1. Introduction

Let be the unit disc in the complex plane , the normalized Lebesgue area measure on , the class of all holomorphic functions on , and the space of bounded holomorphic functions on with the norm

The logarithmic Bloch-type space , was recently introduced in [1]. The space consists of all such that

The norm on is introduced as follows:

When , becomes the -Bloch space . For -Bloch and other Bloch-type spaces, see, for example, [1–9], as well as the related references therein. For , is the logarithmic Bloch space, which appeared in characterizing the multipliers of the Bloch space (see [3, 9]).

The little logarithmic Bloch-type space , consists of all such that

The following theorem summarizes the basic properties of the logarithmic Bloch-type spaces. Here, as usual, for fixed

Theorem 1 A (see [1]). The following statements are true. (a)The logarithmic Bloch-type space is Banach with the norm given in (1.2).(b) is a closed subset of (c)Assume Then if and only if (d)The set of all polynomials is dense in .(e)Assume then for each , . Moreover

A positive continuous function on is called weight.

The Bloch-type space consists of all such that where is a weight. With the norm the Bloch-type space becomes a Banach space.

The little Bloch-type space is a subspace of consisting of all such that

Let be a holomorphic self-map of and . For the corresponding weighted composition operator is defined by It is of interest to provide function-theoretic characterizations for when and induce bounded or compact weighted composition operators on spaces of holomorphic functions. For some classical results mostly on composition operators, see, for example, [10]. For some recent related results, mostly in or related to Bloch-type or weighted-type spaces, see, for example, [4, 10–46] and the references therein.

Here we study the boundedness and compactness of the weighted composition operator from the logarithmic Bloch-type space and the little logarithmic Bloch-type space to the Bloch-type or the little Bloch-type space.

In this paper, constants are denoted by , they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant such that . We say that , if both and hold.

2. Auxiliary Results

In this section we quote several auxiliary results which will be used in the proofs of the main results.

Lemma 2.1. Assume , then the following statements are true. (a)Assume then the function is increasing on the interval (b)The function is increasing on the interval

Proof. (a) We have Since when and , and the function is decreasing on the interval , we have from which this statement follows.
The proof of (b) is similar, hence it is omitted.

The next lemma regarding the point evaluation functional on follows from [1, Lemma  3] and some elementary asymptotic relationship, such as

Lemma 2.2. Let Then for some independent of

The proof of the following lemma is similar to [25, Lemma  2.1], so we omit it.

Lemma 2.3. Assume is a weight. A closed set in is compact if and only if it is bounded and

Remark 2.4. If in Lemma 2.3 we assume that is not closed, then the word compact can be replaced by relatively compact.

The next characterization of compactness is proved in a standard way (see, e.g., the proofs of the corresponding lemmas in [10, 30, 47–49]). Hence we omit it.

Lemma 2.5. Assume that , is a holomorphic self-map of and is a weight. Let be one of the following spaces , and one of the spaces , . Then the operator is compact if and only if is bounded and for every bounded sequence converging to uniformly on compacts of one has

Some concrete examples of the functions belonging to logarithmic Bloch-type spaces can be found in the next lemma.

Lemma 2.6. The following statements are true. (a)Assume that and then where and is a nonconstant function belonging to (b)Assume that and then where and is a nonconstant function belonging to (c)Assume that then where and is a nonconstant function belonging to
Moreover, for each , it holds that belong to the corresponding space, and for fixed and

Proof. (a) Let be fixed. Then we have where in (2.13) we have used that and in (2.14) we have used the fact that the function in (2.1) is increasing on the interval .
From (2.13), since , and by Lemma 2.1(a), we have that as , from which it follows that as desired.
(b) For fixed , we have where in (2.16) we have used the assumption , while in (2.17), as in (a), we have used the fact that the function in (2.1) is increasing on the interval .
From (2.16), and by Lemma 2.1(a), we obtain as . Hence finishing the proof of this statement.
(c) We have where we have used the assumption and the fact that function (2.1) is increasing on .
From (2.19), Lemma 2.1(a), and since we obtain as that is,
Estimations (2.12) follow from (2.14), (2.17), (2.20) and by using the following facts we finish the proof of the lemma.

Remark 2.7. Note that from Lemmas 2.2 and 2.6 the functions defined in (2.9)–(2.11) have maximal growths in the corresponding logarithmic Bloch-type spaces.

3. Boundedness and Compactness of the Operator

This section studies the boundedness and compactness of the weighted composition operator .

Case 1. , .

Theorem 3.1. Assume , , is an analytic self-map of the unit disk, , and is a weight. Then the operator is bounded if and only if

Proof. First assume that (3.1) and (3.2) hold. Then, by Lemma 2.2 and the definition of , we have Applying (3.1) and (3.2) in (3.4), the boundedness of follows.
Now assume the operator is bounded. By taking the test functions and (which obviously belong to ), we obtain From (3.5) and (3.6), and since the function is bounded, it follows that
For set
We have that , and as an easy consequence of Lemma 2.6(a), and for each
Using these facts and the boundedness of , for the test functions , where and , we get
From (3.10) it follows that
On the other hand, by using (3.7) and Lemma 2.1(b), we have Hence, (3.11) and (3.12) imply (3.2).
Let Then and by Lemma 2.6(a) we get , and for every . Using the boundedness of , the test functions , and equalities (3.14) we get for each , .
From (3.2), (3.5), (3.15), and using the fact that condition (3.1) follows.

Theorem 3.2. Assume , , is an analytic self-map of the unit disk, , and is a weight. Then the operator is compact if and only if is bounded