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PureMathematics
n
Ø
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Æ
,2021,11(12),2057-2068
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1112229
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f
•
I
(
n
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g,ϕ
f
(
z
) =
Z
z
0
f
(
n
)
(
ϕ
(
ζ
))
g
(
ζ
)
dζ.
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Bloch
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m
DifferencesofGeneralizedIntegration
Operatorsfrom
α
-BlochSpacesto
β
-BlochSpaces
ZhonghuaHe
SchoolofFinancialMathematicsandStatistics,GuangdongUniversityofFinance,Guangzhou
Guangdong
Received:Nov.8
th
,2021;accepted:Dec.9
th
,2021;published:Dec.17
th
,2021
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[J].
n
Ø
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Æ
,2021,11(12):2057-2068.
DOI:10.12677/pm.2021.1112229
Û
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Abstract
Ageneralizedintegrationoperatorisdefinedby
I
(
n
)
g,ϕ
f
(
z
) =
Z
z
0
f
(
n
)
(
ϕ
(
ζ
))
g
(
ζ
)
dζ
inducedbyholomorphicmaps
g
and
ϕ
oftheunitdisk
D
,where
ϕ
(
D
)
⊂
D
and
n
isa
positiveinteger.In thispaper, weinvestigatetheboundedness andthecompactnessof
thedifferencesoftwo generalizedintegrationoperators from
α
-Blochspacesto
β
-Bloch
spaces.
Keywords
Differences,GeneralizedIntegrationOperator,BlochSpace
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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(
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g
1
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(
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I
(
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g
1
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1
−
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(
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DOI:10.12677/pm.2021.11122292059
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.
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4[27]
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…
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6
= 0,
-
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a
(
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−|
a
|
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(
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+1)
···
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−
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a
n
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−
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,
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+1)
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+
n
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n
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a
|
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a
n
+1
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−
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+
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−
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|
2
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a
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(
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|
2
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−
¯
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+
n
,
k
(
n
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a
(
z
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(
a
−
z
)(1
−|
a
|
2
)
(1
−
¯
az
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+
n
+1
.
3.
Ì
‡
(
Ø
•
•
B
,
P
D
ϕ,g
(
z
) :=
(1
−|
z
|
2
)
β
g
(
z
)
(1
−|
ϕ
(
z
)
|
2
)
α
+
n
−
1
.
¿
-
I
1
(
z
) =
|
D
ϕ
1
,g
1
(
z
)
|
ρ
(
ϕ
1
(
z
)
,ϕ
2
(
z
))
,
I
2
(
z
) =
|
D
ϕ
2
,g
2
(
z
)
|
ρ
(
ϕ
1
(
z
)
,ϕ
2
(
z
))
,
I
3
(
z
) =
|
D
ϕ
1
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1
(
z
)
−
D
ϕ
2
,g
2
(
z
)
|
.
½
n
1
ϕ
1
,
ϕ
2
∈
S
(
D
),
g
1
,g
2
∈
H
(
D
)
…
n
∈
N
.
K
e
ã
^
‡
d
:
(
i
)
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
:
B
α
→B
β
k
.
;
(
ii
)sup
z
∈
D
I
1
(
z
)
<
∞
andsup
z
∈
D
I
3
(
z
)
<
∞
;
(
iii
)sup
z
∈
D
I
2
(
z
)
<
∞
andsup
z
∈
D
I
3
(
z
)
<
∞
.
y
²
(
i
)
⇒
(
ii
).
b
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
:
B
α
→B
β
k
.
.
é
a
∈
D
…
a
6
= 0,
-
f
a
(
z
) =
1
−|
a
|
2
α
(
α
+1)
···
(
α
+
n
−
1)¯
a
n
(1
−
¯
az
)
α
,
k
a
(
z
) =
1
α
(
α
+1)
···
(
α
+
n
)
n
(1
−|
a
|
2
)
¯
a
n
+1
(1
−
¯
az
)
α
+
α
·
(
a
−
z
)(1
−|
a
|
2
)
¯
a
n
(1
−
¯
az
)
α
+1
.
