# Generalized Composition Operators on Zygmund-Orlicz Type Spaces and Bloch-Orlicz Type Spaces

Authors: Congli Yang; Fangwei Chen; Pengcheng Wu
Journal: Journal of Function Spaces
 (2014)

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## Abstract

The boundedness and compactness of  generalized composition
operators on Zygmund-Orlicz type spaces and Bloch-Orlicz type spaces are
established in this paper.


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## Body

## 1. Introduction

LetD be a unit disk in complex plane C, and let H(D) be the space of all holomorphic functions on D with the topology of uniform convergence on compact subsets of D. The Bloch space, B, consists of all functions f∈H(D) for which
(1)fb:=sup⁡z∈D1-z2f′z<∞.B becomes a Banach space when it is equipped with the norm ∥f∥B:=|f(0)|+∥f∥b (see, e.g., [1]).Letα>0, and the α-Bloch space, denoted by Bα, consists of all holomorphic functions f on D such that
(2)fα:=sup⁡z∈D1-z2αf′z<∞.α-Bloch space is introduced and studied by numerous authors. The general theory of α-Bloch function spaces is referred to in [2]. Recently, many authors studied different class of Bloch-type spaces, where the typical weight function, w(z)=1-|z|2(z∈D), is replaced by a bounded continuous positive function μ defined on D. More precisely, a function f∈H(D) is called a μ-Bloch function, denoted by f∈Bμ, if
(3)fμ:=sup⁡z∈Dμzf′z<∞.
Clearly, if μ(z)=w(z)α with α>0, Bμ is just the α-Bloch space Bα. It is readily seen that Bμ is a Banach space with the norm
(4)fBμ:=f0+fμ.Bμ spaces appear in the literature of a natural way when one studies the properties of some operators of holomorphic functions in a certain space. For instance, Attele in [3] proves that the Hankel operator induced by a function f in the Bergman space is bounded if and only if f∈Bμ1 (where μ1=w(z)log⁡⁡(z/wz), with z∈D). The space Bμ1 is known as the log-Bloch space or the weighted Bloch space. The logarithmic Bloch-type space is introduced by Krantz and Stević [4, 5], where some properties of this space are studied. In the last decade, there was a big interest in the investigation of Bloch-type spaces and various concrete linear operators L:X→Y, where at least one of the spaces X and Y is Bloch space. For some other recent results in the areas, see, for example, [4, 6–10] and a lot of references therein.Recently, Fernándz in [8] uses Young’s functions to define the Bloch-Orlicz space as a generalization of Bloch space. More precisely, let φ:[0,∞)→[0,∞) be an N-function; that is, φ is a strictly increasing convex function such that φ(0)=0 and lim⁡t→∞(t/φt)=lim⁡t→0(φt/t)=0. The Bloch-Orlicz space associated with the function φ, denoted by Bφ, is the class of all analytic functions f∈D such that
(5)sup⁡z∈D1-z2φλf′z<∞,
for some λ>0 depending on f. Without loss of generality, we can suppose that φ-1 is continuous and differentiable. In fact, if φ-1 is not differentiable everywhere, we set the function
(6)ψt=∫0t‍φxxdxt≥0;
then φ is differentiable, whence ψ-1 is differentiable everywhere on [0,∞). Furthermore, φ is a strictly increasing and convex function satisfying φ(0)=0; then the function φ(t)/t, t>0, is increasing and
(7)φ(t)≥ψ(t)≥∫t/2t‍φ(x)xdx≥φt2,
for all t≥0. Hence Bφ=Bψ. By the convexity of φ, it is not difficult to see that the Minkowski’s functional
(8)fbφ=inf⁡k>0:Sφf′k≤1
defines a seminorm for Bφ, which in this case is known as Luxemburg’s seminorm, where
(9)Sφf:=sup⁡z∈D1-z2φfz.
In fact, it can be shown that Bφ is a Banach space with the norm
(10)fBφ=f0+fbφ.
