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PureMathematicsnØêÆ,2020,10(9),826-836
PublishedOnlineSeptember2020inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2020.109095
9©†²£æX¼êkn¼êÂñ
•ê
---•••§§§BBB
∗
uHà’ŒÆA^êÆïĤ§2À2²
Email:yangseawell@foxmail.com,
∗
dyang@scau.edu.cn
ÂvFϵ2020c813F¶¹^Fϵ2020c94F¶uÙFϵ2020c911F
Á‡
c´˜‡š"k¡Eê§f´˜‡k¡?‡æX¼ê§R´˜‡š~êkn¼ê,©ï
Äf(z)−R(z),f(z+ c)−R(z)9∆
c
f(z)−R(z)":Âñ•ê†f?ƒm'X"ddí2
Chen,Zhang-Chen,Chen-Zheng<(J"
'…c
æX¼ê§©§²£§Âñ•ê
TheExponentsofConvergenceof
RationalFunctionsofMeromorphic
FunctionsConcerningDifferences
andShifts
ShiweiYang,DeguiYang
∗
InstituteofAppliedMathematics,SouthChinaAgriculturalUniversity,GuangzhouGuangdong
Email:yangseawell@foxmail.com,
∗
dyang@scau.edu.cn
Received:Aug.13
th
,2020;accepted:Sep.4
th
,2020;published:Sep.11
th
,2020
∗ÏÕŠö"
©ÙÚ^:-•,B.9©†²£æX¼êkn¼êÂñ•ê[J].nØêÆ,2020,10(9):826-836.
DOI:10.12677/pm.2020.109095
-•§B
Abstract
Letcbeanonzerofinitecomplexnumber,letfbeatranscendentalmeromorphic
functionoffiniteorder,andletRbeanonconstantrationalfunction.Itisstudied
thattherelationshipbetweentheexponentofconvergenceofzerosoff(z)−R(z),
f(z+c)−R(z),and∆
c
f(z)−R(z)and theorderoff.Thisimproves theresultsofChen,
Zhang-ChenandChen-Zheng.
Keywords
MeromorphicFunctions,Differences,Shifts,TheExponentofConvergence
Copyright
c
2020byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(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][2][3][4]),Ù¥S(r,f)L«?˜¼êf÷vS(r,f) =
o{T(r,f)},r→∞,r6∈E,E´˜‡éêÿÝk¡rŠ8.
a´E²¡þæX ¼ê(Œ±ð•∞),XJa÷vT(r,a)=S(r,f),¡a´f¼ê.©
^ρ(f),λ(f)Úλ(
1
f
)©OL«f?,f":Ú4:Âñ•ê,
ρ(f) =lim
r→∞
log
+
T(r,f)
logr
,
λ(f) =lim
r→∞
log
+
N(r,
1
f
)
logr
,
λ(
1
f
) =lim
r→∞
log
+
N(r,f)
logr
.
d´˜‡Eê,eλ(f−d) <ρ(f),¡d•fBorel~Š.f´˜‡E²¡þš~êæX¼ê,
c´˜‡š"k¡Eê,f©•∆
c
f(z) = f(z+c)−f(z).
•CA›c5,k 'æX ¼ê9©†²£ØÄ:,æX¼ê†Ù²£Ú©ØÄ:†
?'XÑkNõ#ïĤJ,X[5–14].2000c,Fang[12]ïÄæX¼êêØÄ
DOI:10.12677/pm.2020.109095827nØêÆ
-•§B
:Š©Ù,y²
½nA f´˜‡‡æX¼ê,ef¤k":Ú4:-?þ≥2,Kf
0
káõ‡ØÄ:.
2008c,Chen-Shon[10]ïÄ‡¼êÚæX ¼ê":ÚØÄ:.2013c,Chen[8]ïÄ
æX¼ê†Ù²£Ú©ØÄ:‹?ƒm'X,y²
