Về dạng định lý cơ bản thứ hai kiểu cartan cho các đường cong chỉnh hình
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ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC SƯ PHẠM
NGUYỄN TRƯỜNG GIANG
VỀ DẠNG ĐỊNH LÝ CƠ BẢN THỨ HAI KIỂU
CARTAN CHO CÁC ĐƯỜNG CONG CHỈNH HÌNH
LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN HỌC
THÁI NGUYÊN – 2008
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ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC SƯ PHẠM
NGUYỄN TRƯỜNG GIANG
VỀ DẠNG ĐỊNH LÝ CƠ BẢN THỨ HAI KIỂU CARTAN
CHO CÁC ĐƯỜNG CONG CHỈNH HÌNH
Chuyên ngành: GIẢI TÍCH
Mã số: 60.46.01
LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN HỌC
Người hướng dẫn khoa học:
TS. TẠ THỊ HOÀI AN
THÁI NGUYÊN – 2008
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▼ö❝ ❧ö❝
▼ð ✤➛✉
✷
✶ ▲þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤
✻
✶✳✶
❍➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✻
✶✳✷
▲þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤
✳ ✳ ✳ ✳ ✳ ✳
✽
❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳
✽
✶✳✷✳✶
✶✳✷✳✷
▼ët sè ✈➼ ❞ö ✈➲ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛
✳ ✳ ✳ ✳
✶✵
✶✳✷✳✸
▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✳ ✳
✶✸
✶✳✷✳✹
✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t ❝õ❛ ◆❡✈❛♥❧✐♥♥❛ ✳ ✳ ✳
✶✹
✶✳✷✳✺
✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✺
✷ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❦✐➸✉ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❝❤♦
❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
✷✸
✷✳✶
❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ✷✸
✷✳✷
✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❝➢t
❝→❝ s✐➯✉ ♠➦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✻
✷✳✷✳✶
▼ët sè ❜ê ✤➲ q✉❛♥ trå♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✻
✷✳✷✳✷
✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣
❝❤➾♥❤ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶
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✷✾
Header Page 4 of 166.
ỵ tt ố tr ừ ữủ ởt
tr ỳ t tỹ s s ừ t ồ tr t
ữỡ ữủ t tứ ỳ ừ ừ t ỵ
tt õ ỗ ố tứ ỳ ổ tr ừ r
r õ ự ử tr ỹ
ừ t ồ ỵ tt ố tr ờ sỹ tờ qt õ
ỵ ỡ ừ số ỡ ỵ tt ự sỹ
ố tr ừ tứ
C C{} r t
ừ ỵ tt ỗ ỵ ỡ ừ ỵ ỡ
tự t ởt t ừ ổ tự Pss s
ỵ õ r trữ
T (r, a, f )
ổ ử tở
a t s ởt ữủ tr õ a ởt số ự
