Characterizations of Integral Type for Weighted Classes of Analytic Banach Function Spaces

Numerous studies of analytic function spaces by several classes of functions are introduced and intensively studied. (e theory of function spaces provides interesting tools in many active branches of mathematics, especially in mathematical analysis such as in operator theory, measure theory, and differential equations. In the present study, we aim to give the definition of g-Bloch space of holomorphic functions. Using the new class of functions, some essential relations between it and some other known classes are investigated. Next, we report the recent advancements of the concepts of specific-weighted classes of holomorphic function spaces. (e choice of the appropriate functions gives the specific essential properties of the underlying weighted classes of functions that can have an important impact for the study. Specific weighted classes of holomorphic function spaces and concepts are presented. Let U � w: w ∈ C, |w|< 1 { }, and H(U) denote the class of all holomorphic functions that belonging to U. (e known analytic Bloch-type space [1–5] is defined by


Introduction and Preliminaries
Numerous studies of analytic function spaces by several classes of functions are introduced and intensively studied. e theory of function spaces provides interesting tools in many active branches of mathematics, especially in mathematical analysis such as in operator theory, measure theory, and differential equations. In the present study, we aim to give the definition of g-Bloch space of holomorphic functions. Using the new class of functions, some essential relations between it and some other known classes are investigated.
Next, we report the recent advancements of the concepts of specific-weighted classes of holomorphic function spaces. e choice of the appropriate functions gives the specific essential properties of the underlying weighted classes of functions that can have an important impact for the study.
Hereafter, we set and set e modified Green's function is introduced by Motivated by the modified Green's function, the following definitions can be presented. Definition 1. Let 0 < m < ∞ and 0 < n < ∞. For the function f ∈ H(U), we define the analytic g-Bloch space B n,m g as follows: Furthermore, assume that Example 1. Let m ∈ (0, ∞). Suppose that It is very obvious that the function f is a g-Bloch function.
Remark 1. Using Definition 1, relationships between analytic Dirichlet-type functions and analytic Bloch functions can be characterized.
When m � 0 and 0 < n � α < ∞, then we will obtain α-Bloch space. e case m ∈ (− ∞, 0) induces some other types of analytic function spaces with different behaviors and can be studied separately. e little analytic g-Bloch space B n,m,0 g can be considered as a subspace of the analytic g-Bloch space that includes all Remark 2. e symbol B n,m g (f) stands for a seminorm, while the usual norm can be defined as Applying the above norm, the space B n,m g is a complete normed space (Banach space). e next lemma can be applied for some results in this study.

g-Bloch Space and Dirichlet Space
Some essential characterizations between the analytic Dirichlet-type space and the analytic g-Bloch space are given in this section.

Journal of Mathematics
Proof. Because the pseudohyperbolic disk can be symbolized by where w 0 is its center and R is its radius, then we infer that Using the definition of the modified Green's function as well as the inequalities, [25]).

(17)
We can obtain e proof of Proposition 1 is therefore finished.

Remark 3. Corollaries 1 and 2 interpret that the analytic
Dirichlet-type space can be considered as a subspace of the analytic g-Bloch space when m � n � 2 as well as for n ∈ [1, ∞) and m ∈ (0, ∞), respectively.
en, for 0 < m < ∞ and 0 < n < ∞ with m + n > 0, we obtain that Proof. From the definition of g-Bloch space, we have By a change of the variables technique, we infer that Since, erefore, which implies that where 0 < k 1 < ∞. e proof is therefore established.

Journal of Mathematics
Propositions 2 and 3 result in the following fundamental theorem. □ Theorem 1. Let f ∈ H(U); then, we have equivalence between the following statements:

Remark 4.
e obtained results in eorem 1 reflexed the major role of the newly definition of the analytic g-Bloch space which has used to get relations between analytic Dirichlet-type space D and B n,m g space.
Proof. From subharmonicity principle, the following inequality can be easily obtained: Applying the technique of change of variables, the next inequality can be inferred: From [5], we have is yields that Because, en, Using the inequality 4 Journal of Mathematics we can deduce that e proof is therefore completely finished. □ Corollary 3. By Proposition 4, for p > 0, 0 ≤ m < ∞, 1 ≤ n < ∞, and |w 0 | < 1, the following inclusions can be easily proved: where where and λ is an absolute positive constant.
Proof. It is not hard to see that Applying Lemma 1, we obtain Combining Corollary 3 and Proposition 5, we have the following theorem: □ Theorem 2. Let h ∈ H(U). Assume that 0 ≤ m < ∞ and 0 < n < ∞ with m + n + p > 0 as well as 0 < p < ∞. erefore, the next assertions can be equivalent: Proof. (a)⇒(b) can be established using Proposition 5.

A Specific Criteria
e following symbol stands for a non-Euclidean distance of hyperbolic-type between the points w 0 and w in U. [26]).
Now, for 0 < R < ∞, set Let h ∈ H(U) be the nonconstant analytic function, and let S h (w, R) define the concerned area of the Riemannian image P(w, R) of T(w, R) by the function h, and assume that S * h (w, R) defines the area of the image P(w, R) of T(w, R) using the function h. It should be noted that P(w, R) defines the projection to C. e length of the Riemannian image of by M h (z, ρ), and the symbol M h (z, ρ) defines the length of the outer boundary of P(w, R). If Γ defines a specific bounded domain in C, the concept of outer boundary means that we are working in the boundary of C\B, where B may be defined as the unbounded component for the complement C\Γ. Now, we clearly have the following inequalities: where 0 < R < ∞ and each w ∈ U.
Proof. First, we suppose that h ′ (w) ≠ 0. Now, we suppose that . . are the positive constants. Now, which implies that Using eorems 1 and 2 in [27], the following inequalities can be deduced: Since, 0 < |φ w 0 (w)| < ρ < 1, then Also, □ Remark 6. In the proof of Proposition 6, positive coefficients are considered to keep the convergence of Taylor or Fourier power series in its region.

Remark 7.
eorem 3 gives an interesting and global criterion for analytic g-Bloch-type function by the help of the concrete area as well as the concerned length of the images of both non-Euclidean unified discs and unified circles, respectively. e obtained results in this section extended and improved some specific results in the study by Yamashita [26].

Remark 8.
Quite recently, a new study of bicomplex functions was introduced in [28].
An interesting question can be formulated for the newly defined g-Bloch functions as follows. Can we discuss and study the new defined class of analytic g-Bloch type functions in the case of bicomplex functions?

Conclusions
Function spaces theory is developed, extended, and generalized to spaces of several complex variables ( [8][9][10][11][12][13]29]) also using quaternion-valued functions ( [21][22][23][24][30][31][32][33]). e intention of this study is to introduce a new type of analytic function spaces, which plays an interesting and global rule of studying complex function spaces. It should be emphasized that both the worked plane of the study (i.e., U) and the considered holomorphic functions of Bloch-type as well as specific properties of Green's function are extremely needed for the new classes. e holomorphic classes of g-Bloch functions are defined and deeply considered using a modified Green's function. By the new function classes, some specific relations and inclusions for the holomorphic Dirichlet-type space as well as holomorphic Q p spaces are obtained. On the other hand, an extension of Yamashita's result [26] is presented using holomorphic g-Bloch functions.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.