DOI:10.12677/pm.2021.11122292060
n
Ø
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Æ
Û
§
u
K
d
Ú
n
4
Œ
•
,
f
a
,k
a
∈B
α
.
u
´
é
½
w
∈
D
…
÷
v
ϕ
1
(
w
)
6
= 0,
k
∞
>
k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
f
ϕ
1
(
w
)
k
B
β
= sup
z
∈
D
(1
−|
z
|
2
)
β
|
((
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
f
ϕ
1
(
w
)
)
0
(
z
)
|
≥
(1
−|
w
|
2
)
β
g
1
(
w
)(1
−|
ϕ
1
(
w
)
|
2
)
(1
−|
ϕ
1
(
w
)
|
2
)
α
+
n
−
(1
−|
w
|
2
)
β
g
2
(
w
)(1
−|
ϕ
1
(
w
)
|
2
)
(1
−
ϕ
1
(
w
)
ϕ
2
(
w
))
α
+
n
≥|
D
ϕ
1
,g
1
(
w
)
|−
D
ϕ
2
,g
2
(
w
)
(1
−|
ϕ
1
(
w
)
|
2
)(1
−|
ϕ
2
(
w
)
|
2
)
α
+
n
−
1
(1
−
ϕ
1
(
w
)
ϕ
2
(
w
))
α
+
n
,
(1)
Ú
∞
>
k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
k
ϕ
1
(
w
)
k
B
β
= sup
z
∈
D
(1
−|
z
|
2
)
β
|
((
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
k
ϕ
1
(
w
)
)
0
(
z
)
|
≥
(1
−|
w
|
2
)
β
g
2
(
w
)(
ϕ
1
(
w
)
−
ϕ
2
(
w
))(1
−|
ϕ
1
(
w
)
|
2
)
(1
−
ϕ
1
(
w
)
ϕ
2
(
w
))
α
+
n
+1
=
D
ϕ
2
,g
2
(
w
)
(1
−|
ϕ
1
(
w
)
|
2
)(1
−|
ϕ
2
(
w
)
|
2
)
α
+
n
−
1
(1
−
ϕ
1
(
w
)
ϕ
2
(
w
))
α
+
n
·
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
))
.
(2)
(1)
ª
ü
>
Ó
ž
¦
±
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
)),
(
Ü
(2)
Œ
sup
w
∈
D
\
D
1
|
D
ϕ
1
,g
1
(
w
)
|
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
))
<
∞
,
(3)
Ù
¥
D
1
=
{
w
∈
D
:
ϕ
1
(
w
) = 0
}
.
Ó
n
Œ
sup
w
∈
D
\
D
2
|
D
ϕ
2
,g
2
(
w
)
|
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
))
<
∞
,
(4)
Ù
¥
D
2
=
{
w
∈
D
:
ϕ
2
(
w
) = 0
}
.
,
˜
•
¡
,
d
(1)
Œ
±
∞
>
k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
f
ϕ
1
(
w
)
k
B
β
≥
(1
−|
w
|
2
)
β
g
1
(
w
)(1
−|
ϕ
1
(
w
)
|
2
)
(1
−|
ϕ
1
(
w
)
|
2
)
α
+
n
−
(1
−|
w
|
2
)
β
g
2
(
w
)(1
−|
ϕ
1
(
w
)
|
2
)
(1
−
ϕ
1
(
w
)
ϕ
2
(
w
))
α
+
n
≥|
D
ϕ
1
,g
1
(
w
)
−
D
ϕ
2
,g
2
(
w
)
|−|
D
ϕ
2
,g
2
(
w
)
|·
1
−
(1
−|
ϕ
1
(
w
)
|
2
)(1
−|
ϕ
2
(
w
)
|
2
)
α
+
n
−
1
(1
−
ϕ
1
(
w
)
ϕ
2
(
w
))
α
+
n
≥
C
(
|
D
ϕ
1
,g
1
(
w
)
−
D
ϕ
2
,g
2
(
w
)
|−|
D
ϕ
2
,g
2
(
w
)
|
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
)))
,
(5)
l
sup
w
∈
D
\{
D
1
∪
D
2
}
|
D
ϕ
1
,g
1
(
w
)
−
D
ϕ
2
,g
2
(
w
)
|
<
∞
.