tũ ỵ ỵ ỡ tự t ỳ t q t s
s t ừ ỵ tt ố tr ỵ ữ r ố q
ỳ trữ
rt ự ỵ s
f : C Pn(C) ữớ ổ s
t t Hi i = 1, ..., q s tr tờ qt ợ
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♠é✐ ε > 0 t❛ ❝â
q
m(r, Hj , f ) ≤ (n + 1 + ε)T (r, f ),
j=1
tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✤ó♥❣ ✈î✐ ♠å✐ r > 0 ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❝â ✤ë
✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳
❑➳t q✉↔ tr➯♥ ❝õ❛ ❍✳ ❈❛rt❛♥ ❧➔ ❝æ♥❣ tr➻♥❤ ✤➛✉ t✐➯♥ ✈➲ ♠ð rë♥❣ ❧þ
t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤✳ ❙û ❞ö♥❣ ❦➳t q✉↔ ✤â
æ♥❣ ✤➣ ✤÷❛ r❛ ❝→❝ ÷î❝ ❧÷ñ♥❣ sè ❦❤✉②➳t ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤
❤➻♥❤ ❣✐❛♦ ✈î✐ ❝→❝ s✐➯✉ ♣❤➥♥❣ ð ✈à tr➼ tê♥❣ q✉→t✳ ❈æ♥❣ tr➻♥❤ ♥➔② ❝õ❛
æ♥❣ ✤➣ ✤÷ñ❝ ✤→♥❤ ❣✐→ ❧➔ ❤➳t sù❝ q✉❛♥ trå♥❣ ✈➔ ♠ð r❛ ♠ët ❤÷î♥❣
♥❣❤✐➯♥ ❝ù✉ ♠î✐ ❝❤♦ ✈✐➺❝ ♣❤→t tr✐➸♥ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✳ ❇ð✐ ✈➟②✱
❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ s❛✉ ♥➔② ✤÷ñ❝
♠❛♥❣ t➯♥ ❤❛✐ ♥❤➔ t♦→♥ ❤å❝ ♥ê✐ t✐➳♥❣ ❝õ❛ t❤➳ ❦✛ ✷✵✱ ✤â ❧➔ ✏▲þ t❤✉②➳t
◆❡✈❛♥❧✐♥♥❛ ✲ ❈❛rt❛♥✧✳
◆❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙②✱ ✈✐➺❝ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❈❛rt❛♥ ❝❤♦ tr÷í♥❣
❤ñ♣ ❝→❝ s✐➯✉ ♠➦t t❤✉ ❤ót ✤÷ñ❝ sü ❝❤ó þ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝✳ ◆➠♠
✷✵✵✹✱ ▼✳ ❘✉ ❬✶✷❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❣✐↔ t❤✉②➳t ❝õ❛ ❇✳ ❙❤✐❢❢♠❛♥ ❬✶✹❪ ✤➦t r❛
❈❤♦ f : C → Pn(C)
❧➔ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥ ✤↕✐ sè✱ Dj , j = 1, ..., q, ❧➔
❝→❝ s✐➯✉ ♠➦t ❜➟❝ dj ð ✈à tr➼ tê♥❣ q✉→t✳ ❑❤✐ ✤â
✈➔♦ ♥➠♠ ✶✾✼✾✳ ❈ö t❤➸✱ æ♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣✿
q
d−1
j N (r, Dj , f ) + o(T (r, f )),
(q − (n + 1) − ε)T (r, f ) ≤
j=1
tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤ó♥❣ ✈î✐ ♠å✐ r ✤õ ❧î♥ ♥➡♠ ♥❣♦➔✐ ♠ët
t➟♣ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳ ❑➳t q✉↔ tr➯♥ ✤➣ ✤÷ñ❝ ◗✳ ❨❛♥ ✈➔
✸
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rở trữớ ủ t ở
ỏ ồ ửt t q ữủ t ữ s
sỷ f : C Pn(C) ởt ổ s
số Dj 1 j q q s t tr Pn(C) õ dj tữỡ
ự tr tờ qt õ ợ ộ > 0 tỗ t ởt số
ữỡ M s
q
M
d1
j N (r, Dj , f ) + o (T (r, f )) ,
q (n + 1) )T (r, f )
j=1
tr õ t tự tr ú ợ ồ r ừ ợ ởt
t õ ở s ỳ
ự sỹ tỗ t ừ
tổ q ữủ ừ s t ữớ t tữớ sỷ ử
ỵ ỡ tự rt tổ q t
ở r ỵ rt ỏ t
t t s ừ ữớ
ử t ừ tr t q ữủ
ữ r ừ ợ ổ ử ự ừ
ỵ tt rt tứ
C
Pn (C).
