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PureMathematicsnØêÆ,2020,10(12),1167-1175
PublishedOnlineDecember2020inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/pm.2020.1012139
æX¼êMillouxØª²£¦È
9©[
oooŒŒŒ!!!§§§---•••§§§BBB
∗
uHà’ŒÆA^êÆïĤ§2À2²
Email:ql20200220@163.com,yangseawell@foxmail.com,
∗
dyang@scau.edu.cn
ÂvFϵ2020c1116F¶¹^Fϵ2020c1217F¶uÙFϵ2020c1224F
Á‡
f´˜‡k¡?‡æX¼ê,©ïÄæX¼êMillouxØª²£¦È9© [,¼
9²£¦È9©MillouxØª[.éu¤¼9²£¦ÈMillouxØª[,
U?ÇèÚMöö<CÏ(J"
'…c
æX¼ê§MillouxØª§©§²£
AnaloguesofMillouxInequalityof
MeromorphicFunctionsConcerning
ProductsofShiftsandDifferences
KexinLi,ShiweiYang,DeguiYang
∗
InstituteofAppliedMathematics,SouthChinaAgriculturalUniversity,GuangzhouGuangdong
Email:ql20200220@163.com,yangseawell@foxmail.com,
∗
dyang@scau.edu.cn
Received:Nov.16
th
,2020;accepted:Dec.17
th
,2020;published:Dec.24
th
,2020
∗ÏÕŠö"
©ÙÚ^:oŒ!,-•,B.æX¼êMillouxØª²£¦È9©[[J].nØêÆ,2020,10(12):
1167-1175.DOI:10.12677/pm.2020.1012139
oŒ!
Abstract
Letfbeatranscendentalmeromorphicfunctionof 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/
1.Úó9̇(J
©¥§æX¼ê•´3‡E²¡þæX¼ê"±eò¦^Š©ÙØ¥IOP
ÒT(r,f)§m(r,f)§N(r,f)§S(r,f),···ë„[1–4]§Ù¥S(r,f)L«?˜¼êf÷vS(r,f)=
o{T(r,f)}§r→∞§r6∈E§E´˜‡éêÿÝk¡rŠ8"
f´˜‡E²¡þš~êæX¼ê§©^σ(f(z))L«f(z)?½Â•
σ(f(z)) =lim
r→∞
log
+
T(r,f)
logr
,
f´˜‡E²¡þš~êæX¼ê§c´˜‡š"k¡Eê§f©•∆
c
f(z)=f(z+
c)−f(z),n©•∆
n
c
f(z) = ∆
c
(∆
n−1
c
f(z)),n∈N, n>2"
NevanlinnaïáͶNevanlinnanؼXeNevanlinna1˜Ä½nÚ1Ä½
nµ
½nA (1˜Ä½n)f´E²¡þæX¼ê§a´?¿Eê§Kk
DOI:10.12677/pm.2020.10121391168nØêÆ
oŒ!
T(r,
1
f−a
) = T(r,f)+O(1).
½nB(1Ä½n)f´E²¡þš~êæX¼ê§a
i
(16i6q)´*¿E²¡þ
q(q>3)‡OEê§Kk
(q−2)T(r,f) ≤
q
X
i=1
N(r,
1
f−a
i
)+S(r,f).
1940c§Millouxeã(J§
½nC(MillouxØª)n´ê§b•š"k¡Eê§f´E²¡þæX¼ê÷
vf
(n+1)
6≡0§Kk
T(r,f) ≤N(r,f)+N(r,
1
f
)+N(r,
1
f
(n)
−b
)−N(r,
1
f
(n+1)
)+S(r,f).
1989c§Yi[5]y²eãMilloux.Øªš‚5‡©ü‘ªf
0
f(J"
½nD f´E²¡þš~êæX¼ê§b´˜‡š"k¡Eê§Kk
2T(r,f) ≤N(r,f)+2N(r,
1
f
)+N(r,
1
ff
0
−b
)+S(r,f).
•C§ŒþØ©ïÄNevanlinnanØ²£9©["2020c§WuÚXu[6]yeã'
uMillouxØª²£¦È["
½nE f´k¡?‡æX¼ê§m•˜‡ê§c
1
,c
2
,···,c
m
•Ok¡Eê"
Φ(z) = f
d
1
(z+c
1
)f
d
2
(z+c
2
)···f
d
m
(z+c
m
),
Ù¥d
1
,d
2
,···,d
m
´ê§d= d
1
+d
2
+···+d
m
§b´š"k¡Eê"eΦ
0
(z) 6≡0§Kk
dT(r,f) ≤2dN(r,f)+dN(r,
1
f
)+N(r,
1
Φ(z)−b
)+S(r,f).
