Tập duy nhất cho các hàm phân hình với giá trị khuyết

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

❚❡✉✐ ❱❖◆●❉❆▲❆

❚❾P ❉❯❨ ◆❍❻❚ ❈❍❖ ❈⑩❈ ❍⑨▼ P❍❹◆ ❍➐◆❍
❱❰■ ●■⑩ ❚❘➚ ❑❍❯❨➌❚

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✺


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

❚❡✉✐ ❱❖◆●❉❆▲❆

❚❾P ❉❯❨ ◆❍❻❚ ❈❍❖ ❈⑩❈ ❍⑨▼ P❍❹◆ ❍➐◆❍
❱❰■ ●■⑩ ❚❘➚ ❑❍❯❨➌❚
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿
P●❙✳❚❙✳ ❍⑨ ❚❘❺◆ P❍×❒◆●

❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✺

▲í✐ ❝❛♠ ✤♦❛♥

✐

❇↔♥ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ sü ♥❣❤✐➯♥ ❝ù✉ ✤ë❝ ❧➟♣ ❝õ❛ tæ✐ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥
❝õ❛ P●❙✳❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣✱ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤÷❛ tø♥❣
✤÷ñ❝ ❝æ♥❣ ❜è tr♦♥❣ ❝→❝ ❝æ♥❣ tr➻♥❤ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝ ð ❱✐➺t ◆❛♠✳
❍å❝ ✈✐➯♥

❚❡✉✐ ❱❖◆●❉❆▲❆

❳→❝ ♥❤➟♥
❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❚♦→♥

❳→❝ ♥❤➟♥
❝õ❛ ♥❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝

P●❙✳❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣


ớ ỡ



ữủ tỹ t t trữớ ồ ữ
ữợ sỹ ữợ ồ ừ P r
Pữỡ 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 tr ồ t
ổ tr rữớ ồ ữ t
t tr tổ ỳ tự ỡ s tr ữớ
ự ồ
ổ ụ t ỡ t ổ tr Pỏ t
ồ ữ t ồ tổ t
từ tử tổ t
ổ tọ ỏ t ỡ s s tợ ỳ ữớ t tr
ừ ỳ ữớ ổ ở s õ ổ
ọ tổ t ổ
ổ ụ ỷ ớ ỡ tr ợ ồ
ở ú ù tổ tr q tr ồ t
ổ t tr ọ ỳ t sõt t rt
ữủ sỹ t t ừ t ổ ỗ
t




✐✐✐

▼ö❝ ❧ö❝
▼Ð ✣❺❯

✶

✶

✸

▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ tr♦♥❣ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛

✶✳✶✳ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳ ❈→❝ ✤à♥❤ ❧þ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✶✳ ❈æ♥❣ t❤ù❝ ❏❡♥s❡♥ ✈➔ ✤à♥❤ ❧þ ❝ì t❤ù ♥❤➜t
✶✳✷✳✷✳ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳

✳
✳

✸
✽
✽
✶✵

❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈î✐ ✤✐➲✉ ❦✐➺♥ ❝❤ù❛
❣✐→ trà ❦❤✉②➳t

✶✻

✷✳✶✳ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✶✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳ ❳→❝ ✤à♥❤ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❜ð✐ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝❤ù❛ ❣✐→
trà ❦❤✉②➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻
✶✻
✷✵
✷✼

❑➳t ❧✉➟♥

✹✸

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✸




é
ữủ ự tọ ởt tr
t ự C ữủ ởt t ữủ
ổ t ở ừ t tr ổ tr ừ
ữủ ỗ ự t
t s ự sỹ
ữủ ừ ởt t ỳ tỷ t út ữủ sỹ q t ừ
t ồ tr ữợ
rss t ự t
ữủ ừ ởt t ỳ
ự ừ rss ự
trũ ú t S = {z :
z n + az nm + b = 0} tr õ m, n số ữỡ s m
n ổ õ ữợ số n > 2m + 8 (m 2) a, b số
ổ s ữỡ tr zn + aznm + b = 0 ổ õ
ở ự
f g tọ (, f ) >

11
12 , (, g)
6

>

11

12

Ef (S) = Eg (S)

t q tr ừ
t ởt số f g, tr õ
số õ ự số t t s õ t ồ
t tử rở t ữợ ự ợ ố t r
số ợ õ ự số t trũ
r r r r
r
ợ ố t ữủ ởt
t số õ ự tr t ú tổ
ồ t t ợ tr
t f g tr õ S = {z : z 7 z 1 = 0}.


