#### Abstract

In this paper, we obtain some characterizations of composition operators , which are induced by an analytic self-map of the unit disk , from hyperbolic Bloch type space into hyperbolic type space .

#### 1. Introduction and Results

Let be the unit disk in the complex plane , and denote the space of all analytic functions on . Let denote the subspace of consisting of those analytic self-maps for which . For every , the linear composition operator is defined as

An important problem is to characterize the boundedness and compactness of in analytic function spaces by means of the geometric and analytic properties of . The reader is invited to the monographs [1, 2] for an excellent overview of this subject. Composition operators from the Bloch-type space into some other analytic function spaces have been much investigated in recent years, see [3–19] and references therein. Also, there are many works concerning composition operators from hyperbolic Bloch type spaces into some other hyperbolic classes, see [20–23]. This work is intended to characterize the boundedness and compactness of composition operators from hyperbolic Bloch type spaces into hyperbolic type spaces . We start with some notations and definitions.

A positive continuous function on the interval is called normal if there exist constants and such that
(i) is decreasing on [*δ*, 1) and ;(ii) is increasing on [*δ*, 1) and .

It is clear that such a normal weight function is decreasing in the neighborhood of the point and satisfies

It should be pointed out that the normal weight function has been usually used to define the mixed norm spaces [6]. Let be normal, the Bloch type space is defined as the set of all analytic functions so that

The little Bloch-type space consists of all such that

Let , . Then, the Bloch-type space becomes the so-called -Bloch space , which for reduces to the classical Bloch space . If , where and , then, we obtain the so-called logarithmic Bloch-type space which consists of all analytic functions so that

The little logarithmic Bloch-type space consists of all analytic functions and

For more results on space, see, e.g., [24–26] and related references therein.

Hyperbolic function classes are subset of and are defined by using the hyperbolic derivative

It is clear that the hyperbolic derivative of the composition satisfies the chain rule

The hyperbolic Bloch-type space consists of all analytic functions with

The little hyperbolic Bloch-type space is the set of all analytic functions for which and

Replacing the derivative by hyperbolic derivative , we can define the corresponding hyperbolic -Bloch space and hyperbolic logarithmic Bloch-type space and .

For any , the function is a Möbius map of the unit disk and interchanges the points to . The Green function of is defined as

Let and be right continuous and nondecreasing. The space consists of all analytic functions in such that where is Lebesgue measure on , normalized so that the unit disk has Lebesgue measure equals to 1. The definition was introduced in [23] and was motivated by the theory of , , and , see [16, 27–31] and references therein. We note that the case when and is the classic space, which was introduced by Wulan and Wu in [28] and has been much investigated in recent years. We also observe that the case when is the space introduced by Zhao in [31]. For , the spaces are subsets of space . Throughout this paper, we assume that

Otherwise, contains constant functions only (see [32]).

The corresponding hyperbolic space is defined as the set of all analytic functions in such that where is defined as before and satisfies the condition (14). Replacing the derivative by hyperbolic derivative , we can define the corresponding hyperbolic space .

Yang, Xu, and Kotilainen [30] obtained some characterizations of the boundedness and compactness of the composition operators . For an and , set and . We use to denote the characteristic function of a set .

Theorem 1 (see [18]). *Assume that is a normal function and . Let , and be nonnegative and nondecreasing in . Then
*(i)* is bounded if and only if**(ii)** is compact if and only if*

In [11], the authors considered bounded composition operators from hyperbolic -Bloch spaces into hyperbolic space . They proved the following result.

Theorem 2. *Assume that . Let , . Then the following statements are equivalent:
*(i)* is bounded;*(ii)* is bounded;*(iii)* is bounded.*

In this paper, we extend Theorem 1 to the corresponding hyperbolic classes. For a real increasing function defined on , its growth order is defined as

We obtain the following result.

Theorem 3. *Suppose that is a normal function defined on and the growth order of function is strictly less than . Let , , and be nonnegative and nondecreasing in . Then, the following statements are equivalent:
*(1)* is bounded;*(2)* is bounded;*(3)* is bounded.*

It should be pointed out that there are many normal functions which satisfy the condition of Theorem 3. For example, if with , then the growth order of is . As a corollary, we obtain the following result by taking , where and .

Corollary 4. *Let , and . Let , and be nonnegative and nondecreasing in . Then, the following statements are equivalent:
*(1)* is bounded;*(2)* is bounded;*(3)* is bounded.*

We now turn to the compactness of composition operator . It is clear that is not a linear space since the sum of two functions in does not necessarily belong to . It is also known that there is a natural metric in , which is defined by

The classes equipped with the metric is a complete metric space, and is a closed subspace of (see [5, 9]). Also, there is a natural metric in defined as

A similar argument shows that the classes equipped with the metric is a complete metric space (see [21, 22]).