ữủ t ữỡ ũ ợ t
ử t t
ữỡ tr ởt số tự ỡ s
t t ừ r ự
ỵ ỡ tự ừ
ữỡ tr ự ởt ỵ ỡ tự
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❝❤♦ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❝➢t ❝→❝ s✐➯✉ ♠➦t ð ✈à tr➼ tê♥❣ q✉→t✳ ❈❤÷ì♥❣
♥➔② ✤÷ñ❝ ✈✐➳t ❞ü❛ tr➯♥ ❝æ♥❣ tr➻♥❤ ❝õ❛ ◗✳ ❨❛♥✱ ❩✳ ❈❤❡♥ ❬✹❪✳
▲✉➟♥ ✈➠♥ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝ ❝õ❛
❚❙✳ ❚↕ ❚❤à ❍♦➔✐ ❆♥
✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤
✤➳♥ ❚❙ ✈➲ sü ❣✐ó♣ ✤ï ❦❤♦❛ ❤å❝ ♠➔ ❚❙ ✤➣ ❞➔♥❤ ❝❤♦ t→❝ ❣✐↔ ✈➔ ✤➣ t↕♦
♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ♥❤➜t ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳
❚→❝ ❣✐↔ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝
❙÷ ♣❤↕♠ t❤✉ë❝ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ✤➦❝ ❜✐➺t ❧➔ ❚❤➔②
P❤÷ì♥❣
❍➔ ❚r➛♥
✈➔ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✈➔ ❝→❝
t❤➛② ❝æ ❣✐→♦ ❱✐➺♥ ❚♦→♥ ❤å❝ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ ❤♦➔♥
t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥✳
❚→❝ ❣✐↔ ❝ô♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ❈❛♦
✤➥♥❣ ❈æ♥❣ ♥❣❤➺ ✈➔ ❑✐♥❤ t➳ ❈æ♥❣ ♥❣❤✐➺♣✱ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ t↕♦
♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ♥❤➜t ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣✳
✺
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Header Page 8 of 166.
ữỡ
ỵ tt
r ữỡ ú tổ ởt số tự ỡ s
ữủ sỷ ử tr s tự ừ ữỡ
ữủ tr tứ
D
ởt tr t ự
C
f (z) = u(x, y) + iv(x, y) ữủ ồ C t z0 C tỗ
f (z0 + h) f (z0 )
t ợ ỳ lim
h0
h
tr õ ữủ ồ ự ừ f (z) t z0
f (z)
ữủ ồ
C
tr D õ C t ồ
z0 D.
õ
C
f (z)
ữủ ồ
t z0 C
tr ởt õ ừ
f (z)
ữủ ồ
tr D õ t ồ
Footer Page 8 of 166.
z0
Header Page 9 of 166.
✤✐➸♠
z
t❤✉ë❝
D✳
❚➟♣ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ♠✐➲♥
✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳
♣❤ù❝
C
✤÷ñ❝ ❣å✐ ❧➔
f (z)
❍➔♠
D✱
❦➼ ❤✐➺✉ ❧➔
H(D)✳
❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ t♦➔♥ ♠➦t ♣❤➥♥❣
❤➔♠ ♥❣✉②➯♥✳
✶✳✶✳✹ ✣à♥❤ ❧þ✳ ❍➔♠ f (z) = u(x, y) + iv(x, y) ❝❤➾♥❤ ❤➻♥❤ tr➯♥ D ♥➳✉
❝→❝ ❤➔♠ u(x, y) ✈➔ v(x, y) ❧➔ R2 ✲ ❦❤↔ ✈✐ tr➯♥ D ✈➔ tr➯♥ ✤â ❝→❝ ❤➔♠
u(x, y)✱ v(x, y) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❈❛✉❝❤② ✲ ❘✐❡♠❛♥♥✱ tù❝ ❧➔
∂u ∂v ∂u
∂v
=
,
= − , ∀ (x, y) ∈ D.
∂x ∂y ∂y
∂x
✶✳✶✳✺ ✣à♥❤ ❧þ✳ ●✐↔ sû f (z) ❧➔ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ ♠✐➲♥ ❤ú✉
❤↕♥ D ⊂ C✳ ❑❤✐ ✤â tr♦♥❣ ♠é✐ ❧➙♥ ❝➟♥ ❝õ❛ ♠é✐ ✤✐➸♠ z ∈ D✱ ❤➔♠
f (z) ✤÷ñ❝ ❦❤❛✐ tr✐➸♥ t❤➔♥❤ ❝❤✉é✐
(z − z0 )
(z − z0 )2
f (z) = f (z0 ) +
f (z0 ) +
f (z0 ) + . . .