✷

✳ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔
✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ♥➠♠ ✷✵✶✸ ❜ð✐ ❆✳ ❇❛♥❡r❥❡❡ ✈➔ ❙✳ ▼❛❥✉♠❞❡r tr♦♥❣ ❬✶❪
✈➔ ❬✷❪✳ ▲✉➟♥ ✈➠♥ ♥➔② ❣ç♠ ❝â ❤❛✐ ❝❤÷ì♥❣ ♥❤÷ s❛✉✿
❈❤÷ì♥❣ ✶✿ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì sð tr♦♥❣ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✳ ❚r♦♥❣
❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ tr♦♥❣ ❧þ t❤✉②➳t
♣❤➙♥ ❜è ❣✐→ trà ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ♠ët sè ❦❤→✐ ♥✐➺♠
✈➔ ❦➼ ❤✐➺✉ sû ❞ö♥❣ tr♦♥❣ ❈❤÷ì♥❣ ✷✳
❈❤÷ì♥❣ ✷✿ ❚➟♣ ❣✐→ trà ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈î✐ ❣✐→ trà
❦❤✉②➳t✳ ✣➙② ❧➔ ❝❤÷ì♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❧↕✐ ♠ët
sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❆✳ ❇❛♥❡r❥❡❡ ✈➔ ❙✳ ▼❛❥✉♠❞❡r ✈➲ ✤✐➲✉ ❦✐➺♥
✤↕✐ sè ❝â ❝❤ù❛ ❣✐→ trà ❦❤✉②➳t ✤➸ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❧➔ ❜➡♥❣ ♥❤❛✉✳

❦❤✉②➳t✑




ữỡ

ởt số tự ỡ tr ỵ
tt


t t

rữợ t t ởt số ổ ỹ
ừ tữớ ữủ sỷ ử tr ỵ tt ố
f tr t ự C
z0 C ữủ ồ ổ ở k > 0 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 > 0 ừ
f (z) tr U ừ z0 f ữủ ữợ
1
.h(z) tr õ h(z) ổ
f (z) =
(z z0 )k
trt t tr U ừ z0.
ợ ộ số tỹ x > 0

log+ x = max{log x, 0}.


✹

❑❤✐ ✤â log x = log+ x − log+(1/x).
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C✱ r > 0✱ ✈î✐ ♠é✐ ϕ ∈ [0; 2π], t❛
❝â
log |f (reiϕ )| = log+ |f (reiϕ )| − log+

♥➯♥
2π

2π

1
log f (reiϕ ) dϕ =
2π

1
2π

1
,
f (reiϕ )

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| |r|}✳
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❍➔♠
r

N (r, ∞; f ) = N (r, f ) =

n(t, f ) − n(0, f )
dt + n(0, f ) log r
t

0

✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ ❝õ❛ f ✭❝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

t→0


✺

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳

❍➔♠
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✐ ❧➔ ❝→❝ ❤➔♠
◆❡✈❛♥❧✐♥♥❛✳
❚✐➳♣ 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(t,

) − n(0,
)
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♦♥❣


✻

✈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
Dr

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

❑➼ ❤✐➺✉ 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 (r, a; f, g) =

nE (t, a; f, g) − nE (0, a; f, g)
dt + nE (0, a; f, g) log r,
t

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✳
✣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)

m(r, fν );
ν=1
p

fν ) ≤

N (r,
ν=1
p

(4)

m(r, fν ) + log p;

N (r, fν );
ν=1
p

fν ) ≤

N (r,
ν=1

N (r, fν );
ν=1


✽

p

(5)

p

fν ) ≤

T (r,
ν=1
p

(6)

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

✈➔

ν=1

p

log

+

p

aν

+

log+ |aν | + log p.

