Articulo 2.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 02, (19–27). June 2010

Differences of weighted composition operators between
weighted Banach spaces of holomorphic functions and

weighted Bloch type spaces

Elke Wolf

Mathematical Institute,

University of Paderborn,

D-33095 Paderborn, Germany.

email: lichte@math.uni-paderborn.de

ABSTRACT

We consider analytic self-maps φ1, φ2 of the open unit disk as well as analytic maps ψ1,ψ2.

These maps induce differences of weighted composition operators acting between weighted

Banach spaces of holomorphic functions and weighted Bloch type spaces. In this article we

give necessary and sufficient conditions for such a difference to be bounded resp. compact.

RESUMEN

Nosotros consideramos auto aplicaciones φ1, φ2 del disco unitario abierto bien como apli-

caciones anaĺıticas ψ1,ψ2. Estas aplicaciones inducen diferencias de composición de op-

eradores con peso actuando entre espacios de Banach pesados de funciones holomorfas y

espacios de tipo Bloch con peso. En este art́ıculo damos condiciones necesarias y suficientes

para que tal diferencia sea acotada, respectivamente, compacta.

Key words and phrases: weighted composition operators, weighted Bloch type spaces, weighted

Banach spaces of holomorphic functions.

Math. Subj. Class. 2000: 47B33, 47B38.



20 Elke Wolf CUBO
12, 2 (2010)

1 Introduction

For analytic self-maps φ1,φ2 of D and analytic maps ψ1,ψ2 the corresponding weighted composition

operators ψiCφi are defined by ψiCφif = ψif ◦ φi, i = 1, 2. Composition operators and weighted

composition operators acting on various spaces of analytic functions have recently been of much

interest, see for example [14], [8], [12], [2], [4], [13]. Differences of them have been studied e.g. in [3],

[9], [16], [17], [18].

Let v and w be strictly positive, continuous and bounded functions (weights) on D and H(D) be the

set of all analytic functions on D. In this article we are interested in differences ψ1Cφ1 −ψ2Cφ2 acting

between weighted Banach spaces of holomorphic functions

H∞v := {f ∈ H(D); ‖f‖v := sup
z∈D

v(z)|f(z)| < ∞}

and the weighted Bloch type spaces Bw of functions f ∈ H(D) satisfying

‖f‖Bw := supz∈D v(z)|f
′(z)| < ∞.

Our aim is to give necessary and sufficient conditions for a difference ψ1Cφ1 − ψ2Cφ2 : H
∞
v → Bw to

be bounded resp. compact in terms of the involved weights and the analytic maps φ1,φ2,ψ1,ψ2.

2 Notation and auxiliary results

An introduction to the concept of composition operators can be found in the monographs [5] and [15].

In this article we are especially interested in radial weights (i.e. weights with v(z) = v(|z|) for every

z ∈ D) which satisfy additionally the Lusky condition (L1) (due to Lusky [10])

(L1) inf
k∈N

v(1 − 2−k−1)

v(1 − 2−k)
> 0.

When dealing with differences of weighted composition operators we need some geometric data.

Recall that for any z ∈ D, ϕz is the Möbius transformation which interchanges the origin and z,

namely ϕz(w) =
z−w
1−zw

,w ∈ D. The pseudohyperbolic distance ρ(z,w) for z,w ∈ D is defined by

ρ(z,w) = |ϕz(w)|. Moreover, we have ϕ
′
z(w) =

|z|2−1
(1−zw)2

for z,w ∈ D. Let us recall some auxiliary

results.

The next lemma is taken from [3], see also [6].

Lemma 1. (Bonet-Lindström-Wolf [3]) Let v be a radial weight satisfying the Lusky condition (L1)

and let f ∈ H∞v . Then there exists a constant Cv > 0 (depending on the weight v) such that

|f(z) − f(p)| ≤ Cv‖f‖v max

{

1

v(z)
,

1

v(p)

}

ρ(z,p)

for all z,p ∈ D.

