Vấn đề duy nhất cho hàm phân hình chung nhau một hàm nhỏ
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❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❍⑩■ ◆●❯❨➊◆
❑❍❖❆ ❚❖⑩◆
❈❍❆◆❚❍❖◆❊ ❑❊❖▼❆◆■❙❆❨
❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ❈❍❖ ❍⑨▼ P❍❹◆ ❍➐◆❍
❈❍❯◆● ◆❍❆❯ ▼❐❚ ❍⑨▼ ◆❍➘
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❍⑩■ ◆●❯❨➊◆
❑❍❖❆ ❚❖⑩◆
❈❍❆◆❚❍❖◆❊ ❑❊❖▼❆◆■❙❆❨
❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ❈❍❖ ❍⑨▼ P❍❹◆ ❍➐◆❍
❈❍❯◆● ◆❍❆❯ ▼❐❚ ❍⑨▼ ◆❍➘
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿
P●❙✳❚❙✳ ❍⑨ ❚❘❺◆ P❍×❒◆●
❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼
▲í✐ ❝❛♠ ✤♦❛♥
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ sü ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ tæ✐
❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ P●❙✳❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤
tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤÷❛ tø♥❣ ✤÷ñ❝ ❝æ♥❣ ❜è tr♦♥❣ ❝→❝ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛
❝→❝ t→❝ ❣✐↔ ❦❤→❝ ð ❱✐➺t ◆❛♠✳
❍å❝ ✈✐➯♥
❈❤❛♥t❤♦♥❡ ❑❡♦♠❛♥✐s❛②
❳→❝ ♥❤➟♥
❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❚♦→♥
❳→❝ ♥❤➟♥
❝õ❛ ♥❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝
P●❙✳❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣
ớ ỡ
t tổ ổ ữủ sỹ ữợ
ú ù t t ừ P r Pữỡ rữớ ồ
ữ ồ ổ tọ ỏ t ỡ ổ
tợ P r Pữỡ ữớ t t t tổ tứ ỳ
ữợ ỳ t tr ữớ ự ồ ợ tt
s ồ t t ừ ữớ t
ổ ụ t ỡ t ổ tr Pỏ t
ở q ỵ t ồ tở rữớ ồ ữ
ồ t ồ tổ t từ
tử tổ t
ổ t ỡ t ổ rữớ
ồ ữ ồ rữớ ồ ữ
ở t t tr tổ ỳ tự ỡ s
tr ữớ ự ồ
ổ ụ ỷ ớ ỡ tr ợ ồ
ở ú ù tổ tr q tr ồ t
ố ũ tổ tọ ỏ t ỡ s s tợ ỳ ữớ t
tr ừ ỳ ữớ ổ ở s õ
ổ ọ tổ t ổ
ổ t tr ọ ỳ t sõt t rt
ữủ sỹ t t ừ t ổ ỗ
t
t s
▼ö❝ ❧ö❝
▼Ð ✣❺❯
✶
✶
❑✐➳♥ t❤ù❝ ❝ì sð ❝❤✉➞♥ ❜à
✸
✶✳✶✳ ❍❛✐ ✤à♥❤ ❧þ ❝ì ❜↔♥ tr♦♥❣ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✳ ✳ ✳ ✳ ✳ ✳
✸
✶✳✶✳✶✳ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸
✶✳✶✳✷✳ ❍❛✐ ✤à♥❤ ❧þ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✺
✶✳✷✳ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ❤❛② ❤➔♠ ♥❤ä ✳ ✳ ✶✸
✶✳✷✳✶✳ ❑❤→✐ ♥✐➺♠ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✶✳✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷
❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤✐ ✤❛ t❤ù❝ ❝❤ù❛
✤↕♦ ❤➔♠ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä
✷✶
✷✳✶✳ ❚r÷í♥❣ ❤ñ♣ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝➜♣ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✷✳ ❚r÷í♥❣ ❤ñ♣ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
❑➳t ❧✉➟♥
✹✶
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
✹✶
é
ởt tr ỳ ữợ ự q trồ ừ ỵ tt
ự t ừ
ữủ ự tọ
tr r t ổ ở t s trũ ổ tr ừ
ữủ ỗ ự t
ừ s t tr ự t ữợ
t út ữủ sỹ q t ừ t ồ tr
ữợ rss t t
t ồ
rss r t ủ T = {z C|ez + z = 0} t
t ở tr C tự ợ
f g Ef (T ) = Eg (T ) t f g. ú ỵ
t T ữ tr ự ổ số tỷ t
t ủ SY = {z C|z n + az m + b = 0} tr õ n 15, n > m 5
a, b số ổ s z n +az m +b = 0 ổ õ
ở ự SY t t
tr C r rs r ởt ử
t t tr C
ởt
ợ ồ tr ữỡ n t
t r tt
f tọ f nf = 1
ự tt
ự ởt ỵ t
tự ự t
ởt tr ứ õ ỳ ự t ữợ
ữủ t tr ổ tr ừ t tr
ữợ ữ ss
ssst
P
ợ ố t ữủ ởt
t số ừ tự ự ú
tổ ồ t t
ởt ọ
ử ừ tr
ởt số t q ữủ ự P
ởt số t q ỗ õ ữỡ ữ s
ữỡ tự ỡ s r ữỡ ú tổ
tr ởt số tự ỡ tr ỵ tt ố tr
ởt số t q sỷ
ử tr ữỡ
ữỡ t ừ tự ự
ởt ọ ữỡ ừ
ú tổ tr ởt số t q ự ừ
P ởt số t q ừ t ổ ố
tr tớ
ữỡ
tự ỡ s
ỵ ỡ tr ỵ tt
t t
rữợ t t ởt số tữớ ữủ sỷ ử tr
ỵ tt ố tr
f tr t ự C
z0 C ữủ ồ ổ ở k N ừ f (z) tỗ t
ởt h(z) ổ trt t tr U ừ z0 s
tr õ f ữủ ữợ
f (z) = (z z0 )k h(z).
f (n) (z0 ) = 0, ợ ộ n = 1, ..., k 1 f (k) (z0 ) = 0 z0
ữủ ồ ỹ ở k N ừ f (z) õ ổ
1
.