log (p max |aν |)
ν=1,...,p

ν=1

✶✳✷✳

log+ |aν |

aν

ν=1

❈→❝ ✤à♥❤ ❧þ ❝ì ❜↔♥

✶✳✷✳✶✳ ❈æ♥❣ t❤ù❝ ❏❡♥s❡♥ ✈➔ ✤à♥❤ ❧þ ❝ì t❤ù ♥❤➜t
✣à♥❤ ❧þ ✶✳✷

✭❈æ♥❣ t❤ù❝ P♦✐ss♦♥ ✲ ❏❡♥s❡♥✮✳ ❈❤♦ f (z) ≡ 0 ❧➔ ♠ët ❤➔♠

♣❤➙♥ ❤➻♥❤ tr♦♥❣ ❤➻♥❤ trá♥ {|z| ≤ R} ✈î✐ 0 < R < ∞✳ ●✐↔ sû a1 , ..., ap
❧➔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✱ b1 , ..., bq ❧➔ ❝→❝ ❝ü❝ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐ tr♦♥❣
❤➻♥❤ trá♥ ✤â✳ ❑❤✐ ✤â ✈î✐ ♠é✐ z tr♦♥❣ {|z| < R} ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❦❤æ♥❣
✤✐➸♠ ❤❛② ❝ü❝ ✤✐➸♠ ❝õ❛ f ✱ t❛ ❝â
2π

1
log |f (z)| =
2π

R2 − |z|2
iϕ
log
|f
(Re
)|dϕ
|Reiϕ − z|2

0
p

−
i=1

R 2 − ai z
+
log

R(z − ai )

q

R2 − bj z
log
.
R(z
−
b
)
j
j=1

❈æ♥❣ t❤ù❝ P♦✐ss♦♥✲❏❡♥s❡♥ ❝❤➾ r❛ r➡♥❣✱ ♥➳✉ ❜✐➳t ❣✐→ trà ❝õ❛
♠♦❞✉❧✉s f (z) tr➯♥ ❜✐➯♥✱ t↕✐ ❝→❝ ❝ü❝ ✤✐➸♠ ✈➔ t↕✐ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛
❤➔♠ f (z) tr♦♥❣ |z| < R t❤➻ t❛ ❝â t❤➸ t➻♠ ✤÷ñ❝ ❣✐→ trà ❝õ❛ ♠♦❞✉❧✉s f (z)
t↕✐ ❝→❝ ❣✐→ trà z ❦❤→❝ ❜➯♥ tr♦♥❣ ✤➽❛ |z| < R.
◆❤➟♥ ①➨t✳




q

ợ |z| < R t õ
2

d
1

=
.
R2 |z|2
|R2 ei z|2 2
1

0

q

ổ tự s 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
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 ord0 f + ...., ord0 f Z, cf số ổ ọ
t tr tr rt ừ f t 0

ỵ s ởt t ừ ổ tự s ữủ ồ
ỵ ỡ tự t
ỵ ỵ ỡ tự t f 0 ởt
tr C õ ợ ộ r > 0 t õ

1
f

+ N r,

(1)

T (r, f ) = m r,

(2)

ợ ộ số ự a C,

T (r, f ) m r,

1
f a

1
f

+ log |cf | ;

N r,

log

1
f a

c1
+ log+ |a| + log 2,
f a

tr õ cf số 0 ọ t tr tr r ừ

f tr 0 c1 /(f a) số 0 ọ t tr
tr r ừ 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 → ∞✳

✶✳✷✳✷✳ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐

❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ r > 0✳ ❑➼ ❤✐➺✉
Nram (r, f ) = N r,

1
f

+ 2N (r, f ) − N (r, f )

✈➔ ❣å✐ ❧➔ ❤➔♠ ❣✐→ 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➟♣ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳

❑➼ ❤✐➺✉ 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) − aj | ≤ |f (z) − aj |, ✈î✐ ♠å✐ 1 ≤ j ≤ q.
❑❤✐ ✤â✱ ✈î✐ j = j0✱ t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ t❛ ❝â |f (z) − aj | ≥ δ/2✳

❈❤ù♥❣ ♠✐♥❤✳

0


✶✶

◆❤÷ ✈➟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=1

j=j0

+ (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 − 1) log f (z) ≤

q

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
0
q

log(
0

log |f (reiθ )|

j=1

|f (reiθ )| dθ
+ O(1).

