Abstract

All entire functions which transform a class of holomorphic Zygmund-type spaces into weighted analytic Bloch space using the so-called -generalized superposition operator are characterized in this paper. Moreover, certain specific properties such as boundedness and compactness of the newly defined class of generalized integral superposition operators are discussed and established by using the concerned entire functions.

1. Fundamental Materials

Let be the known concerned open unit disk in . Assume that and are two concerned metric spaces of holomorphic-type functions both defined on the complex unit disk . Let be a concerned complex-valued function defined on . The known superposition operator on the concerned metric space is here defined by

When the operator for , we call that is induced by the concerned superposition operator from into . When the concerned metric space contains specific concerned linear functions, thus, must be a concerned entire-type function. The symbol is supposed to be the class of all concerned holomorphic functions on . For various emerging research studies on the superposition-type operators, we refer to [1–14] and others.

The motivation of this current study arises from the recent research of the theory of operators on the theory of complex-type function spaces.

The nonlinear -generalized superposition operator can be defined in this current paper by

The operator will be called the -generalized superposition operator, where

Remark 1. When and , the superposition operator is obtained because the difference is a positive constant. Therefore, is a generalization of the superposition-type operator. For good knowledge, the new operator is defined and discussed in the present concerned manuscript for the first time.

Remark 2. It should be noted that the graph of the -generalized superposition operator is almost closed, but because the new operator is a nonlinear operator, then there is no guarantee to establish its boundedness. Nonetheless, there are some spaces and , for instance, holomorphic-type Bergman, holomorphic Bloch, and holomorphic-type Dirichlet spaces, which implies that the specific function must belong to a specific concerned class of entire-type functions, resulting in the boundedness-type property.

Definition 3 [9]. For a given nondecreasing bounded and continuous function , the concerned function is said to be in the weighted concerned space when

Remark 4. In the above definition, suppose that ; then, we obtain the definition of -Bloch space, where .

The following new definition can also be introduced.

Definition 5. Let ; then, is said to be in the weighted little -type concerned Bloch space when

Due to the result of Zygmund in [15] as well as the closed graph theorem, we can deduce that

It is obvious to see that the space is a Banach-type space with the specific norm , where

Here, the concerned Zygmund-type space on , denoted by , is a concerned closed subspace of the holomorphic-type space that consists of all holomorphic-type functions , for which

Using (7), it is very obvious to deduce that

Some important and essential auxiliary results shall be proved in the next lemmas.

Lemma 6. Let and ; then, the following inequality can be obtained: for some which is independent of , with

Proof. Let , , and . Then, we infer that Additionally, we deduce Let ; then, applying the known mean value theorem for the function (cf. [16]) and considering Jensen’s inequality, we deduce that where the homogeneous representation expansion of the function results in the second inequality.
Therefore, Combining (13) and (15), we deduce The proof is therefore established completely.

Hereafter, the letter can be used to denote a concerned planar domain and defines its boundary.

In view of the known Riemann mapping theorem [16], for any , there exists a holomorphic function (that can be called a concerned Riemann mapping) which maps onto and the concerned origin to a prescribed concerned specific point. The known Euclidean metric of the specific point to the concerned boundary is denoted by ; also, the concerned Riemann-type mapping has the next interesting concerned property: which is comparatively similar to the concerned constructions of the concerned connected domains as the concerned images for specific classes of functions in different analytic-type function spaces can be obtained in [4].

2. Auxiliary Results

The next lemmas establish some elementary properties of the -Bloch-type spaces of analytic functions.

Lemma 7. Let , and assume that (11) holds. Then, for the -Bloch space , we have the following: (i)Every bounded concerned sequence is uniformly bounded on concerned compact sets(ii)For any concerned sequence on with , then and the convergence is of uniform type on concerned compact sets

Proof. If , ; thus, the following inequalities can be deduced

Hence, the result follows.

Lemma 8. Assume that and . Thus, for the -generalized superposition operator which is a concerned compact operator and which is actually bounded for any bounded concerned sequence with uniformly bounded on concerned compact sets if , we obtain as .