½nBf´E²¡þ˜‡÷vλ(
1
f
)<ρ(f)k¡?æX¼ê,c´˜‡š"k¡Eê,÷
vf(z+c) 6≡f(z)+c,Kk
max{τ(f(z)),τ(∆
c
f(z))}= ρ(f),
max{τ(f(z)),τ(f(z+c))}= ρ(f),
max{τ(∆
c
f(z)),τ(f(z+c))}= ρ(f).
Ù¥τ(f)L«fØÄ:Âñ•ê(eÓ).
2016c,Zhang-Chen[14]•Ñ½nB¥^‡λ(
1
f
) <ρ(f)ØUK,¿ò^‡U•λ(f(z)−d) <
ρ(f),d´˜‡k¡Eê,½nB•,¤á,y²
½nCd´˜‡k¡Eê,f´E²¡þ˜‡÷vλ(f(z)−d)<ρ(f)k¡?æX¼ê,
c´˜‡š"k¡Eê,Kk
max{τ(f(z)),τ(∆
c
f(z))}= ρ(f),
max{τ(f(z)),τ(f(z+c))}= ρ(f),
max{τ(∆
c
f(z)),τ(f(z+c))}= ρ(f).
2003c,Bergweiler-Pang[6]í2¿U?½nA,y²
½nDf´˜‡k¡?‡æX¼ê,P6≡0´˜‡õ‘ª,ef¤k":-?≥2,Kk
•‡~,Kf
0
−Pkáõ‡":.
½nEf´˜‡‡æX¼ê,R6≡0´˜‡kn¼ê,ef¤k":Ú4:-?þ≥2,
Kk•‡~,Kf
0
−Rkáõ‡":.
2019c,Chen-Zheng[7]í2½nC,y²
½nFc´˜‡š"k¡Eê,f´˜‡k¡?‡æX¼ê,m∈N
+
,P(z)=p
m
z
m
+
p
m−1
z
m−1
+···+ p
0
´˜‡š~êõ‘ª,p
i
∈C,i=0,1,···,m,…p
m
6=0,efk˜‡Borel~
Šd∈C,Kk
max{λ(f(z)−P(z)),λ(∆
c
f(z)−P(z))}= ρ(f),
max{λ(f(z)−P(z)),λ(f(z+c)−P(z))}= ρ(f),
max{λ(∆
c
f(z)−P(z)),λ(f(z+c)−P(z))}= ρ(f).
g,¬¯,éuk¡?‡æX¼ê†kn¼ê":Âñ•ê´Ä•kaqu½nF(Ø.
DOI:10.12677/pm.2020.109095828nØêÆ
-•§B
©‰Ñ’½£‰,y²
½n1c´˜‡š"k¡Eê,f´˜‡k¡?‡æX¼ê,
R(z) =
a
p
z
p
+a
p−1
z
p−1
+···+a
0
b
q
z
q
+b
q−1
z
q−1
+···+b
0
´˜‡š~êkn¼ê,Ù¥a
p
6=0,a
p−1
,···,a
0
,b
q
6=0,b
q−1
,···,b
0
´k¡Eê,p,q´šKê,
…p+q≥1.efk˜‡Borel~Šd∈C,Kk
max{λ(f(z)−R(z)),λ(∆
c
f(z)−R(z))}= ρ(f),
max{λ(f(z)−R(z)),λ(f(z+c)−R(z))}= ρ(f),
max{λ(∆
c
f(z)−R(z)),λ(f(z+c)−R(z))}= ρ(f).
½n2c´˜‡š"k¡Eê,f´˜‡k¡?‡æX¼ê,
R(z) =
a
p
z
p
+a
p−1
z
p−1
+···+a
0
b
q
z
q
+b
q−1
z
q−1
+···+b
0
´˜‡š~êkn¼ê,Ù¥a
p
6=0,a
p−1
,···,a
0
,b
q
6=0,b
q−1
,···,b
0
´k¡Eê,p,q´šKê,
…p+q≥1.efk˜‡Borel~Šd= ∞,∆
c
f(z)´‡æX¼ê,Kk
max{λ(f(z)−R(z)),λ(∆
c
f(z)−R(z))}= ρ(f),
max{λ(f(z)−R(z)),λ(f(z+c)−R(z))}= ρ(f),
max{λ(∆
c
f(z)−R(z)),λ(f(z+c)−R(z))}= ρ(f).
~1f(z)=e
z
+ R(z),R(z)´kn¼ê,c´÷ve
c
=1š"~ê.K∆
c
f(z)=R(z+
c) −R(z),ρ(f)=1,λ(f(z) −R(z))=0,λ(∆
c
f(z) −R(z))=0,λ(f(z+ c) −R(z))=0.Ï
dmax{λ(f(z)−R(z)),λ(f(z+c)−R(z)),λ(∆
c
f(z)−R(z))}<ρ(f).
~1`²½n2¥^‡∆
c
f(z)´‡æX¼ê´7I.
2.˜Ún
•y²©(J,I‡XeA‡Ún.
Ún2.1[11][14]f´E²¡þ˜‡k¡?æX¼ ê,c´˜‡‰½š"k¡Eê,d´˜
‡k¡Eê,Kéu?¿ε>0,k
T(r,f(z+c)) = T(r,f)+O(r
ρ(f)−1+ε
)+O(logr),
N(r,f(z+c)) = N(r,f)+O(r
ρ(f)−1+ε
)+O(logr),
N