ở k ừ
f (z)
ợ ộ số tỹ x > 0
log+ x = max{log x, 0}.
õ log x = log+ x log+ (1/x).
f ởt tr C r > 0 ợ ộ [0; 2], t
✹
❝â
log |f (reiϕ )| = log+ |f (reiϕ )| − log+
1
,
f (reiϕ )
♥➯♥
2π
2π
1
log f (reiϕ ) dϕ =
2π
1
2π
0
2π
log+
1
f (reiϕ ) dϕ−
2π
0
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳
log+
1
dϕ.
f (reiϕ )
0
❍➔♠
2π
1
m(r, f ) =
2π
log+ f (reiϕ ) dϕ
0
✤÷ñ❝ ❣å✐ ❧➔
❤➔♠ ①➜♣ ①➾ ❝õ❛ ❤➔♠ f ✳
❇➙② ❣✐í t❛ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❤➔♠ ✤➳♠✳ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔
r > 0✳ ❑➼ ❤✐➺✉ n(r, 1/f ) ❧➔ sè ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✱ n(r, 1/f ) ❧➔ sè
❦❤æ♥❣ ✤✐➸♠ ❦❤æ♥❣ ❦➸ ❜ë✐ ❝õ❛ f ✱ n(r, f ) ❧➔ sè ❝ü❝ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✱ n(r, f )
❧➔ sè ❝ü❝ ✤✐➸♠ ❦❤æ♥❣ ❦➸ ❜ë✐ ❝õ❛ f tr♦♥❣ Dr = {z ∈ C : |z|
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳
|r|}✳
❍➔♠
r
N (r, ∞; f ) = N (r, f ) =
n(t, f ) − n(0, f )
dt + n(0, f ) log r
t
0
❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ ❝õ❛ f
✤÷ñ❝ ❣å✐ ❧➔
✭❝á♥ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ t↕✐ ❝→❝ ❝ü❝
✤✐➸♠✮✳ ❍➔♠
r
N (r, ∞; f ) = N (r, f ) =
n(t, f ) − n(0, f )
dt + n(0, f ) log r
t
0
✤÷ñ❝ ❣å✐ ❧➔
❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐✳ ❚r♦♥❣ ✤â
n(0, f ) = lim n(t, f ), n(0, f ) = lim n(t, f ).
t→0
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳
t→0
❍➔♠
T (r, f ) = m(r, f ) + N (r, f )
✺
❣å✐ ❧➔
❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ f ✳
❈→❝ ❤➔♠ ✤➦❝ tr÷♥❣ T (r, f )✱ ❤➔♠ ①➜♣ ①➾ m(r, f ) ✈➔ ❤➔♠ ✤➳♠ N (r, f )
❧➔ ❜❛ ❤➔♠ ❝ì ❜↔♥ tr♦♥❣ ❧þ t❤✉②➳t ♣❤➙♥ ❜è ❣✐→ trà✱ ♥â ❝á♥ ❣å✐ ❧➔ ❝→❝ ❤➔♠
◆❡✈❛♥❧✐♥♥❛✳ ✣à♥❤ ❧þ s❛✉ ✤➙② ❝❤♦ t❤➜② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠
♥➔②✳
✣à♥❤ ❧þ ✶✳✶✳
❈❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f1, f2, · · · , fp✱ ❦❤✐ ✤â✿
p
(1)
p
fν ) ≤
m(r,
ν=1
p
(2)
ν=1
p
fν ) ≤
m(r,
ν=1
p
(3)
(5)
fν ) ≤
N (r,
N (r, fν );
ν=1
p
fν ) ≤
N (r,
N (r, fν );
ν=1
ν=1
p
p
fν ) ≤
T (r,
ν=1
p
(6)
m(r, fν );
ν=1
p
ν=1
p
(4)
m(r, fν ) + log p;
T (r, fν ) + log p;
ν=1
p
fν ) ≤
T (r,
ν=1
T (r, fν ).
ν=1
❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t ♥➔② ❧➔ ✤ì♥ ❣✐↔♥✱ t❛ ❝❤➾ ❝➛♥ ❞ü❛ t❤❡♦
t➼♥❤ ❝❤➜t ✿ ♥➳✉ a1 , . . . , ap ❧➔ ❝→❝ sè ♣❤ù❝ ♣❤➙♥ ❜✐➺t t❤➻
p
log
p
+
ν=1
✈➔
log+ |aν |
aν
ν=1
p
log
+
p
aν
+
log+ |aν | + log p.
log (p max |aν |)
ν=1
ν=1,...,p
ν=1
✶✳✶✳✷✳ ❍❛✐ ✤à♥❤ ❧þ ❝ì ❜↔♥
❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❤❛✐ ✤à♥❤ ❧þ ❝ì ❜↔♥ tr♦♥❣ ❧þ t❤✉②➳t
♣❤➙♥ ❜è ❣✐→ trà ◆❡✈❛♥❧✐♥♥❛✱ tr÷î❝ ❤➳t ❧➔ ❝æ♥❣ t❤ù❝ P♦✐ss♦♥ ✲ ❏❡♥s❡♥✳
f (z) 0 ởt
tr trỏ {|z| R} ợ 0 < R < sỷ a1, ..., ap
ổ ở b1, ..., bq ỹ ở tr
trỏ õ õ ợ ộ z tr {|z| < R} ổ ổ
ỹ ừ f t õ
ỵ
ổ tự Pss s
2
1
log |f (z)| =
2
R2 |z|2
i
log
|f
(Re
)|d
|Rei z|2
0
p
i=1
t
R 2 ai z
+
log
R(z ai )
q
R2 bj z
log
.