Theorem 2. (Harutyunyan-Lusky, [7] Theorem 2.1) Let v and w be radial weights which are contin-

uously differentiable with respect to |z| with lim|z|→1 v(z) = lim|z|→1 w(z) = 0 and such that H
∞
w is

isomorphic to l∞. If lim supr→1

(

−
w

′(r)
v(r)

)

< ∞, then D : H∞v → H
∞
w ,f → f

′ is bounded.



CUBO
12, 2 (2010)

Differences of weighted composition operators ... 21

For conditions when H∞w is isomorphic to l∞ we refer the reader to [11] and [7]. By [7] we know

that the following weights have the desired properties:

w(z) = (1 − |z|)α,α > 0,w(z) = e
− 1

1−|z| , z ∈ D.

For the study of the compactness of the difference ψ1Cφ1 − ψ2Cφ2 we need the following result.

Proposition 3. (Cowen-MacCluer, [5] Proposition 3.11) Let X and Y be H∞v or Bw. Then ψ1Cφ1 −

ψ2Cφ2 : X → Y is compact if and only if for every bounded sequence (fn)n∈N in X such that fn → 0

uniformly on the compact subsets of D, then (ψ1Cφ1 − ψ2Cφ2 )fn → 0 in Y .

3 Main Result

In the sequel we consider weights v of the following type: Let ν be a holomorphic function on D,

non-vanishing and strictly positive on [0, 1[. Moreover we assume that ν is decreasing on [0, 1[ and

satisfies limr→1 ν(r) = 0. Then we define the corresponding weight v by v(z) := ν(|z|
2) for every

z ∈ D. Furthermore, we suppose the boundedness of the function ν′ on D. Next, we give some

illustrating examples of weights of this type:

(i) Consider ν(z) = (1−z)α, α ≥ 1. Then the corresponding weight is the so-called standard weight

v(z) = (1 − |z|2)α.

(ii) Selecting ν(z) = e
− 1

(1−z)α , α ≥ 1, we obtain the weight v(z) = e
− 1

(1−|z|2)α .

(iii) Choose ν(z) = sin(1 − z) and the corresponding weight is given by v(z) = sin(1 − |z|2).

Fix a point p ∈ D and an analytic self-map φ of D. We introduce a function

vφ(p)(z) := ν(φ(p)z) for every z ∈ D.

Since ν is holomorphic on D, the function vφ(p) is also holomorphic on D. Furthermore, vφ(p)(φ(p)) =

ν(|φ(p)|2) = v(φ(p)) and v′
φ(p)

(z) = φ(p)ν′(φ(p)z) for every z ∈ D, i.e. v′
φ(p)

(φ(p)) = φ(p)ν′(|φ(p)|2).

We start with considering boundedness of operators ψ1Cφ1 − ψ2Cφ2 : H
∞
v → Bw and give first a

necessary condition in terms of the involved weights and then a sufficient condition.

Proposition 4. Let w be a weight and v be a weight as described in the beginning of this section. Let

ψ1,ψ2 ∈ H(D) and φ1,φ2 be analytic self-maps of D. If ψ1Cφ1 − ψ2Cφ2 : H
∞
v → Bw is bounded, then

the following conditions are satisfied

(a) supz∈D w(z)

∣

∣

∣

∣

ψ
′
1(z)

v(φ1(z))
1
2
ϕ2
φ2(z)

(φ1(z)) + 2
ψ1(z)

v(φ1(z))
1
2
ϕφ2(z)(φ1(z))ϕ

′
φ2(z)

(φ1(z))

∣

∣

∣

∣

< ∞,

(b) supz∈D w(z)

∣

∣

∣

∣

ψ
′
2(z)

v(φ2(z))
1
2
ϕ2
φ1(z)

(φ2(z)) + 2
ψ2(z)

v(φ2(z))
1
2
ϕφ1(z)(φ2(z))ϕ

′
φ1(z)

(φ2(z))

∣

∣

∣

∣

< ∞,

(c) supz∈D

∣

∣

∣

ψ1(z)w(z)φ1(z)ν
′(|φ1(z)|

2)
v(φ1(z))

∣

∣

∣
ρ(φ1(z),φ2(z)) < ∞,

(d) supz∈D

∣

∣

∣

ψ2(z)w(z)φ2(z)ν
′(|φ2(z)|

2)
v(φ2(z))

∣

∣

∣
ρ(φ1(z),φ2(z)) < ∞,



22 Elke Wolf CUBO
12, 2 (2010)

Proof (a) Fix a point p ∈ D and put

fφ1(p)(z) :=

(

2

vφ1(p)(z)
−
vφ1(p)(φ1(p))

vφ1(p)(z)
2

)

1
2

and

gφ1(p)(z) := fφ1(p)(z)ϕ
2
φ2(p)

(z) for every z ∈ D.

