Some Characterizations of Weighted Holomorphic Function Classes by Univalent Function Classes

Some characterizations of QK ,ωðp, qÞ − type classes of holomorphic functions by Schwarzian derivatives with known conformal-type mappings are introduced in the present manuscript. Moreover, the action of the pre-Schwarzian derivatives on QK ,ωðp, qÞ − type classes, typically the univalent ones by using concerned Carleson-type measures, is investigated. In addition, we reveal important characterizations of some concerned weighted analytic-type spaces with the known Schwarzian derivatives evolving certain Q − type of concerned function class for a high utility toward practical and feasible application of concerned domains.


Introduction
Some concepts and terminologies related to weighted holomorphic function spaces are briefly recalled in this concerned section.
Suppose that D = fw ∈ ℂ : 0<|w|<1g defines the open unit disc in ℂ: The class which involves all holomorphic functions in D is symbolized by H ðDÞ: Assume that dσ w = dx dy defines the usual normalized area measure on D: Suppose that the known Green's function on D is given by gðw, aÞ = log ð1/|φ a ðwÞ | Þ, with φ a ðwÞ = w − a/1 − aw, for w, a ∈ D: The specific class of all known univalent functions in D will be symbolized by U (see [1][2][3] and others). If ∈U, Q = gðDÞ, and ∂Q stands for the Jordan curve, so g : D ⟶ Q will be called conformal map [2]. Throughout this manuscript, the function ω : ð0, 1 ⟶ ½0,∞Þ with ω ≠ 0: In addition, the concerned function K : ½0,∞Þ ⟶ ½0,∞Þ stands for a right-continuous and nondecreasing specific function.
Next, we report the recent advancements of the concepts of specific weighted classes of holomorphic function spaces. The choice of the appropriate functions give the specific essential properties of the underlying weighted classes of functions can have an important impact for the study.
In this manuscript, we dealt with discussing Schwarzian derivative class on Q K,ω ðp, qÞ classes impulsive problems of conformal mappings which is due to the derivative terms in the definitions of both weighted Bloch and Q K,ω ðp, qÞ classes of analytic-type functions. The characterizations involved Carleson measures of K, ω-type with a combination of the behavior of conformal mappings between the weighted Q-type functions. For a holomorphic function in the unit disc, the Schwarzian derivative is introduced early in 1873. This derivative was given to generalize and extend the Schwarz-Christoffel derivative type to study some properties of mappings that preserve certain angles. Recently, some authors have used Schwarzian derivative to characterize some results for holomorphic classes of function spaces (see [1,[18][19][20][21][22][23] and others).
Hereafter, the holomorphic function gðwÞ stands for a conformal map, and we shall set f ðwÞ≕ ln ðg ′ ÞðwÞ: The symbol N g ðwÞ denotes the pre-Schwarzian differentiability of gðwÞ, that is The Schwarzian differentiability for the function g is defined by Some essential basic properties of N g ðwÞ and S g ðwÞ are stated in [2].
(1) When the function gðwÞ is univalent on D, thus ð1 − jwj 2 Þ | N g ðwÞ | ≤6 < e 2 and ð1 − jwj 2 Þ 2 | S g ðwÞ | ≤6 then g is univalent function on D (3) For all concerned functions f ∈ HðDÞ, we have that f ∈ B ⟺ , we can find w ∈ ℂ and a concerned univalent function g, with f = w ln g ′ (4) The concerned Schwarzian-type derivative is a Möbius invariant using the equality S φ a ∘g = S g , Let the symbol I ⊂ ∂D define a subarc of the boundary of D: Assume that when |I | ≥1, thus by letting S 1 ðIÞ = D: The specific positive measure μ is said to be an actually bounded K, ω -Carleson-type measure on the disc D when Moreover, when thus μ is an actually compact Carleson-type measure of K, ω-type.
The next generic concerned result can be proved as the corresponding result in [25].
Some general concepts are defined in the following.