R(z
b
)
j
j=1
ổ tự Psss r r t tr ừ
|f (z)| tr t ỹ ổ ừ f (z) tr
|z| < R t t õ t t ữủ tr ừ ổ |f (z)| t tr
z tr |z| < R.
q
ợ |z| < R t õ
2
1
1
d
=
.
R2 |z|2
|R2 e z|2 2
i
0
f (z) 0 ởt
tr trỏ {|z| R} ợ 0 < R < sỷ a1, ..., ap ổ
ở b1, ..., bq ỹ ở tr trỏ õ
trứ ọ 0 õ
q
ổ tự s
2
1
log |cf | =
2
p
i
log |f (Re )|d
i=1
0
R
log
+
ai
q
log
j=1
R
bj
(ord0 f ) log R,
tr õ f (z) = cf z ord f + ...., ord0f Z, cf số ổ ọ
t tr tr rt ừ f t 0
0
ỵ s ởt t ừ ổ tự s ữủ ồ
ỵ ỡ tự t
✼
❈❤♦ f ≡ 0 ❧➔ ♠ët ❤➔♠ ♣❤➙♥
❤➻♥❤ tr➯♥ C✳ ❑❤✐ ✤â✱ ✈î✐ ♠é✐ r > 0✱ t❛ ❝â✿
✣à♥❤ ❧þ ✶✳✸
✭✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t✮✳
1
f
(1)
T (r, f ) = m r,
(2)
❱î✐ ♠é✐ sè ♣❤ù❝ a ∈ C,
T (r, f ) − m r,
+ N r,
1
f −a
1
f
+ log |cf | ;
− N r,
≤ log
1
f −a
c1
+ log+ |a| + log 2,
f −a
tr♦♥❣ ✤â cf ❧➔ ❤➺ sè ❦❤→❝ 0 ♥❤ä ♥❤➜t tr♦♥❣ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛ ❤➔♠
f tr♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ 0✱ c1 /(f − a) ❧➔ ❤➺ sè ❦❤→❝ 0 ♥❤ä ♥❤➜t tr♦♥❣ ❦❤❛✐
tr✐➸♥ ❚❛②❧♦r ❝õ❛ ❤➔♠ 1/(f − a) tr♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ 0✳
◆❤➟♥ ①➨t ✶✳✶✳
❞↕♥❣
❚❛ t❤÷í♥❣ ❞ò♥❣ (2) ❝õ❛ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t ❞÷î✐
T (r,
1
) = T (r, f ) + O(1),
f −a
tr♦♥❣ ✤â O(1) ❧➔ ✤↕✐ ❧÷ñ♥❣ ❜à ❝❤➦♥ ❦❤✐ r → ∞✳
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ r > 0✳ ❑➼ ❤✐➺✉
Nram (r, f ) = N r,
✈➔ ❣å✐ ❧➔ ❤➔♠
1
f
+ 2N (r, f ) − N (r, f )
❣✐→ trà ♣❤➙♥ ♥❤→♥❤ ❝õ❛ ❤➔♠ f ✳ ❍✐➸♥ ♥❤✐➯♥ Nram(r, f ) ≥ 0
✣à♥❤ ❧þ ✶✳✹ ✭✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐✮✳ ●✐↔ sû f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝
❤➡♥❣ tr➯♥ C, a1, · · · , aq ∈ C, (q > 2) ❧➔ ❝→❝ ❤➡♥❣ sè ♣❤➙♥ ❜✐➺t✱ ❦❤✐ ✤â
✽
✈î✐ ♠é✐ ε > 0✱ ❜➜t ✤➥♥❣ t❤ù❝
q
(q − 1)T (r, f ) ≤
N r,
j=1
1
f − aj
+ N (r, f ) − Nram (r, f ) + log T (r, f )
+ (1 + ε) log+ log T (r, f ) + O(1)
q
≤
N r,
j=1
1
f − aj
+ N (r, f ) + log T (r, f )
+ (1 + ε) log+ log T (r, f ) + O(1)
✤ó♥❣ ✈î✐ ♠å✐ r ≥ r0 ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ E ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳
❈❤ù♥❣ ♠✐♥❤✳
❑➼ ❤✐➺✉ δ = mini=j {|ai −aj |, 1}✳ ❱î✐ ♠é✐ z ♠➔ f (z) = ∞
✈➔ f (z) = aj , j = 1, ..., q ✱ ❣å✐ j0 ❧➔ ♠ët ❝❤➾ sè tr♦♥❣ t➟♣ {1, ..., q}✱ s❛♦
❝❤♦
|f (z) − aj0 | ≤ |f (z) − aj |, ✈î✐ ♠å✐ 1 ≤ j ≤ q.
❑❤✐ ✤â✱ ✈î✐ j = j0 ✱ t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ t❛ ❝â |f (z) − aj | ≥ δ/2✳
◆❤÷ ✈➟②✱ ✈î✐ j = j0 ✱
log+ f (z) ≤ log+ |f (z) − aj | + log+ |aj | + log 2
≤ log |f (z) − aj | + log+ 2/δ + log+ |aj | + log 2.