We observe that, for any f∈Bφ∖{0}, the following relation
(11)Sφf′fBφ≤1
holds.The inequality above allows us to obtain(12)f′z≤φ-111-z2fBφ,
for all f∈Bφ and for all z∈D. This last inequality implies that the evaluation function is defined as Tz(f):=f(z), where z∈D is fixed and f∈Bφ is continuous on Bφ. In fact, let z∈D be fixed and any f∈Bφ; we have
(13)Tzf=fz≤f0+∫0,z‍f′sds≤1+∫01‍φ-111-z2t2dzfBφ.
From the definition of Luxemburg seminorm and the expression (11), we have
(14)Sφ(f′)≤1⟺fBφ≤1,
for any f∈Bφ.As an easy consequence of (11), we have that the Bloch-Orlicz space is isometrically equal to μ-Bloch space when
(15)μ(z)=1φ-1(1/(1-z2)),
with z∈D. Thus, for any f∈Bφ, we have
(16)fBφ=sup⁡z∈Dμzf′z.Denote byZ the class of all f∈H(D)∩C(D¯) such that
(17)f=sup⁡feiθ+h+feiθ-h-2feiθh<∞,
where the supremum is taken over all eiθ∈∂D and h>0. From the theorem of Zygmund (see [11, Theroem 5.3]) and the closed graph theorem, we see that f∈Z if and only if sup⁡z∈D(1-|z|2)|f′′(z)|<∞. It is easy to see that Z is a Banach space under the norm ∥·∥Z, where
(18)fZ=f0+f′0+sup⁡z∈D1-z2f′′z.
From (18) it is easy to obtain
(19)f′z-f′0≤12fZlog⁡1+z1-zforf∈Z.
For some other information and operators on this space, see, for example, [6, 12, 13].Inspired by the way Bloch-Orlicz spaces were defined (see [8, 14]), we define the Zygmund-Orlicz space, which is denoted by Zφ, as the class of all analytic functions f in D such that
(20)sup⁡z∈D1-z2φλf′′z≤∞,
for some λ>0 depending on f. Same as the Bloch-Orlicz space, since φ is convex, it is not difficult to see that the Minkowski functional
(21)fzφ=inf⁡k>0:Sφf′′k≤1
defines a seminorm for Zφ. Furthermore, it can be shown that Zφ is a Banach space with the norm
(22)fZφ=f0+f′0+fzφ.
The following useful lemmas are easily obtained.Lemma 1 (see [15]).
Iff∈Z, then
(23)(i)fz≤fZforeveryz∈D,(24)(ii)f′z≤log⁡(e1-z2)fZforeveryz∈D.Also we can observe the following property.Lemma 2.
For anyf∈Zφ∖{0}, the following relation
(25)Sφf′′fZφ≤1
holds.Proof.
In the same way as the case of Bloch-Orlicz space, so the details are omitted here.Lemma2 allows us to obtain that
(26)f′′z≤φ-111-z2fZφ,
for all f∈Zφ and all z∈D.From the definition of Luxemburg seminorm and expression (25), we have
(27)Sφ(f′′)≤1⟺fZφ≤1.
Also, as an easy consequence of (25), we have that Zygmund-Orlicz space is isometrically equal to μ-Zygmund space, where μ(z)=1/φ-11/(1-z2) with z∈D. For more information about Zμ, see [7, 10, 16].Specially, ifφ is an N-function such that
(28)Iφ(z)=∫01‍φ-111-z2t2dt
is bounded for all z∈D, then we get Zφ⊂H∞, the space of all bounded analytic functions on D. However, there exists N-functions for which Iφ(z) is not a bounded function; for instance, consider φ(t)=tlog⁡(1+t) with t≥0.Letϕ be an analytic self-map of D; then the composition operator on H(D) is given by
(29)Cϕf=f∘ϕ.