(6)
DOI:10.12677/pm.2021.11122292061
n
Ø
ê
Æ
Û
§
u
ϕ
1
(
w
) =
ϕ
2
(
w
) = 0
ž
,
-
f
0
(
z
) =
z
n
n
!
,
K
d
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
:
B
α
→B
β
k
.
5
Œ
sup
w
∈
D
1
∩
D
2
|
D
ϕ
1
,g
1
(
w
)
−
D
ϕ
2
,g
2
(
w
)
|
=sup
w
∈
D
1
∩
D
2
µ
(
|
w
|
)
|
g
1
(
w
)
−
g
2
(
w
)
|
≤k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
f
0
k
B
β
<
∞
,
(7)
sup
w
∈
D
1
∩
D
2
|
D
ϕ
1
,g
1
(
w
)
|
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
)) = 0
,
(8)
…
sup
w
∈
D
1
∩
D
2
|
D
ϕ
2
,g
2
(
w
)
|
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
)) = 0
.
(9)
ϕ
2
(
w
) = 0
,ϕ
(
w
)
1
6
= 0
ž
,
-
P
ϕ
1
(
w
)
(
z
) =
1
α
···
(
α
+
n
)
n
ϕ
1
(
w
)
n
+1
(1
−
ϕ
1
(
w
)
z
)
α
+
α
(
ϕ
1
(
w
)
−
z
)
ϕ
1
(
w
)
n
(1
−
ϕ
1
(
w
)
z
)
α
+1
!
.
K
∞
>
k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
P
ϕ
1
(
w
)
k
B
β
= sup
z
∈
D
|
((
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
P
ϕ
1
(
w
)
)
0
(
z
))
|
≥
(1
−|
w
|
2
)
β
g
2
(
w
)(
ϕ
1
(
w
)
−
ϕ
2
(
w
))
(1
−
ϕ
1
(
w
)
ϕ
2
(
w
))
α
+
n
+1
=(1
−|
w
|
2
)
β
|
ϕ
1
(
w
)
g
2
(
w
)
|
=
|
D
ϕ
2
,g
2
(
w
)
|
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
))
,
u
´
sup
w
∈
D
2
\
D
1
|
D
ϕ
2
,g
2
(
w
)
|
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
))
<
∞
.
(10)
Ï
d
,
(
Ü
(5)
Ú
(10)
Œ
±
sup
w
∈
D
2
\
D
1
|
D
ϕ
1
,g
1
(
w
)
−
D
ϕ
2
,g
2
(
w
)
|
<
∞
.
(11)
Ó
n
Œ
,
ϕ
1
(
w
) = 0
,ϕ
(
w
)
2
6
= 0
ž
,
k
sup
w
∈
D
1
\
D
2
|
D
ϕ
1
,g
1
(
w
)
−
D
ϕ
2
,g
2
(
w
)
|
<
∞
(12)
sup
w
∈
D
1
\
D
2
|
D
ϕ
2
,g
2
(
w
)
|
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
))
<
∞
.
(13)
DOI:10.12677/pm.2021.11122292062
n
Ø
ê
Æ
Û
§
u
,
d
(3),(8)
Ú
(13)
Œ
sup
z
∈
D
I
1
(
z
)
<
∞
;
d
(6),(7),(11)
Ú
(12)
Œ
sup
z
∈
D
I
3
(
z
)
<
∞
.
