Abstract
Some characterizations of 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 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 type of concerned function class for a high utility toward practical and feasible application of concerned domains.
1. Introduction
Some concepts and terminologies related to weighted holomorphic function spaces are briefly recalled in this concerned section.
Suppose that defines the open unit disc in The class which involves all holomorphic functions in is symbolized by Assume that defines the usual normalized area measure on Suppose that the known Green’s function on is given by with for The specific class of all known univalent functions in will be symbolized by (see [1–3] and others). If , and stands for the Jordan curve, so will be called conformal map [2]. Throughout this manuscript, the function with In addition, the concerned function 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.
Suppose that
Definition 1 [4–6]. Let Then, the function is including in weighted Bloch-type space when
Definition 2 [4–6]. Let The function is included in the little specific weighted Bloch-type space when When let be the concerned analytic space that consists of all specific functions ([5, 6]), for which
Further, on the boundary of we can say that belongs to the concerned class when
Remark 3. For more research on we refer to [7–11] and others. When we obtain the classes and (see [12, 13] and others). Also, there are some recent research in Clifford setting (see [14–17] and others).
In this manuscript, we dealt with discussing Schwarzian derivative class on classes impulsive problems of conformal mappings which is due to the derivative terms in the definitions of both weighted Bloch and classes of analytic-type functions. The characterizations involved Carleson measures of -type with a combination of the behavior of conformal mappings between the weighted -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, 18–23] and others).
Hereafter, the holomorphic function stands for a conformal map, and we shall set The symbol denotes the pre-Schwarzian differentiability of that is
The Schwarzian differentiability for the function is defined by
Some essential basic properties of and are stated in [2]. (1)When the function is univalent on thus
and (2)If or then is univalent function on (3)For all concerned functions we have that we can find and a concerned univalent function with (4)The concerned Schwarzian-type derivative is a Möbius invariant using the equality and it is also satisfying for every Möbius transformation .
Let the symbol define a subarc of the boundary of Assume that
when , thus by letting The specific positive measure is said to be an actually bounded -Carleson-type measure on the disc when
Moreover, when
thus is an actually compact Carleson-type measure of -type.
Remark 4. It is very clearly to see that, if and , then is an actually bounded -Carleson-type measure on and an actually bounded -Carleson-type measure on (see [24]).
The next generic concerned result can be proved as the corresponding result in [25].
Lemma 5. A concerned positive measure on is a concerned Carleson measure of -type
Some general concepts are defined in the following.
Let then the concerned specific Carleson boxes of dyadic-type are then defined by: of side-length and their inner half
For the concerned univalent function assume that and are small enough. If is a Carleson-type box of dyadic-type, then it is called that is bad when
Also, is said to be a specific maximal-type bad square when the next specific bigger concerned dyadic square including either has the specific equality or has the concerned inequality
Lemma 6 [22]. Let be a concerned univalent function on the disc and assume that there exists with Thus, we can find a specific positive constant for which whenever
Next, some estimations concerning will be given. For this result, suppose that
Lemma 7. Assume that are positive constants. Hence, for we have that
Proof. Suppose that is a maximal-type square such that Thus, is a maximal-type bad square; hence, we can find with
Therefore, using Lemma 6, we can find a disc such that
Therefore, we infer that
Because the half may be represented by two times only, then there are two specific square types only, such that ; thus (17) is verified. Then, the concerned lemma is therefore established.
2. Carleson Measure of -Type and Spaces
Certain equivalent concerned general conditions for the th derivatives of analytic classes are obtained in the next generalized result.
Theorem 8. Let and Assume that and . Hence, the next general concerned quantities are equivalent: (1)(2) is a Carleson measure of -type(3)(4)
Proof. (1) (2). This can be obtained as a corresponding concerned part in Theorem 2 (see [13]).
(1) (3). This can be deduced as the proof of Theorem 1 in [13] by very simple and minor specific modifications
(2) (4). Using Lemma 5 for then is a concerned Carleson measure of -type if and only if
Thus, the concerned proof is completely finished.
For the compact Carleson measure of -type, the following interesting result can be obtained.
