# Some Characterizations of Weighted Holomorphic Function Classes by Univalent Function Classes

Authors: A. El-Sayed Ahmed; S. Omran
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Some characterizations ofQK,ωp,q− 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,ωp,q− 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.


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## Body

## 1. Introduction

Some concepts and terminologies related to weighted holomorphic function spaces are briefly recalled in this concerned section.Suppose thatD=w∈ℂ:0<∣w∣<1 defines the open unit disc in ℂ. The class which involves all holomorphic functions in D is symbolized by HD. Assume that dσw=dxdy defines the usual normalized area measure on D. Suppose that the known Green’s function on D is given by gw,a=log1/∣φaw∣, with φaw=w−a/1−a¯w, for w,a∈D. The specific class of all known univalent functions in D will be symbolized by U (see [1–3] and others). If ∈U,Q=gD, and ∂Q stands for the Jordan curve, so g:D⟶Q will be called conformal map [2]. Throughout this manuscript, the function ω:0,1⟶0,∞ with ω≠0. In addition, the concerned function K:0,∞⟶0,∞ 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(1)Tα,ω=1−∣w∣αω1−w,TKgw,a,ω=Kgw,aω1−∣w∣.Definition 1 [4–6].
Let0<α<∞. Then, the function g∈HD is including in α−weighted Bloch-type space Bωα when
(2)∥g∥Bωα=supw∈DTα,ω∣g′w∣<∞.Definition 2 [4–6].
Let0<α<∞. The function g∈HD is included in the little specific weighted Bloch-type space Bω,0α when
(3)∥g∥Bω,0α=lim∣w∣→1−Tα,ω∣g′w∣=0.
Whenp∈0,∞,q∈−2,∞, let QK,ωp,q be the concerned analytic space that consists of all specific functions g∈HD ([5, 6]), for which
(4)∥g∥QK,ωp,qp≔supa∈D∫Dg′wp1−w2qTKgw,a,ωdσw<∞.Further, on the boundary ofD, we can say that g∈QK,ωp,q belongs to the concerned class QK,ω,0p,q when
(5)lim∣a∣→1∫Dg′wp1−w2qTKgw,a,ωdσw=0.Remark 3.
For more research onQK,ωp,q, we refer to [7–11] and others. When ωt≡1, we obtain the classes QKp,q and QK,0p,q (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 onQK,ωp,q classes impulsive problems of conformal mappings which is due to the derivative terms in the definitions of both weighted Bloch and QK,ωp,q 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, 18–23] and others).Hereafter, the holomorphic functiongw stands for a conformal map, and we shall set fw≕lng′w. The symbol Ngw denotes the pre-Schwarzian differentiability of gw, that is
(6)Ngw≕dfwdw=g′′wg′w.The Schwarzian differentiability for the functiong is defined by
(7)Sgw=Ng′w−12Ngw2=g′′wg′w′−12g′′wg′w2.Some essential basic properties ofNgw and Sgw are stated in [2].
(1)
When the functiongw is univalent on D, thus1−w2∣Ngw∣≤6<e2 and 1−w22∣Sgw∣≤6<e2(2)
If1−w2∣Ngw∣≤1, or 1−w22∣Sgw∣≤2, then g is univalent function on D(3)