r,
1
f(z+c)−d

= N

r,
1
f(z)−d

+O(r
ρ(f)−1+ε
)+O(logr).
DOI:10.12677/pm.2020.109095829nØêÆ
-•§B
Ún2.2 [2]f´E²¡þ˜‡æX¼ê,Kéuf¤kØŒkn¼ê
R(z,f(z)) =
p
P
i=0
a
i
(z)f
i
(z)
q
P
j=0
b
j
(z)f
j
(z)
,
Ù¥a
i
(z),i= 0,1,···,p,b
j
(z),j= 0,1,···,q´fæX¼ê,Kk
T(r,R(z,f(z))) = max{p,q}T(r,f)+S(r,f).
Ún2.3 [14]f´E²¡þ˜‡æX¼ê,÷vN(r,f)+N(r,
1
f
) = S(r,f).-
F(z) =
a
0
(z)f
p
(z)+a
1
(z)f
p−1
(z)+···+a
p
(z)
b
0
(z)f
q
(z)+b
1
(z)f
q−1
(z)+···+b
q
(z)
,
Ù¥a
i
(z),i=0,1,···,p,b
j
(z),j=0,1,···,q´fæX¼ê,…a
0
b
0
a
p
6≡0,eq≤p,T(r,F)≥
T(r,f)+S(r,f),Kk
λ(F) = ρ(f).
Ún2.4[3]f
1
(z),f
2
(z),···,f
n
(z)´E²¡þæX¼ê,g
1
(z),g
2
(z),···,g
n
(z)´¼ê,
÷v±e^‡
(1)
n
P
j=1
f
j
(z)e
g
j
(z)
≡0;
(2)éu1 ≤j<k≤n,g
j
(z)−g
k
(z)Ø•~ê;
(3)éu1 ≤j≤n,1 ≤h<k≤n,
T(r,f
j
) = o{T(r,e
g
h
−g
k
)},r→∞.
Kéuj= 1,2,···,n,kf
j
(z) ≡0.
Ún2.5 [7] [14]H´E²¡þ˜‡æX¼ê,h´÷vdegh≥1õ‘ª, c´˜‡š"k¡
Eê,eρ(H) <ρ(e
h
),Kk
T(r,H) = S(r,e
h
),T(r,H(z+c)) = S(r,e
h
),T(r,e
h(z+c)−h(z)
) = S(r,e
h
).
éuj=1,2,···,n,T(r,H(z+jc))=S(r,e
h
).éuk∈N
+
,s∈N,k>s,T(r,e
h(z+kc)−h(z+sc)
)=
S(r,e
h
).
3.½ny²
½n1y²
DOI:10.12677/pm.2020.109095830nØêÆ
-•§B
bλ(f(z)−R(z)) <ρ(f),eyλ(∆
c
f(z)−R(z)) = λ(f(z+c)−R(z)) = ρ(f).-
F
1
(z) =
f(z)−R(z)
f(z)−d
.(3.1)
dÚn2.2T(r,F
1
) = T(r,f)+S(r,f),Kρ(F
1
) = ρ(f) <∞,Ïdk
λ(
1
F
1
) = λ(f(z)−d) <ρ(f) = ρ(F
1
),
λ(F
1
) = λ(f(z)−R(z)) <ρ(f) = ρ(F
1
),
ù¿›X0Ú∞´F
1
Borel~Š.dHadamardÏf©)½n,
F
1
(z) = A
1
(z)e
B
1
(z)
.(3.2)
Ù¥A
1
(z)6≡0´÷vρ(A
1
)<ρ(F
1
)=ρ(f)æX¼ê.B
1
(z)´÷vρ(f)=ρ(F
1
)=degB
1
(z)≥
1õ‘ª.
d(3.1)(3.2)
f(z) =
R(z)−d
1−F
1
(z)
+d=
R(z)−d
1−A
1
(z)e
B
1
(z)
+d.(3.3)
d(3.3)
∆
c
f(z) =f(z+c)−f(z)
=
R(z+c)−d
1−A
1
(z+c)e
B
1
(z+c)
−
R(z)−d
1−A
1
(z)e
B
1
(z)
=