❉♦ ✤â
q
+
log+ |aj |
log |f (z) − aj | +
(q − 1) log f (z) ≤
j=j0
j=1
+ (q − 1)(log+ 2/δ + log 2).
❇➙② ❣✐í t❛ ÷î❝ ❧÷ñ♥❣ tê♥❣ ✤➛✉ t✐➯♥ tr♦♥❣ ✈➳ ♣❤↔✐ ❝õ❛ ❜✐➸✉ t❤ù❝ tr➯♥✱
t❛ ❝â
q
log |f (z) − aj | =
log |f (z) − aj | − log |f (z)| + log
j=1
q
j=j0
≤
|f (z)|
|f (z) − aj0 |
q
log |f (z) − aj | − log |f (z)| + log(
j=1
j=1
|f (z)|
).
|f (z) − aj |
✾
◆❤÷ ✈➟②
q
q
+
(q − 1) log f (z) ≤
log+ |aj |
log |f (z) − aj | − log |f (z)| +
j=1
j=1
q
+ log(
j=1
|f (z)|
) + (q − 1)(log+ 2/δ + log 2).
|f (z) − aj |
❚✐➳♣ t❤❡♦✱ ✤➦t z = reiθ ✈➔ ❧➜② t➼❝❤ ♣❤➙♥ t❤❡♦ ❜✐➳♥ θ t❛ ✤÷ñ❝
2π
log+ |f (reiθ )|
(q − 1)m(r, f ) = (q − 1)
dθ
2π
0
2π
q
≤
j=1 0
2π
+
2π
dθ
log |f (reiθ ) − aj | −
2π
log |f (reiθ )|
dθ
2π
0
q
log(
j=1
0
|f (reiθ )| dθ
+ O(1).
|f (reiθ ) − aj |) 2π
❚❛ ❝â
2π
log |f (reiθ ) − aj |
0
✈➔
1
dθ
= N (r,
) − N (r, f ) + log |cf −aj |
2π
f − aj
2π
log |f (reiθ )|
dθ
= N (r, 1/f ) − N (r, f ) + log |cf |.
2π
0
◆❤÷ ✈➟②
q
(q − 1)m(r, f ) −
N (r,
j=1
2π
≤
1
) + qN (r, f ) + N (r, 1/f ) − N (r, f )
f − aj
q
log(
0
j=1
|f (reiθ )| dθ
+
|f (reiθ ) − aj |) 2π
q
log+ |aj |
j=1
q
+
+ (q − 1)(log 2/δ + log 2) +
log |cf −aj | − log |cf |.
j=1
✶✵
▼➦t ❦❤→❝✱ t❤❡♦ ✤à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t ✈➔ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ Nram (r, f )✱
✈➳ tr→✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤÷ñ❝ ✈✐➳t ❧↕✐ ❧➔
q
(q − 1)T (r, f ) −
N (r,
j=1
1
) − N (r, f ) + Nram (r, f ).
f − aj
✣➸ ❤♦➔♥ t➜t ❝❤ù♥❣ ♠✐♥❤ t❛ ❝➛♥ ÷î❝ ❧÷ñ♥❣
2π
q
log(
j=1
0
|f (reiθ )| dθ
.
|f (reiθ ) − aj |) 2π
●å✐ α ∈ (0, 1)✱ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❝õ❛ log ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ (
dj )α ≤
dαj ❝❤♦ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ dj ✱ t❛ ❝â
2π
q
iθ
log(
j=1
0
|f (re )| dθ
= 1/α
|f (reiθ ) − aj |) 2π
2π
≤
0
2π
q
log(
j=1
0
|f (reiθ )|
dθ
iθ
α
|f (re ) − aj |) 2π
q
dθ
f (reiθ )
|
log(
|
f (reiθ ) − aj )α 2π
j=1
q
2π
≤ log(
j=1 0
f (reiθ )
dθ
|
|
f (reiθ ) − aj )α ) 2π
❙û ❞ö♥❣ log+ (x + y) ≤ log+ x + log+ y ✱ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤÷ñ❝ ✈✐➳t ❧↕✐
♥❤÷ s❛✉
q
2π
log(
j=1 0
dθ
ρ
f (reiθ )
+
|
≤
log
|
f (reiθ ) − aj )α ) 2π
r(ρ − r)
q
+ log
+
2T (ρ, f − aj ) + C(α),
j=1
tr♦♥❣ ✤â C(α) ❧➔ ❤➡♥❣ sè ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ α✳ ✣➦t
ρ=r+
1
,
log1+ε T (r, f )
✈î✐ r ≤ r0 ✈➔ r ∈
/ E t❛ ❝â
log T (ρ, f ) ≤ log T (r, f ) + 1
✶✶
✈➔
ρ
≤ (1 + ε) log+ T (r, f ) + log 2.