(
ii
)
⇒
(
iii
).
e
(
ii
)
¤
á
,
K
sup
z
∈
D
I
2
(
z
)=sup
z
∈
D
|
D
ϕ
2
,g
2
(
z
)
|
ρ
(
ϕ
1
(
z
)
,ϕ
2
(
z
))
≤
sup
z
∈
D
|
D
ϕ
1
,g
1
(
z
)
|
ρ
(
ϕ
1
(
z
)
,ϕ
2
(
z
))
+sup
z
∈
D
|
D
ϕ
1
,g
1
(
z
)
−
D
ϕ
2
,g
2
(
z
)
|
ρ
(
ϕ
1
(
z
)
,ϕ
2
(
z
))
≤
sup
z
∈
D
I
1
(
z
)+sup
z
∈
D
I
3
(
z
)
<
∞
,
l
(
iii
)
•
¤
á
.
(
iii
)
⇒
(
i
).
e
(
iii
)
¤
á
,
K
d
Ú
n
1
Œ
,
é
f
∈B
α
…
k
f
k
B
α
≤
1,
k
k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
f
k
B
β
=sup
z
∈
D
|
(1
−|
z
|
2
)
β
f
(
n
)
(
ϕ
1
(
z
))
g
1
(
z
)
−
(1
−|
z
|
2
)
β
f
(
n
)
(
ϕ
2
(
z
))
g
2
(
z
)
|
=sup
z
∈
D
|
D
ϕ
1
,g
1
(
z
)(1
−|
ϕ
1
(
z
)
|
2
)
α
+
n
−
1
f
(
n
)
(
ϕ
1
(
z
))
−
D
ϕ
2
,g
2
(
z
)(1
−|
ϕ
2
(
z
)
|
2
)
α
+
n
−
1
f
(
n
)
(
ϕ
2
(
z
))
|
≤
sup
z
∈
D
|
D
ϕ
1
,g
1
(
z
)
−
D
ϕ
2
,g
2
(
z
)
|
+
C
sup
z
∈
D
|
D
ϕ
2
,g
2
(
z
)
|
ρ
(
ϕ
1
(
z
)
,ϕ
2
(
z
))
<
∞
,
¤
±
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
:
B
α
→B
β
k
.
.
3
?
Ø
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
:
F
(
p,q,s
)
→B
µ
;
5
ƒ
c
,
·
‚
Ú
\
e
P
Ò
:
Γ(
ϕ
) =
{{
z
k
}⊂
D
:
|
ϕ
(
z
k
)
|→
1
}
,
D
(
g,ϕ
) :=
{{
z
k
}⊂
D
:
|
ϕ
(
z
k
)
|→
1
,
|
D
ϕ,g
(
z
k
)
|
9
0
}
.
½
n
2
ϕ
1
,
ϕ
2
∈
S
(
D
),
g
1
,g
2
∈
H
(
D
).
e
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
:
B
α
→B
β
k
.
,
I
(
n
)
g
1
,ϕ
1
Ú
I
(
n
)
g
2
,ϕ
2
Ñ
Ø
´
;
Ž
f
,
K
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
:
B
α
→B
β
´
;
Ž
f
…
=
(
i
)
D
(
g
1
,ϕ
1
) =
D
(
g
2
,ϕ
2
)
6
= 0
,D
(
g
1
,ϕ
1
)
⊂
Γ(
ϕ
2
),
(
ii
)
é
z
k
∈
Γ(
ϕ
1
)
∩
Γ(
ϕ
2
),
lim
k
→∞
I
1
(
z
k
) =lim
k
→∞
I
2
(
z
k
) =lim
k
→∞
I
3
(
z
k
) = 0
.
DOI:10.12677/pm.2021.11122292063
n
Ø
ê
Æ
Û
§
u
y
²
¿
©
5
.
{
f
k
}
´
B
α
¥
S
,
Ù
3
D
þ
?
¿
;
f
8
þ
Ñ
˜
—
Â
ñ
u
0,
…
k
f
k
k
B
α
≤
1.
e
k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
f
k
k
B
β
9
0,
K
•
3
ε>
0,
¦
é
?