Next, we get

‖gφ1(p)‖v ≤ sup
z∈D

∣

∣

∣

∣

v(z)2
2

vφ1(p)(z)
− v(z)2

vφ1(p)(φ1(p))

vφ1(p)(z)
2

∣

∣

∣

∣

1
2

≤ (3M)
1
2

where M = supz∈D v(z) and therefore the constant does not depend on the choice of p. Thus, gφ1(p) ∈

H∞v and g
′
φ1(p)

(z) = f′
φ1(p)

(z)ϕ2
φ2(p)

(z)+2fφ1(p)(z)ϕ
′
φ2(p)

(z)ϕφ2(p)(z) for every z ∈ D, where f
′
φ1(p)

(z) =
(

−
v
′
φ1(p)

(z)

vφ1(p)(z)
2 +

v
′
φ1(p)

(z)vφ1(p)(φ1(p))

vφ1(p)(z)
3

)(

2
vφ1(p)(z)

−
vφ1(p)(φ1(p))

vφ1(p)(z)
2

)− 1
2

and ϕ′
φ2(p)

(z) =
|φ2(p)|

2−1

(1−φ2(p)z)2
for every

z ∈ D. We get fφ1(p)(φ1(p)) =
1

v(φ1(p))
1
2

and f′
φ1(p)

(φ1(p)) = 0 and hence gφ1(p)(φ1(p)) =
ϕ

2
φ2(p)

(φ1(p))

v(φ1(p))
1
2

as well as g′
φ1(p)

(φ1(p)) = 2
ϕ

′
φ2(p)

(φ1(p))ϕφ2(p)(φ1(p))

v(φ1(p))
1
2

. Now,

w(p)

∣

∣

∣

∣

∣

ψ′1(p)ϕ
2
φ2(p)

(φ1(p))

v(φ1(p)))
1
2

+ 2
ψ1(p)ϕφ2(p)(φ1(p))ϕ

′
φ2(p)

(φ1(p))

v(φ1(p)))
1
2

∣

∣

∣

∣

∣

= w(p)
∣

∣

∣
ψ
′
1(p)gφ1(p)(φ1(p)) + ψ1(p)φ

′
1(p)g

′
φ1(p)

(φ1(p)) − ψ
′
2(p)gφ1(p)(φ2(p)) − ψ2(p)φ

′
2(p)g

′
φ1(p)

(φ2(p))
∣

∣

∣

≤ ‖ψ1Cφ1 − ψ2Cφ2‖‖gφ1(p)‖v < ∞.

Thus, (a) follows, and we can show (b) analogously. For the proof of condition (c) we fix a point

p ∈ D and put

fφ1(p)(z) :=
vφ1(p)(φ1(p))

vφ1(p)(z)
−

(

vφ1(p)(φ1(p))

vφ1(p)(z)

)

1
2

=
v(φ1(p))

vφ1(p)(z)
−

(

v(φ1(p))

vφ1(p)(z)

)
1
2

and

gφ1(p)(z) := fφ1(p)(z)ϕ
2
φ2(p)

(z) for every z ∈ D.