|f (reiθ ) − aj |) 2π

dθ
2π


✶✷

❚❛ ❝â
2π

log |f (reiθ ) − aj |
0

✈➔

dθ
1
= N (r,
) − N (r, f ) + log |cf −aj |
2π
f − aj

2π

log |f (reiθ )|

◆❤÷ ✈➟②

dθ

= N (r, 1/f ) − N (r, f ) + log |cf |.
2π

0

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

|f (reiθ )| dθ
+
|f (reiθ ) − aj |) 2π

log(
j=1

0

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(
0

j=1

|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

log(
j=1

0

|f (reiθ )| dθ
= 1/α
|f (reiθ ) − aj |) 2π
2π

≤
0

2π

q

log(
j=1

0

|f (reiθ )|
dθ
|f (reiθ ) − aj |)α 2π

q

dθ
f (reiθ )
|
log(
|
f (reiθ ) − aj )α 2π
j=1

q

2π

≤ log(
j=1 0

f (reiθ )
dθ
|
|
iθ
α
f (re ) − aj ) ) 2π

❙û ❞ö♥❣ log+(x + y) ≤ log+ x + log+ y✱ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤÷ñ❝ ✈✐➳t ❧↕✐
♥❤÷ s❛✉
q

2π

|

log(
j=1 0

dθ
f (reiθ )
ρ
+

|
≤
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)
r ≥ r0 , r = E t❛ ❝â

log+

◆❤÷ ✈➟② ✱ ✈î✐

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,

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳

δ(a, f ) ✤÷ñ❝ ❣å✐ ❧➔ sè ❦❤✉②➳t✱ Θ(a, f ) ❣å✐ ❧➔ sè ❦❤✉②➳t

❦❤æ♥❣ ❦➸ ❜ë✐✱ θ(a, f )

❣å✐ ❧➔ ❜➟❝ ❝õ❛ ❜ë✐ ❝õ❛ sè ❦❤✉②➳t✳

✶✳ ◆➳✉ f (z) = a ✈æ ♥❣❤✐➺♠ t❤➻ N (r, f −1 a ) = 0 ✈î✐ ♠å✐ r
s✉② r❛ δ(a, f ) = 1✳ ❈❤➥♥❣ ❤↕♥ f (z) = ez t❤➻ δ(0, f ) = 1.
✷✳ ◆➳✉ N (r, f −1 a ) = o(T (r, f )) ❦❤✐ ✤â δ(a, f ) = 1✳ ◆❤÷ ✈➟② sè
❦❤✉②➳t ❜➡♥❣ 1 ❦❤✐ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ q✉→ ➼t s♦ ✈î✐ ❝➜♣ t➠♥❣
❝õ❛ ♥â✳
◆❤➟♥ ①➨t✳


✶✺

✸✳ ❱î✐ ♠é✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ✈➔ a ∈ C✱ t❛ ❧✉æ♥ ❝â
0

δ(a, f )

Θ(a, f )

1.

✣à♥❤ ❧þ s❛✉ ❝❤♦ t❛ ♠ët t➼♥❤ ❝❤➜t ❝õ❛ sè ❦❤✉②➳t✱ t❤÷í♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ❇ê
✤➲ q✉❛♥ ❤➺ sè ❦❤✉②➳t✳
✣à♥❤ ❧þ ✶✳✺✳

❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ C✳ ❑❤✐ ✤â t➟♣

❤ñ♣ ❝→❝ ❣✐→ trà ❝õ❛ a ♠➔ Θ(a, f ) > 0 ❝ò♥❣ ❧➢♠ ❧➔ ✤➳♠ ✤÷ñ❝✱ ✤ç♥❣ t❤í✐
t❛ ❝â

δ(a, f ) + θ(a, f )
a∈C

Θ(a, f )
a∈C

2.


✶✻

❈❤÷ì♥❣ ✷

❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤
✈î✐ ✤✐➲✉ ❦✐➺♥ ❝❤ù❛ ❣✐→ trà ❦❤✉②➳t

✷✳✶✳

❍➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà

✷✳✶✳✶✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉

❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tæ✐ s➩ ❣✐î✐ t❤✐➺✉ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❝❤ù♥❣
♠✐♥❤ ♠ët sè ❜ê ✤➲ sû ❞ö♥❣ tr♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ✈➲
t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t✳ ❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ t❛ ❦þ ❤✐➺✉
T (r) = max{T (r, f ), T (r, g)} ✈➔ S(r, f ), S(r) ❧➔ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ①→❝ ✤à♥❤
♥❤÷ s❛✉✿
S(r, f ) = o(T (r, f )),

S(r) = o(T (r))