¿
k
k
k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
f
k
k
B
β
>ε
.
u
´
é
z
‡
k
,
•
3
z
k
∈
D
¦
|
D
ϕ
1
,g
1
(
z
k
)(1
−|
ϕ
1
(
z
k
)
|
2
)
α
+
n
−
1
f
(
n
)
k
(
ϕ
1
(
z
k
))
−
D
ϕ
2
,g
2
(
z
k
)(1
−|
ϕ
2
(
z
k
)
|
2
)
α
+
n
−
1
f
(
n
)
k
(
ϕ
2
(
z
k
))
|
>ε.
(14)
l
|
ϕ
1
(
z
k
)
|→
1
½
ö
|
ϕ
2
(
z
k
)
|→
1.
e
|
ϕ
1
(
z
k
)
|→
1,
w
∈
D
•
{
ϕ
2
(
z
k
)
}
4
•
.
K
•
3
f
Â
ñ
u
w
,
Ø
”
Ò
ϕ
2
(
z
k
)
→
w
.
X
J
|
w
|
<
1,
@
o
z
k
6∈
Γ(
ϕ
1
)
∩
Γ(
ϕ
2
).
u
´
d
D
(
g
1
,ϕ
1
)
⊂
Γ(
ϕ
1
)
∩
Γ(
ϕ
2
)
Œ
D
ϕ
1
,g
1
(
z
k
)
→
0.
,
˜
•
¡
,
du
I
(
n
)
g
2
,ϕ
2
k
.
,
u
´
|
D
ϕ
2
,g
2
(
z
k
)
|
(1
−|
ϕ
2
(
z
k
)
|
2
)
α
+
n
−
1
= (1
−|
z
k
|
2
)
β
|
g
2
(
z
k
)
|
<
∞
,
l
|
w
|
<
1
í
Ñ
f
(
n
)
k
(
ϕ
2
(
z
k
))
→
0,
†
(14)
g
ñ
,
|
w
|
= 1.
Ï
d
,
|
ϕ
1
(
z
k
)
|→
1
…
|
ϕ
2
(
z
k
)
|→
1.
d
b
Œ
±
|
D
ϕ
1
,g
1
(
z
k
)(1
−|
ϕ
1
(
z
k
)
|
2
)
α
+
n
−
1
f
(
n
)
k
(
ϕ
1
(
z
k
))
−
D
ϕ
2
,g
2
(
z
k
)(1
−|
ϕ
2
(
z
k
)
|
2
)
α
+
n
−
1
f
(
n
)
k
(
ϕ
2
(
z
k
))
|
≤|
D
ϕ
1
,g
1
(
z
k
)
−
D
ϕ
2
,g
2
(
z
k
)
|
+sup
z
∈
D
|
D
ϕ
2
,g
2
(
z
k
)
|
ρ
(
ϕ
1
(
z
k
)
,ϕ
2
(
z
k
))
→
0
,k
→∞
.
†
(14)
g
ñ
.
7
‡
5
.
d
b
Œ
•
,
e
I
(
n
)
g
1
,ϕ
1
š
;
,
K
•
3
S
{
z
k
}⊂
D
(
g
1
,ϕ
1
)
¦
|
ϕ
1
(
z
k
)
|→
1
ž
|
D
ϕ
1
,g
1
(
z
k
)
|
9
0.
é
w
k
=
ϕ
1
(
z
k
),
†
½
n
1
a
q
/½
Â
f
w
k
Ú
k
w
k
,
K
{
f
w
k
}
Ú
{
k
w
k
}
´
B
α
¥
k
.
S
,
…
3
D
z
‡
;
f
8
þ
Ñ
˜
—
Â
ñ
u
0.
Ï
d
,
d
Ú
n
2
Œ
,
k
→∞
ž
0
←k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
f
ϕ
1
(
z
k
)
k
B
β
≥
|
D
ϕ
1
,g
1
(
z
k
)
|−
D
ϕ
2
,g
2
(
z
k
)
(1
−|
ϕ
1
(
z
k
)
|
2
)(1
−|
ϕ
2
(
z
k
)
|
2
)
α
+
n
−
1
(1
−
ϕ
1
(
z
k
)
ϕ
2
(
z
k
))
α
+
n
!