(d−R(z+c))A
1
(z)+(R(z)−d)A
1
(z+c)e
B
1
(z+c)−B
1
(z)

e
B
1
(z)
+R(z+c)−R(z)
[A
1
(z)A
1
(z+c)e
B
1
(z+c)−B
1
(z)
]e
2B
1
(z)
+[−A
1
(z+c)e
B
1
(z+c)−B
1
(z)
−A
1
(z)]e
B
1
(z)
+1
=
D
1
(z)e
B
1
(z)
+R(z+c)−R(z)
D
3
(z)e
2B
1
(z)
+D
2
(z)e
B
1
(z)
+1
,(3.4)
Ù¥
D
1
(z) = (d−R(z+c))A
1
(z)+(R(z)−d)A
1
(z+c)e
B
1
(z+c)−B
1
(z)
,
D
2
(z) = −A
1
(z+c)e
B
1
(z+c)−B
1
(z)
−A
1
(z),
D
3
(z) = A
1
(z)A
1
(z+c)e
B
1
(z+c)−B
1
(z)
6≡0.
R(z+c)−R(z) =
a
p
(z+c)
p
+a
p−1
(z+c)
p−1
+···+a
0
b
q
(z+c)
q
+b
q−1
(z+c)
q−1
+···+b
0
−
a
p
z
p
+a
p−1
z
p−1
+···+a
0
b
q
z
q
+b
q−1
z
q−1
+···+b
0
=
a
p
b
q
(C
1
p
−C
1
q
)c·z
p+q−1
+···+(a
p
c
p
+···+a
1
c)b
0
−(b
q
c
q
+···+b
1
c)a
0
[b
q
(z+c)
q
+b
q−1
(z+c)
q−1
+···+b
0
][b
q
z
q
+b
q−1
z
q−1
+···+b
0
]
6≡0.
DOI:10.12677/pm.2020.109095831nØêÆ
-•§B
d(3.4)
∆
c
f(z)−R(z) =
D
1
(z)e
B
1
(z)
+R(z+c)−R(z)
D
3
(z)e
2B
1
(z)
+D
2
(z)e
B
1
(z)
+1
−R(z)
=
−R(z)D
3
(z)e
2B
1
(z)
+(D
1
(z)−R(z)D
2
(z))e
B
1
(z)
+R(z+c)−2R(z)
D
3
(z)e
2B
1
(z)
+D
2
(z)e
B
1
(z)
+1
=
−R(z)D
3
(z)e
2B
1
(z)
+D
4
(z)e
B
1
(z)
+R(z+c)−2R(z)
D
3
(z)e
2B
1
(z)
+D
2
(z)e
B
1
(z)
+1
,(3.5)
Ù¥
D
4
(z) = D
1
(z)−R(z)D
2
(z) = (d−R(z+c)+R(z))A
1
(z)+(2R(z)−d)e
B
1
(z+c)−B
1
(z)
.
R(z+c)−2R(z) =
a
p
(z+c)
p
+a
p−1
(z+c)
p−1
+···+a
0
b
q
(z+c)
q
+b
q−1
(z+c)
q−1
+···+b
0
−2
a
p
z
p
+a
p−1
z
p−1
+···+a
0
b
q
z
q
+b
q−1
z
q−1
+···+b
0
=
−a
p
b
q
z
p+q
+···+(a
p
c
p
+···+a
1
c)b
0
−2(b
q
c
q
+···+b
1
c)a
0
−a
0
b
0
[b
q
(z+c)
q
+b
q−1
(z+c)
q−1
+···+b
0
][b
q
z
q
+b
q−1
z
q−1
+···+b
0
]
6≡0.
d(3.4),R(z+c) −R(z)6≡0†(3.5),R(z)D
3
(z)6≡0Œ±wÑ∆
c
f(z)†∆
c
f(z)−R(z)´e
B
1
(z)
kn¼ê.Ï•ρ(A
1
)<ρ(e
B
1
),dÚn2.5,XêD
i
(z)(i=1,2,3,4)´e
B
1
(z)