©òU?½nEXe§
½n1 f´k¡?‡æX¼ê§m•ê§c
1
,c
2
,···,c
m
•Ok¡Eê"
Φ(z) = f
d
1
(z+c
1
)f
d
2
(z+c
2
)···f
d
m
(z+c
m
),
Ù¥d
1
,d
2
,···,d
m
´ê,d= d
1
+d
2
+···+d
m
§b´š"k¡Eê"eΦ
0
(z) 6≡0,Kk
dT(r,f) ≤dN(r,f)+dN(r,
1
f
)+N(r,
1
Φ(z)−b
)+S(r,f).
d½n1=
íØ1 f´k¡?‡æX¼ê§b,c•š"k¡Eê§Kk
DOI:10.12677/pm.2020.10121391169nØêÆ
oŒ!
T(r,f) ≤N(r,f)+N(r,
1
f
)+N(r,
1
f(z+c)−b
)+S(r,f).
íØ2 f´k¡?‡æX¼ê§b,c•š"k¡Eê§Kk
2T(r,f) ≤2N(r,f)+2N(r,
1
f
)+N(r,
1
f(z)f(z+c)−b
)+S(r,f).
©¥§·‚„¼½nC†½nD©["
½n2n•˜‡ê§ b,c•š"k ¡Eê§f´k¡?‡æX¼ê÷v∆
n+1
c
f6≡0§
Kk
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).
íØ3f´˜‡‡¼ê§n•˜‡ê§c•˜‡š"k¡Eê§Kéu?¿k¡E
êaÚb(6= 0)½öf(z)−akÃê‡":½ö∆
n
c
f−bkÃê‡":"
½n3 b,c´ü‡š"k¡Eê§f´E²¡þ÷v(f∆
c
f)
0
6≡0æX¼ê§Kk
2T(r,f) ≤2N(r,f)+2N(r,
1
f
)+N(r,
1
f∆
c
f−b
)+S(r,f).
2.˜Ún
•y²©(J§I‡XeA‡Ún"
Ún1[7–10]f´k¡?‡æX¼ê§c´˜‡š"k¡Eê§Kk
m(r,
f(z+c)
f(z)
) = S(r,f).
Ún2[11,12]f´k¡?‡æX¼ê§c´˜‡š"k¡Eê§Kk
N(r,f(z+c)) = N(r,f)+S(r,f),
N(r,f(z+c)) = N(r,f)+S(r,f),
T(r,f(z+c)) = T(r,f)+S(r,f).
Ún3[8,11]f´k¡?‡æX¼ê§Ké?¿ên§k
m(r,
∆
n
c
f
f
) = S(r,f).
Ún4[3,4]f´‡æX¼ê§a
0
(6= 0),a
1
,···,a
n
´k¡Eê§Kk
DOI:10.12677/pm.2020.10121391170nØêÆ
oŒ!
T(r,a
0
f
n
+a
1
f
n−1
+···+a
n−1
f+a
n
) = nT(r,f)+S(r,f).
Ún5[1–4]f´‡æX¼ê§Kk
lim
r→∞
T(r,f)
logr
= ∞.
3.½ny²
½n1y²µdÚn1§Ún2ÚNevanlinna1˜Ä½n
m(r,
1
f
)+m(r,
1
Φ−b
)
≤m(r,
1
f
d
)+m(r,
1
Φ−b
)
≤m(r,
1
Φ
)+m(r,
1
Φ−b
)+m(r,
Φ
f
d
)
≤m(r,
Φ
0
Φ
+
Φ
0
Φ−b
)+m(r,
1
Φ
0
)+S(r,f)
≤T(r,Φ
0
)−N(r,
1
Φ
0
)+S(r,f)
=m(r,Φ
0
)+N(r,Φ
0
)−N(r,
1
Φ
0
)+S(r,f)
≤m(r,Φ)+m(r,
Φ
0
Φ
)+N(r,Φ)+N(r,Φ)−N(r,
1
Φ
0
)+S(r,f)
=T(r,Φ)+dN(r,f)−N(r,
1
Φ
0
)+S(r,f).
u´k
m(r,
1
f
)+m(r,
1
Φ−b
) ≤T(r,Φ)+dN(r,f)−N(r,
1
Φ
0
)+S(r,f).