Recall that an operator is compact if it maps any ball in onto a related relatively compact set in . This is equivalent to say that the operator is completely continuous. We obtain the following result.

Theorem 5. *Suppose that is a normal function defined on and the growth order of function is strictly less than . Let , , and be nonnegative and nondecreasing in . Then, the following statements are equivalent:
*(1)* is compact*

Similarly, if we take , where and , then we have

Corollary 6. *Let , and . Let , and be nonnegative and nondecreasing in . Then, the following statements are equivalent:
*(1)* is compact*

The paper is organized as follows. We shall prove Theorem 3 in section 2 and prove Theorem 5 in section 3. Throughout this paper, we use the notation to denote that there is a constant such that , and the notation to indicate that .

#### 2. Proof of Theorem 1.1

To prove Theorem 3, we need the following Lemma [30], which gives some lower estimate of the norm of some functions in .

Lemma 7 (see [18]). *Suppose that is a normal function defined on . Then, there exist two functions such that for all ,
for some positive constant .*

We shall prove a hyperbolic version of Lemma 7.

Lemma 8. *Suppose that is a normal function defined on and the growth order of function is strictly less than . Then, there exist two functions such that for all ,
**for some positive constant .*

*Proof. *By Lemma 7, there are two functions so that
Consequently, we get
Let and note that the growth order of is less than , we deduce that for positive number , there exists with , such that for all ,
On the other hand, since is a positive continuous function, there exists positive number such that for all , . Thus, combining (26) and (27) yields that
hold for all . This implies that the functions are bounded.

Let and . Then, and This gives the following
The proof follows.

We now start our proof of Theorem 1.1.

*Proof of Theorem 1.1. *Since is a closed subset of , it is obvious that implies . Thus, it is enough to show that and .

We first show that . Assume that (1) holds, that is is bounded. Then, for any , it follows from Theorem 1 that
uniformly for all . This implies that is bounded, and we complete the proof of .

We next show that . By Lemma 8, there exist two functions such that for all ,
for some positive constant . Consequently, by the assumption that is bounded, we have
The proof of Theorem 1.1 is completed.

#### 3. Proof of Theorem 1.3

In this section, we shall prove Theorem 5. First, we find some test functions in . Let be a normal weight function defined in , be large and , consider a family analytic functions defined by where , , and as .

It is proved in [21] that if we take the nature number large enough, then, the sequence weakly converges to zero in .

Lemma 9 (see [33]). *Let be the functions defined in (33). Then, there are natural number such that and positive constant such that for all and . Moreover, set , then
where *

Now, we are in a position to prove Theorem 5.

*Proof of Theorem 1.3. *We first show that . Let be the analytic functions defined as in Lemma 9. Therefore, there is a positive constant such that
for all and . Observe that the growth order of function is less than . Arguing as we have done in Lemma 8, we can find a constant such that for all and . Take , where and . Then, and

On the other hand, by the definition of , it is clear that the analytic function sequence converges uniformly on compact subset of to . Consequently, by Lemma 9, the sequence belongs to and tends to weakly as .

Since is compact, we have
Thus, it follows from Lemma 9 that
This completes the proof of .

We now prove that . Suppose that (2) holds. Let , where and , and be any sequence. By the definition of compactness of , it is sufficient to show that its image has a convergent subsequence in .

Since is a normal family, there is a subsequence which converges uniformly on compact subset of to an analytic function . It is clear that also converges uniformly on compact subset of to . Consequently, the sequences and converge uniformly on compact subsets of to and .

We claim that . Indeed, for any fixed with , we have

Letting gives the desired result.

For any , by the uniform convergence, there is such that for all ,

By Theorem 1, the condition (4) implies that the composition operator is compact. Therefore, by passing once more to a subsequence and adjusting the notations, we can assume that for all , for some .

On the other hand, since (2) holds, we may find with so that

Observe that and , it follows that

Consequently, we have

Finally, by the uniform convergence on compact subset of , we can find so that for all and for all with ,

Note again that the condition (4) implies that the composition operator is compact. Thus, is also bounded, and we get from Theorem 1 that

Therefore, we have for all ,

Combining (39), (40), (43), and (46) yields

The proof of Theorem 1.3 is completed.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors would like to thank the referees for their careful reading and valuable suggestions, which improve the quality of this paper. This work was supported by the National Natural Science Foundation of China (grant no. 11601100) and the joint foundation of the Guizhou Provincial Science and Technology Department (grant no. [2017]7337, [2017]5726).