Hence ‖gφ1(p)‖v ≤ 2M and we get

g′φ1(p)(z) = f
′
φ1(p)

(z)ϕ2φ2(p)(z) + 2fφ1(p)(z)ϕφ2(p)(z)ϕ
′
φ2(p)

(z) for every z ∈ D,

where

f′φ1(p)(z) = −
vφ1(p)(φ1(p))v

′
φ1(p)

(z)

vφ1(p)(z)
2

+
1

2

vφ1(p)(φ1(p))
1
2 v′
φ1(p)

(z)

vφ1(p)(z)
3
2

Thus, we obtain fφ1(p)(φ1(p)) = 0 and f
′
φ1(p)

(φ1(p)) = −
1
2
φ1(p)ν

′(|φ1(p)|
2)

v(φ1(p))
. Hence gφ1(p)(φ1(p)) = 0

and g′
φ1(p)

(φ1(p)) = −
1
2

φ1(p)ν
′(|φ1(p)|

2)ϕ2
φ2(p)

(φ1(p))

v(φ1(p))
. Finally,

1

2
w(p)

∣

∣

∣

∣

∣

φ1(p)ν
′(|φ1(p)|

2)ϕ2
φ2(p)

(φ1(p))

vφ2(p)(φ1(p))

∣

∣

∣

∣

∣

= w(p)
∣

∣

∣
ψ′1(p)gφ1(p)(φ1(p)) + ψ1(p)φ

′
1(p)g

′
φ1(p)

(φ1(p)) − ψ
′
2(p)gφ1(p)(φ2(p)) − ψ2(p)φ

′
2(p)g

′
φ1(p)

(φ2(p))
∣

∣

∣

≤ ‖ψ1Cφ1 − ψ2Cφ2‖‖gφ1(p)‖v < ∞.



CUBO
12, 2 (2010)

Differences of weighted composition operators ... 23

The claim follows. We can show (d) analogously.

Proposition 5. Let v and w be weights. If

(a) there is a weight u such that the operator D : H∞v → H
∞
u ,f → f

′ is bounded and additionally

supz∈D max
{

w(z)
u(φ1(z))

,
w(z)

u(φ2(z))

}

ρ(φ1(z),φ2(z)) < ∞ as well as

supz∈D max
{

w(z)
u(φ1(z))

,
w(z)

u(φ2(z))

}

|φ′1(z)ψ1(z) − φ
′
2(z)ψ2(z)| < ∞,

(b) supz∈D max
{

1
v(φ1(z))

, 1
v(φ2(z))

}

w(z)|ψ′1(z) − ψ
′
2(z)| < ∞,

(c) supz∈D max
{

1
v(φ1(z))

, 1
v(φ2(z))

}

w(z) max{|ψ′1(z)|, |ψ
′
2(z)|}ρ(φ1(z),φ2(z)) < ∞

then ψ1Cφ1 − ψ2Cφ2 : H
∞
v → Bw is bounded.

Proof Let f ∈ H∞v . Using Lemma 1 we obtain

sup
z∈D

w(z)|((ψ1Cφ1 − ψ2Cφ2 )f)
′(z)| ≤ sup

z∈D
w(z)|ψ′1(z) − ψ

′
2(z)||f(φ1(z))|

+ sup
z∈D

w(z)|ψ′2(z)||f(φ1(z)) − f(φ2(z))|

+ sup
z∈D

w(z)|f′(φ1(z))||φ
′
1(z)ψ1(z) − φ

′
2(z)ψ2(z)|

+ sup
z∈D

w(z)|φ′2(z)ψ2(z)||f
′(φ1(z)) − f

′(φ2(z))|

≤ sup
z∈D

w(z)|ψ′1(z) − ψ
′
2(z)| max

{

1

v(φ1(z))
,

1

v(φ2(z))

}

‖f‖v

+ sup
z∈D

max{|ψ′1(z)|, |ψ
′
2(z)|} max

{

1

v(φ1(z))
,

1

v(φ2(z))

}

ρ(φ1(z),φ2(z))‖f‖v

+ sup
z∈D

w(z)

u(φ1(z))
‖f′‖u|φ

′
1(z)ψ1(z) − φ

′
2(z)ψ2(z)|

+ sup
z∈D

max

{

w(z)

u(φ1(z))
,

w(z)

u(φ2(z))

}

ρ(φ1(z),φ2(z))‖f
′‖u

≤ sup
z∈D

w(z)|ψ′1(z) − ψ
′
2(z)| max

{

1

v(φ1(z))
,

1

v(φ2(z))

