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PureMathematicsnØêÆ,2021,11(12),2057-2068
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1112229
α-Bloch˜mβ-Bloch˜m2ÂÈ©Žf
©
ÛÛÛ§§§uuu
2À7KÆ7KêƆÚOÆ§2À2²
ÂvFϵ2021c118F¶¹^Fϵ2021c129F¶uÙFϵ2021c1217F
Á‡
ϕ,g´E²¡C¥ü Dþ)ÛN§…ϕ(D) ⊂D,n∈N"½Â2ÂÈ©Žf•
I
(n)
g,ϕ
f(z) =
Z
z
0
f
(n)
(ϕ(ζ))g(ζ)dζ.
©‘3&Äα-Bloch˜mβ-Bloch˜mþ2ÂÈ©Žfk.5Ú;5¯K"
'…c
©§2ÂÈ©Žf§Bloch˜m
DifferencesofGeneralizedIntegration
Operatorsfromα-BlochSpacesto
β-BlochSpaces
ZhonghuaHe
SchoolofFinancialMathematicsandStatistics,GuangdongUniversityofFinance,Guangzhou
Guangdong
Received:Nov.8
th
,2021;accepted:Dec.9
th
,2021;published:Dec.17
th
,2021
©ÙÚ^:Û§u.α-Bloch˜mβ-Bloch˜m2ÂÈ©Žf©[J].nØêÆ,2021,11(12):2057-2068.
DOI:10.12677/pm.2021.1112229
Û§u
Abstract
Ageneralizedintegrationoperatorisdefinedby
I
(n)
g,ϕ
f(z) =
Z
z
0
f
(n)
(ϕ(ζ))g(ζ)dζ
inducedbyholomorphicmapsgandϕoftheunitdiskD,whereϕ(D)⊂Dandnisa
positiveinteger.In thispaper, weinvestigatetheboundedness andthecompactnessof
thedifferencesoftwo generalizedintegrationoperators fromα-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/
1.Úó
PD•E²¡C¥ü m,H(D)•Dþ)Û¼êN,H
∞
(D)•Dþk.)Û¼ê
N.é0 <α<∞,Bloch.˜m(½α-Bloch˜m)B
α
´•¤k÷v
kfk
α
= sup
z∈D
(1−|z|
2
)
α
|f
0
(z)|<∞
)Û¼êN.´•,3‰êkfk
B
α
= |f(0)|+kfk
α
¿Âe,B
α
¤˜‡Banach˜m.
3©¥,PS(D)•Dg)ÛNN.éf∈H(D),½Â±ϕ∈S(D)•ÎÒEÜŽ
f•C
ϕ
f=f◦ϕ.EÜŽfnØ´Cc5ïÄ9:ƒ˜,AO´ϕ5Ÿ†C
ϕ
'X¯K.éu
²;)Û¼ê˜mþEÜŽfnØ,•„©z[1]Ú[2].
g: D→C´ü mþ)ÛN,éf∈H(D),z∈D,¡Žf
I
g
f(z) =
Z
z
0
f(ζ)g
0
(ζ)dζ(z∈D)
DOI:10.12677/pm.2021.11122292058nØêÆ
Û§u
•Riemann -StieltjesŽf(½2ÂCes`aro Žf).Ch.Pommerenke3©z[3]¥ÄgïÄRiemann
-Stieltjes Žf,Ùy²I
g
3H
2
þk.…=g∈BMOA.©z[4]Ú[5]rT(Øí2Ù¦
H
p
(1 ≤p<∞)˜m,ÙŠâÎÒg„•xI
g
3H
p
;5ÚI
g
3H
2
þSchattena.
3©p,·‚•ÄÈ©Žf
I
(n)
g,ϕ
f(z) =
Z
z
0
f
(n)
(ϕ(ζ))g(ζ)dζ,z∈D.
