Linear Differential Equations on Some Classes of Weighted Function Spaces

: Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new deﬁned classes, the speciﬁc growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefﬁcients of certain analytic-type functions, that is the equation h ( k ) + K k − 1 ( υ ) h ( k − 1 ) + . . . + K 1 ( υ ) h (cid:48) + K 0 ( υ ) h = 0, will be discussed in this current research, whereas the considered coefﬁcients K 0 ( υ ) , . . . , K k − 1 ( υ ) are holomorphic in the disc Γ R = { υ ∈ C : | υ | < R } , 0 < R ≤ ∞ . In addition, some non-trivial speciﬁc examples are illustrated to clear the roles of the obtained results with some sharpness sense. Hence, the obtained results are strengthen to some previous interesting results from the literature.


Introduction
The known theory of Function spaces is one of the most interesting and active research areas with various crucial applications in many branches of mathematics. This important theory has a major role in both mathematical analysis (e.g., approximation theory, theory differential and integral equations, measure theory, operator theory) and engineering sciences (e.g., recent computer-aided geometric design, the known image processing). This joyful theory has numerus generalizations including, among the others, C n extensions as well as Clifford analysis generalizations. Recently, the theory has been intensively researched in differential equations. Some radial growth of certain types linear differential equations have been introduced and studied with the help of some certain classes of analytic function spaces.
Suppose that Γ = Γ υ = {υ : |υ| < 1 stands for the open disc in C. Let H(Γ υ ) denote the class of all holomorphic functions on Γ υ . For a ∈ Γ υ the known Möbius transformations ϕ a (υ) is given by For a point a ∈ Γ υ and r ∈ (0, 1), the specific pseudo-hyperbolic concerned disk Γ(a, r) with specific pseudo-hyperbolic concerned center a and specific pseudo-hyperbolic radius r is defined by Γ(a, r) = ϕ a (rD).
This type of pseudo-concerned disk Γ(a, r) is a considered Euclidean-type disk, for which its Euclidean concerned supposed center, as well as its concerned Euclideantype radius are (1−r 2 )a 1−r 2 |a| 2 and (1−|a| 2 )r 1−r 2 |a| 2 , respectively (see [1]). The considered normalized Lebesgue-type area measure on Γ υ will be symbolized here by dm(υ). The following known identity can be stated as follows (see [1]): For a ∈ Γ υ , by the assumption υ = ϕ a (w), we easily have dm(w) = |ϕ a (υ)| 2 dm(υ). In addition, we have ϕ a (ϕ a (υ)) = υ, then ϕ −1 a (υ) = ϕ a (υ). For a, υ ∈ Γ υ and r ∈ (0, 1), the specific supposed pseudo-disc Γ(a, r) is given by Γ(a, r) = {υ ∈ Γ υ : |ϕ a (υ)| < r}. Further, the known analytic Green's function on the disc Γ υ with a specific singularity at the point a, is defined by For two specific quantities K h and C h , these specific quantities are depending on the holomorphic-type function h on Γ υ , we say that these quantities are equivalent and write K h ≈ C h , when we can obtain a constant N * that is finite, positive and not depend on the holomorphic function h, for which In the case of equivalence between K h and C h , thus we obtain that It should be noted: The symbol F 1 F 2 means that, we can find a specific constant γ > 0, for which F 1 ≤ γF 2 , whereas F 1 and F 2 are two holomorphic functions on Γ υ .
The authors in [2] have introduced an interesting active class of holomorphic functions, which is denoted by holomorphic Q p -spaces as follows: where dm(υ) is the known considered Euclidean area element on Γ υ and p ∈ (0, ∞). Here the considered weighted-type function g(υ, a) = log 1−āυ a−υ is introduced and studied as the well defined composition of the known Möbius-type transformation ϕ a (υ) with the essential concerned solution of the two dimensional-type Laplacian. Also, the considered weighted-type holomorphic function g(υ, a) is an actually the holomorphic Green's-type function in Γ υ , that has a specific pole at the point a ∈ Γ υ .
From now on, we will let the function : Γ υ → R+, to be bounded and continuous function. With the help of this function, we introduce the following new holomorphic function classes: Definition 2. Let h ∈ H(Γ υ ), for α ∈ (0, ∞), we say that h is in the little exponential -Bloch space B α ,n;0 , when Remark 1. It should be noted here that the exponential -weighted Bloch space B α ;n , is more general than the holomorphic Bloch space from algebraic point of view but for geometric meaning it needs more study on connections of Bergman metric and the exponential -weighted Bloch space B α ;n ,. When (υ) ≡ 1 and n = 1, thus the definition of known α-Bloch is obtained (see [3]).
The following general analytic function spaces can be introduced as follows: Definition 3. Suppose that 0 < p < ∞. The holomorphic function h ∈ H(Γ υ ) is said to belong to the space P p, e ,n (ϕ), when h 2 e ,p,n = sup The holomorphic function h ∈ H(Γ υ ) is said to belong to the space P p, ,n (g), when Definition 6 ( [1,4]). Let h ∈ H(Γ υ ) be holomorphic-type function in Γ υ and let p ∈ (1, ∞). When, thus h belongs to the holomorphic-type Besov space B p .