Composition operators acting on various spaces of analytic functions have been the object for recent years. In particular, the problems of relating operator-theoretic properties of Cϕ to function-theoretic properties of ϕ are interesting and have been widely discussed. See the book of Cowen and MacCluer [17] and Shapiro [9] for discussions of composition operators classical spaces of analytic functions.Assume thatg:D→C is a holomorphic map of the disk D, for f∈H(D); we define a linear operator as follows:
(30)(Cϕgf)(z)=∫0z‍f′(ϕ(ξ))g(ξ)dξ,z∈D.
The operator Cϕg is called the generalized composition operator, when g=ϕ′. We see that this operator is essentially composition operator, since the difference Cϕg-Cϕ is constant. Therefore, Cϕg is a generalization of the composition operator, which was introduced in [6].Recall that ifX and Y are Banach spaces and L:X→Y is a linear operator, then L is said to be compact if, for every bounded sequence {xn} in X, the sequence (L{xn}) has a convergent subsequence. The book [17] contains plenty of information on this topic. By the standard arguments (see, e.g., Proposition  3.11 in [17]), the following lemma follows.Lemma 3.
Letφ:[0,∞)→[0,∞) be an N-function, g∈H(D), and ϕ an analytic self-map of D. Let X=Z and Y=BφorZφ. Then Cϕg:X→Y is compact if and only if Cϕg:X→Y is bounded and, for any bounded sequence (fk)k∈N in X which converges to zero uniformly on D as k→∞, one has ∥Cϕgfk∥Y→0 as k→∞.Some characterization of the boundedness and compactness of the composition operator, as well as Volterra type operator, on Bloch-Orlicz-type space and Zygmund space can be found in [2, 18–22]. In [6], the boundedness and compactness of the generalized composition operator on Zygmund space and Bloch-type spaces are characterized by Li and Stević.In this paper, we are devoted to investigating the boundedness and compactness of generalized composition operators between Zygmound-Orlicz type spaces and Bloch-Orlicz type spaces. The paper is organized as follows. In Section2 we give the necessary and sufficient conditions for the boundedness and compactness of the operator Cϕg:Z→Bφ. In Section 3 we obtain the necessary and sufficient conditions for the boundedness and compactness of the operator Cϕg on Zygmund type spaces. Throughout this paper, we use the letter C to denote a generic positive constant that can change its value at each occurrence. The notation a≤b means that there is a positive constant C such that a≤Cb. If a≤b and b≤a hold, then one says that a≍b.


## 2. The Boundedness and Compactness ofCϕg:Z→Bφ

Now, we are ready to state and prove the main results in this section.Theorem 4.
Letφ:[0,∞)→[0,∞) be an N-function, g∈H(D), and ϕ an analytic self-map of  D. Then Cϕg:Z→Bφ is bounded if and only if
(31)L≔sup⁡z∈Dφ-111-z2-1gzlog⁡e1-ϕz2.<∞.Proof.
Suppose that (31) holds. For arbitrary z∈D and f∈Z, by Lemma 1, we have the following estimate:
(32)SφCϕgf′zLfZ=sup⁡z∈D1-z2φf′ϕzgzLfZ≤sup⁡z∈D(1-z2)φ×φ-1(11-z2)f′ϕzlog⁡(e/(1-ϕz2))fZ≤sup⁡z∈D(1-z2)φφ-111-z2≤1,
where we use Lemma 1 (relation (ii)) in the last inequality. From here, we can conclude that ∥Cϕg∥φ≤L∥f∥Z. Since Cϕgf(0)=0, it follows that the generalized composition operator Cϕg:Z→Bφ is bounded.
Now, suppose that there exists a constantC>0 such that
(33)Cϕgφ≤CfZ∀f∈Z.