Theorem 9. Let and Suppose also and the concerned analytic function . Then, the next essential specific statements are equivalent: (a)(b) is a concerned compact Carleson measure of -type(c)(d)
Now, we give the following interesting lemma which connects Carleson measure of -type, Schwarzian derivative and the pre-Schwarzian derivative.
Lemma 10. Let , and suppose that with and assume that Then, if is a concerned -Carleson measure, we have that is a concerned -Carleson measure too.
Proof. Since then by Theorem 8, we deduce is a concerned Carleson measure of -type is a Carleson measure of -type, and that for every Thus, for any we obtain that
Suppose that is continuous function on the closed unit disc ; because if the claim is not verified, we will lose the dilatations of ; then, the proof can be ended by letting
Because for any , there exists for which ; we thus obtain that with
Thus, for some we infer that
Because for every we then obtain that letting be so small such that Then,
Since is a concerned Carleson measure of -type, after considering the supremum over of (29), we deduce that
Using Theorem 8, we infer that is a concerned Carleson measure of -type too. The proof is therefore completely obtained.
The following essential and interesting proposition shall be given.
Proposition 11. Let and When then is a Carleson measure of -type.
Proof. Since In view of Theorem 8 for and we then infer that is a Carleson measure of .
Moreover, using Theorem 8 with we deduce that is a concerned Carleson measure of .
Therefore, for we have that
Hence,
Using (33) and (34), we have is a concerned Carleson measure of -type.
The proof is therefore completely finished.
3. Results by Logarithmic Characterizations
We give some logarithmic characterizations for univalent functions belonging to the classes where is the weighted holomorphic class or the weighted holomorphic class.
Theorem 12. Let and also suppose that or and Thus, the following fundamental concerned statements are equivalent: (I)(II) and is a concerned Carleson measure of -type.
Proof. Assume that the extended choice of the parameters Then, if with The concerned proof of (I) (II) can be deduced by Lemma 10 as well as Proposition 11.
Remark 13. When and that is, the case of the weighted holomorphic Besov-type spaces it can be proved clearly by remarking that each of these classes is also inclusive in the little Bloch class This case was presented in [3] too.
Theorem 14. Let and and suppose that Thus, is a concerned Carleson measure of -type.
Proof. Suppose that ; then, by the help of Proposition 11, we deduce that is a concerned Carleson measure of -type.
Because the equality we then obtain that Now, the aim is to prove that when is a concerned Carleson measure of -type, then Thus, we need to prove that implies that
This is equivalent to
Since then we obtain that
Let us now suppose that is a specific concerned continuous function on which hence occupy that for the end of the concerned proof.
Calculating the following specific integral
For suppose that
Using Theorem 8, we can find a specific constant for which
Then,
Since we infer that
Then,
Let us consider the sequence of Carleson bad boxes, then
For such that we infer that
By Lemma 7, we deduce that
Letting appear as a global concerned positive constant, we may infer that
Therefore, which gives that
This is equivalent to , and this finishes the proof.
4. Conclusions
Revealing studies on the theory of weighted function spaces evolving the operator theory which includes important and useful studies could enhance research on both theories. In this concerned manuscript, the investigations of emission evolving a class of operators on a general space of weighted functions are then studied. Typically, the possibility of acting the Schwarzian derivative evolving certain -type classes is explored. We have revealed that the majority of proof methods supply a snapshot of the role of the Schwarzian derivatives under properties of the weighted class of function spaces. As the definition of the weighted holomorphic -type of function class structure, some changes in weights and in the derivative part, the amount of required results entirely is large, and it leads to an efficient clear utility.
To this end, we propose interesting characterizations, a novel of Schwarzian differentially private evolving -type of function class releasing the derivative algorithm which reduces the scales and tool up a clear utility.
Data Availability
We have not apply or consider any data during this concerned current research.
Conflicts of Interest
The authors of this concerned study declare completely that they have no any competing of interests.
Acknowledgments
The first author thanks the Taif University researchers for the support (Project number TURSP-2020/159), Taif University, Saudi Arabia. S. Omran thanks the Academy of Scientific Research and Technology (ASRT), Egypt, for supporting him with the project(6407).