r(ρ − r)
◆❤÷ ✈➟②✱ ✈î✐ r ≥ r0 , r ∈
/ E t❛ ❝â
log+
q
ρ
log
+ log+
2T (ρ, f − aj ) + C(α)
r(ρ − r)
j=1
+
≤ (1 + ε) log+ log T (r, f ) + log+ max (2T (ρ, f − aj )) + C(α)
1≤j≤q
≤ (1 + ε) log+ log T (r, f ) + log(2T (ρ, f )) + C(α)
≤ (1 + ε) log+ log T (r, f ) + log(T (ρ, f )) + C(α),
❑➳t ❤ñ♣ ❝→❝ ÷î❝ ❧÷ñ♥❣ tr➯♥ t❛ ❝â
q
(q − 1)T (r, f ) +
N (r,
j=1
1
) − N (r, f ) + Nram (r, f )
f − aj
≤ (1 + ε) log+ log T (r, f ) + log T (r, f ) + C(α),
❜➜t ✤➥♥❣ t❤ù❝ t❤ù ♥❤➜t ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳ ◆❣♦➔✐ r❛
q
1
N r,
f − aj
j=1
q
+ N (r, f ) − Nram (r, f ) ≤
N r,
j=1
1
f − aj
+ N (r, f ),
♥➯♥ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ ✤÷ñ❝ s✉② r❛ trü❝ t✐➳♣ tø ❜➜t ✤➥♥❣ t❤ù❝ t❤ù
♥❤➜t✳ ✣à♥❤ ❧þ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
●✐↔ sû f (z) ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C✱ a ∈ C ∪ {∞} ✈➔ k ❧➔ ♠ët sè
♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❛ ❦➼ ❤✐➺✉
1
1
N (r,
)
)
f −a
f −a
= 1 − lim sup
;
T (r, f )
T (r, f )
r→∞
m(r,
δ(a, f ) = lim inf
r→∞
1
)
f −a
Θ(a, f ) = 1 − lim sup
;
T (r, f )
r→∞
1
1
N (r,
) − N (r,
)
f −a
f −a
θ(a, f ) = lim inf
.
r→∞
T (r, f )
N (r,
(a, f ) ữủ ồ số
t (a, f ) ồ số t
ổ ở (a, f ) ồ ừ ở ừ số t
1
) = 0 ợ ồ r
f a
s r (a, f ) = 1 f (z) = ez t (0, f ) = 1.
1
) = o(T (r, f )) õ (a, f ) = 1 ữ số
N (r,
f a
t 1 số ừ ữỡ tr q t s ợ t
t
f (z) = a ổ t N (r,
ừ õ
ợ ộ f a C t ổ õ
0
(a, f )
(a, f )
1.
ỵ s t ởt t t ừ số t tữớ ữủ ồ ờ
q số t
f tr C õ t
ủ tr ừ a (a, f ) > 0 ũ ữủ ỗ tớ
t õ
ỵ
(a, f ) + (a, f )
aC
2.
aC
sỷ f tr C
ổ tr a1, a2, a3 C {} t f
q
f
(a, f )
ỵ Pr
t f ổ tr a t ữỡ tr
1
f (z) = a ổ t N (r,
) = 0, s r
f a
(a, f ) = 1. ứ ờ q số t f số
ự
t tỗ t ổ q tr a C {} s (a, f ) = 1
t t ừ q
✶✸
✶✳✷✳
❍➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ❤❛② ❤➔♠ ♥❤ä
✶✳✷✳✶✳ ❑❤→✐ ♥✐➺♠ ♠ð ✤➛✉
❚✐➳♣ t❤❡♦ t❛ ✤➲ ❝➟♣ ✤➳♥ ♠ët sè ❤➔♠ ✤➳♠ ♠ð rë♥❣ t❤÷í♥❣ ❞ò♥❣ tr♦♥❣
❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤à♥❤ ❧þ ✈➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ❈❤♦ f
❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ r > 0✱ ❦➼ ❤✐➺✉ nk (r, f ) ❧➔ sè ❝ü❝ ✤✐➸♠ ❜ë✐ ❝➢t ❝öt
❜ð✐ k tr♦♥❣ Dr ❝õ❛ f ✭tù❝ ❧➔ ❝→❝ ❝ü❝ ✤✐➸♠ ❜ë✐ l > k ❝❤➾ ✤÷ñ❝ t➼♥❤ k ❧➛♥
tr♦♥❣ tê♥❣ nk (r, f )✮✳ ❍➔♠
r
Nk (r, f ) =
nk (r, f ) − nk (0, f )
dt + nk (0, f ) log r
t
0
✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❜ë✐ ❝➢t ❝öt ❜ð✐ k ✱ tr♦♥❣ ✤â nk (0, f ) = limt→0 nk (r, f )✳
❙è k tr♦♥❣ nk (r, f ) ✤÷ñ❝ ❣å✐ ❧➔ ❝❤➾ sè ❜ë✐ ❝➢t ❝öt✳
❈❤♦ a ∈ C ∪ {∞}✱ ❦➼ ❤✐➺✉ n(r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔
❜ë✐✱ n(r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ f − a tr♦♥❣ Dr
r
N (r, 0; f ) = N (r,
1
)=
f −a
n(t,
1
1
) − n(0,
)
f −a
f −a
dt
t
0
1
) log r,
f −a
1
1
) − n(0,
)
n(t,
f −a
f −a
dt
t
+ n(0,
r
1
N (r, 0; f ) = N (r,
)=
f −a
0
+ n(0,
1
) log r.
f −a
❈❤♦ a ∈ C ∪ {∞}✱ ❦➼ ❤✐➺✉ nk) (r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔
❜ë✐✱ nk) (r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ f − a tr♦♥❣
Dr ✈î✐ ❜ë✐ ❦❤æ♥❣ ✈÷ñt q✉→ k ❀ n(k (r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸
❝↔ ❜ë✐✱ n(k (r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ f − a tr♦♥❣
✶✹
Dr ✈î✐ ❜ë✐ ➼t ♥❤➜t ❜➡♥❣ k ✳ ✣➦t
Nk) (r,
r
nk) (t,
r
nk) (t,
r
n(k (t,
r
n(k (t,
1
)=
f −a
1
1
) − nk) (0,
)
1
f −a
f −a
dt + nk) (0,
) log r,
t
f −a
0
N k) (r,
1
)=
f −a
1
1
) − nk) (0,
)
1
f −a
f −a
dt + nk) (0,
) log r,
t
f −a
0
N(k (r,
1
)=
f −a
1
1
) − n(k (0,
)
1
f −a
f −a
dt + n(k (0,
) log r,
t
f −a
0
N (k (r,
1
)=
f −a
1
1
) − n(k (0,
)
1
f −a
f −a
dt + n(k (0,
) log r,
t
f −a
0
tr♦♥❣ ✤â
nk) (0,
1
1
1
1
) = lim nk) (t,
), nk) (0,
) = lim nk) (t,
),
t→0
t→0
f −a
f −a
f −a
f −a
n(k (0,
1
1
1
1
) = lim n(k (t,
), n(k (0,
) = lim n(k (t,
).