Let
(34)h(z)=(z-1)1+log⁡e1-z2+1,
and put
(35)fa(z)=h(a¯z)a¯log⁡e1-a2-1,
for any a∈D such that 1/2<|a|<1. Then we have
(36)fa′(z)=log⁡e1-a¯z2log⁡e1-a2-1,fa′′z=2a¯1-a¯zlog⁡e1-a¯zlog⁡e1-a2-1,
which implies that
(37)fa′′z≤21-zC+log⁡e1-alog⁡e1-a2-1≤21-z,
for 1/2<|a|<1 and sup⁡1/2<|a|<1∥fa∥Z<∞. Therefore we have
(38)Cϕgfaφ≤CfaZ≤L<∞.
It follows that
(39)SφCφgfa′(z)L≤1.
That is,
(40)1-z2φCφgfa′(z)L≤1,
for any a∈D. But, on the other hand,
(41)φgzLfa′ϕz≤11-z2,
and then
(42)gzLfa′ϕz≤φ-111-z2.
So we have
(43)φ-111-z2-1gzfa′ϕz≤L,
for all a∈D. In particular, for a=ϕ(z), we have
(44)φ-111-z2-1gzlog⁡e1-ϕz2≤L.
This concludes the proof of the theorem.Theorem 5.
Letφ:[0,∞)→[0,∞) be an N-function, g∈H(D), and ϕ an analytic self-map of D. Then Cϕg:Z→Bφ is compact if and only if Cϕg:Z→Bφ is bounded and
(45)lim⁡|ϕz|→1μzgzlog⁡e1-ϕz2=0.Proof.
First assume thatCϕg:Z→Bφ is bounded and (45) holds. By the boundedness of Cφg with f(z)=z, we see that
(46)L:=sup⁡z∈Dμzgz<∞.
Let {fn} be a sequence in Z such that sup⁡n∈N∥fn∥Z≤K and fn→0 uniformly on compact subsets of D as n→∞. Then, by Lemma 3, it suffices to show that ∥Cϕgfn∥μ→0 as n→∞. By (45), we have that, for every ɛ>0, there is a constant r∈(0,1), such that r<|ϕ(z)|≤1, which implies
(47)μzgzlog⁡e1-ϕz2<ɛk,
for any z∈D. From here, we have that
(48)μzCϕgfn′z=μgzfn′ϕz≤μzgzlog⁡e1-ϕz2fnZ≤ɛ,
where r<|ϕ(z)|<1.
On the other hand, letΩ={ω∈D:|ω|≤r}; by the cauchy estimate, if {fn}n∈N is a sequence converging to zero on compact subsets of D, then the sequence {fn′}n∈N also converges to zero on compact subsets of D as n→∞. In particular, since Ω is compact, it follows that lim⁡n→∞sup⁡ω∈Ω|fn′(ω)|=0. Using these facts and letting n→∞, we obtain
(49)sup⁡|ϕz|≤rμzCϕgfn′z=sup⁡|ϕz|≤rμzgzfn′ϕz≤Lsup⁡ω∈Ωfn′ω⟶0,asn⟶∞.
Since (Cϕgfn)(0)→0 as n→∞, for given ɛ>0, there exists an n∈N, such that
(50)Cϕgfnμ=Cϕgfn0+sup⁡z∈DμzCϕgfn′z≤3ɛ,
whenever n≥N, which means that Cϕg:Z→Bφ is a compact operator.
To prove the converse, suppose that there exists anε0>0 such that
(51)sup⁡|ϕz|≥rμzgzlog⁡e1-ϕz2≥ε0,
for any r∈(0,1). Given a sequence of real number {rn}⊂(0,1) such that rn→1 as n→∞, we can find a sequence {zn}⊂D such that |ϕ(zn)|≥rn and
(52)μzngznlog⁡e1-ω2≥12ε0,
where ωn=ϕ(zn), and, if necessary, we may suppose that |ωn|→1 as n→∞.
Note thatfa, defined by (35), converges to zero uniformly on compact subset of D as |a|→1 and
(53)fa′(a)=log⁡e1-a2,fora∈D∖{0}.
