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PureMathematicsnØêÆ,2021,11(1),41-46
PublishedOnlineJanuary2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.111007
NÚ¼êpSchwarzianê
444ˆˆˆÕÕÕ§§§ÖÖÖÀÀÀ
®Ê˜ÊUŒÆêƉÆÆ§®
ÂvFϵ2020c1212F¶¹^Fϵ2021c111F¶uÙFϵ2021c118F
Á‡
©½ÂNÚ¼êpSchwarzianê/ª§¿y²ÙEäkM¨obiusØC5"Ùg§
©‰ÑNÚ¼êpSchwarzianê˜«d•x"
'…c
NÚ¼ê§pSchwarzianê
HigherOrderofSchwarzianDerivativeof
theHarmonicFunctions
YutongLiu,YiQi
SchoolofMathematicalScience,BeihangUniversity,Beijing
Received:Dec.12
th
,2020;accepted:Jan.11
th
,2021;published:Jan.18
th
,2021
Abstract
Inthispaper,wedefinethehigherorderofSchwarzianderivativeoftheharmonic
©ÙÚ^:4ˆÕ,ÖÀ.NÚ¼êpSchwarzianê[J].nØêÆ,2021,11(1):41-46.
DOI:10.12677/pm.2021.111007
4ˆÕ§ÖÀ
functions.Wealsoprovethatit isstillM¨obiusinvariant.Finally, wegiveanequivalent
characterizationofthehigherorderofSchwarzianderivativeoftheharmonicfunctions.
Keywords
HarmonicFunction,HigherOrderofSchwarzianDerivative
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
eNÚ¼êf3ü SÜÛÜü“§KfŒL«•f=h+g§Ù¥hÚgþ•ü S)Û
¼ê"ω=g
0
/h
0
•NÚ¼êf1EA§J
f
=|h
0
|
2
−|g
0
|
2
•fäŒ'"Lewy[1]y²N
Ú¼êf3ü SÛÜü“…=J
f
6= 0"ü SÛÜü“NÚ¼êf§XJJ
f
>0§·‚
¡f´•¶ XJJ
f
<0§K¡f´‡•"eNÚ¼êf=h+g•ü S)Û¼ê§w,
Œf= h§Ù¥g= 0"¿…§f1EAω= 0"
˜‡3üëÏ«•SÛÜü“)Û¼êfpre-SchwarzianêP(f)ÚSchwarzianêS(f)©
O½Â•
P(f) =
f
00
f
0
,S(f) = P
0
(f)−
1
2
P
2
(f) = (
f
00
f
0
)
0
−
1
2
(
f
00
f
0
)
2
.(1)
3ƒ?Ø¥§·‚=•Äü S)ۼꧧpre-SchwarzianêÚSchwarzian
ê÷v±e5Ÿµ
(1)ψ◦fÚfÛÜü“)Ûž§P(ψ◦f)=

P(ψ)◦f

·f
0
+P(f)§S(ψ◦f)=

S(ψ)◦f

·
(f
0
)
2
+S(f)"
(2)ψ= A(w) = aw+b,a6= 0§=A(w)••C†ž§P(A◦f) = P(f)"
(3)ψ= T(w) =
aw+b
cw+d
,ad−bc6= 0§=T(w)•M¨obiusC†ž§S(T◦f) = S(f)"
Hern´andez-Martin[2][3]‰ÑƒANÚ¼êfpre-SchwarzianêP
H
(f)ÚSchwarzian
êS
H
(f)½Â"eNÚ¼êf=h+g3ü SÛÜü“§ω=g
0
/h
0
´§1EA§
J
f
= |h
0
|
2
−|g
0
|
2
§K
DOI:10.12677/pm.2021.11100742nØêÆ
4ˆÕ§ÖÀ
P
H
(f) =
∂
∂z
logJ
f
=
h
00
h
0
−
ωω
0
1−|ω|
2
(2)
-S(h) = (
h
00
h
0
)
0
−
1
2
(
h
00
h
0
)
2
•)Û¼êhSchwarzian꧅
S
H
(f) =
∂
∂z
P
H
(f)−
1
2
P
2
H
(f) = S(h)+
ω
1−|ω|
2
(
h
00
ω
0
h
0
−ω
00
)−
3
2
(
ωω
0
1−|ω|
2
)
2
(3)
N´yNÚ¼êpre-SchwarzianêÚSchwarzianê÷v±e5Ÿµ
(1)f)Ûž§P
H
(f)ÚS
H
(f)÷v(1)ª"
(2)P
H
(f) = P
H
(f)ÚS
H
(f) = S
H
(f)§Ù¥f•fݼê"
(3)ψ◦fÛÜü“NÚ§fÛÜü“)Ûž§
P
H
(ψ◦f) =