¼ê.w
,D
i
(z) 6≡0(i= 1,2,3,4),dÚn2.4
D
1
(z)e
B
1
(z)
+R(z+c)−R(z) 6≡0.
Kd(3.4)(3.5)
T(r,∆
c
f(z)) ≥T(r,e
B
1
(z)
)+S(r,e
B
1
(z)
),
T(r,∆
c
f(z)−R(z)) ≥T(r,e
B
1
(z)
)+S(r,e
B
1
(z)
).(3.6)
d(3.5)(3.6)†Ún2.3
λ(∆
c
f(z)−R(z)) = ρ(e
B
1
(z)
) = ρ(f),(3.7)
=
max{λ(f(z)−R(z)),λ(∆
c
f(z)−R(z))}= ρ(f).
d(3.3)
f(z+c)−R(z) =
R(z+c)−d
1−A
1
(z+c)e
B
1
(z+c)
+d−R(z)
=
(R(z)−d)A
1
(z+c)e
B
1
(z+c)
+R(z+c)−R(z)
1−A
1
(z+c)e
B
1
(z+c)
.(3.8)
DOI:10.12677/pm.2020.109095832nØêÆ
-•§B
Ï•(R(z)−d)A
1
(z+c)+(R(z+c)−R(z))A
1
(z+c) = (R(z+c)−d)A
1
(z+c) 6≡0,¤±f(z+c)−R(z)Œ
±wŠe
B
1
(z+c)
ØŒkn¼ê.Ï•ρ(A
1
(z+c)) = ρ(A
1
) <ρ(e
B
1
) = ρ(e
B
1
(z+c)
),dÚn2.5
T(r,A
1
(z+c)) = S(r,e
B
1
(z+c)
).(3.9)
d(3.8)(3.9)†Ún2.2
T(r,f(z+c)−R(z)) = T(r,e
B
1
(z+c)
)+S(r,e
B
1
(z+c)
).
dÚn2.3
λ(f(z+c)−R(z)) = ρ(e
B
1
(z+c)
) = ρ(e
B
1
) = ρ(f),(3.10)
=
max{λ(f(z)−R(z)),λ(f(z+c)−R(z))}= ρ(f).
bλ(f(z+c)−R(z)) <ρ(f),eyλ(∆
c
f(z)−R(z)) = ρ(f).-
F
2
(z) =
f(z+c)−R(z)
f(z+c)−d
.(3.11)
d(3.11)
f(z) =
R(z−c)−d
1−F
2
(z−c)
+d.(3.12)
d(3.11)(3.12)†Ún2.1
ρ(F
2
) = ρ(F
2
(z−c)) = ρ(f).
dÚn2.1†λ(f(z)−d) <ρ(f),
λ(
1
F
2
) = λ(f(z+c)−d) = λ(f(z)−d) <ρ(f) = ρ(F
2
),
λ(F
2
) = λ(f(z+c)−R(z)) <ρ(f) = ρ(F
2
),
ù¿›X0Ú∞´F
2
borel~Š.aqu(3.2)-(3.7)L§Œ
λ(∆
c
f(z)−R(z)) = ρ(f).(3.13)
=
max{λ(∆
c
f(z)−R(z)),λ(f(z+c)−R(z))}= ρ(f).
y..
DOI:10.12677/pm.2020.109095833nØêÆ
-•§B
½n2y²
bλ(f(z)−R(z))<ρ(f),eyλ(∆
c
f(z)−R(z))=λ(f(z+c)−R(z))=ρ(f).Ï•λ(
1
f
)<
ρ(f),R´˜‡š~êkn¼ê,dHadamardÏf©)½n,k
f(z)−R(z) = α(z)e
p(z)
,(3.14)
Ù¥α(z)´÷vρ(α) <ρ(f)æX¼ê,p(z)´÷vdegp= ρ(f)š~êõ‘ª.Ïd
T(r,α) = S(r,e
p
),T(r,f) = T(r,e
p
)+S(r,f).(3.15)
d(3.14)
∆
c
f(z) = f(z+c)−f(z)
= R(z+c)+α(z+c)e
p(z+c)
−R(z)−α(z)e
p(z)
=