❦❤✐ r → ∞, r ∈/ E ✱ tr♦♥❣ ✤â E ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❝→❝ sè t❤ü❝
❞÷ì♥❣ ❝â ✤ë ✤♦ t✉②➳♥ t➼♥❤ ❤ú✉ ❤↕♥✳
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ sè ✈➔ a
❧➔ ♠ët sè ♣❤ù❝ ❤ú✉ ❤↕♥✳ ❚❛ ♥â✐ r➡♥❣ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà a ❈▼
♥➳✉ f − a ✈➔ g − a ❝â ❝❤✉♥❣ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ✈î✐ ❜ë✐ ♥❤÷ ♥❤❛✉✳ ▼ët ❝→❝❤
t÷ì♥❣ tü✱ ❝❤ó♥❣ t❛ ♥â✐ r➡♥❣ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà a ■▼ ♥➳✉ f − a
✈➔ g − a ❝â ❝❤✉♥❣ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ✭❦❤æ♥❣ ❝➛♥ ✤➸ þ ✤➳♥ ❜ë✐✮✳ ◆❣♦➔✐ r❛✱
t❛ ♥â✐ r➡♥❣ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ ∞ ❈▼ ✭❤♦➦❝ ■▼✮✱ ♥➳✉ 1/f ✈➔ 1/g ❝❤✉♥❣


✶✼

♥❤❛✉ ❣✐→ trà 0 ❈▼ ✭t÷ì♥❣ ù♥❣ ■▼✮✳
❈❤♦ S ❧➔ t➟♣ ❝→❝ ♣❤➙♥ tû ♣❤➙♥ ❜✐➺t ❝õ❛ C ∪ {∞}✱ ❦þ ❤✐➺✉
{z ∈ C : f (z) = a},

Ef (S) =
a∈S

tr♦♥❣ ✤â ♠é✐ ❦❤æ♥❣ ✤✐➸♠ ✤÷ñ❝ ✤➳♠ ❜➡♥❣ sè ❜ë✐ ❝õ❛ ♥â✳ ❚❛ ❝ô♥❣ ❦þ
❤✐➺✉
{z ∈ C : f (z) = a},

E f (S) =
a∈S

tr♦♥❣ ✤â ♠é✐ ❦❤æ♥❣ ✤✐➸♠ ✤÷ñ❝ ✤➳♠ ✶ ❧➛♥✳
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ❚❛ ♥â✐ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ t➟♣ S
❈▼ ✭■▼✮ ♥➳✉ Ef (S) = Eg (S) ✭t÷ì♥❣ ù♥❣ E f (S) = E g (S)✮✳
✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ❚➟♣ S ⊂ C ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝❤♦ ❝→❝
❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭❤➔♠ ♥❣✉②➯♥✮ ❦➸ ❝↔ ❜ë✐✱ ❦➼ ❤✐➺✉ ❧➔ ❯❘❙▼✲❈▼ ✭t÷ì♥❣
ù♥❣ ❧➔ ❯❘❙❊✲❈▼✮✱ ♥➳✉ ✈î✐ ♠é✐ ❝➦♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭❤➔♠ ♥❣✉②➯♥✮ ❦❤→❝
❤➡♥❣ f ✈➔ g t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ Ef (S) = Eg (S)✱ t❛ ❧✉æ♥ ❝â f ≡ g. ❚➟♣
S ⊂ C ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭❤➔♠
♥❣✉②➯♥✮ ❦❤æ♥❣ ❦➸ ❜ë✐✱ ❦➼ ❤✐➺✉ ❧➔ ❯❘❙▼✲■▼ ✭t÷ì♥❣ ù♥❣ ❧➔ ❯❘❙❊✲■▼✮✱
♥➳✉ ✈î✐ ♠é✐ ❝➦♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭❤➔♠ ♥❣✉②➯♥✮ ❦❤→❝ ❤➡♥❣ f ✈➔ g t❤ä❛
♠➣♥ ✤✐➲✉ ❦✐➺♥ E f (S) = E g (S)✱ t❛ ❧✉æ♥ ❝â f ≡ g. ❈→❝ t➟♣ ❯❘❙▼✲❈▼✱
❯❘❙▼✲■▼✱ ❯❘❙❊✲❈▼✱ ❯❘❙❊✲■▼ ✤÷ñ❝ ❣å✐ ❝❤✉♥❣ ❧➔ t➟♣ ①→❝ ✤à♥❤ ❞✉②
♥❤➜t✳
✣à♥❤ ♥❣❤➽❛ ✷✳✹✳ ▼ët ✤❛ t❤ù❝ P tr♦♥❣ C ✤÷ñ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ ❞✉② ♥❤➜t
♠↕♥❤ ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭❤❛② ❤➔♠ ♥❣✉②➯♥✮ ♥➳✉ ✈î✐ ♠é✐ ❝➦♣ ❝→❝
❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭t÷ì♥❣ ù♥❣ ❤➔♠ ♥❣✉②➯♥✮ f, g✱ ✤✐➲✉ ❦✐➺♥ P (f ) = cP (g)
❦➨♦ t❤❡♦ f = g✱ tr♦♥❣ ✤â c ❧➔ ♠ët ❤➡♥❣ sè t❤➼❝❤ ❤ñ♣ ❦❤→❝ 0✳ ❚r♦♥❣
tr÷í♥❣ ❤ñ♣ ♥➔② t❛ ❣å✐ P ❧➔ ❙❯P▼ ✭❙❯P❊✮✳ ▼ët ❝→❝❤ ❦❤→❝✱ ♠ët ✤❛
t❤ù❝ P tr♦♥❣ C ✤÷ñ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤

✭❤❛② ❤➔♠ ♥❣✉②➯♥✮ ♥➳✉ ✈î✐ ♠é✐ ❝➦♣ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭t÷ì♥❣ ù♥❣ ❤➔♠


✶✽

♥❣✉②➯♥✮ f, g✱ ✤✐➲✉ ❦✐➺♥ P (f ) = P (g) ❦➨♦ t❤❡♦ f = g✱ tr♦♥❣ tr÷í♥❣ ❤ñ♣
♥➔② t❛ ❣å✐ P ❧➔ ❯P▼ ✭❯P❊✮✳
❈❤♦ P ❧➔ ✤❛ t❤ù❝ ❜➟❝ n tr♦♥❣ C ❝❤➾ ❝â ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ✤ì♥ ✈➔ S ❧➔
t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ P ✳ ◆➳✉ S ❧➔ ❯❘❙▼ ✭❯❘❙❊✮ t❤➻ tø
✤à♥❤ ♥❣❤➽❛ t❛ s✉② r❛ P ❧➔ ❯P▼ ✭❯P❊✮✳ ❚✉② ♥❤✐➯♥✱ ✈➜♥ ✤➲ ♥❣÷ñ❝ ❧↕✐✱
tr♦♥❣ tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t ❦❤æ♥❣ ✤ó♥❣✳ ❈❤➥♥❣ ❤↕♥
P (z) = az + b (a = 0)

❧➔ ♠ët ❯P▼ ♥❤÷♥❣ ✈î✐ f = (−b/a)ez ✈➔ g = (−b/a)e−z t❤➻ t❛ t❤➜②
Ef (S) = Eg (S)✱ tr♦♥❣ ✤â S = {−b/a} ❧➔ t➟♣ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ P.
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ k ❧➔ ♠ët sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ ❤♦➦❝ ✈æ
❝ò♥❣✳ ❱î✐ a ∈ C ∪ {∞}✱ ❦þ ❤✐➺✉ Ek (a; f ) ❧➔ ❝→❝ t➟♣ ❝õ❛ a✲✤✐➸♠ ❝õ❛ f ✱
tr♦♥❣ ✤â ♠é✐ a✲✤✐➸♠ ❜ë✐ m ✤÷ñ❝ t➼♥❤ m ❧➛♥ ♥➳✉ m ≤ k ✈➔ k + 1 ❧➛♥
♥➳✉ m > k✳
✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ◆➳✉ Ek (a; f ) = Ek (a; g)✱ t❛ ♥â✐ r➡♥❣ f, g ❝❤✉♥❣ ♥❤❛✉
❣✐→ trà a ✈î✐ trå♥❣ sè k✳
❚❛ ✈✐➳t f, g ❝❤✉♥❣ (a, k) ♥❣❤➽❛ ❧➔ f, g ❝❤✉♥❣ ❣✐→ trà a ✈î✐ trå♥❣ sè k✳
❍✐➸♥ ♥❤✐➯♥✱ ♥➳✉ f, g ❝❤✉♥❣ (a, k) t❤➻ f, g ❝❤✉♥❣ (a, p) ✈î✐ ♠ët sè ♥❣✉②➯♥
p : 0 ≤ p < k ✳ ❍ì♥ ♥ú❛ ❞➵ t❤➜② r➡♥❣ f, g ❝❤✉♥❣ ❣✐→ trà a ■▼ ❤♦➦❝ ❈▼
♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f, g ❝❤✉♥❣ (a, 0) ❤♦➦❝ (a, ∞) t÷ì♥❣ ù♥❣✳
✣à♥❤ ♥❣❤➽❛ ✷✳✻✳ ❈❤♦ S ❧➔ t➟♣ ❝→❝ ♣❤➛♥ tû ♣❤➙♥ ❜✐➺t ❝õ❛ C ∪ {∞} ✈➔
k ❧➔ ♠ët sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ ❤♦➦❝ ∞✳ ❑þ ❤✐➺✉
Ef (S, k) =