Ïd§dNevanlinna1˜Ä½n
T(r,f
d
)−N(r,
1
f
d
)+T(r,Φ)−N(r,
1
Φ−b
)
≤T(r,Φ)+dN(r,f)−N(r,
1
Φ
0
)+S(r,f).
DOI:10.12677/pm.2020.10121391171nØêÆ
oŒ!
u´k
dT(r,f) ≤dN(r,f)+N(r,
1
f
d
)+N(r,
1
Φ−b
)−N(r,
1
Φ
0
)+S(r,f)
≤dN(r,f)+dN(r,
1
f
)+N(r,
1
Φ−b
)+S(r,f).
½n1y"
½n2y²µdÚn1§Ún2ÚNevanlinna1˜Ä½n
m(r,
1
f
)+m(r,
1
∆
n
c
f−b
) ≤m(r,
1
∆
n
c
f
)+m(r,
1
∆
n
c
f−b
)+m(r,
∆
n
c
f
f
)
≤m(r,
1
∆
n
c
f
+
1
∆
n
c
f−b
)+S(r,f) ≤m(r,
1
∆
n+1
c
f
)+m(r,
∆
n+1
c
f
∆
n
c
f
+
∆
n+1
c
f
∆
n
c
f−b
)+S(r,f)
=m(r,
1
∆
n+1
c
f
)+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).
Ïd§dNevanlinna1˜Ä½n
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).
½n2y"
íØ3y²µbXíØ3(ØØé§K•3k¡EêaÚb(6= 0)¦f(z)−aÚ∆
n
c
f−bþ•
DOI:10.12677/pm.2020.10121391172nØêÆ
oŒ!
kk•‡":"u´d½n2
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(logr)+S(r,f).
u´k
T(r,f) ≤O(logr),r6∈E,
Ù¥E´˜‡éêÿÝk¡rŠ8"dÚn5lim
r→∞
T(r,f)
logr
= ∞§gñ"
½n3y²µdÚn1§Ún2ÚNevanlinna1˜Ä½n
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)+2N(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)+2N(r,f)−N(r,
1
(f∆
c
f)
0
)+S(r,f).
Ïd§dNevanlinna1˜Ä½n
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)
+2N(r,f)−N(r,
1
f(∆
c
f)
0
)+S(r,f).
DOI:10.12677/pm.2020.10121391173nØêÆ
oŒ!
k
2T(r,f) ≤2N(r,f)+2N(r,
1
f
)+N(r,
1
f∆
c
f−b
)−N(r,
1
(f∆
c
f)
0
)+S(r,f).
u´=
2T(r,f) ≤2N(r,f)+2N(r,
1
f
)+N(r,
1
f∆
c
f−b
)+S(r,f).
½n3y"
—
Šöé"v<JÑBïÆL«©%aœ
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(NNSF11701188)"
ë•©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)ValueDistributionoff
0
f.ChineseScienceBulletin,34,727-730.
https://doi.org/10.1360/csb1989-34-10-727
[6]Wu,Z.J.andXu,H.Y.(2020)MillouxInequalityofNonlinearDifferenceMonomialsandIts
Application.JournalofMathematicalInequalities,14,819-827.
https://doi.org/10.7153/jmi-2020-14-52
[7]Halburd,R.G.andKorhonen,R.J.(2007)MeromorphicSolutionsofDifferenceEquations,
IntegrabilityandtheDiscretePainleveEquations.JournalofPhysicsA:Mathematicaland
Theoreticalis,40,1-38.https://doi.org/10.1088/1751-8113/40/6/R01
DOI:10.12677/pm.2020.10121391174nØêÆ
oŒ!
[8]Halburd,R.G.andKorhonen,R.J.(2006)NevanlinnaTheoryfortheDifferenceOperator.
AnnalesAcademiæScientiarumFennicæ,31,463-478.
[9]Chiang, Y.M.andFeng, S.J.(2009)On theGrowthofLogarithmicDifference, DifferenceEqua-
tionsandLogarithmicDerivativesofMeromorphicFunctions.TransactionsoftheAmerican
MathematicalSociety,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.ScienceChinaMathematics,42,1115-1130.(InChinese)
https://doi.org/10.1360/012011-760
DOI:10.12677/pm.2020.10121391175nØêÆ

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