Proof. Now, the concerned assertions (7), (9), and (11) of Lemma 3.7 in [17] can be satisfied for the analytic-type space. In view of Lemma 7, thus, it is clear to show that the concerned assertions (i) and (ii) hold. To clear that (ii) holds, assume that the specific sequence can be taken as a concerned sequence in the specific closed unit ball of . Therefore, using Lemma 7 implies that can be seen as uniformly bounded on concerned compact sets. Hence, applying the theorem of Montel [18], we can get a concerned subsequence , for which which indeed is uniformly bounded on concerned compact sets, for some concerned functions . Then, it is enough to clear that . From Fatou’s theorem [16] and the concerned assumptions, we deduce that Therefore, in view of Lemma 3.7 in [17], we infer that the -generalized superposition operator is compact for any bounded concerned sequence with uniformly bounded on compact sets as , , as . Thus, the proof of Lemma 8 is clearly obtained.

Lemma 9. Let . The concerned closed set is compact iff it is bounded and satisfies the next concerned condition:

Proof. Assume that is compact, and let ; thus, the specific balls that centered at the specific elements of the concerned with radii cover ; then, by the help of compactness, we can find functions , for which the analytic function yields that for some ; hence, Now, for each , we can find , for which By letting , we obtain This establishes that (20) holds.
On the other hand, assume that is closed and bounded and satisfies (20).
Then, the set can be a normal family. When is a concerned sequence in the set , then passing to a specific subsequence, we can suppose that with the concerned uniformly converging on the concerned compact subsets of . Hence, we can clear that in . Then, for a given , using (20), we can find , for which Since the convergence is uniformly bounded on the concerned compact subsets of the disk , it yields that the type of convergence for is pointwise on . Additionally, we deduce that Therefore, Because the convergence is uniformly bounded on , then we can find a specific positive integer , for which Hence, Hence, in . Because is specifically closed, then we can infer that . Therefore, we deduce that the concerned set is a compact set. The proof of Lemma 9 is therefore completely established.

3. Superpositions on Bloch and Zygmund Spaces

In this section, some interesting and important properties of the new class of the -generalized superposition acting from the analytic-type Zygmund classes to the analytic -Bloch classes are clearly established.

Theorem 10. Let and . Then, the -generalized superposition operator is bounded

Proof. Assume that (29) holds. Hence, for arbitrary and , we deduce that From the above inequality (29) and because , we deduce that the operator is a bounded operator.
On the other hand, suppose that is bounded. Set and also set the function where with . Then, the easy calculation gives which yields that with and . Hence, the following inequality can be deduced: Hence, by using the maximum modulus principle, we obtain (29). The proof is therefore completely established.

For the next theorem, the analytic -Bloch-type space is considered as follows:

Theorem 11. Assume that , where . Suppose that and . Then, the operator is a compact operator iff is bounded, and the following condition holds:

Proof. Assume the boundedness for the operator holds as well as condition (38) holds too. Therefore, since the operator is bounded with , we deduce that Let be a concerned sequence in the Zygmund space , for which and is uniformly bounded on a concerned compact subset of as . Using (38), we obtain that for every , there is a constant , such that , which implies that Now, set . By (9), we deduce that Applying the known Cauchy’s estimate, when we have the specific sequence that has a uniform type that converges to zero on a specific compact concerned subset of , we obtain that the specific sequence converges to zero on a specific compact concerned subset of as . Because is actually compact, it yields that Letting in the last inequality, we deduce that Because the number is an arbitrary and positive, this yields that The concerned specific implication is therefore proved.
On the other hand, assume that is a compact operator. Remarking that the function given by (33) has a uniform type that converges to zero on a specific compact concerned subset of as . In addition, we have Let be a concerned specific sequence in the disk , for which when . The test function can be defined by From the proof steps in Theorem 10, we deduce that . Furthermore, the specific function converges to zero uniformly on a concerned compact subset of the disk . Then, using Lemma 7, we obtain that the operator , when . Moreover, Therefore, we can obtain Thus, the concerned proof of our result is obtained clearly.