For all concerned functionsf∈HD, we have that f∈B⟺, we can find w∈ℂ and a concerned univalent function g, with f=wlng′(4)
The concerned Schwarzian-type derivative is a Möbius invariant using the equalitySφa∘g=Sg, and it is also satisfying 1−w22∣Sg∘φaw∣=1−φaw22∣Sgφaw, for every Möbius transformation φaw,a∈D.Let the symbolI⊂∂D define a subarc of the boundary of D. Assume that
(8)S1I=rζ∈D:1−I<r<1,ζ∈I,when∣I∣≥1, thus by letting S1I=D. The specific positive measure μ is said to be an actually bounded K,ω-Carleson-type measure on the disc D when
(9)supI⊂∂D∫S1IK1−∣z∣/∣I∣ω1−∣z∣dμz<∞.Moreover, when(10)lim∣I∣→0∫SIK1−∣z∣/∣I∣ω1−∣z∣dμz=0,thusμ is an actually compact Carleson-type measure of K,ω-type.Remark 4.
It is very clearly to see that, ifKt=tp, and ω1−∣z∣≡1, then μ is an actually bounded p-Carleson-type measure on D⟺1−w2dμ and an actually bounded p-Carleson-type measure on D (see [24]).The next generic concerned result can be proved as the corresponding result in [25].Lemma 5.
A concerned positive measureμ on D is a concerned Carleson measure of K,ω-type ⟺(11)supa∈D∫DK1−φaz2ω1−∣z∣dμz<∞.Some general concepts are defined in the following.Letn=1,2,⋯, then the concerned specific Carleson boxes of dyadic-type are then defined by:
(12)Bn,m=z=reiθ∈D:1−12n≤∣w∣<1,m2n+1≤θπ<m+12n+1,0≤m≤2n+1,of side-length ℓBn,m=1/2n and their inner half
(13)TBn,m=Bn,m∩w∈Bn,m:1−12n≤∣w∣<12ℓBn,m.For the concerned univalent functiong, 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
(14)supz∈TB1−w2∣Ngw∣≥εalsosupw∈TB1−w22∣Sgw∣≤δ.Also,B is said to be a specific maximal-type bad square when the next specific bigger concerned dyadic square B~ including B either has the specific equality ℓB~=1/2 or has the concerned inequality supw∈TB1−w22∣Sgw∣>δ.Lemma 6 [22].
Letg be a concerned univalent function on the disc D, and assume that there exists g0∈D with Sgw021−w02>δ. Thus, we can find a specific positive constant k=kδ<1, for which
(15)Sgw21−w2>δ32,whenever w∈Dw0,k1−w02.Next, some estimations concerning∑mℓKB will be given. For this result, suppose that
(16)Tφa,ω=K1−φaw2ω1−∣w∣.Lemma 7.
Assume thatp,ε,δ are positive constants. Hence, for λ1,λ2>0, we have that
(17)∑mℓKBQm≤λ1+λ2∫DSgwp1−w22p−2Tφa,ωdσw.Proof.
Suppose thatB is a maximal-type square such that ℓB≠1/2. Thus, B~ is a maximal-type bad square; hence, we can find w0∈TB~ with
(18)1−w22∣Sgw0∣>δ.Therefore, using Lemma6, we can find a disc Dw0=Dw0,c1−w02⊂TB~ such that
(19)1−w22∣Sgw∣>δ32,∀w∈Dw0.Therefore, we infer that(20)ℓKB≈ℓKB~≤∫Dw01−w2−2Tφa,ωdσw≤C∫Dw0Sgwp1−w2p−2Tφa,ωdσw.Because the halfTB~m may be represented by two times only, then there are two specific square types only, B′ such that ℓB′=1/2; thus (17) is verified. Then, the concerned lemma is therefore established.