Now, we choose functions {fn}n∈N defined by
(54)fn(z)=w¯nz-1w¯1+log⁡e1-wn¯z2+1log⁡e1-wn2-1.
From the proof of Theorem 4, we see that
(55)sup⁡n∈NfZ≤C.
Moreover, we can see that {fn} converges to zero uniformly on compact subsets of D and satisfies
(56)Cϕgfnμ≥μzngznfn′wn=μzngznlog⁡e1-wn2≥12ε0>0.
Therefore, Cϕg:Z→Bφ is not a compact operator. This completes the proof of the theorem.


## 3. The Boundedness and Compactness ofCϕg:Z→Zφ

In this section, we characterize the boundedness and compactness of the operatorCϕg on Zygmund-Orlicz type spaces.Theorem 6.
Letφ:[0,∞)→[0,∞) be an N-function, g∈H(D), and ϕ an analytic self-map of D. Then Cϕg:Z→Zφ is bounded if and only if
(57)k1=sup⁡z∈Dϕ′zgzφ-11/(1-z2)(1-ϕz2)<∞,k2=sup⁡z∈Dg′zlog⁡e/(1-ϕz2)φ-11/(1-z2)(1-ϕz2)<∞.Proof.
Suppose first that (57) holds. For each f∈Z, we have the following estimate:
(58)SφCϕgf′′(z)fZ≤sup⁡z∈D1-z2φ×f′′ϕzgz+f′ϕzg′zCfZ=sup⁡z∈D1-z2φ×φ-11/1-ϕz2f′′ϕzϕ′zgzφ-11/(1-z2)CfZhhihh+φ-11/1-ϕz2f′ϕzg′zφ-11/1-z2CfZ≤sup⁡z∈D1-z2φ×k1φ-11/1-z21-ϕz2f′′ϕzCfZhhhhh+k2φ-11/1-z2f′ϕzlog⁡e/1-ϕz2CfZ=sup⁡z∈D(1-z2)φk1C1+k2C2Cφ-111-z2≤sup⁡z∈D(1-z2)φφ-111-z2=1,
where C1 and C2 are constants, such that k1C1+k2C2≤C. Here we use relations (26) and (24) in the last seconded inequality. Now, we can conclude that
(59)CϕgfZϕ≤CfZ,
and the generalized composition operator Cϕg:Z→Z is bounded.
Now assume thatCϕg:Z→Z is bounded; that is, there exists a constant C such that ∥Cϕgf∥Zϕ≤C∥f∥Z for all f∈Z. Taking the functions f(z)=z and f(z)=z2, respectively, we obtain
(60)1-z2φg′zC≤1,
which implies
(61)sup⁡z∈Dg′zφ-11/(1-z2)<∞,(62)(1-(z2))φϕ′zgz+ϕzg′zC(1-z2)≤1.
Then
(63)ϕ′zgz+ϕzg′zφ-11/(1-z2)≤C.
So we obtain
(64)sup⁡z∈Dϕ′zgz+ϕzg′zφ-11/(1-z2)≤∞.
Using these facts and the boundedness of the function ϕ(z) we get
(65)sup⁡z∈Dϕ′zgzφ-11/(1-z2)<∞.
Let h(z)=(z-1)1+log⁡⁡(e/1-z)2+1 and put fa(z)=(ha¯z/a¯)log⁡⁡(e/(1-a2))-1 for a∈D such that |a|>1/2. We have
(66)(1-λ2)φg′λlog⁡⁡e1-ϕz2ϕλ1-ϕz2hhhhhhhhhhhh-2ϕ′λgλϕλ1-ϕz2hhhhhhhhhhi×ϕλ1-ϕz2C||Cϕg||Z→Zφ-1≤1.
Then
(67)(g′λlog⁡⁡e1-ϕz2-2ϕ′λgλ|ϕ(λ)|1-ϕz2)|ϕ(λ)|1-ϕz2×CCϕgZ→Zφ-1≤φ-111-λ2,g′λlog⁡⁡(e/1-ϕλ2)φ-11/(1-λ2)-2ϕ′λgλϕλφ-11/(1-λ2)(1-ϕλ2)≤CCϕgZ→Zφ.