1!
2!
❍ì♥ ♥ú❛✱ ❝❤✉é✐ tr➯♥ ❤ë✐ tö ✤➲✉ ✤➳♥ ❤➔♠
|z − z0 | ≤ ρ tò② þ ♥➡♠ tr♦♥❣ D.
❈❤✉é✐ ✭✶✳✶✮ ✤÷ñ❝ ❣å✐ ❧➔
❝õ❛ ✤✐➸♠
✣✐➸♠
✭❤❛② ❦❤æ♥❣✲✤✐➸♠ ❝➜♣
n = 1,..., m − 1
❝❤✉é✐ ❚❛②❧♦ ❝õ❛ ❤➔♠ f (z) tr♦♥❣ ❧➙♥ ❝➟♥
♥➳✉
f=
g
h
z0 ∈ C
m > 0✮
✈➔
✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳
D⊂C
tr♦♥❣ ❤➻♥❤ trá♥
z0 .
✶✳✶✳✻ ✣à♥❤ ♥❣❤➽❛✳
♠å✐
f (z)
✭✶✳✶✮
✤÷ñ❝ ❣å✐ ❧➔
❝õ❛ ❤➔♠
♥➳✉
f (n) (z0 ) = 0,
❝❤♦
f (m) (z0 ) = 0.
❍➔♠
tr♦♥❣ ✤â
f (z)
g, h
✤÷ñ❝ ❣å✐ ❧➔
❤➔♠ ♣❤➙♥ ❤➻♥❤
❧➔ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣
✼
Footer Page 9 of 166.
f (z)
❦❤æ♥❣ ✤✐➸♠ ❜➟❝ m > 0
tr♦♥❣
D.
Header Page 10 of 166.
◆➳✉
❧➔
D = C t❤➻ t❛ ♥â✐ f (z) ♣❤➙♥ ❤➻♥❤ tr➯♥ C ❤❛② ✤ì♥ ❣✐↔♥ ❧➔ f (z)
❤➔♠ ♣❤➙♥ ❤➻♥❤✳
✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳
✣✐➸♠
z0
✤÷ñ❝
❣å✐
❝ü❝ ✤✐➸♠ ❝➜♣
❧➔
1
.h(z)✱
(z − z0 )m
z0 ✈➔ h(z0 ) = 0✳
m > 0 ❝õ❛ ❤➔♠ f (z) ♥➳✉ tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛ z0 ❤➔♠ f (z) =
tr♦♥❣ ✤â
h(z)
❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛
✶✳✶✳✾ ✣à♥❤ ❧þ ✭❈æ♥❣ t❤ù❝ P♦✐✐s♦♥ ✲ ❏❡♥s❡♥✮✳ ●✐↔ sû f (z) ≡ 0 ❧➔
♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr♦♥❣ ❤➻♥❤ trá♥ {|z| ≤ R} ✈î✐ 0 < R < ∞✳ ●✐↔
sû aµ✱ µ = 1, ..., M, ❧➔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✱ bν , ν = 1, 2, ..., N,
❧➔ ❝→❝ ❝ü❝ ✤✐➸♠ ❝õ❛ f tr♦♥❣ ❤➻♥❤ trá♥ ✤â✱ ❝ô♥❣ ❦➸ ❝↔ ❜ë✐✳ ❑❤✐ ✤â✱
♥➳✉ z = reiθ (0 < r < R), f (z) = 0, f (z) = ∞ t❤➻
2π
1
log |f (z)| =
2π
log f (Reiφ )
0
M
+
R2 − r 2
dφ
R2 − 2Rr cos(θ − φ) + r2
✭✶✳✷✮
N
log
µ=1
R(z − aµ )
R(z − bν )
−
log
.