,
(15)
Ú
0
←k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
k
ϕ
1
(
z
k
)
k
B
β
≥
D
ϕ
2
,g
2
(
z
k
)
(1
−|
ϕ
1
(
z
k
)
|
2
)(1
−|
ϕ
2
(
z
k
)
|
2
)
α
+
n
−
1
(1
−
ϕ
1
(
z
k
)
ϕ
2
(
z
k
))
α
+
n
·
ρ
(
ϕ
1
(
z
k
)
,ϕ
2
(
z
k
))
.
(16)
u
´
(
Ü
(15)
Ú
(16)
lim
k
→∞
I
1
(
z
k
) =lim
k
→∞
|
D
ϕ
1
,g
1
(
z
k
)
|
ρ
(
ϕ
1
(
z
k
)
,ϕ
2
(
z
k
)) = 0
.
(17)
DOI:10.12677/pm.2021.11122292064
n
Ø
ê
Æ
Û
§
u
d
|
D
ϕ
1
,g
1
(
z
k
)
|
9
0
Ú
(17)
Œ
lim
k
→∞
ρ
(
ϕ
1
(
z
k
)
,ϕ
2
(
z
k
)) = 0.
l
lim
k
→∞
I
2
(
z
k
) =lim
k
→∞
|
D
ϕ
2
,g
2
(
z
k
)
|
ρ
(
ϕ
1
(
z
k
)
,ϕ
2
(
z
k
)) = 0
.
(18)
?
˜
Ú
,
é
?
¿
{
z
k
}⊂
D
(
g
1
,ϕ
1
),
k
lim
k
→∞
|
ϕ
1
(
z
k
)
−
ϕ
2
(
z
k
)
|
= 0.
Ï
d
D
(
g
1
,ϕ
1
)
⊂
Γ(
ϕ
2
)
.
(19)
d
,
k
→∞
ž
,
k
|
D
ϕ
1
,g
1
(
z
k
)
−
D
ϕ
2
,g
2
(
z
k
)
|−|
D
ϕ
2
,g
2
(
z
k
)
|
ρ
(
ϕ
1
(
z
k
)
,ϕ
2
(
z
k
))
→
0
.
u
´
d
(18)
Œ
±
lim
k
→∞
I
3
(
z
k
) =lim
k
→∞
|
D
ϕ
1
,g
1
(
z
k
)
−
D
ϕ
2
,g
2
(
z
k
)
|
= 0
.
(20)
Ï
d
,
d
(19)
Ú
(20)
Œ
D
(
g
1
,ϕ
1
)
⊂
D
(
g
2
,ϕ
2
).
Ó
n
Œ
D
(
g
2
,ϕ
2
)
⊂
D
(
g
1
,ϕ
1
).
,
D
(
g
1
,ϕ
1
)=
D
(
g
2
,ϕ
2
).
é
?
¿
S
{
z
k
}
,
|
ϕ
1
(
z
k
)
|→
1
,
|
ϕ
2
(
z
k
)
|→
1
…
|
D
ϕ
1
,g
1
(
z
k
)
|→
0
ž
,
k
lim
k
→∞
I
1
(
z
k
) =lim
k
→∞
|
D
ϕ
1
,g
1
(
z
k
)
|
ρ
(
ϕ
1
(
z
k
)
,ϕ
2
(
z
k
)) = 0
.
(21)
,
˜
•
¡
,
k
→∞
ž
,
k
0
←k
(
I
(
n
)
g
1
,ϕ
1
−
I
(
n
)
g
2
,ϕ
2
)
k
ϕ
2
(
z
k
)
k
B
β
≥
C
(
|
D
ϕ
1
,g
1
(
w
)
−
D
ϕ
2
,g
2
(
w
)
|−|
D
ϕ
2
,g
2
(
w
)
|
ρ
(
ϕ
1
(
w
)
,ϕ
2
(
w
)))
.
u
´
,
Œ
±
lim
k
→∞
I
3
(
z
k
) =lim
k
→∞
|
D
ϕ
1
,g
1
(
z
k
)
−
D
ϕ
2
,g
2
(
z
k
)
|
= 0
.