¡TŽf•2ÂÈ©Žf,Ù3[6]¥ÄgÚ^¿3©z[7]Ú[8]¥?˜ÚïÄ.Óž,Ž
fI
(n)
g,ϕ
Œ±w¤´dg¤pRimann-StieltjesŽfI
g
í2.¯¢þ,ŽfI
(n)
g,ϕ
Œ±NõÙ•
Žf.'X,n=1ž,I
(n)
g,ϕ
Ò´S.Stevi´c,S.Li,X.ZhuÚW.Yang3©z[9–15]¥¤ïÄÈ
©Žf.n= 1,g(z) = ϕ
0
(z)ž,K•EÜŽfC
ϕ
.ePD•‡©Žf,n= m+1,g(z) = ϕ
0
(z),K
•ŽfC
ϕ
D
m
f(z) = f
(m)
(ϕ(z))−f
(m)
(ϕ(0))(„©z[16–18]).
•ïÄH
2
˜mþEÜŽf8C(H
2
)ÿÀ((Žf‰êÿÀ¿Âe),©z[19]¥Äg&
Äü‡EÜŽf©5Ÿ.‘,ØïÄö••x(\)EÜŽf©.MacCluer,
OhnoÚZhao3[20]¥y²C
ϕ
−C
ψ
:H
∞
→H
∞
;5†C
ϕ
−C
ψ
:B→H
∞
;5
d.Óž,H
∞
þEÜŽfC
ϕ
ÚC
ψ
3Ó˜‡ëÏ©|þ…=C
ϕ
−C
ψ
:B→H
∞
k..
HosokawaÚOhno3[21]¥Ø=éBH
∞
þü‡\EÜŽf©k.5Ú;5‰Ñ#
(J,„‰Ñ¦‚5‰ê.8c•Ž,'uü‡(\)EÜŽf©©zkéõ,
X[22–25],Ù¦•õ©z3dؘ˜Ñ.
©¥,·‚ò&Äα-Bloch˜mβ-Bloch˜mþü‡2ÂÈ©Žfk.5Ú;5 ¯K.
Óž,3d`²:©¥~êC3ØÓ/•ŠØ˜½ƒÓ.
2.˜Ún
éa∈D,σ
a
•Dþr0N¤agÓN.=,σ
a
(z) =
a−z
1−¯az
,z∈D.Óž,Dþ–V-ål
P•
ρ(z,a) = |σ
a
(z)|=|
a−z
1−¯az
|,z,a∈D.
e5,·‚‰Ñ©¥Ì‡(Øy²žI‡^A‡(Ø.
Ún1[26]éα>0,f∈B
α
,k
|f
(n)
(z)|≤C
kfk
B
α
(1−|z|
2
)
α+n−1
,
Ù¥C†fÃ'.
†([1],Pro.3.11)y²aq,Œ±eÚn2.
Ún2ϕ
1
,ϕ
2
∈S(D),g
1
,g
2
∈H(D).KI
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
:B
α
→B
β
´;Žf…=
I
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
: B
α
→B
β
´k.Žf,…éB
α
¥?¿k.S{f
k
},Ù3D?¿;f8þ˜
—Âñu0ž,kk(I
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
)f
k
k
B
β
→0.
DOI:10.12677/pm.2021.11122292059nØêÆ
Û§u
Ún3[27]0<α<∞.Ké?¿z,w∈D,•3~êC>0,¦é¤kf∈B
α
÷
vkfk
B
α
≤1ž,k
|(1−|z|
2
)
α+n−1
f
(n)
(z)−(1−|w|
2
)
α+n−1
f
(n)
(w)|≤Cρ(z,w).
Ún4[27]éz,w∈D…a6= 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

.
Kf
a
,k
a
∈B
α
…
f
(n)
a
(z) =
1−|a|
2
(1−¯az)
α+n
,
k
(n)
a
(z) =
(a−z)(1−|a|
2
)
(1−¯az)
α+n+1
.
3.̇(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
,g
1
(z)−D
ϕ
2
,g
2
(z)|.