P
H
(ψ)◦f

·f
0
+P(f),S
H
(ψ◦f) =

S
H
(ψ)◦f

·(f
0
)
2
+S(f).
(4)A(w) = aw+bw+c,a
2
+b
2
6= 0§=A(w)••NÚC†§P
H
(A◦f) = P
H
(f)"
(5)T(w) =
aw+b
cw+d
,ad−bc6= 0§=T(w)•M¨obiusC†§S
H
(T◦f) = S
H
(f)"
3)Û¼êïÄ¥§Schwarzianêp/ª´˜‡-‡ïÄ••§NõÆölØÓ
݉ÑNõ-‡(J[4][5][6][7]"
ef3ü S)Û§ÙpSchwarzianêσ
n
(f) (n≥3)§½Â•
σ
3
(f) = S(f),σ
n+1
(f) = σ
0
n
(f)−(n−1)P(f)σ
n
(f)(4)
´•σ
n
(f)äk,«M¨obiusØC5§=eR´ü þM¨obiusC†§K
σ
n
(f◦R) = (σ
n
(f)◦R)·(R
0
)
n−1
¿…ÏLOŽ§·‚Œ•§ew= f(z)§

∂
n−3
∂w
n−3
σ
3
(f
−1
)

◦f=
σ
n
(f)
(f
0
)
n−1
(5)
Ïd§·‚aq/½Âü SÛÜü“NÚ¼êfpSchwarzianêσ
n
(f)/ª§¿
…·‚òy²σ
n
(f)E,äkM¨obiusØC5Úþã5Ÿ"
2.̇SN
eNÚ¼êf3ü SÛÜü“"fŒL«•f= h+g§Ù¥hÚgþ´ü S)Û¼§
DOI:10.12677/pm.2021.11100743nØêÆ
4ˆÕ§ÖÀ
K§pSchwarzianêσ
n
(f)½ÂXe(n≥3)µ
σ
3
(f) = S
H
(f),σ
n+1
(f) =
∂
∂z
σ
n
(f)−(n−1)P
H
(f)σ
n
(f).(6)
N´yNÚ¼êσ
n
(f)÷v±e5Ÿµ
(1)f)Ûž§σ
n
(f)÷v(4)ª"
(2)σ
n
(f) = σ
n
(f)"
(3)A(w) = aw+bw+c,a
2
+b
2
6= 0§=A(w)••NÚC†§σ
n
(A◦f) = σ
n
(f)"
(4)T(w)=
aw+b
cw+d
,ad−bc6=0§=T(w)•M¨obiusC†§ f3üëÏ«•SÛÜü“NÚž§
σ
n
(T◦f) = σ
n
(f)"
5Ÿ2.1.eR´ü þM¨obiusC†§f´ü SÛÜü“NÚ¼ê§K
σ
n
(f◦R) = (σ
n
(f)◦R)·(R
0
)
n−1
y²:
n= 3ž§w,¤á"
n>3ž§¦^êÆ8B{y²"bn= kž§(ؤá"
n= k+1ž§
σ
k+1
(f◦R) =
∂
∂z
σ
k
(f◦R)−(k−1)P
H
(f◦R)σ
k
(f◦R)
=
∂
∂z

(σ
k
(f)◦R)·(R
0
)
k−2

−(k−1)

(P
H
(f)◦R)·R
0
+
R
00
R
0

·

(σ
k
(f)◦R)·(R
0
)
k−1

=

∂
∂w
σ
k
(f)◦R

·(R
0
)
k
+(k−1)(σ
k
(f)◦R)·(R
0
)
k−2
R
00
−(k−1)(σ
k
(f)◦R)(P
H
(f)◦R)(R
0
)
k
−(k−1)(σ
k
(f)◦R)·(R
0
)
k−1
R
00
= (σ
k+1
(f)◦R)·(R
0
)
k
.
·‚Eò¦^êÆ8B{y²±e5Ÿ"
5Ÿ2.2.ef=h+g´ü SÛÜü“NÚ¼ê"b¼êf•3‡¼êf
−1
§ …f
−1
•NÚ
¼ê"K

∂
n−3
∂w
n−3
σ
3
(f
−1
)