## 2. Carleson Measure ofK,ω-Type and QK,ωp,q Spaces

Certain equivalent concerned general conditions for thenth derivatives of analytic QK,ωp,q classes are obtained in the next generalized result.Theorem 8.
Let0<p<∞ and −2<q<∞. Assume that n∈ℕ and f∈HD. Hence, the next general concerned quantities are equivalent:
(1)
f∈QK,ωp,q(2)
fnwp1−w2np−p+qdσw is a Carleson measure of K,ω-type(3)
supa∈D∫Dfnwp1−w2np−p+qTK(gw,a,ωdσw<∞(4)
supa∈D∫Dfnwp1−w2np−p+qTφa,ωdσw<∞.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 dμw=fnwp1−w2np−p+qdσw, then μ is a concerned Carleson measure of K,ω-type if and only if
(21)supa∈D∫DTφa,ωdμw=supa∈D∫Dfnwp1−w2np−p+qTφa,ωdσw<∞.Thus, the concerned proof is completely finished.For the compact Carleson measure ofK,ω-type, the following interesting result can be obtained.Theorem 9.
Letp∈0,∞ and q∈−2,∞. Suppose also n∈ℕ and the concerned analytic function f∈HD. Then, the next essential specific statements are equivalent:
(a)
f∈QK,ω,0p,q(b)
fnwp1−w2np−p+qdσw is a concerned compact Carleson measure of K,ω-type(c)
lim∣a∣→1∫Dfnwp1−w2np−p+qTKgw,ωdσw=0(d)
lim∣a∣→1∫Dfnwp1−w2np−p+qTφa,ωdσw=0.Now, we give the following interesting lemma which connects Carleson measure ofK,ω-type, Schwarzian derivative and the pre-Schwarzian derivative.Lemma 10.
Let1≤p<∞, and suppose that q∈−2,∞ with q−p≤−2 and assume that f=lng′∈B0. Then, if
(22)Sgwp1−w2p+qdσwis a concerned K,ω-Carleson measure, we have that Ngwp1−w2qdσw is a concerned K,ω-Carleson measure too.Proof.
SinceSgw=Ng′w−1/2Ng′w2, then by Theorem 8, we deduce
(23)Ng′wp1−w2qdσwis a concerned Carleson measure of K,ω-type ⟺(24)Ng′wp1−w2p+qdσwis a Carleson measure of K,ω-type, and that 1−w2∣Ngw∣≤e2 for every w∈D. Thus, for any 1≤p<∞, we obtain that
(25)Ia=∫DNg′wp1−w2p+qTφa,ωdσw≤2p−1∫DSgwp1−w2p+qTφa,ωdσw+12∫DNg′w2p1−w2p+qTφa,ωdσw.Suppose thatNg is continuous function on the closed unit disc D¯; because if the claim is not verified, we will lose the dilatations of Ngrz=Ngrw; then, the proof can be ended by letting r⟶1.Becauseh=lng′∈B0, for any ε>0, there exists rε for which ∣w∣>rε; we thus obtain that ∣Ngw∣1−w2<ε, with
(26)∫DNgw2p1−w2p+qTφa,ωdσw=∫∣w∣>rεNgw2p1−w2p+qTφa,ωdσw+∫∣w∣≤rεNgz2p1−w2p+qTφa,ωdσw=J1a+J2a.Thus, for somek=kp,q, we infer that
(27)J1a=∫∣w∣>rεNg′z2p1−w2p+qTφa,ωdσw≤εp∫DNg′zp1−w2qTφa,ωdσw≤Cεp∫DNg′zp1−w2p+qTφa,ωdσw=CεpJa.Becauseq≤p−2, for every a∈D, we then obtain that
(28)J2a=∫∣w∣≤rεNgz2p1−w2p+qTφa,ωdσw≤e4p∫∣w∣≤rε1−w2q−pTφa,ωdσw≤e2p1−rε2p−q.letting ε be so small such that 1−Cεp/2>0. Then,
(29)1−Cεp2Ja≤2p−1∫DSgwp1−w2p+qTφa,ωdσw+e4p21−rε2p−q,SinceSgwp1−w2p+qdσw is a concerned Carleson measure of K,ω-type, after considering the supremum over a∈D of (29), we deduce that
(30)supa∈DJa=supa∈D∫DN′gwp1−w2p+qTφa,ωdσw<∞.Using Theorem8, we infer that Ngwp1−w2qdσw is a concerned Carleson measure of K,ω-type too. The proof is therefore completely obtained.The following essential and interesting proposition shall be given.Proposition 11.
Let1≤p<∞ and−2<q<∞. When f=lng′∈QK,ωp,q, then
(31)Sgwp1−w2p+qdσwis a Carleson measure of K,ω-type.Proof.
Since(32)∥lng′∥B=supa∈D1−w2∣Ngw∣≤e3.
In view of Theorem8 for n=1 and f=lng′, we then infer that
(33)f=lng′∈QK,ωp,q⟺Ngwp1−w2qdσwis a Carleson measure of K,ω−type.Moreover, using Theorem8 with n=2, we deduce that
(34)Ng′wp1−w2p+qdσwis a concerned Carleson measure of K,ω−type.Therefore, forp≥1, we have that
(35)Sgwp≤2p−1Ng′wp+12Ng′w2p.Hence,(36)Sgwp1−w2p+q≤2p−1Ng′wp1−w2p+q+12∥logg′∥BNgwp1−w2q.Using (33) and (34), we have
(37)Sgwp1−w2p+qdσwis a concerned Carleson measure of K,ω-type.The proof is therefore completely finished.


