}(z)\bigr|+ \Bigl( \sup_{z\in K}+\sup_{z\in {\mathbb{D}}\setminus K} \Bigr) \mu_{\Psi}(z) \bigl|\psi_{1}(z)\varphi '(z)+ \psi_{2}'(z) \bigr| \bigl|f_{i}'\bigl(\varphi (z) \bigr) \bigr| \\ &\qquad{} + \Bigl(\sup_{z\in K}+\sup_{z\in {\mathbb{D}}\setminus K} \Bigr)\mu _{\Psi}(z) \bigl|\varphi '(z) \bigr| \bigl|\psi_{2}(z) \bigr| \bigl|f_{i}''\bigl(\varphi (z)\bigr) \bigr| \\ &\quad\leq2\varepsilon+M_{1}\sup_{z\in {\mathbb{D}}}\bigl|f_{i}(z)\bigr|+L_{1} \sup_{|z|\leq\eta }\bigl|f_{i}'(z)\bigr| +L_{2} \sup_{|z|\leq\eta}\bigl|f_{i}''(z)\bigr|. \end{aligned}$$
(25)

Since \(f_{i}\to\) uniformly on compact subsets of \({\mathbb{D}}\) as \(i\to\infty\) implies that for each \(k\in {\mathbb{N}}\), \(f_{i}^{(k)}\to0\) uniformly on compact subsets of \({\mathbb{D}}\) as \(i\to\infty\), from (25) and Lemma 3 we get

$$\lim_{i\to\infty}\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl|(T_{\psi _{1},\psi_{2},\varphi }f_{i})'(z) \bigr|=0. $$

It is clear that

$$\begin{aligned} \lim_{i\to\infty} \bigl|T_{\psi_{1},\psi_{2},\varphi }f_{i}(0) \bigr|=0. \end{aligned}$$
(26)

From (25) and (26) we obtain

$$\begin{aligned} \lim_{i\to\infty}\|T_{\psi_{1},\psi_{2},\varphi }f_{i} \|_{\mathcal{B}^{\Psi}}=0. \end{aligned}$$
(27)

Hence from (27) and Lemma 1, we see that \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is compact. The proof is finished. □