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PureMathematics
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,2020,10(12),1167-1175
PublishedOnlineDecember2020inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/pm.2020.1012139
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AnaloguesofMillouxInequalityof
MeromorphicFunctionsConcerning
ProductsofShiftsandDifferences
KexinLi,ShiweiYang,DeguiYang
∗
InstituteofAppliedMathematics,SouthChinaAgriculturalUniversity,GuangzhouGuangdong
Email:ql20200220@163.com,yangseawell@foxmail.com,
∗
dyang@scau.edu.cn
Received:Nov.16
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,2020;accepted:Dec.17
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,2020;published:Dec.24
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,2020
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1167-1175.DOI:10.12677/pm.2020.1012139
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Abstract
Let
f
beatranscendentalmeromorphicfunctionof finiteorder.Inthispaper, we stud-
ied the analoguesof Milloux Inequality of meromorphic functions concerning products
ofshiftsanddifferencesandobtainedtheanaloguesofMillouxInequalityofmero-
morphicfunctionsconcerningproductsofshiftsanddifferences.Fortheanaloguesof
Milouxinequalityconcerningproductsofshifts,weimprovedtheresultofWuand
Xu.
Keywords
MeromorphicFunctions,MillouxInequality,Differences,Shifts
Copyright
c
2020byauthor(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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DOI:10.12677/pm.2020.10121391170
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n
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f
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)+
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.
DOI:10.12677/pm.2020.10121391171
n
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u
´
k
dT
(
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≤
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N
(
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1
f
d
)+
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(
r,
1
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−
b
)
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Φ
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)+
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(
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≤
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½
n
m
(
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c
f
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b
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+1
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f
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+1
c
f
∆
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f
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∆
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)+
S
(
r,f
)
≤
T
(
r,
∆
n
+1
c
f
)
−
N
(
r,
1
∆
n
+1
c
f
)+
S
(
r,f
)
≤
m
(
r,
∆
n
+1
c
f
)+
N
(
r,
∆
n
+1
c
f
)
−
N
(
r,
1
∆
n
+1
c
f
)+
S
(
r,f
)
≤
m
(
r,
∆
n
c
f
)+
N
(
r,
∆
n
c
f
(
z
+
c
))+
N
(
r,
∆
n
c
f
(
z
))
−
N
(
r,
1
∆
n
+1
c
f
)+
m
(
r,
∆
n
+1
c
f
∆
n
c
f
)+
S
(
r,f
)
=
T
(
r,
∆
n
c
f
)+(
n
+1)
N
(
r,f
)
−
N
(
r,
1
∆
n
+1
c
f
)+
S
(
r,f
)
.
u
´
k
m
(
r,
1
f
)+
m
(
r,
1
∆
n
c
f
−
b
)
≤
T
(
r,
∆
n
c
f
)+(
n
+1)
N
(
r,f
)
−
N
(
r,
1
∆
n
+1
c
f
)+
S
(
r,f
)
.
Ï
d
§
d
Nevanlinna
1
˜
Ä
½
n
T
(
r,f
)
−
N
(
r,
1
f
)+
T
(
r,
∆
n
c
f
)
−
N
(
r,
1
∆
n
c
f
−
b
)
≤
T
(
r,
∆
n
c
f
)+(
n
+1)
N
(
r,f
)
−
N
(
r,
1
∆
n
+1
c
f
)+
S
(
r,f
)
.
u
´
k
T
(
r,f
)
≤
(
n
+1)
N
(
r,f
)+
N
(
r,
1
f
)+
N
(
r,
1
∆
n
c
f
−
b
)
−
N
(
r,
1
∆
n
+1
c
f
)+
S
(
r,f
)
.
½
n
2
y
"
í
Ø
3
y
²
µ
b
X
í
Ø
3
(
Ø
Ø
é
§
K
•
3
k
¡
E
ê
a
Ú
b
(
6
= 0)
¦
f
(
z
)
−
a
Ú
∆
n
c
f
−
b
þ
•
DOI:10.12677/pm.2020.10121391172
n
Ø
ê
Æ
o
Œ
!
kk
•
‡
"
:
"
u
´
d
½
n
2
T
(
r,f
)
≤
(
n
+1)
N
(
r,f
)+
N
(
r,
1
f
−
a
)+
N
(
r,
1
∆
n
c
f
−
b
)
−
N
(
r,
1
∆
n
+1
c
f
)+
S
(
r,f
)
≤
O
(log
r
)+
S
(
r,f
)
.