Linear-type Complex Differential Equations
The specific growth of the solutions of the differential-type complex equations of the form: will be studied in this current research. Meanwhile, the considered coefficients K 0 (υ), . . . , For further details on the numerous studies of the theory of complex linear differential-type equations on some various classes of complex function spaces, the following citations can be used [6][7][8][9][10][11][12][13][14][15][16].
Pommerenke [16] studied the second-type order differential complex equation, that given by: whereas B(υ) is a holomorphic function in Γ υ . Heittokangas [10] improved the research due to Pommerenke by using the equation of the form: whereas B(υ) is a holomorphic-type function in Γ υ and k ∈ N. This study followed by an interesting research in [12], which considered a joyful research with the help of the complex linear differential equation of the form (1) with all obtained solutions are belonging to the classes H ∞ q (e )-space or the Dirichlet D p (e ( ) )-space. Recall now some equivalent statements for holomorphic-type functions to be in the holomorphic exponential -Bloch spaces B α (e , n) and the little holomorphic exponential little -Bloch spaces B α (e , n, 0).

Lemma 3 ([19]
). Let f be an analytic function in Γ υ and let 0 < p < ∞, −1 < q < ∞. Then there exist two positive constant C 1 and C 2 , depending only on p and q, such that
Proof. The proof can be obtained simply as given in Lemma 4.1 (see [21]), thus it will be removed.
In view of (ii), the next specific inequality can be obtained Thus, whereas c is a specific positive constant. Hence using Theorem 2.1, we obtain which gives (i). This completes the proof of Theorem 2.
A collocation of interesting approach for solving a class of certain complex linear-type differential equations in the unit disc will be studied in the next section. By using a concerned collocation of holomorphic functions defined in some general weighted spaces of the analytic-type, some joyful solutions of a complex linear differential equations can be investigated. Such solutions transforms the considered linear complex differential equations into certain weighted holomorphic function spaces. For some certain classes of holomorphic function spaces, some concerned coefficients have important roles in the obtained results with the help of a specific system of complex linear equations. The used method results the analytic solution if the exact solutions are in certain weighted holomorphic functions. An interesting example is also illustrated to investigate the validity and applicability of the given technique and the comparisons which made with some obtained results. The given examples have reflexed and demonstrated the importance of the current research.

Entire Solutions
Some few decades ago, many interesting techniques have been developed and evolved for the solutions of the concerned complex-type differential Equations (1). While quite a good number portion of the solutions is useful for certain research purposes, there are special some which are so important by the complex function spaces solutions. In this current manuscript, very general classes of complex function spaces for solving complex-type differential equations are considered and deeply discussed. After that, the research results are established with two basic methods commonly proved by the help of holomorphic norms with the defined spaces and by introducing certain entire solutions for the complextype linear differential equations. The obtained results are pursued by the corresponding results in literature.

Remark 2.
From Proposition 2, we can deduce that the norm of the exponential Bloch-type space is equivalent to the following norm Using Proposition 3.1 and mathematical induction, the next result can be deduced easily.

Proposition 3.
Let h ∈ H(Γ υ ) and let either α ∈ (0, ∞) for n ∈ N or α > 1 and n = 0. Thus Further, For the n-th derivatives of holomorphic-type functions in H ∞ q (e ) and P p, ;n , the next interesting proposition can be deduced.

Proposition 4. Let
: Γ υ → (0, ∞), be a bounded continuous function and let h ∈ H(Γ υ ), 1 < α < ∞ and n ∈ N. Hence, the next concerned specific statements are comparable: Proof. Let Y be a specific Banach space of holomorphic functions on Γ. Assume that L(Y) denotes the space of all known defined point-wise multipliers of Y. Since we have the fact, L(Y) ⊂ H ∞ q , the proof of (a 1 ) can be obtained directly from Propositions 2, 3 and this fact. For the proof of (a 2 ), we can use the fact that L(Y) ⊂ Y. Proof of (a 3 ) is clearly from the definition of the exponential Bloch functions. Now, we give the following interesting result, which gives some solutions of the complex differential Equation (1).

Theorem 3.
Let : Γ υ → (0, ∞), be a bounded continuous nondecreasing function. For every q > 0, thus we can find a specific constant β = β(q, k) > 0, for which that when the specific coefficients K j (υ) of (1) hold thus all specific solutions of (1) are belonging to the holomorphic space H ∞ q (e ).
The following interesting result can be deduced from Theorem 3.
Proof. The proof can be followed by setting ≡ 1 in Theorem 3.
Proof. Suppose that β ≤ 1 and r ∈ (δ, 1). As in [11], we can find a specific constant λ 1 > 0, depending only some initial values of the analytic function h, for which for all 0 ≤ θ ≤ 2π, then the specific assumption (4) gives Choosing β = ( q k 2 ) k , we deduce that the proof is completely established.