Ek (a; f ).

a∈S

❘ã r➔♥❣ Ef (S) = Ef (S, ∞) ✈➔ E f (S) = Ef (S, 0)✳
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ a ∈ C ∪ {∞}✳ ❑þ ❤✐➺✉ N (r, a; f | = 1)
❧➔ ❤➔♠ ✤➳♠ t↕✐ ❝→❝ a−✤✐➸♠ ✤ì♥ ❝õ❛ f ✳ ❱î✐ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ m✱
❝❤ó♥❣ t❛ ❦þ ❤✐➺✉
N (r, a; f | ≤ m) (N (r, a; f | ≥ m))


✶✾

❧➛♥ ❧÷ñt ❧➔ ❤➔♠ t↕✐ ❝→❝ a−✤✐➸♠ ❝õ❛ f ♠➔ ❜ë✐ ❝õ❛ ♥â ❦❤æ♥❣ ❧î♥ ❤ì♥
✭t÷ì♥❣ ù♥❣✱ ❦❤æ♥❣ ❜➨ ❤ì♥✮ m✱ tr♦♥❣ ✤â ♠é✐ a−✤✐➸♠ ✤÷ñ❝ t➼♥❤ ❜➡♥❣ sè
❜ë✐ ❝õ❛ ♥â✳ ❈→❝ ❤➔♠ N (r, a; f | ≤ m)(N (r, a; f | ≥ m)) ✤÷ñ❝ ①→❝ ✤à♥❤
♠ët ❝→❝❤ t÷ì♥❣ tü✱ tr♦♥❣ ✤â ❝→❝ a−✤✐➸♠ ❝õ❛ f s➩ ❦❤æ♥❣ t➼♥❤ ❜ë✐✳ ◆❣♦➔✐
r❛✱
N (r, a; f | < m), N (r, a; f | > m), N (r, a; f | < m), N (r, a; f | > m)

❧➔ ✤÷ñ❝ ①→❝ ✤à♥❤ t÷ì♥❣ tü✳
✣à♥❤ ♥❣❤➽❛ ✷✳✼✳ ❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ ❝❤✉♥❣
♥❤❛✉ (a, 0)✳ ●å✐ z0 ❧➔ a−✤✐➸♠ ❝❤✉♥❣ ❝õ❛ f ✈➔ g ✈î✐ sè ❜ë✐ ❧➛♥ ❧÷ñt ❧➔ p
✈➔ q✳ ❑þ ❤✐➺✉ ❜ð✐ N L(r, a; f ) ❧➔ ❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐ t↕✐ ❝→❝ a−✤✐➸♠
❝❤✉♥❣ ♥❤÷ t❤➳ ❝õ❛ f ✈➔ g ❦❤✐ p > q✱ ❦➼ ❤✐➺✉ NE1)(r, a; f ) ❧➔ ❤➔♠ ✤➳♠
(2
a−✤✐➸♠ ❝❤✉♥❣ ✤â ❝õ❛ f ✈➔ g ❦❤✐ p = q = 1✱ ❦➼ ❤✐➺✉ N E (r, a; f ) ❧➔ ❤➔♠
✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐ t↕✐ ❝→❝ a✲✤✐➸♠ ❝❤✉♥❣ ✤â ❝õ❛ f ✈➔ g ❦❤✐ p = q ≥ 2✳
❇➡♥❣ ❝→❝❤ t÷ì♥❣ tü t❛ ✤à♥❤ ♥❣❤➽❛ ✤÷ñ❝ ❝→❝ ❤➔♠
1)

(2

N L (r, a; g), NE (r, a; g), N E (r, a; g).

❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ t÷ì♥❣ tü✱ t❛ ❝â t❤➸ ①→❝ ✤à♥❤ ✤÷ñ❝ N L(r, a; f ) ✈➔
N L (r, a; g) ✤è✐ ✈î✐ a ∈ C ∪ {∞}✳
❑❤✐ f ✈➔ g ❝❤✉♥❣ (a, m), m ≥ 1 t❤➻ NE1)(r, a; f ) = N (r, a; f | = 1).
✣à♥❤ ♥❣❤➽❛ ✷✳✽✳ ❑þ ❤✐➺✉ N (r, a; f | = k) ❧➔ ❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐ t↕✐
❝→❝ a−✤✐➸♠ ❝õ❛ f ♠➔ ❜ë✐ ✤ó♥❣ ❜➡♥❣ k✱ tr♦♥❣ ✤â k ≥ 2 ❧➔ sè ♥❣✉②➯♥✳
✣à♥❤ ♥❣❤➽❛ ✷✳✾✳ ❈❤♦ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ (a, 0)✳ ❚❛ ❦þ ❤✐➺✉ N ∗ (r, a; f, g)
❧➔ ❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐ t↕✐ ❝→❝ a− ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ f ✈➔ g s❛♦ ❝❤♦ ❜ë✐
t↕✐ ❝→❝ a−✤✐➸♠ ♥➔② ✤è✐ ✈î✐ f ✈➔ g ❧➔ ❦❤→❝ ♥❤❛✉✳
❘ã r➔♥❣
N ∗ (r, a; f, g) ≡ N ∗ (r, a; g, f )

✈➔
N ∗ (r, a; f, g) = N L (r, a; f ) + N L (r, a; g).


✷✵

✷✳✶✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t

❈❤♦ f, g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✳ ●å✐ F ✈➔ G ❧➔ ❤❛✐ ❤➔♠
♣❤➙♥ ❤➻♥❤ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
F =

g n−1 (g + a)
f n−1 (f + a)
,G =
.

−b
−b

✭✷✳✶✮

❚ø ♥❛② ✈➲ s❛✉ ❝❤ó♥❣ t❛ ❧✉æ♥ ❦þ ❤✐➺✉ ❜ð✐ ❤➔♠ H ♥❤÷ s❛✉✿
F
2F
G
2G
−
−
−
.
F
F −1
G
G−1

H=
❇ê ✤➲ ✷✳✶

✭❬✽❪✮✳ ❈❤♦ f

✭✷✳✷✮

❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ ✈➔

R(f ) =

n
k
k=1 ak f
m
j
j=0 bj f

❧➔ ♠ët ❤➔♠ ❤ú✉ t✛ ✈î✐ ❝→❝ ❤➺ sè ❤➡♥❣ sè ak ✈➔ bj ✱ tr♦♥❣ ✤â an = 0 ✈➔

bm = 0✳ ❑❤✐ ✤â
T (r, R(f )) = dT (r, f ) + S(r, f ),
tr♦♥❣ ✤â d = max{n, m}.
❇ê ✤➲ ✷✳✷

✭❬✶✷❪✮✳ ◆➳✉ F, G ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ s❛♦ ❝❤♦

F ✈➔ G ❝❤✉♥❣ (1, 0) ✈➔ H ≡ 0✱ ❦❤✐ ✤â
1)

1)

NE (r, 1; F | = 1) = NE (r, 1; G| = 1) ≤ N (r, ∞; H) + S(r, F ) + S(r, G).
❇ê ✤➲ ✷✳✸

✭❬✶❪✮✳ ❈❤♦
S = {z : z n + az n−1 + b = 0},

tr♦♥❣ ✤â a, b ❧➔ ❝→❝ ❤➡♥❣ sè ❦❤→❝ ❦❤æ♥❣ t❤ä❛ ♠➣♥ z n + az n−1 + b = 0
❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ❧➦♣✱ n (≥ 3) ❧➔ sè ♥❣✉②➯♥ ✈➔ F, G ❧➔ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✮✳
◆➳✉ ✈î✐ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f ✈➔ g ✱ Ef (S, 0) = Eg (S, 0) ✈➔