◦f= (
h
0
J
f
)
n−1
·σ
n
(f)(7)
y²µ
DOI:10.12677/pm.2021.11100744nØêÆ
4ˆÕ§ÖÀ
Äk§w= f(z)§·‚•§
f
−1
w
=
h
0
J
f
,f
−1
w
= −
g
0
J
f
,J
f
−1
= |f
−1
w
|
2
−|f
−1
w
|
2
=
1
J
f
.
¿…½Â
∂z
∂w
= f
−1
w
"
n= 3ž§
σ
3
(f) =
∂
2
∂z
2
logJ
f
−
1
2
(
∂
∂z
logJ
f
)
2
=
∂
2
∂z
2
J
f
J
f
−
3
2
(
∂
∂z
J
f
)
2
J
2
f
Ù¥§
∂
∂z
J
f
= h
00
h
0
−g
00
g
0
,
∂
2
∂z
2
J
f
= h
000
h
0
−g
000
g
0
.
aq/
σ
3
(f
−1
) =
∂
2
∂w
2
J
f
−1
J
f
−1
−
3
2
(
∂
∂w
J
f
−1
)
2
J
2
f
−1
K
σ
3
(f
−1
)◦f= J
f
·(
∂
2
∂z
2
1
J
f
)·(
∂z
∂w
)
2
+J
f
·(
∂
∂z
1
J
f
)·(
∂z
∂w
)·
∂
∂z
(
∂z
∂w
)−
3
2
J
2
f
(
∂
∂z
1
J
f
)
2
·(
∂z
∂w
)
2
=
(h
0
)
2
J
f
·
(
∂
2
∂z
2
J
f
)J
2
f
−2(
∂
∂z
J
f
)
2
J
f
J
4
f
−(
h
0
∂
∂z
J
f
J
2
f
)
2
+
3
2
(h
0
)
2
(
∂
∂z
J
f
J
2
f
)
2
=
(h
0
)
2
J
2
f
·(
∂
2
∂z
2
J
f
J
f
−
3
2
(
∂
∂z
J
f
)
2
J
2
f
).
¤±§n= 3ž§(7)ª¤á"
n>3ž§¦^êÆ8B{y²"bn= kž§(ؤá"
n= k+1ž§

∂
k−2
∂w
k−2
σ
3
(f
−1
)

◦f=
∂
∂z

(
h
0
J
f
)
k−1
·σ
k
(f)

∂z
∂w
= −(k−1)σ
k
(f)(
h
0
J
f
)
k−1
h
0
(
∂
∂z
J
f
)
J
2
f
+(
h
0
J
f
)
k
(
∂
∂z
σ
k
(f))
= −(k−1)σ
k
(f)(
h
0
J
f
)
k
∂
∂z
J
f
J
f
+(
h
0
J
f
)
k

σ
k+1
(f)+(k−1)P
H
(f)σ
k
(f)

= (
h
0
J
f
)
k
·σ
k+1
(f).
Ù¥P
H
(f) =
∂
∂z
J
f
J
f
"
·‚I‡5¿´3[8]¥§ÜÚ4‰Ñü“NÚNì‡¼êENÚd•x"´¢
Sþ§eü“¼êf•3f
z
Úf
z
§·‚þŒdäŒ'¼êJ
f
= |f
z
|
2
−|f
z
|
2
§½ÂÙpre-Schwarzian
DOI:10.12677/pm.2021.11100745nØêÆ
4ˆÕ§ÖÀ
êÚSchwarzian꧿…§pSchwarzianê/ª"Ùg§Œ±y²p/ª÷v5
Ÿ2.1Ú5Ÿ2.2"
Ä7‘8
I[g,‰ÆÄ7]Ï‘8No.11871085"
ë•©z
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BulletinoftheAmericanMathematicalSociety,42,689-692.
https://doi.org/10.1090/S0002-9904-1936-06397-4
[2]Hern´andez,R.andMart´ın,M.J.(2015)Pre-SchwarzianandSchwarzianDerivativesofHar-
monicMappings.JournalofGeometricAnalysis,25,64-91.
https://doi.org/10.1007/s12220-013-9413-x
[3]Hern´andez,R.andMart´ın,M.J.(2017)OntheharmonicM¨obiusTransformations.eprint
arXiv:1710.05952.
[4]Kim,S.-A. andSugawa, T.(2011)Invariant Schwarzian Derivatives ofHigherOrder.Complex
AnalysisandOperatorTheory,5,659-670.https://doi.org/10.1007/s11785-010-0081-6
[5]Donaire,J.J.(2019)AShimorin-TypeEstimateforHigher-OrderSchwarzianDerivatives.
ComputationalMethodsandFunctionTheory,19,315-322.
https://doi.org/10.1007/s40315-019-00265-0
[6]Tamanoi, H.(1996)HigherSchwarzianOperatorsandCombinatoricsof theSchwarzianDeriva-
tive.MathematischeAnnalen,305,127-151.https://doi.org/10.1007/BF01444214
[7]Cho,N.E.,Kumar,V.andRavichandran,V.(2018)SharpBoundsontheHigherOrder
SchwarzianDerivativesforJanowskiClasses.Symmetry,10,348.
https://doi.org/10.3390/sym10080348
[8]Üîõ,4r.ü“NÚNì‡¼ê[J].êÆ?Ð,1996,25(3):270-276.
DOI:10.12677/pm.2021.11100746nØêÆ

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