(22)
l
k
lim
k
→∞
|
D
ϕ
1
,g
1
(
z
k
)
|
=lim
k
→∞
|
D
ϕ
2
,g
2
(
z
k
)
|
= 0
.
Ï
d
,
k
lim
k
→∞
I
2
(
z
k
) =lim
k
→∞
|
D
ϕ
2
,g
2
(
z
k
)
|
ρ
(
ϕ
1
(
z
k
)
,ϕ
2
(
z
k
)) = 0
.
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
]
Ï
‘
8
(No.11971123).
DOI:10.12677/pm.2021.11122292065
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Æ
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u
ë
•
©
z
[1]Cowen,C.C.andMacCluer,B.D.(1995)CompositionOperatorsonSpacesofAnalyticFunc-
tions(StudiesinAdvancedMathematics).CRCPress,BocaRaton,FL,xii+388p.
[2]Shapiro,J.H.(1993)CompositionOperatorsandClassicalFunctionTheory(Universitext:
TractsinMathematics).Springer-Verlag,NewYork,xvi+223p.
https://doi.org/10.1007/978-1-4612-0887-7
[3]Pommerenke,Ch.(1977)SchlichteFunktionenundanalytischeFunktionenvonbeschrankter
mittlererOszillation.
CommentariiMathematiciHelvetici
,
52
,591-602.(InGerman)
https://doi.org/10.1007/BF02567392
[4]Aleman, A. and Siskakis, A.G. (1995) An Integral Operator on
H
p
.
ComplexVariables,Theory
andApplication
,
28
,149-158.https://doi.org/10.1080/17476939508814844
[5]Aleman, A.andCima, J.A.(2001)AnIntegralOperator onHpand Hardy’sInequality.
Journal
d’AnalyseMath´ematique
,
85
,157-176.https://doi.org/10.1007/BF02788078
[6]Sharma,S.D.andSharma,A.(2011)GeneralizedIntegrationOperatorsfromBlochType
SpacestoWeightedBMOASpaces.
DemonstratioMathematica
,
44
,373-390.
https://doi.org/10.1515/dema-2013-0306
[7]He,Z.H. andCao,G.F. (2013)GegeralizedIntegration OperatorsbetweenBloch-Type Spaces
and
F
(
p,q,q
)Spaces.
TaiwaneseJournalofMathematics
,
17
,1211-1225.
https://doi.org/10.11650/tjm.17.2013.2658
[8]Stevi´c,S., Sharma,A.K. andSharma,S.D.(2012)GeneralizedIntegration Operators from the
SpaceofIntegral TransformsintoBloch-TypeSpaces.
JournalofComputationalAnalysisand
Applications
,
14
,1139-1147.
[9]Li, S. and Stevi´c, S. (2008) Generalized Composition Operators on Zygmund Spaces and Bloch
TypeSpaces.
JournalofMathematicalAnalysisandApplications
,
338
,1282-1295.
https://doi.org/10.1016/j.jmaa.2007.06.013
[10]Li,S.andStevi´c,S.(2008)ProductsofCompositionandIntegralTypeOperatorsfrom
H
∞
totheBlochSpace.
ComplexVariablesandEllipticEquations
,
53
,463-474.
https://doi.org/10.1080/17476930701754118
[11]Li,S.andStevi´c,S.(2008)ProductsofVolterraTypeOperatorandCompositionOpera-
torfrom
H
∞
andBlochSpacestoZygmundSpaces.
JournalofMathematicalAnalysisand
Applications
,
345
,40-52.https://doi.org/10.1016/j.jmaa.2008.03.063
[12]Li,S.andStevi´c,S.(2009)ProductsofIntegral-TypeOperatorsandCompositionOperators
betweenBloch-TypeSpaces.
Journalof Mathematical Analysis and Applications
,
349
, 596-610.
https://doi.org/10.1016/j.jmaa.2008.09.014
[13]Stevi´c, S.(2008) GeneralizedCompositionOperatorsfrom LogarithmicBlochSpacestoMixed-
NormSpaces.