t→0
t→0
f −a
f −a
f −a
f −a
❉➵ t❤➜②
Nk r,
1
f −a
= N r,
N r,
1
h
1
1
1
+N (2 r,
+· · ·+N (k r,
,
f −a
f −a
f −a
+ N (2 r,
1
h
= N2 r,
1
h
≤ N r,
1
h
✈➔
N2 (r, f ) = N (r, f ) + N (2 (r, f ).
❑➼ ❤✐➺✉ nE (r, a; f, g), (nE (r, a; f, g)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐
✭❦❤æ♥❣ ❦➸ ❜ë✐✮ t↕✐ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝ò♥❣ ❜ë✐ ❝õ❛ f − a ✈➔ g − a
✈➔ n0 (r, a; f, g)✱ (n0 (r, a; f, g)) sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐ ✭❦❤æ♥❣ ❦➸
❜ë✐✮ t↕✐ t➜t ❝↔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ f − a ✈➔ g − a✳ ✣➦t
✶✺
r
nE (t, a; f, g) − nE (0, a; f, g)
dt + nE (0, a; f, g) log r,
t
NE (r, a; f, g) =
0
r
N E (r, a; f, g) =
0
r
N0 (r, a; f, g) =
nE (t, a; f, g) − nE (0, a; f, g)
dt + nE (0, a; f, g) log r,
t
n0 (t, a; f, g) − n0 (0, a; f, g)
dt + n0 (0, a; f, g) log r,
t
0
r
N 0 (r, a; f, g) =
n0 (t, a; f, g) − n0 (0, a; f, g)
dt + n0 (0, a; f, g) log r.
t
0
tr♦♥❣ ✤â
nE (0, a; f, g) = lim nE (t, a; f, g), nE (0, a; f, g) = lim nE (t, a; f, g),
t→0
t→0
n0 (0, a; f, g) = lim n0 (t, a; f, g), n0 (t, a; f, g) = lim n0 (t, a; f, g).
t→0
t→0
❈→❝ ❤➔♠ NE (r, a; f, g), (N E (r, a; f, g)) ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐
✭❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐✮ t↕✐ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝ò♥❣ ❜ë✐ ❝õ❛ f − a
✈➔ g − a✱ N0 (r, a; f, g)❀ (N 0 (r, a; f, g)) ❧➔ ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ ✭❤➔♠ ✤➳♠
❦❤æ♥❣ ❦➸ ❜ë✐✮ t↕✐ t➜t ❝↔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ f − a ✈➔ g − a✳
◆❣♦➔✐ r❛✱ t❛ ✤à♥❤ ♥❣❤➽❛
δk (0, f ) = 1 − lim sup
r−→∞
Nk (r, 1/f )
,
T (r, f )
tr♦♥❣ ✤â k ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ tò② þ✳
❚✐➳♣ t❤❡♦ t❛ ✤➲ ❝➟♣ ✤➳♥ ❦❤→✐ ♥✐➺♠ ❝→❝ ❤➔♠ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➡♥❣
sè ❤❛② ❤➔♠ ♥❤ä✳ ❈❤♦ f, g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝
C ✈➔ a ∈ C ∪ {∞}✳ ❚❛ ♥â✐ r➡♥❣ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà a−❈▼ ♥➳✉
f − a ✈➔ g − a ❝â ❝❤✉♥❣ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✳
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C✱ t❛ ❦➼ ❤✐➺✉ Mf (C) ❧➔ t➟♣ ❝→❝
❤➔♠ ♥❤ä ✤è✐ ✈î✐ f ✳ ❑❤✐ f ❧➔ ❤➔♠ ♥❣✉②➯♥✱ t❛ ❦➼ ❤✐➺✉ Af (C) t❤❛② ❝❤♦
✶✻
Mf (C). ❚❛ ♥â✐ r➡♥❣ f ✱ g ∈ M(C)
❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ α ❈▼ ♥➳✉
f − α ✈➔ g − α ❝â ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐ ❧➔ ♥❤÷ ♥❤❛✉✳
❚❛ ♥❤➢❝ ❧↕✐ r➡♥❣ ♠ët ✤❛ t❤ù❝ P ∈ C[x] ✤÷ñ❝ ❣å✐ ❧➔
✤❛ t❤ù❝ ❞✉② ♥❤➜t
❝❤♦ ❧î♣ ❤➔♠ F ♥➳✉ ✈î✐ ♠é✐ ❝➦♣ ❤➔♠ f, g ∈ F ✱ ✤✐➲✉ ❦✐➺♥ P (f ) = P (g)
❦➨♦ t❤❡♦ f = g.
✶✳✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t
❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✳ ◆➳✉
✈➔ g ❝❤✉♥❣ ♥❤❛✉ 1 ❈▼ t❤➻ ♠ët tr♦♥❣ ❜❛ ❦❤↔♥❣ ✤à♥❤ s❛✉ ✤ó♥❣✿
✭❬✶✵❪✮✳
▼➺♥❤ ✤➲ ✶✳✶
f
(i) max{T (r, f ), T (r, g)}
1
1
N2 (r, f ) + N2 (r, ) + N2 (r, g) + N2 (r, )
f
g
+ S(r, f ) + S(r, g);
(ii) f ≡ g;
(iii) f g ≡ 1.
