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

Authors: Amnah E. Shammaky; A. El-Sayed Ahmed
Journal: Journal of Mathematics
 (2022)

---

## Abstract

The aim of this study is to give some new definitions of Banach spaces of holomorphic functions. Some holomorphic characterizations of integral type for some classes of Banach spaces of holomorphic functions are established in the unit discU.


---

## Body

## 1. Introduction and Preliminaries

Numerous studies of analytic function spaces by several classes of functions are introduced and intensively studied. The 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 ofg-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. The 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. LetU=w:w∈C,w<1, and HU denote the class of all holomorphic functions that belonging to U.The known analytic Bloch-type space [1–5] is defined by(1)ℬ=f:f∈HUandsupw∈U1−w2f′w<∞.The analytic little Bloch-type spaceℬ0 is symbolized by ℬ0, for which(2)ℬ0=f:f∈HUandlimw⟶1−1−w2f′w=0.For numerous global studies on Bloch-type spaces, we refer to [6–13]  and others.The analytic Dirichlet-type space [1, 14] is given by(3)D=f:f∈HUand∫Uf′w2dσw<∞,where dσw=dxdy.Using Green’s functiongw,w0=ln1−w¯0w/w0−w=ln1/ϕw0w as a weight function, the analytic Qp-spaces is defined in [1] by(4)Qp=f:f:f∈HUandsupw0∈U∫Uf′w2gpw,w0dσw<∞.By analyticQp-spaces, some interesting relationships between analytic Dirichlet-type space and the analytic Bloch-type space are obtained [1].For intensive research on analyticQp spaces, we may refer to [1, 15–18] and others. Also, there are certain specific generalizations of these weighted classes of analytic functions in Cn [12, 19, 20]. On the other hand, there are some interesting extensions using quaternion-valued functions setting ([21–24]).Hereafter, we set(5)φw0w=w0−w1−w¯0w,w≠w0,and set(6)φw0w=C<1,whenw=w0.The modified Green’s function is introduced by(7)gw,w0=ln1−w¯0ww0−w=ln1φw0w.Motivated by the modified Green’s function, the following definitions can be presented.Definition 1.
Let0<m<∞and0<n<∞. For the function f∈HU, we define the analytic g-Bloch space ℬgn,m as follows:(8)ℬgn,m=f:supw,w0∈U1−w2ngmw,w0f′w<∞.Furthermore, assume that(9)ℬgn,mf=supw,w0∈U1−w2ngmw,w0f′w.Example 1.
Letm∈0,∞. Suppose that(10)fw=∫|w|<1log1φawmdw.It is very obvious that the functionf is a g-Bloch function.Remark 1.
Using Definition1, relationships between analytic Dirichlet-type functions and analytic Bloch functions can be characterized.
Whenm=0 and 0<n=α<∞, then we will obtain α-Bloch space. The case m∈−∞,0 induces some other types of analytic function spaces with different behaviors and can be studied separately.The little analyticg-Bloch space ℬgn,m,0 can be considered as a subspace of the analytic g-Bloch space that includes all f∈ℬgn,m, with(11)limw⟶1−limw0⟶1−1−w2ngmw,w0f′w=0.Remark 2.
The symbolℬgn,mf stands for a seminorm, while the usual norm can be defined as(12)fℬgn,m=f0+ℬgn,mf.
Applying the above norm, the spaceℬgn,m is a complete normed space (Banach space).The next lemma can be applied for some results in this study.Lemma 1 (See [14]).
Let0<s<∞ and assume that w0<1. Then,(13)∫Γw11−w0¯w2sdΓw≤k1−w0s,where k>0, and Γw defines the boundary of U.