UtilitasMathematica
,
77
,167-172.
DOI:10.12677/pm.2021.11122292066
n
Ø
ê
Æ
Û
§
u
[14]Yang,W.(2011)CompositionOperatorsfrom
F
(
p,q,s
)Spacestothe
n
thWeighted-Type
SpacesontheUnitDisc.
AppliedMathematicsandComputation
,
218
,1443-1448.
https://doi.org/10.1016/j.amc.2011.06.027
[15]Zhu,X.(2009)GeneralizedCompositionOperatorsfromGeneralizedWeightedBergmanS-
pacestoBlochTypeSpaces.
JournaloftheKoreanMathematicalSociety
,
46
,1219-1232.
https://doi.org/10.4134/JKMS.2009.46.6.1219
[16]Hibschweiler,R.A.andPortnoy,N.(2005)CompositionFollowedbyDifferentiationbetween
BergmanandHardySpaces.
RockyMountainJournalofMathematics
,
35
,843-855.
https://doi.org/10.1216/rmjm/1181069709
[17]Ohno,S.(2006)ProductsofCompositionandDifferentiation betweenHardySpaces.
Bulletin
oftheAustralianMathematicalSociety
,
73
,235-243.
https://doi.org/10.1017/S0004972700038818
[18]Zhu,X.(2007)ProductsofDifferentiation,CompositionandMultiplicationfromBergman
TypeSpacestoBersTypeSpaces.
IntegralTransformsandSpecialFunctions
,
18
,223-231.
https://doi.org/10.1080/10652460701210250
[19]Shapiro,J.H.andSundberg,C.(1990)IsolationamongsttheCompositionOperators.
Pacific
JournalofMathematics
,
145
,117-152.https://doi.org/10.2140/pjm.1990.145.117
[20]MacCluer, B., Ohno, S.and Zhao,R. (2001)Topological Structureof theSpace ofComposition
Operatorson
H
∞
.
IntegralEquationsandOperatorTheory
,
40
,481-494.
https://doi.org/10.1007/BF01198142
[21]Hosokawa, T.andOhno,S. (2007) DifferencesofComposition OperatorsontheBloch Spaces.
TheJournalofOperatorTheory
,
57
,229-242.
[22]Bonet,J.,Lindstr¨om,M.andWolf,E.(2008)DifferencesofCompositionOperatorsbetween
WeightedBanachSpacesofHolomorphicFunctions.
JournaloftheAustralianMathematical
Society
,
84
,9-20.https://doi.org/10.1017/S144678870800013X
[23]Lindstr¨om,M. andWolf, E.(2008) Essential Normof the Differenceof Weighted Composition
Operators.
Monatsheftef¨urMathematik
,
153
,133-143.
https://doi.org/10.1007/s00605-007-0493-1
[24]Wolf,E.(2008)CompactDifferencesofCompositionOperators.
BulletinoftheAustralian
MathematicalSociety
,
77
,161-165.https://doi.org/10.1017/S0004972708000166
[25]Zhou, Z.H.and Liang,Y.X.(2012) DifferencesofWeightedComposition OperatorsfromHardy
SpacetoWeighted-Type Spaceson theUnit Ball.
Czechoslovak Mathematical Journal
,
62
, 695-
708.https://doi.org/10.1007/s10587-012-0040-7
[26]Zhu,K.(1993)BlochTypeSpacesofAnalyticFunctions.
RockyMountainJournalofMathe-
matics
,
23
,1143-1177.https://doi.org/10.1216/rmjm/1181072549
DOI:10.12677/pm.2021.11122292067
n
Ø
ê
Æ
Û
§
u
[27]Wang,C.andZhou,Z.H.(2019)DifferencesofWeightedDifferentiationCompositionOpera-
torsfrom
α
-BlochSpaceto
H
∞
Space.
Filomat
,
33
,761-772.
https://doi.org/10.2298/FIL1903761W
DOI:10.12677/pm.2021.11122292068
n
Ø
ê
Æ