So we obtain
(68)g′λlog⁡⁡(e/(1-ϕλ2))φ-11/(1-λ2)≤CCϕgZ→Zφ+2ϕ′λgλϕλφ-11/1-λ21-ϕλ2.
Set
(69)ha(z)=h(a¯z)a¯log⁡11-a2-1-∫0z‍log⁡11-a¯wdw,
for a∈D, such that |a|>1/2. Then,
(70)ha′(z)=log⁡11-a¯z2log⁡11-a2-1-log⁡e1-a¯z,ha′′(z)=2a¯1-a¯zlog⁡11-a¯zlog⁡11-a2-1-a¯1-a¯z.
Similar to the case of fa, we have ha∈Z and M1=sup⁡1/2<|a|<1∥ha∥Z<∞. From this and by the facts that ha′(a)=0 and ha′′(a)=a¯/(1-|a|2), it follows that
(71)CCϕgZ→Zφ≥ϕ′λgλϕλφ-11/(1-λ2)(1-ϕλ2).
From (71) we have
(72)sup⁡|ϕ(λ)|>1/2ϕ′λgλϕλφ-11/(1-λ2)(1-ϕλ2)<2ϕ′λgλϕλφ-11/(1-λ2)(1-ϕλ2)≤sup⁡|ϕ(λ)|>1/2CCϕgZ→Zφ<∞.
By (65) we see that
(73)sup⁡|ϕ(λ)|≤1/2ϕ′λgλφ-11/(1-λ2)(1-ϕλ2)≤sup⁡|ϕ(λ)|≤1/243ϕ′λgλφ-11/(1-λ2)<∞.
From (72) and (73) we obtain the first inequality of (57). Similarly, from (61) and (68) the second inequality in (57) follows as desired.Theorem 7.
Letφ:[0,∞)→[0,∞) be an N-function, g∈H(D), and ϕ an analytic self-map of D. Then Cϕg:Z→Zφ is compact if and only if Cϕg:Z→Zφ is bounded and
(74)lim⁡|ϕ(z)|→1ϕ′zgzφ-11/(1-z2)(1-ϕz2)=0,lim⁡|ϕ(z)|→1g′zlog⁡⁡(e/1-ϕz2)φ-11/(1-z2)=0.Proof.
Suppose thatCϕg:Z→Zφ is compact. It is clear that Cϕg:Z→Z is bounded. Let {Zn}n∈N be a sequence in D such that |ϕ(zn)|→1 as n→∞ and (fn)n∈N defined by
(75)fnz=ϕ(zn)z-1ϕ¯(zn)×1+log⁡e1-ϕ(zn)¯zlog⁡e1-ϕzn2-1.
Then sup⁡n∈N∥fn∥Z<∞, and fn→0 uniformly on compact subsets of D as n→∞. Since Cϕg:Z→Zφ is compact, it gives lim⁡n→∞∥Cϕgfn∥Zφ=0. Note that
(76)fn′(ϕ(zn))=log⁡e1-ϕzn2,fn′′(ϕ(zn))=2ϕ(zn)¯1-ϕzn2.
We have
(77)CϕgfnZφ≥ϕ′zngznϕznφ-11/(1-zn2)(1-ϕzn2)-g′znlog⁡⁡e/1-ϕzn2φ-11/1-zn2,
and, consequently,
(78)lim⁡|ϕ(zn)|→1ϕ′zngznϕznφ-11/(1-zn2)(1-ϕzn2)=lim⁡|ϕ(zn)|→1g′znlog⁡⁡(e/1-ϕzn2)φ-11/(1-zn2).