## 2.g-Bloch Space and Dirichlet Space

Some essential characterizations between the analytic Dirichlet-type space and the analyticg-Bloch space are given in this section.Proposition 1.
Letf∈HU and assume that |w0|<1. Then,(14)1−w22gmw,w0f′w02≤4πR2∫Uf′w02dσw,where 0<R<1.Proof.
Because the pseudohyperbolic disk can be symbolized by(15)Dw0,R=w:ρw,w0=φw0w=w−w01−w¯0w<R,where w0 is its center and R is its radius, then we infer that(16)∫Uf′w2dσw≥∫Dw0,Rf′w2dσw=πR21−w022f′w02≥πR21−w022f′w02.
Using the definition of the modified Green’s function as well as the inequalities,(17)0<φw0w0=C<1,1−φw0w2≤2gw,w0,see25.
We can obtain(18)∫Uf′w2dσw≥πR21−w0221−φw0w022f′w02>2πR21−w022C=g2w0,w0f′w02.
The proof of Proposition1 is therefore finished.Corollary 1.
In view of Proposition1, for w0<1, we have the following interesting inclusion:(19)D⊂ℬg2,2.Proposition 2.
Letf∈HU and assume that w0<1. Also, we suppose that n∈1,∞ and m∈0,∞, then(20)1−w022ngmw0,w0f′w02≤2mπR2∫Uf′w2dσw,where 0<R<1.Proof.
The proof of Proposition2 can be obtained as in the proof of Proposition 1, with the following changes:(21)∫Uf′w2dσw≥πR21−w0221−φw0w02mf′w02≥π2mR21−w022ngmw0,w0f′w02.Corollary 2.
Proposition2 results that(22)D⊂ℬg2n,m,with w0<1, n∈1,∞, and m∈0,∞.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.Proposition 3.
Letf∈HU and f∈ℬgn,m. Suppose that(23)Im,n=∫01ln1/rm1−r21−nrdr<∞.Then, for0<m<∞ and 0<n<∞ with m+n>0, we obtain that(24)supw∈U∫Uf′w2dσw≤24−mπIm,nℬgn,mf2.Proof.
From the definition ofg-Bloch space, we have(25)1−w022ngmw0,w0f′w≤ℬgn,mf.
By a change of the variables technique, we infer that(26)∫Uf′w2dσw≤ℬgn,mf2∫Uln1/w0m1−φw0w02n1−φw0w0221−w022dσw.
Since,(27)1−φw0w02=1−w021−w21−w0¯w2.
Therefore,(28)∫Uf′w2dσw≤ℬgn,mf2∫Uln1/w0m1−φw0w02ndσw≤π24−nℬgn,mf2∫01ln1/rm1−r21−nrdr,which implies that(29)∫Uf′w2dσw≤ℬgn,mf2k1,where 0<k1<∞. The proof is therefore established.
Propositions2 and 3 result in the following fundamental theorem.Theorem 1.
Letf∈HU; then, we have equivalence between the following statements:(i)
f∈ℬgn,m with 0≤m<∞ and 0<n<∞ with m+n>0(ii)
f∈D.Remark 4.
The obtained results in Theorem1 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 ℬgn,m space.


## 3.g-Bloch Functions and Qp Functions

Proposition 4.
For0≤m<∞, 1≤n<∞, let f∈HU and w0<1; hence,(30)1−w022ngmw0,w0f′w02≤22m+4ρ21−ρ2p∫Uf′w21−φw0w2pdσw,where 0<ρ<1 and p∈0,∞.Proof.
From subharmonicity principle, the following inequality can be easily obtained:(31)h02≤1ρ2∫D0,ρhz2dσw,
Sincef∈HU is holomorphic on Dw0,ρ, then use (31) to the function(32)h=f′∘φw0.
Applying the technique of change of variables, the next inequality can be inferred:(33)f′w02≤1ρ2∫D0,ρf′φw0w2dσw=1ρ2∫Dw0,ρf′w21−φw0w21−w22dσw.
From [5], we have(34)1−φw0w21−w22≤41−w022.
This yields that(35)f′w02≤4/ρ21−w022∫Dw0,ρf′w2dσw.
Because,(36)1−φw0w2p≥1−ρ2p,φw0w0=0.
Then,(37)∫Dw0,ρf′w21−φw0w2pdσw≥ρ21−ρ2p161−w022f′w02≥ρ21−ρ2p161−w0221−φw0w022f′w02≥ρ21−ρ2p22m+41−w022ng2mw,w0f′w02.
Using the inequality(38)∫Uf′w21−φw0w2pdσw≥∫Dw0,ρf′w21−φw0w02pdσw,we can deduce that(39)1−w022ng2mw0,wf′w02≤22m+4ρ21−ρ2p∫Uf′w21−φw0w2pdσw.
The proof is therefore completely finished.Corollary 3.
By Proposition4, for p>0, 0≤m<∞, 1≤n<∞, and w0<1, the following inclusions can be easily proved:(40)ℬg2n,2mf≤22m+4ρ21−ρ2pQp2f,where(41)Qpf=supw0∈U∫Uf′w21−φw0w2pdσz≈supw0∈U∫Uf′w2gw,w0pdσw<∞.Proposition 5.
Letf∈HU. Assume that 0≤m<∞ and 0<n<∞ with m+n+p>0. Then, we have that(42)∫Uf′w2gw,w0pdσw≤Im,n,pℬgm,n2f,where(43)Im,n,p=λ∫01ln1/rm+p1−r1−nrdr,and λ is an absolute positive constant.Proof.
It is not hard to see that(44)∫Uf′w2gpw0,wdσw≤4ℬgm,n2f∫Uln1/wm+p1−w22−2ndσw.
Applying Lemma1, we obtain(45)∫Uf′w2gpw0,wdσw≤8πℬgm,n2f∫01ln1/rm+p1−r1−nrdr=Im,n,pℬgm,n2f.
Combining Corollary3 and Proposition 5, we have the following theorem:Theorem 2.
Leth∈HU. Assume that 0≤m<∞ and 0<n<∞ with m+n+p>0 as well as 0<p<∞. Therefore, the next assertions can be equivalent:(a)
h∈ℬgm,n(b)
h∈Qp(c)
h∈Qp for some 0<p<∞.Proof.
a⇒b can be established using Proposition 5. b⇒c is quite obvious. c⇒a can be proved by Corollary 3.Remark 5.
Theorem2 investigates relations between the new type of holomorphic g-Bloch functions and the weighted holomorphic Dirichlet-type functions.