2−b z
R 2 − aµ z
R
ν
ν=1
✶✳✷ ▲þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤
✶✳✷✳✶ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤
●✐↔ sû
f
❑þ ❤✐➺✉
❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr♦♥❣ ✤➽❛ ❜→♥ ❦➼♥❤
r.
✈➔
r < R✳
n(r, ∞, f ) ✭t÷ì♥❣ ù♥❣✱ n(r, ∞, f ), ❧➔ sè ❝→❝ ❝ü❝ ✤✐➸♠ t➼♥❤
❝↔ ❜ë✐✱ ✭t÷ì♥❣ ù♥❣✱ ❦❤æ♥❣ t➼♥❤ ❜ë✐✮✮✱ ❝õ❛ ❤➔♠
❦➼♥❤
R
●✐↔ sû
a ∈ C✱
t❛ ✤à♥❤ ♥❣❤➽❛
n(r, a, f ) = n r, ∞,
✽
Footer Page 10 of 166.
f
1
,
f −a
tr♦♥❣ ✤➽❛ ✤â♥❣ ❜→♥
Header Page 11 of 166.
n(r, a, f ) = n r, ∞,
1
.
f −a
✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ✤➳♠ t➼♥❤ ❝↔ ❜ë✐ N (r, a, f ),
❦❤æ♥❣ t➼♥❤ ❜ë✐
N (r, a, f )✮✱
❝õ❛ ❤➔♠
f
t↕✐ ❣✐→ trà
a
✭t÷ì♥❣ ù♥❣✱
✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛
♥❤÷ s❛✉
r
N (r, a, f ) = n(0, a, f ) log r +
n(t, a, f ) − n(0, a, f )
dt
,
t
n(t, a, f ) − n(0, a, f )
dt
).
t
0
✭t÷ì♥❣ ù♥❣✱
r
N (r, a, f ) = n(0, a, f ) log r +
0
❱➻ t❤➳✱ ♥➳✉
a=0
t❛ ❝â
(♦r❞+
z f ) log |
N (r, 0, f ) = (♦r❞+
0 f ) log r +
z∈D(r)
r
|,
z
z=0
tr♦♥❣ ✤â
D(r)
❧➔ ✤➽❛ ❜→♥ ❦➼♥❤
+
r
✈➔ ♦r❞z
f = max{0, ♦r❞z f }
❧➔ ❜ë✐
❝õ❛ ❦❤æ♥❣ ✤✐➸♠✳
✶✳✷✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ①➜♣ ①➾
a∈C
m(r, a, f )
❝õ❛ ❤➔♠
f
t↕✐ ❣✐→ trà
✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉
2π
log+
m(r, a, f ) =
0
dθ
1
,
f (reiθ ) − a 2π
✈➔
2π
log+ | f (reiθ ) |
m(r, ∞, f ) =
0
tr♦♥❣ ✤â
❍➔♠
dθ
,
2π
+
log x = max{0, log x}.
mf (r, ∞)
✤♦ ✤ë ❧î♥ tr✉♥❣ ❜➻♥❤ ❝õ❛
|z| = r✳
✾
Footer Page 11 of 166.
log |f |
tr➯♥ ✤÷í♥❣ trá♥
Header Page 12 of 166.
✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ✤➦❝ tr÷♥❣ T (r, a, f )
a∈C
❝õ❛ ❤➔♠
f
t↕✐ ❣✐→ trà
✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉
T (r, a, f ) = m(r, a, f ) + Nf (r, a, f ),
T (r, f ) = m(r, ∞, f ) + N (r, ∞, f ).