u
´
k
T
(
r,f
)
≤
O
(log
r
)
,r
6∈
E,
Ù
¥
E
´
˜
‡
é
ê
ÿ
Ý
k
¡
r
Š
8
"
d
Ú
n
5
lim
r
→∞
T
(
r,f
)
log
r
=
∞
§
g
ñ
"
½
n
3
y
²
µ
d
Ú
n
1
§
Ú
n
2
Ú
Nevanlinna
1
˜
Ä
½
n
m
(
r,
1
f
2
)+
m
(
r,
1
f
∆
c
f
−
b
)
≤
m
(
r,
1
f
∆
c
f
)+
m
(
r,
1
f
∆
c
f
−
b
)+
m
(
r,
∆
c
f
f
)
≤
m
(
r,
1
f
∆
c
f
+
1
f
∆
c
f
−
b
)+
S
(
r,f
)
≤
m
(
r,
1
(
f
∆
c
f
)
0
)+
m
(
r,
(
f
∆
c
f
)
0
f
∆
c
f
+
(
f
∆
c
f
)
0
f
∆
c
f
−
b
)+
S
(
r,f
)
=
m
(
r,
1
(
f
∆
c
f
)
0
)+
S
(
r,f
)
≤
T
(
r,
(
f
∆
c
f
)
0
)
−
N
(
r,
1
(
f
∆
c
f
)
0
)+
S
(
r,f
)
≤
m
(
r,
(
f
∆
c
f
)
0
)+
N
(
r,
(
f
∆
c
f
)
0
)
−
N
(
r,
1
f
∆
c
f
)
0
)+
S
(
r,f
)
≤
m
(
r,f
∆
c
f
)+
N
(
r,f
∆
c
f
)+
N
(
r,f
∆
c
f
)
−
N
(
r,
1
(
f
∆
c
f
)
0
)+
m
(
r,
(
f
∆
c
f
)
0
f
∆
c
f
)+
S
(
r,f
)
=
T
(
r,f
∆
c
f
)+2
N
(
r,f
)
−
N
(
r,
1
(
f
∆
c
f
)
0
)+
S
(
r,f
)
.
u
´
k
m
(
r,
1
f
2
)+
m
(
r,
1
f
∆
c
f
−
b
)
≤
T
(
r,f
∆
c
f
)+2
N
(
r,f
)
−
N
(
r,
1
(
f
∆
c
f
)
0
)+
S
(
r,f
)
.
Ï
d
§
d
Nevanlinna
1
˜
Ä
½
n
T
(
r,f
2
)
−
N
(
r,
1
f
2
)+
T
(
r,f
∆
c
f
)
−
N
(
r,
1
f
∆
c
f
−
b
)
≤
T
(
r,
∆
c
f
)
+2
N
(
r,f
)
−
N
(
r,
1
f
(∆
c
f
)
0
)+
S
(
r,f
)
.
DOI:10.12677/pm.2020.10121391173
n
Ø
ê
Æ
o
Œ
!
k
2
T
(
r,f
)
≤
2
N
(
r,f
)+2
N
(
r,
1
f
)+
N
(
r,
1
f
∆
c
f
−
b
)
−
N
(
r,
1
(
f
∆
c
f
)
0
)+
S
(
r,f
)
.
u
´
=
2
T
(
r,f
)
≤
2
N
(
r,f
)+2
N
(
r,
1
f
)+
N
(
r,
1
f
∆
c
f
−
b
)+
S
(
r,f
)
.
½
n
3
y
"
—
Š
ö
é
"
v
<
J
Ñ
B
ï
Æ
L
«
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œ
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©
z
[1]Hayman,W.K.(1964)MeromorphicFunctions.ClarendonPress,Oxford.
[2]Laine,I.(1993)NevanlinnaTheoryandComplexDifferentialEquations.WalterdeGruyter,
Berlin.
[3]Yang,L.(1993)ValueDistributionTheory.Springer-Verlag,Berlin.
[4]Yang,C.C.andYi,H.X.(2003)UniquenessTheoryofMeromoprhicFunctions.KluwerAca-
demicPublishers,Dordrecht,TheNetherlands.
[5]Yi,H.X.(1989)ValueDistributionof
f
0
f
.
ChineseScienceBulletin
,
34
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https://doi.org/10.1360/csb1989-34-10-727
[6]Wu,Z.J.andXu,H.Y.(2020)MillouxInequalityofNonlinearDifferenceMonomialsandIts
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JournalofMathematicalInequalities
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14
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https://doi.org/10.7153/jmi-2020-14-52
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IntegrabilityandtheDiscretePainleveEquations.
JournalofPhysicsA:Mathematicaland
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Ø
ê
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o
Œ
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[8]Halburd,R.G.andKorhonen,R.J.(2006)NevanlinnaTheoryfortheDifferenceOperator.
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,463-478.
[9]Chiang, Y.M.andFeng, S.J.(2009)On theGrowthofLogarithmicDifference, DifferenceEqua-
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TransactionsoftheAmerican
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361
,3767-3791.https://doi.org/10.1090/S0002-9947-09-04663-7
[10]Halburd,R.G. andKorhonen,R.J.(2006)DifferenceAnalogueofthe LemmaontheLogarith-
micDerivativewithApplicationstoDifferenceEquations.
JournalofMathematicalAnalysis
andApplications
,
314
,477-487.https://doi.org/10.1016/j.jmaa.2005.04.010
[11]Chiang, Y.M.andFeng, S.J.(2008)OntheNevanlinnaCharacteristicof
f
(
z
+
η
)andDifference
EquationsintheComplexPlane.
TheRamanujanJournal
,
16
,105-129.
https://doi.org/10.1007/s11139-007-9101-1
[12]Zheng,R.R.and Chen,Z.X. (2012)ValueDistributionofDifference Polynomials ofMeromor-
phicFunctions.
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,
42
,1115-1130.(InChinese)
https://doi.org/10.1360/012011-760
DOI:10.12677/pm.2020.10121391175
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Ø
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