α(z+c)e
p(z+c)−p(z)
−α(z)

e
p(z)
+R(z+c)−R(z)
= D
5
(z)e
p(z)
+R(z+c)−R(z),(3.16)
Ù¥D
5
(z)=α(z+ c)e
p(z+c)−p(z)
−α(z).d(3.15)T(r,D
5
)=S(r,f).Ï•∆
c
f(z)´‡æX¼
ê,KD
5
(z) 6≡0.d½n1y²,R(z+c)−R(z) 6≡0.d(3.15)(3.16)†Nevanlinna1Ä½n,

T(r,∆
c
f) = T(r,e
p
)+S(r,f),
N

r,
1
∆
c
f−R

= T(r,e
p
)+S(r,f).
Ïd
λ(∆
c
f(z)−R(z)) = ρ(f).(3.17)
=
max{λ(∆
c
f(z)−R(z)),λ(f(z)−R(z))}= ρ(f).
d(3.14)
f(z+c)−R(z) = R(z+c)+α(z+c)e
p(z+c)
−R(z)
=

α(z+c)e
p(z+c)−p(z)

e
p(z)
+R(z+c)−R(z)
= D
6
(z)e
p(z)
+R(z+c)−R(z),(3.18)
Ù¥D
6
(z) = α(z+c)e
p(z+c)−p(z)
.aqu(3.16)(3.17)L§Œ
λ(f(z+c)−R(z)) = ρ(f).
DOI:10.12677/pm.2020.109095834nØêÆ
-•§B
=
max{λ(f(z+c)−R(z)),λ(f(z)−R(z))}= ρ(f).
bλ(f(z+c)−R(z)) <ρ(f),eyλ(∆
c
f(z)−R(z)) = ρ(f).aqu(3.14)-(3.17)L§Œ
λ(∆
c
f(z)−R(z)) = ρ(f).
=
max{λ(∆
c
f(z)−R(z)),λ(f(z+c)−R(z))}= ρ(f).
y..
—
Šöé"v<JÑBïÆL«©%a!
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(NNSF11701188)"
ë•©z
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[2]Laine,I.(1993)NevanlinnaTheoryandComplexDifferentialEquations.WalterdeGruyter,
Berlin.
[3]Yang,L.(1993)ValueDistributionTheory.Springer-Verlag,Berlin.
[4]Yi,H.X. andYang,C.C. (1995)Uniqueness Theoryof MeromorphicFunctions.SciencePress,
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ematicalProceedingsoftheCambridgePhilosophicalSociety,142,133-147.
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https://doi.org/10.1016/S0022-247X(02)00349-9
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OrderDifferencesandShifts.OpenMathematics,17,677-688.
https://doi.org/10.1515/math-2019-0054
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-•§B
[8]Chen,Z.X.(2013)FixedPointsofMeromorphicFunctionsandTheirDifferencesandShifts.
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ferenceEquationsintheComplexPlane.TheRamanujanJournal,16,105-129.
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DOI:10.12677/pm.2020.109095836nØêÆ

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