If one of these two limits exists, set
(79)hnz=h(ϕ(zn)¯z)ϕ(zn)log⁡e1-ϕzn2-1-log⁡e1-ϕzn2-2∫0z‍log⁡3e1-ϕzn¯wdw;
then hn′(ϕ(zn))=0, sup⁡n∈N∥hn∥Z≤C, and hn converges to 0 uniformly on compact subsets of D as n→∞. Since Cϕg:Z→Zφ is compact, we have
(80)lim⁡n→∞CϕghnZφ=0.
On the other hand,(81)ϕ′zngznϕznφ-11/(1-zn2)(1-ϕzn2)≤CϕghnZφ.
Hence
(82)lim⁡n→∞ϕ′zngznϕznφ-11/(1-zn2)(1-ϕzn2)=0.
Therefore
(83)lim⁡|ϕ(zn)|→1ϕ′zngznϕznφ-11/(1-zn2)(1-ϕzn2)=lim⁡k→∞ϕ′zngznϕznφ-11/(1-zn2)(1-ϕzn2)=0.
This together with (78) implies
(84)lim⁡|ϕ(zn)|→1g′znlog⁡⁡(e/1-ϕzn2)φ-11/(1-zn2)=0.
The implication follows from the last two equalities.
Conversely, assume thatCϕg:Z→Zφ is bounded and (74) holds. From the proof of Theorem 6 we have
(85)C1=sup⁡z∈Dg′zφ-11/(1-z2)<∞,C2=sup⁡z∈Dϕ′zg′zφ-11/(1-z2)<∞.
On the other hand, from (74), we have that, for every ɛ>0, there is a δ∈(0,1), such that
(86)ϕ′zgzφ-11/(1-zn2)(1-ϕzn2)<ɛ,g′zlog⁡⁡(e/1-ϕz2)φ-11/(1-zn2)<ɛ,
when δ<|ϕ(z)|<1.
Assume that{fn}n∈N is a sequence in Z such that sup⁡n∈N∥fn∥Z≤1 and fn converges to 0 uniformly on compact subsets of D as n→∞. Let k={z∈D:|ϕ(z)≤δ|}. Then by (85) and (86) it follows that
(87)CϕgfnZφ=sup⁡z∈DCϕgfn′′zφ-11/1-z2=sup⁡z∈Dfn′′ϕzϕ′zgz+fn′ϕzg′zφ-11/(1-z2)≤sup⁡z∈Kfn′′ϕzϕ′zgz+fn′ϕzg′zφ-1(1/(1-z2))+sup⁡z∈D∖Kfn′′ϕzϕ′zgz+fn′ϕzg′zφ-11/(1-z2)≤C2sup⁡z∈Kfn′′ϕz+C1sup⁡z∈Kfn′ϕz+Csup⁡z∈D∖Kϕ′zgzφ-11/1-zn21-ϕzn2fnZ+Csup⁡z∈D∖Kg′zlog⁡⁡(e/1-ϕz2)φ-11/(1-zn2)fnZ≤C2sup⁡w≤δf′′w+C1sup⁡w≤δfn′w+2CεL.
So we obtain
(88)CϕgfnZφ=fn′ϕ0f(0)+sup⁡z∈DCϕgfn′′zφ-11/(1-z2)≤C2sup⁡w≤δfn′′w+C1sup⁡w≤δfn′w+2CɛfnZ+fn′ϕ0g0.
By the Cauchy estimate, if {fn}n∈N is a sequence converging to zero on compact subsets of D, then the sequence {fn}n∈N and {fn′′}n∈N also converges to zero on compact subsets of D as n→∞. In particular, since K is compact it follows that lim⁡n→∞sup⁡w∈K|fn′(w)|=0 and lim⁡n→∞sup⁡w|f′′(w)|=0. Using these facts and letting n→∞ in the last inequality, we obtain that
(89)lim⁡n→∞CϕgZφ≤2CɛL.
Since ɛ is an arbitrary positive number it follows that the last limit is equal to zero. Employing Lemma 3 the implication follows.



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*Source: 549370-2014-11-30.xml*