Journal of Function Spaces
Let n = 1, 2, ⋯, then the concerned specific Carleson boxes of dyadic-type are then defined by: of side-length ℓðB n,m Þ = 1/2 n and their inner half For the concerned univalent function g, assume that δ and ε are small enough. If B is a Carleson-type box of dyadic-type, then it is called that B is bad when Also, B is said to be a specific maximal-type bad square when the next specific bigger concerned dyadic squareB including B either has the specific equality ℓðBÞ = 1/2 or has the concerned inequality sup w∈TðBÞ ð1 − jwj 2 Þ 2 | S g ðwÞ | >δ: Lemma 6 [22]. Let g be a concerned univalent function on the disc D, and assume that there exists g 0 ∈ D with jS g ðw 0 Þj 2 ð1 − jw 0 j 2 Þ > δ: Thus, we can find a specific positive constant k = kðδÞ < 1, for which whenever w ∈ Dðw 0 , kð1 − jw 0 j 2 ÞÞ: Next, some estimations concerning ∑ m ℓ½KðBÞ will be given. For this result, suppose that Lemma 7. Assume that p, ε, δ are positive constants. Hence, for λ 1 , λ 2 > 0, we have that Proof. Suppose that B is a maximal-type square such that ℓðBÞ ≠ 1/2: Thus,B is a maximal-type bad square; hence, we can find w 0 ∈ TðBÞ with Therefore, using Lemma 6, we can find a disc D w 0 = Dðw 0 , cð1 − jw 0 j 2 ÞÞ ⊂ TðBÞ such that Therefore, we infer that Because the half TðB m Þ may be represented by two times only, then there are two specific square types only, B′ such that ℓðB′Þ = 1/2; thus (17) is verified. Then, the concerned lemma is therefore established.

Carleson Measure of K, ω-Type and Q K,ω ðp, qÞ Spaces
Certain equivalent concerned general conditions for the nth derivatives of analytic Q K,ω ðp, qÞ classes are obtained in the next generalized result.

Journal of Function Spaces
For the compact Carleson measure of K, ω-type, the following interesting result can be obtained.
Proof. Since S g ðwÞ = N g ′ ðwÞ − 1/2ðN g ′ ðwÞÞ 2 , then by Theorem 8, we deduce is a concerned Carleson measure of K, ω-type ⟺ is a Carleson measure of K, ω-type, and that ð1 − jwj 2 Þ | N g ðwÞ | ≤e 2 for every w ∈ D: Thus, for any 1 ≤ p < ∞, we obtain that Suppose that N g is continuous function on the closed unit disc D; because if the claim is not verified, we will lose the dilatations of ðN g Þ r ðzÞ = N g ðr wÞ; then, the proof can be ended by letting r ⟶ 1: Because h = ln g′ ∈ B 0 , for any ε > 0, there exists r ε for which |w | >r ε ; we thus obtain that |N g ðwÞ | ð1 − jwj 2 Þ < ε, with Thus, for some k = kðp, qÞ, we infer that Because q ≤ p − 2, for every a ∈ D, we then obtain that letting ε be so small such that 1 − Cε p /2 > 0: Then, Since jS g ðwÞj p ð1 − jwj 2 Þ p+q dσ w is a concerned Carleson measure of K, ω-type, after considering the supremum over a ∈ D of (29), we deduce that Using Theorem 8, we infer that jN g ðwÞj p ð1 − jwj 2 Þ q dσ w is a concerned Carleson measure of K, ω-type too. The proof is therefore completely obtained.

Journal of Function Spaces
The following essential and interesting proposition shall be given. Proposition 11. Let 1 ≤ p < ∞ and−2 < q < ∞: When f = l ng ′ ∈ Q K,ω ðp, qÞ, then is a Carleson measure of K, ω-type.
Proof. Since In view of Theorem 8 for n = 1 and f = ln g′, we then infer that is a Carleson measure of K, ω − type.
Moreover, using Theorem 8 with n = 2, we deduce that is a concerned Carleson measure of K, ω − type. Therefore, for p ≥ 1, we have that Hence, Using (33) and (34), we have is a concerned Carleson measure of K, ω-type. The proof is therefore completely finished.

Results by Logarithmic Characterizations
We give some logarithmic characterizations for univalent functions belonging to the classes where W is the weighted holomorphic Q K,ω ðp, qÞ class or the weighted holomorphic Q K,ω,0 ðp, qÞ class.
Remark 13. When q = p − 2 and KðtÞ ≡ 1, ωðtÞ ≡ 1, that is, the case of the weighted holomorphic Besov-type spaces B p , 1 < p < ∞, it can be proved clearly by remarking that each of these classes is also inclusive in the little Bloch class B 0 : This case was presented in [3] too.
Proof. Suppose that ln g ′ ∈ Q K,ω ðp, qÞ; then, by the help of Proposition 11, we deduce that is a concerned Carleson measure of K, ω-type.
Because the equality q = p − 2, we then obtain that Q K,ω ðp, qÞ = Q K,ω ðp, p − 2Þ: Now, the aim is to prove that when is a concerned Carleson measure of K, ω-type, then ln g ′ ∈ Q K,ω ðp, p − 2Þ: Thus, we need to prove that implies that This is equivalent to