✭✶✳✸✮
❳➨t ✈➲ ♠➦t ♥➔♦ ✤â✱ ❤➔♠ ✤➦❝ tr÷♥❣ ◆❡✈❛♥❧✐♥♥❛ ✤è✐ ✈î✐ ❧þ t❤✉②➳t
❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝â ✈❛✐ trá t÷ì♥❣ tü ♥❤÷ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ tr♦♥❣ ❧þ
t❤✉②➳t ✤❛ t❤ù❝✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➦❝ tr÷♥❣ t❛ ❝â
T (r, a, f ) ≥ N (r, a, f ) + O(1),
tr♦♥❣ ✤â
O(1)
❧➔ ✤↕✐ ❧÷ñ♥❣ ❜à ❝❤➦♥ ❦❤✐
r→∞
✳
❱î✐ ❝→❝❤ ✤à♥❤ ♥❣❤➽❛ ♥➔② t❤➻ ❝æ♥❣ t❤ù❝ P♦✐✐s♦♥✲❏❡♥s❡♥ ✭✣à♥❤ ❧þ ✶✳✶✳✾✮
✤÷ñ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉
T (r, f ) = T (r, a, f ) + log |f (0)|.
✭✶✳✹✮
✶✳✷✳✷ ▼ët sè ✈➼ ❞ö ✈➲ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛
✶✳✷✳✹ ❱➼ ❞ö✳
❳➨t ❤➔♠ ❤ú✉ t➾
f (z) = c
tr♦♥❣ ✤â
c = 0.
✣➛✉ t✐➯♥ ❣✐↔ sû
m(r, a, f ) = 0(1)
❝â
p
z p + ... + ap
,
z q + ... + bp
p > q✳
❦❤✐
❑❤✐ ✤â
z→∞
❝❤♦
f (z) → ∞✱
a
z → ∞✳
◆❤÷ ✈➟②
❤ú✉ ❤↕♥✳ P❤÷ì♥❣ tr➻♥❤
f (z) = a
❦❤✐
♥❣❤✐➺♠ t➼♥❤ ❝↔ ❜ë✐✱ ❞♦ ✤â
r
N (r, a, f ) =
n(t, a)
a
✶✵
Footer Page 12 of 166.
dt
= p log r + O(1)
t
Header Page 13 of 166.
❦❤✐
r → ∞.
◆❤÷ ✈➟②✱
T (r, f ) = p log r + O(1),
✈➔
N (r, a, f ) = p log r + O(1), m(r, a) = O(1)
tr➻♥❤
f (z) = ∞
q
❝â
✈î✐
a = ∞.
P❤÷ì♥❣
♥❣❤✐➺♠✱ ✈➻ t❤➳
N (r, ∞, f ) = q log r + O(1),
✈➔ ❜ð✐ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t
m(r, ∞, f ) = (p − q) log r + O(1).
◆➳✉
p < q✱
t❤➻ t÷ì♥❣ tü t❛ ❝â
T (r, f ) = q log r + O(1),
m(r, a, f ) = O(1),
❑❤✐
✈î✐
N (r, a, f ) = q log r + O(1),
a = 0.
a = 0✱
N (r, 0, f ) = p log r + O(1),
❈✉è✐ ❝ò♥❣✱ ♥➳✉
m(r, a, f ) = (q − p) log r + O(1).
p = q,
T (r, f ) = q log r + O(1),
✈➔
N (r, a) = q log r + O(1),
tr✐➺t t✐➯✉ ❝õ❛
f −c
t↕✐
∞✱
✈î✐
a = c.
❍ì♥ ♥ú❛✱ ♥➳✉ ❦þ ❤✐➺✉
N (r, c, f ) = (q − k) log r + O(1).
❱➟② tr♦♥❣ ♠å✐ tr÷í♥❣ ❤ñ♣
T (r, f ) = d log r + O(1),
d = max(p, q).
✶✶
Footer Page 13 of 166.
❧➔ ❜➟❝
❦❤✐ ✤â
m(r, c, f ) = k log r + O(1),
tr♦♥❣ ✤â
k
Header Page 14 of 166.
✶✳✷✳✺ ❱➼ ❞ö✳
❳➨t ❤➔♠
f (z) = ez .
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✱
π
2
2π
iθ
log+ ere
m(r, f ) =
dθ
=
2π
f
❧➔ ❤➔♠ ♥❣✉②➯♥ ♥➯♥
❱î✐
a = 0, ∞,
t❤➻
dθ
r
= .