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✱ p ✈➔ k ❧➔
❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❑❤✐ ✤â
✭❬✽❪✮✳
▼➺♥❤ ✤➲ ✶✳✷
Np (r,
Np (r,
1
f
)
(k)
1
f (k)
)
1
T (r, f (k) ) − T (r, f ) + Np+k (r, ) + S(r, f );
f
1
kN (r, f ) + Np+k (r, ) + S(r, f ).
f
❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♥❣✉②➯♥ ❦❤→❝ ❤➡♥❣✱ n ✈➔
k ❧➔ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈î✐ n > k ✳ ✣➦t
▼➺♥❤ ✤➲ ✶✳✸
✭❬✶✷❪✮✳
P (z) = am z m + am−1 z m−1 + · · · + a1 z + a0
❧➔ ♠ët ✤❛ t❤ù❝ ❦❤→❝ ✵✱ tr♦♥❣ ✤â a0, a1, . . . , am−1, am ❧➔ ❝→❝ ❤➡♥❣ sè ♣❤ù❝✳
◆➳✉ [f nP (f )](k)[gnP (g)](k) ≡ 1 t❤➻ P (z) q✉② ✈➲ ✤÷ñ❝ ♠ët ✤ì♥ t❤ù❝
❦❤→❝ ✵✱ tù❝ ❧➔✱ P (z) = aiz i ≡ 0 tr♦♥❣ ✤â i ∈ {0, 1, . . . , m}✳ ◆❣♦➔✐ r❛✱
✶✼
√
√
f (z) = c1 / n ai ecz ✱ g(z) = c2 / n ai e−cz ✱
tr♦♥❣ ✤â c1✱ c2 ✈➔ c ❧➔ ❜❛ ❤➡♥❣ sè
t❤ä❛ ♠➣♥ (−1)k (c1c2)n+i[(n + i)c]2k = 1.
l
▼➺♥❤ ✤➲ ✶✳✹ ✭❬✶❪✮✳ ❈❤♦ P (x) = b(x − a1 )n i=2 (x − ai )k ∈ C[x]
✭ai = aj , ∀i = j) ✈î✐ l 2 ✈➔ n max{k2, . . . , kl } ✈➔ ✤➦t k = li=2 ki✳
●å✐ f ✱ g ∈ M(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ ✤➦t θ = P (f )f P (g)g ✳ ◆➳✉
θ ❧➔ ♠ët ❤➔♠ t❤✉ë❝ Mf (C) ∩ Mg (C)✱ t❤➻ t❛ ❝â✿
◆➳✉ l = 2 t❤➻ n t❤✉ë❝ {k, k + 1, 2k, 2k + 1, 3k + 1};
◆➳✉ l = 3 t❤➻ n t❤✉ë❝ { k2 , k + 1, 2k + 1, 3k2 − k, 3k3 − k};
◆➳✉ l 4 t❤➻ n = k + 1.
❍ì♥ ♥ú❛✱ ♥➳✉ f ✱ g t❤✉ë❝ A(C)✱ t❤➻ θ ❦❤æ♥❣ t❤✉ë❝ t➟♣ Af (C).
i
❈❤ó þ✳
❚ø ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ▼➺♥❤ ✤➲ ✶✳✹✱ t❛ ❞➵ ❞➔♥❣ s✉② t❛ ❝→❝ ❦❤➥♥❣
✤à♥❤ ❝õ❛ ▼➺♥❤ ✤➲ ✶✳✹ ✈➝♥ ❝á♥ ✤ó♥❣ ❦❤✐ θ = 1 ✈➔ f ✱ g ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥
❤➻♥❤ ❦❤→❝ ❤➡♥❣✳
▼➺♥❤ ✤➲ ✶✳✺
✭❬✼❪✮✳
❈❤♦ f
∈ M(C)✱
1
T (r, f ) − N (r, )
f
❦❤✐ ✤â t❛ ❝â
T (r, f ) − N (r,
1
) + S(r, f ).
f
❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t✱ α(z)
❧➔ ♠ët ❤➔♠ ♥❤ä ❝õ❛ f ✈➔ g✱ k ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ◆➳✉ f (k) ✈➔ g(k)
❝❤✉♥❣ ♥❤❛✉ α(z) ❈▼ ✈➔
▼➺♥❤ ✤➲ ✶✳✻
✭❬✾❪✮✳
2Θ(∞, g) + δk+2 (0, g) + (k + 2)Θ(∞, f ) + δk+2 (0, f ) > k + 5,
t❤➻ f ≡ g ❤♦➦❝ f (k)g(k) = α(z)2✳ ◆❣♦➔✐ r❛✱ tr♦♥❣ tr÷í♥❣ ❤ñ♣ k = 1✱
❦❤↔♥❣ ✤à♥❤ ❝õ❛ ♠➺♥❤ ✤➲ ♥➔② ✤ó♥❣ ❦❤✐ f ✈➔ g ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤
❦❤→❝ ❤➡♥❣✳
❈❤ù♥❣ ♠✐♥❤✳
g (k)
f (k)
✈➔
❝❤✉♥❣ ♥❤❛✉ 1 ❈▼✱ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✱
❱➻
α(z)
α(z)
✶✽
t❛ ❣✐↔ t❤✐➳t r➡♥❣ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❧➔ ✤ó♥❣
f (k)
g (k)
max{T (r,
), T (r,
)}
α(z)
α(z)
N2 (r, f (k) ) + N2 (r,
+ N2 (r,
1
g (k)
1
f (k)
) + N2 (r, g (k) )
) + S(r, f (k) ) + S(r, g (k) )
+ S(r, f ) + S(r, g).