}

‖f‖v

+ sup
z∈D

max{|ψ′1(z)|, |ψ
′
2(z)|} max

{

1

v(φ1(z))
,

1

v(φ2(z))

}

ρ(φ1(z),φ2(z))‖f‖v

+ sup
z∈D

max

{

w(z)

u(φ1(z))
,

w(z)

u(φ2(z))

}

‖D‖‖f‖v|φ
′
1(z)ψ1(z) − φ

′
2(z)ψ2(z)|

+ sup
z∈D

max

{

w(z)

u(φ1(z))
,

w(z)

u(φ2(z))

}

ρ(φ1(z),φ2(z))‖D‖‖f‖v

and the claim follows.

Next, we turn our attention to compactness of ψ1Cφ1 − ψ2Cφ2 : H
∞
v → Bw.

Proposition 6. Let w be a weight and v be a weight as described in the beginning of this section. Let

ψ1,ψ2 ∈ H(D) and φ1,φ2 be analytic self-maps of D. If ψ1Cφ1 − ψ2Cφ2 : H
∞
v → Bw is bounded, then

the following conditions are satisfied



24 Elke Wolf CUBO
12, 2 (2010)

(a) lim sup|φ1(z)|→1 w(z)

∣

∣

∣

∣

ψ
′
1(z)

v(φ1(z))
1
2
ϕ2
φ2(z)

(φ1(z)) + 2
ψ1(z)

v(φ1(z))
1
2
ϕφ2(z)(φ1(z))ϕ

′
φ2(z)

(φ1(z))

∣

∣

∣

∣

= 0,

(b) lim sup|φ2(z)|→1 w(z)

∣

∣

∣

∣

ψ
′
2(z)

v(φ2(z))
1
2
ϕ2
φ1(z)

(φ2(z)) + 2
ψ2(z)

v(φ2(z))
1
2
ϕφ1(z)(φ2(z))ϕ

′
φ1(z)

(φ2(z))

∣

∣

∣

∣

= 0,

(c) lim sup|φ1(z)|→1

∣

∣

∣

ψ1(z)w(z)φ1(z)ν
′(|φ1(z)|

2)
v(φ1(z))

∣

∣

∣
ρ(φ1(z),φ2(z)) = 0,

(d) lim sup|φ2(z)|→1

∣

∣

∣

ψ2(z)w(z)φ2(z)ν
′(|φ2(z)|

2)
v(φ2(z))

∣

∣

∣
ρ(φ1(z),φ2(z)) = 0,

Proof (a) Consider a sequence (zn)n ⊂ D such that |φ1(zn)| → 1 if n → ∞. We set

fφ1(zn)(z) := vφ1(zn)(φ1(zn))
1
6

(

3

2

1

vφ1(zn)(z)
2
−
vφ1(zn)(φ1(zn))

vφ1(zn)(z)
3

)

1
3

and

gφ1(zn)(z) := fφ1(zn)(z)ϕ
2
φ2(zn)

(z) for every z ∈ D.

Thus ‖gφ1(zn)‖v ≤ supz∈D vφ1(zn)(φ1(zn))
1
6

∣

∣

∣

3
2

v(z)3

vφ1(zn)(z)
2 −

v(z)3vφ1(zn)(φ1(zn))

vφ1(zn)(z)
3

∣

∣

∣

1
3

≤ M
1
6

(

5
2
M
)

1
3 for ev-

ery n ∈ N, where M := supz∈D v(z). Thus, (gφ1(zn))n∈N is a bounded sequence in H
∞
v which converges

to zero uniformly on the compact subsets of D. Moreover,

g′φ1(zn)(z) = f
′
φ1(zn)

(z)ϕ2φ2(zn)(z) + 2fφ1(zn)(z)ϕφ2(zn)(z)ϕ
′
φ2(zn)

(z) for every z ∈ D,

where

f
′
φ1(zn)

(z) = vφ1(zn)(φ1(zn))
1
6

(

3

2

1

vφ1(zn)(z)
2
−
vφ1(zn)(φ1(zn))

vφ1(zn)(z)
3

)− 2
3

·

·

(

−
v′
φ1(zn)

(z)

vφ1(zn)(z)
3

+
vφ1(zn)(φ1(zn))

vφ1(zn)(z)
4

v′φ1(zn)(z)

)

for every n ∈ N. By Proposition 3, the fact that ψ1Cφ1 − ψ2Cφ2 is compact yields

‖(ψ1Cφ1 − ψ2Cφ2 )gφ1(zn)‖Bw → 0 if n → ∞.