2π
π
− π2
0
❉♦
r cos θ
N (r, ∞, f ) = 0
f (z) = a
✈➔ ❞♦ ✤â
T (r, f ) = r/π.
❝â ♥❣❤✐➺♠ ✈î✐ ❝❤✉ ❦ý
2t
2π ♥❣❤✐➺♠ tr♦♥❣ ✤➽❛ ❝â ❜→♥ ❦➼♥❤
t,
2πi✳
❉♦ ✈➟②✱ ❝â
✈➔ ❞♦ ✤â
r
t dt
r
+ O(log r) = + O(log r).
π t
π
N (r, a, f ) =
o
m(r, a, f ) = O(log r).
❉♦ ✈➟②✱
✶✳✷✳✻ ❱➼ ❞ö✳
❱î✐ ♠å✐
a
❳➨t ❤➔♠
sin z
✈➔ ❤➔♠
cos z ✳
❤ú✉ ❤↕♥
N (r, a, sin z) + O(1) = N (r, a, cos z) + O(1) =
❚ø
sin z
e−iz ✱
✈➔
cos z
2r
+ O(1).
π
✤÷ñ❝ ❜✐➸✉ ❞✐➵♥ ❜➡♥❣ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛
t❛ ❝â
T (r, sin z) + O(1) = T (r, cos z) + O(1) ≤
2r
+ O(1).
π
✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦
T (r, sin z) + O(1) = T (r, cos z) + O(1) =
2r
+ O(1)
π
✈➔
m(r, a, sin z) + O(1) = m(r, a, cos z) + O(1) = O(1).
✶✷
Footer Page 14 of 166.
eiz
✈➔
Header Page 15 of 166.
✶✳✷✳✸ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛
❈❤ó♥❣ t❛ t✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ✤ì♥ ❣✐↔♥ ❝õ❛ ❝→❝
❤➔♠ ◆❡✈❛♥❧✐♥♥❛✳ ❈❤ó þ r➡♥❣ ♥➳✉
p
p
log+
ν=1
p
log
log+ |aν |
aν ≤
+
a1 , a2 , ..., ap
❧➔ ❝→❝ sè ♣❤ù❝ t❤➻
✈➔
ν=1
q
aν ≤ log
+
p max |aν |
ν=1,..,p
ν=1
log+ |aν | + log p✳
≤
ν=1
⑩♣ ❞ö♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦
p ❤➔♠ ♣❤➙♥ ❤➻♥❤ f1 (z), f2 (z), ..., fp (z)
✈➔ sû ❞ö♥❣ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✱ ❝❤ó♥❣ t❛ t❤✉ ✤÷ñ❝
❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉
p
✶✳
✷✳
✸✳
✹✳
p
m r,
fν (z)
N
m (r, fν (z)) + log p✳
ν=1
ν=1
p
p
m r,
N
≤
≤
fν (z)
m (r, fν (z))✳
ν=1
ν=1
p
p
r,
fν (z)
≤
N (r, fν (z))✳
ν=1
ν=1
p
p
r,
fν (z)
≤
ν=1
N (r, fν (z)).
ν=1
❙û ❞ö♥❣ ✭✶✳✸✮ t❛ t❤✉ ✤÷ñ❝
p
✺✳
✻✳
T
T
p
r,
fν (z)
≤
T (r, fν (z)) + log p.
ν=1
ν=1
p
p
r,
fν (z)
ν=1
≤
T (r, fν (z)).
ν=1
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t ❦❤✐
❤➡♥❣ sè✮✱ t❛ s✉② r❛
p = 2✱ f1 (x) = f (z), f2 (z) = a✭a
T (r, f + a) ≤ T (r, f ) + log+ |a| + log 2✳
✶✸
Footer Page 15 of 166.
❧➔
❱➔ tø ✤â
Header Page 16 of 166.