½n1ϕ
1
,ϕ
2
∈S(D),g
1
,g
2
∈H(D)…n∈N.Keã^‡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) <∞.
y²(i) ⇒(ii).bI
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
: B
α
→B
β
k..éa∈D…a6= 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.2021.11122292060nØêÆ
Û§u
KdÚn4Œ•,f
a
,k
a
∈B
α
.u´é½w∈D…÷vϕ
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)ªü>Óž¦±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}.
ÓnŒ
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.2021.11122292061nØêÆ
Û§u
ϕ
1
(w) = ϕ
2
(w) = 0ž,-f
0
(z) =
z
n
n!
,KdI
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
: B
α
→B
β
k.5Œ
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= 0ž,-
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)
Ïd,(Ü(5)Ú(10)Œ±
sup
w∈D
2
\D
1
|D
ϕ
1
,g
1
(w)−D
ϕ
2
,g
2
(w)|<∞.(11)
ÓnŒ,ϕ
1
(w) = 0,ϕ(w)
2
6= 0ž,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.2021.11122292062nØêÆ
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,KdÚn1Œ,éf∈B
α
…kfk
B
α
≤1,k
k(I
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
)fk
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)|
+Csup
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?ØI
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
: F(p,q,s) →B
µ
;5ƒc,·‚Ú\ePÒ:
Γ(ϕ) = {{z
k
}⊂D: |ϕ(z
k
)|→1},
D(g,ϕ) := {{z
k
}⊂D: |ϕ(z
k
)|→1,|D
ϕ,g
(z
k
)|90}.
½n2ϕ
1
,ϕ
2
∈S(D),g
1
,g
2
∈H(D).eI
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
: B
α
→B
β
k.,I
(n)
g
1
,ϕ
1
ÚI
(n)
g
2
,ϕ
2
Ñ
Ø´;Žf,KI
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
: B
α
→B
β
´;Žf…=
(i)D(g
1
,ϕ
1
) = D(g
2
,ϕ
2
) 6= 0,D(g
1
,ϕ
1
) ⊂Γ(ϕ
2
),
(ii)éz
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.2021.11122292063nØêÆ
Û§u
y²¿©5.{f
k
}´B
α
¥S,Ù3Dþ?¿;f8þј—Âñu0,…kf
k
k
B
α
≤1.
ek(I
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
)f
k
k
B
β
90,K•3ε>0,¦é?¿kkk(I
(n)
g
1
,ϕ
1
−I
(n)
g
2
,ϕ
2
)f
k
k
B
β
>ε.u´éz
‡k,•3z
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
)|→1½ö|ϕ
2
(z
k
)|→1.e|ϕ
1
(z
k
)|→1,w∈D•{ϕ
2
(z
k
)}4•.K•3fÂñ
uw,Ø”Òϕ
2
(z
k
) →w.XJ|w|<1,@oz
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.,˜•¡,duI
(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|<1íÑf
(n)
k
(ϕ
2
(z
k
)) →0,†(14)gñ,|w|= 1.Ïd,|ϕ
1
(z
k
)|→1…|ϕ
2
(z
k
)|→1.db
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)gñ.
7‡5.dbŒ•,eI
(n)
g
1
,ϕ
1
š;,K•3S{z
k
}⊂D(g
1
,ϕ
1
)¦|ϕ
1
(z
k
)|→1
ž|D
ϕ
1
,g
1
(z
k
)|90.éw
k
=ϕ
1
(z
k
),†½n1aq/½Âf
w
k
Úk
w
k
,K{f
w
k
}Ú{k
w
k
}´B
α
¥
k.S,…3Dz‡;f8þј—Âñu0.Ïd,dÚn2Œ,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.2021.11122292064nØêÆ
Û§u
d|D
ϕ
1
,g
1
(z
k
)|90Ú(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
),klim
k→∞
|ϕ
1
(z
k
)−ϕ
2
(z
k
)|= 0.Ïd
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)
Ïd,d(19)Ú(20)Œ D(g
1
,ϕ
1
)⊂D(g
2
,ϕ
2
).ÓnŒ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
)|→0ž,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.
Ïd,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.
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(No.11971123).
DOI:10.12677/pm.2021.11122292065nØêÆ
Û§u
ë•©z
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