Finally,

‖(ψ1Cφ1 − ψ2Cφ2 )gφ1(zn)‖Bw ≥

w(zn)

∣

∣

∣

∣

∣

ψ′1(zn)

v(φ1(zn))
1
2

ϕ2φ2(zn)(φ1(zn)) + 2
ψ1(zn)ϕφ2(zn)(φ1(zn))ϕ

′
φ2(zn)

(φ1(zn))

v(φ1(zn))
1
2

∣

∣

∣

∣

∣

Thus, (a) follows and we can show (b) analogously. Consider now

fφ1(zn)(z) :=
vφ1(zn)(φ1(zn))

vφ1(zn)(z)
−

(

vφ1(zn)(φ1(zn))

vφ1(zn)(z)

)

1
2

=
v(φ1(zn))

vφ1(zn)(z)
−

(

v(φ1(zn))

vφ1(zn)(z)

)
1
2

and

gφ1(zn)(z) := fφ1(zn)(z)ϕ
2
φ2(zn)

(z) for every z ∈ D.



CUBO
12, 2 (2010)

Differences of weighted composition operators ... 25

Then ‖gφ1(zn)‖v ≤ supz∈D v(z)

∣

∣

∣

∣

vφ1(zn)(φ1(zn))

vφ1(zn)(z)
−
(

vφ1(zn)(φ1(zn))

vφ1(zn)(z)

)
1
2

∣

∣

∣

∣

≤ 2M for every n ∈ N. Thus

(gφ1(zn))n is a bounded sequence in H
∞
v and gφ1(zn) → 0 uniformly on every compact subset of D.

Moreover gφ1(zn)(φ1(zn)) = 0 and g
′
φ1(zn)

(φ1(zn)) = −
1
2

v
′
φ1(zn)

(φ1(zn))

v(φ1(zn))
ϕ2
φ2(zn)

(φ1(zn)). Since ψ1Cφ1 −

ψ2Cφ2 is compact, by Proposition 3 we have ‖(ψ1Cφ1 − ψ2Cφ2 )gn‖Bw → 0 if n → ∞. Thus,

‖(ψ1Cφ1 − ψ2Cφ2 )gφ1(zn)‖Bw = sup
z∈D

w(z)|((ψ1Cφ1 − ψ2Cφ2 )gφ1(zn))
′(z)|

≥
1

2
w(zn)|ψ1(zn)φ

′
1(zn)φ1(zn)|ρ(φ1(zn),φ2(zn))

2 |ν
′(|φ1(zn)|

2)|

v(φ1(zn))
.

Finally, lim sup|φ1(z)|→1 w(z)|ψ(z)||φ
′
1(z)||φ1(z)|

|ν′(|φ1(z)|
2)|

v(φ1(z))
= 0, and (c) holds. (d) follows analo-

gously.

Proposition 7. Let v and w be weights. If

(a) there is a weight u such that the operator D : H∞v → H
∞
u ,f → f

′ is bounded and additionally

lim supmax{|φ1(z)|,|φ2(z)|}→1 max
{

w(z)
u(φ1(z))

,
w(z)

u(φ2(z))

}

ρ(φ1(z),φ2(z)) = 0 as well as

lim supmax{|φ1(z)|,|φ2(z)|}→1 max
{

w(z)
u(φ1(z))

,
w(z)

u(φ2(z))

}

|φ′1(z)ψ1(z) − φ
′
2(z)ψ2(z)| = 0,

(b) lim supmax{|φ1(z)|,φ2(z)|}→1 max
{

1
v(φ1(z))

, 1
v(φ2(z))

}

w(z)|ψ′1(z) − ψ
′
2(z)| = 0,

(c) lim supmax{|φ1(z)|,|φ2(z)|}→1 max
{

1
v(φ1(z))

, 1
v(φ2(z))

}

w(z) max{|ψ′1(z)|, |ψ
′
2(z)|}ρ(φ1(z),φ2(z)) = 0

then ψ1Cφ1 − ψ2Cφ2 : H
∞
v → Bw is compact.