## 4. A Specific Criteria

The following symbol stands for a non-Euclidean distance of hyperbolic-type between the pointsw0 and w in U.(46)dw0,w=ln1−w0¯w+w0−w1−w0¯w−w0−wsee26.Now, for0<R<∞, set(47)Tw,R=w0∈U:dw0,w<R,T1w,R=w0∈U:dw0,w=R.Leth∈HU be the nonconstant analytic function, and let Shw,R define the concerned area of the Riemannian image Pw,R of Tw,R by the function h, and assume that Sh∗w,R defines the area of the image Pw,R of Tw,R using the function h. It should be noted that Pw,R defines the projection to C. The length of the Riemannian image of T1w,R by h is denoted by Mhz,ρ, and the symbol ℳhz,ρ defines the length of the outer boundary of Pw,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:(48)Shw,R≥Sh∗w,R,Mhw,R≥ℳhw,R,where 0<R<∞ and each w∈U.Proposition 6.
Leth∈HU be a nonconstant analytic function. Suppose that 1≤n<∞ and 0≤m<∞, then the following inequalities can be deduced:(49)1−w2ngmw,w0h′w≤1θR2mSh∗w,Rπ1−ρ2m.
Also,(50)1−w2ngmw,w0h′w≤2m1−w2n1−φw0w2mh′w≤2m−1ℳhw,R1−ρ2mπθR,where θR=e2R−1/e2R+1.Proof.
First, we suppose thath′w≠0. Now, we suppose that(51)fw0=hw0+w1+w¯w0=b0+b1w0+b2w02+b3w03+⋯,with w0<1, where b0,b1,b2,b3,… are the positive constants. Now,(52)1−w2nh′w≤b1=1−w2h′w≠0,which implies that(53)1−w2ngmw,w0h′w≤1−w2gmw,w0h′w≠0<∞.
Using Theorems1 and 2 in [27], the following inequalities can be deduced:(54)πθ2Rb12≤Sh∗w,R,2πθRb1≤ℳhw,R.
Since,0<φw0w<ρ<1, then(55)1−w2ngmw,w0h′w≤1θR2mShw,Rπ1−ρ2m.
Also,(56)1−w2ngmw,w0h′w≤2m1−w2n1−φw0w2mh′w≤2m−1ℳhw,R1−ρ2mπθR.Remark 6.
In the proof of Proposition6, positive coefficients are considered to keep the convergence of Taylor or Fourier power series in its region.Theorem 3.
Leth∈HU be the nonconstant analytic function. Suppose that 1≤n<∞ and 0≤m<∞; therefore, the equivalence between next assertions can be mutually obtained:(1)
The functionh∈ℬgn,m(2)
The constant0<R<∞ can be obtained, for which(57)supw0∈UShw,R<∞.(3)
The constant0<R<∞ can be obtained, for which(58)supw0∈USh∗w,R<∞.(4)
The constant0<R<∞ can be obtained, for which(59)supw0∈UMhw,R<∞.(5)
The constant0<R<∞ can be obtained, for which(60)supw0∈Uℳhw,R<∞.Proof.
The assertions2⇒3 and 4⇒5 can be deduced clearly. Additionally, 1⇒2 and 1⇒4 are obvious. Next, suppose that (1) with(61)supw0∈U1−w2ngmw,w0h′w=ℬgn,mh<∞.
Hence,(62)Shw,R=∫Tw,Rh′w02dσw0≤ℬgn,mh2∫w0<θRgmw,w01−w2ndσw0=η1Rℬgn,mh,where η1R is a positive constant depending on R. In addition, using the same way, we deduce that(63)ℬgn,mh2θR2n1−θR22m=Mhw,R=ηR∫T1w,Rf′w0dσw0≤ℬgn,mh∫T1w,Rgmw,w01−w2ndσw0=η2Rℬgn,mh,where ηR and η2R are the positive constants depending on R.
Additionally, the assertion3⇒1 as well as 5⇒1 can be proved in view of Proposition 6.Remark 7.
Theorem3 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. The 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 definedg-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?


## 5. Conclusions

Function spaces theory is developed, extended, and generalized to spaces of several complex variables ([8–13, 29]) also using quaternion-valued functions ([21–24, 30–33]). The 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.The holomorphic classes ofg-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 Qp spaces are obtained. On the other hand, an extension of Yamashita’s result [26] is presented using holomorphic g-Bloch functions.



---

*Source: 6371343-2022-01-28.xml*