T (r, f (k) ) = T (r,
f (k)
.α(z))
α(z)
T (r,
f (k)
) + S(r, f ).
α(z)
◆❤÷ ✈➟②
T (r, f (k) )
N2 (r, f (k) ) + N2 (r,
1
f
) + N2 (r, g (k) ) + N2 (r,
(k)
1
g (k)
)
✭✶✳✶✮
+ S(r, f (k) ) + S(r, g (k) ) + S(r, f ) + S(r, g).
❚÷ì♥❣ tü t❛ ❝â✱
T (r, g (k) )
N2 (r, f (k) ) + N2 (r,
1
1
f
) + N2 (r, g (k) ) + N2 (r,
(k)
g (k)
+ S(r, f (k) ) + S(r, g (k) ) + S(r, f ) + S(r, g).
)
✭✶✳✷✮
❚❛ ❜✐➳t r➡♥❣
T (r, f (k) ) = m(r, f (k) ) + N (r, f (k) )
m(r,
f (k)
) + m(r, f ) + N (r, f ) + kN (r, f )
f
(k + 1)T (r, f ) + S(r, f ).
✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ S(r, f (k) ) = S(r, f )✱ S(r, g (k) ) = S(r, g). ❚ø ✭✶✳✶✮✱
❜➡♥❣ ✈✐➺❝ sû ❞ö♥❣ ▼➺♥❤ ✤➲ ✶✳✷✱ t❛ ❝â
T (r, f (k) )
1
N2 (r, f (k) ) + T (r, f (k) ) − T (r, f ) + Nk+2 (r, ) + N2 (r, g (k) )
f
1
+ kN (r, g) + Nk+2 (r, ) + S(r, f ) + S(r, g).
g
ữ
T (r, f )
1
2N (r, f ) + Nk+2 (r, ) + (k + 2)N (r, g)
f
1
+ Nk+2 (r, ) + S(r, f ) + S(r, g).
g
ữỡ tỹ tứ t õ
T (r, g)
1
2N (r, g) + Nk+2 (r, ) + (k + 2)N (r, f )
g
1
+ Nk+2 (r, ) + S(r, f ) + S(r, g).
f
ổ t t tờ qt t õ t tt tỗ t ởt t I ợ
ở ổ s T (r, f )
T (r, g) ợ r I. õ tứ t
t r
T (r, g)
2(1 (, g))T (r, g) + (1 k+2 (0, g))T (r, g) + S(r, g)
+ (k + 2)(1 (, f ))T (r, g) + (1 k+2 (0, f ))T (r, g).
ữ t õ
(2(, g) + k+2 (0, g) + (k + 2)(, f )
+ k+2 (0, f ) (k + 5))T (r, g)
S(r, g)
ợ r I t ợ
2(, g) + k+2 (0, g) + (k + 2)(, f ) + k+2 (0, f ) > k + 5.
t õ f (k) g (k) f (k) g (k) = (z)2 f (k)
g (k) t f (z) = g(z)+ P (z) tr õ P (z) ởt tự ợ
t k 1 r trữớ ủ k = 1 t õ f = g + c tr õ c
ởt số P (z) 0 t t ỵ ỡ tự
✷✵
♥❤ä t❛ ❝â
T (r, g)
1
1
N (r, g) + N (r, ) + N (r,
) + S(r, g)
g
g + P (z)
1
1
N (r, g) + Nk+2 (r, ) + Nk+2 (r, ) + S(r, g)
g
f
(3 − (Θ(∞, g) + δk+2 (0, g) + δk+2 (0, f )))T (r, g) + S(r, g)
✭✶✳✻✮
❚ø ✭✶✳✻✮✱ t❛ ❝â
(Θ(∞, g) + δk+2 (0, g) + δk+2 (0, f ) − 2)T (r, g)
S(r, g)
✭✶✳✼✮
✈î✐ r ∈ I ✳
▼➦t ❦❤→❝✱ tø ✤✐➲✉ ❦✐➺♥ t❛ ❝â
2Θ(∞, g) + δk+2 (0, g) + (k + 2)Θ(∞, f ) + δk+2 (0, f ) > k + 5
✈î✐ r ∈ I, t❛ s✉② r❛
(Θ(∞, g) + δk+2 (0, g) + δk+2 (0, f )) > (k + 5) − (k + 3) = 2
✈î✐ r ∈ I. ❚ø ✭✶✳✼✮✱ t❛ s✉② r❛ ♠➙✉ t❤✉➝♥✳ ◆❤÷ ✈➟②✱ t❛ t❤✉ ✤÷ñ❝ P (z) ≡ 0✱
tù❝ ❧➔ f ≡ g. ▼➺♥❤ ✤➲ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
▼➺♥❤ ✤➲ s❛✉ ❧➔ ♠ët ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ▼➺♥❤ ✤➲ ✶✳✻✿
❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♥❣✉②➯♥ s✐➯✉ ✈✐➺t✱ α(z) ❧➔
♠ët ❤➔♠ ♥❤ä ❝õ❛ f ✈➔ g✱ k ❧➔ ♠ët sè ♥❣✉②➯♥✳ ◆➳✉ f (k) ✈➔ g(k) ❝❤✉♥❣
♥❤❛✉ α(z) CM ✈➔
▼➺♥❤ ✤➲ ✶✳✼
✭❬✾❪✮✳
δk+2 (0, g) + δk+2 (0, f ) > 1,
❦❤✐ ✤â f ≡ g ❤♦➦❝ f (k)g(k) = α(z)2✳