Proof Let (fn)n∈N be a sequence in H
∞
v with ‖fn‖v ≤ 1 and fn → 0 uniformly on compact subsets

of D. By the assumption, for any ε > 0 there is 0 < δ < 1 such that δ < max{|φ1(z)|, |φ2(z)|} < 1

implies

max

{

w(z)

u(φ1(z))
,

w(z)

u(φ2(z))

}

ρ(φ1(z),φ2(z)) < ε

max

{

w(z)

u(φ1(z))
,

w(z)

u(φ2(z))

}

|φ′1(z)ψ1(z) − φ
′
2(z)ψ2(z)| < ε

max

{

1

v(φ1(z))
,

1

v(φ2(z))

}

w(z)|ψ′1(z) − ψ
′
2(z)| < ε

max

{

1

v(φ1(z))
,

1

v(φ2(z))

}

w(z) max{|ψ′1(z)|, |ψ
′
2(z)|}ρ(φ1(z),φ2(z)) < ε



26 Elke Wolf CUBO
12, 2 (2010)

Then applying Lemma 1

sup
z∈D

w(z)|((ψ1Cφ1 − ψ2Cφ2 )fn)
′(z)| ≤ sup

z∈D
w(z)|ψ′1(z) − ψ

′
2(z)||fn(φ1(z))|

+ sup
z∈D

w(z)|ψ′2(z)||fn(φ1(z)) − fn(φ2(z))|

+ sup
z∈D

w(z)|f′n(φ1(z))||φ
′
1(z)ψ1(z) − φ

′
2(z)ψ2(z)|

+ sup
z∈D

w(z)|φ′2(z)ψ2(z)||f
′
n(φ1(z)) − f

′
n(φ2(z))|

≤ sup
{z; max{|φ1(z)|,|φ2(z)|}>δ}

w(z)|ψ′1(z) − ψ
′
2(z)| max

{

1

v(φ1(z))
,

1

v(φ2(z))

}

‖fn‖v

+ sup
{z; max{|φ1(z)|,|φ2(z)|}>δ}

max{|ψ′1(z)|, |ψ
′
2(z)|} max

{

1

v(φ1(z))
,

1

v(φ2(z))

}

ρ(φ1(z),φ2(z))‖fn‖v

+ sup
{z; max{|φ1(z)|,|φ2(z)|}>δ}

w(z)

u(φ1(z))
‖f′‖u|φ

′
1(z)ψ1(z) − φ

′
2(z)ψ2(z)|

+ sup
{z; max{|φ1(z)|,|φ2(z)|}>δ}

max

{

w(z)

u(φ1(z))
,

w(z)

u(φ2(z))

}

ρ(φ1(z),φ2(z))‖f
′
n‖u

+ sup
{z; max{|φ1(z)|,φ2(z)|}≤δ}

w(z)|ψ′1(z) − ψ
′
2(z)||fn(φ1(z))|

+ sup
{z; max{|φ1(z)|,φ2(z)|}≤δ}

w(z)|ψ′2(z)||fn(φ1(z))|

+ sup
{z; max{|φ1(z)|,φ2(z)|}≤δ}

w(z)|ψ′2(z)||fn(φ2(z))|

+ sup
{z; max{|φ1(z)|,φ2(z)|}≤δ}

w(z)|f′n(φ1(z))||φ
′
1(z)ψ1(z) − φ

′
2(z)ψ2(z)|

+ sup
{z; max{|φ1(z)|,φ2(z)|}≤δ}

w(z)|φ′2(z)ψ2(z)||f
′
n(φ1(z))|

+ sup
{z; max{|φ1(z)|,φ2(z)|}≤δ}

w(z)|φ′2(z)ψ2(z)||f
′
n(φ2(z))|.

The claim follows.

Received: October 2008. Revised: February 2009.

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