## 3. Results by Logarithmic Characterizations

We give some logarithmic characterizations for univalent functions belonging to the classes(38)SW=lng′:g∈U,lng′∈W,where W is the weighted holomorphic QK,ωp,q class or the weighted holomorphic QK,ω,0p,q class.Theorem 12.
Letp∈1,∞ and q∈−2,∞ also suppose that q+2<p or q=p−2 and Kt≡1. Thus, the following fundamental concerned statements are equivalent:
(I)
lng′∈QK,ωp,q(II)
lng′∈Bω,0 and Sgwp1−w2p+qdσw is a concerned Carleson measure of K,ω-type.Proof.
Assume that the extended choice of the parametersp,qlng′∈QK,ωp,q⊂Bωq+2/p. Then, if q+2<p,QK,ωp,q⊂Bωα, with 0<α<1. The concerned proof of (I) ⟺ (II) can be deduced by Lemma 10 as well as Proposition 11.Remark 13.
Whenq=p−2 and Kt≡1,ωt≡1, that is, the case of the weighted holomorphic Besov-type spaces Bp,1<p<∞, it can be proved clearly by remarking that each of these classes is also inclusive in the little Bloch class B0. This case was presented in [3] too.Theorem 14.
Letp∈1,∞ and q∈−2,∞ and suppose that q+2=p. Thus,
(39)lng′∈QK,ωp,q⟺Sgwp1−w2p+qdσw
is a concerned Carleson measure ofK,ω-type.Proof.
Suppose thatlng′∈QK,ωp,q; then, by the help of Proposition 11, we deduce that
(40)Sgwp1−w2p+qdσwis a concerned Carleson measure of K,ω-type.Because the equalityq=p−2, we then obtain that QK,ωp,q=QK,ωp,p−2. Now, the aim is to prove that when
(41)Sgwp1−w22p−2dσwis a concerned Carleson measure of K,ω-type, then lng′∈QK,ωp,p−2. Thus, we need to prove that
(42)∫DSgwp1−w22p−2Tφa,ωdσw<∞implies that
(43)∫DNgwp1−w2p−2Tφa,ωdσw<∞.This is equivalent to(44)∫DNg′wp1−w22p−2Tφa,ωdσw<∞.SinceNg′wp≤2p−1Sgwp+1/2Ngz2p, then we obtain that
(45)∫DNg′wp1−w22p−2Tφa,ωdσw≤2p−1∫DSgwp1−w22p−2Tφa,ωdσw+12∫DNgw2p1−w22p−2Tφa,ωdσw.Let us now suppose thatNg is a specific concerned continuous function on D¯, which hence occupy that r⟶1 for the end of the concerned proof.Calculating the following specific integral(46)INg2D=∫DNgw2p1−w22p−2Tφa,ωdσw.Forε,δ>0, suppose that
(47)D=w∈D:Ngw1−w2<ε,D1=w∈D:Sgw1−w22>δ,Δ=D\D∪D1=w∈D:∣Ngw∣1−w2≥ε,Sgw1−w22≤δ.Using Theorem8, we can find a specific constant λ>0, for which
(48)∫DNg′wp1−w2p−2Tφa,ωdσw≤λ∫DNg′wp1−w22p−2Tφa,ωdσw,Then,(49)INg2D=∫DNg′w2p1−w22p−2Tφa,ωdσw<εp∫DNg′wp1−w2p−2Tφa,ωdσw≤εp∫DNg′wp1−w2p−2Tφa,ωdσw≤λεp∫DNg′wp1−w22p−2Tφa,ωdσw.Since∣Ngw∣1−w2<6<e2, we infer that
(50)INg2D1=∫D1Ngw2p1−w22p−2Tφa,ωdσw<e4p∫D11−w2−2Tφa,ωdσw<e4pδp∫D1Sgwp1−w22p−2Tφa,ωdσw≤e4pδp∫DSgzp1−w22p−2Tφa,ωdσw.Then,(51)INg2Δ=∫ΔNgw2p1−w22p−2Tφa,ωdσw.Let us consider the sequenceBm of Carleson bad boxes, then
(52)INg2Δ=∫ΔNgz2p1−w22p−2Tφa,ωdσw<e4p∫ΔTφa,ωdσw1−w22≤e4p∑m∫TBmTφa,ωdσw1−w22≤e4pC∑mℓKBm.Forε>0 such that λεp<1, we infer that
(53)∫DNg′wp1−w22p−2Tφa,ωdσw≤2p−1∫DSgzp1−w22p−2Tφa,ωdσw+λεp2∫DNg′zp1−w22p−2Tφa,ωdσw+e4p2δp∫DSgzp1−w22p−2Tφa,ωdσw+e4pC∑mℓKBm.By Lemma7, we deduce that
(54)∑mℓKBm≤C1+C2∫DSφwp1−w2p−2Tφa,ωdσw.LettingC appear as a global concerned positive constant, we may infer that
(55)1−λεp2∫DNg′wp1−w22p−2Tφa,ωdσw≤C∫DSgwp1−w22p−2Tφa,ωdσw.Therefore,(56)∫DSgwp1−w22p−2Tφa,ωdσw<∞,which gives that
(57)∫DNg′wp1−w22p−2Tφa,ωdσw<∞.This is equivalent tolng′∈QK,ωp,q, 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 certainQ-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 Q-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 evolvingQ-type of function class releasing the derivative algorithm which reduces the scales and tool up a clear utility.



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