❝❤ó♥❣ t❛ ❝â t❤➸ t❤❛② t❤➳
f + a, f
❜ð✐
f, f − a
✈➔
a
❜ð✐
−a✱
s✉② r❛
|T (r, f ) − T (r, f − a)| ≤ log+ |a| + log 2.
✭✶✳✺✮
✶✳✷✳✹ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t ❝õ❛ ◆❡✈❛♥❧✐♥♥❛
✶✳✷✳✼ ✣à♥❤ ❧þ✳ ●✐↔ sû f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ a ❧➔ ♠ët sè ♣❤ù❝ tò② þ✳
❑❤✐ ✤â t❛ ❝â
m r,
1
f −a
+N
r,
1
f −a
= T (r, f ) − log |f (0) − a| + ε(a, r),
tr♦♥❣ ✤â |ε(a, r)| ≤ log+ |a| + log 2.
❚❛ t❤÷í♥❣ ❞ò♥❣ ✤à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t ❞÷î✐ ❞↕♥❣
m r,
1
f −a
+N
r,
1
f −a
= T (r, f ) + O(1),
tr♦♥❣ ✤â ❖✭✶✮ ❧➔ ♠ët ✤↕✐ ❧÷ñ♥❣ ❣✐î✐ ♥ë✐✳
Þ ♥❣❤➽❛
❱➳ tr→✐ tr♦♥❣ ❝æ♥❣ t❤ù❝ ❝õ❛ ✤à♥❤ ❧þ ✤♦ sè ❧➛♥
❱➳ ♣❤↔✐ ❧➔ ❤➔♠
T (r, f )
❦❤æ♥❣ ♣❤ö t❤✉ë❝
a✱
f −a
✈➔
f
❣➛♥
s❛✐ ❦❤→❝ ♠ët ✤↕✐ ❧÷ñ♥❣
❣✐î✐ ♥ë✐✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ✭✶✳✸✮ ✈➔ ✭✶✳✹✮ t❛ ❝â✿
m r,
1
f −a
+N
r,
1
f −a
=T
r,
1
f −a
= T (r, f − a) + log |f (0) − a| .
❚ø ✭✶✳✺✮ t❛ s✉② r❛
T (r, f − a) = T (r, f ) + ε(a, r),
✶✹
Footer Page 16 of 166.
a✳
Header Page 17 of 166.
✈î✐
|ε(a, r)| ≤ log+ |a| + log 2.
❚ø ✤â t❛ ❝â
m r,
1
f −a
+N
r,
1
f −a
= T (r, f ) + log |f (0) − a| + ε(a, r),
|ε(a, r)| ≤ log+ |a| + log 2.
tr♦♥❣ ✤â
✣à♥❤ ❧þ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳
✶✳✷✳✺ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐
✣➸ ✤ì♥ ❣✐↔♥✱ ❝❤ó♥❣ t❛ s➩ ✈✐➳t
m(r, ∞)
t❤❛② ❝❤♦
m(r, a)
t❤❛② ❝❤♦
m r,
1
f −a
✈➔
m(r, f ).
✶✳✷✳✽ ✣à♥❤ ❧þ✳ ●✐↔ sû ❢✭③✮ ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ sè tr♦♥❣
|z| ≤ r✳
●✐↔ sû a1, a2, ..., aq ✈î✐ q > 2 ❧➔ ❝→❝ sè ♣❤ù❝ ❤ú✉ ❤↕♥✱ r✐➯♥❣
❜✐➺t✱ δ > 0 ✈➔ ❣✐↔ sû r➡♥❣ |aµ − aν | ≥ δ ✈î✐ 1 ≤ µ < ν ≤ q✳ ❑❤✐ ✤â
q
m(r, ∞) +
m(r, aν ) ≤ 2T (r, f ) − N1 (r) + S(r),
ν=1
tr♦♥❣ ✤â N1(r) ❧➔ ❞÷ì♥❣ ✈➔ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
1
N1 (r) = N r,
+ 2N (r, f ) − N (r, f ) ✈➔
f
f
S(r) = m r,
f
▲÷ñ♥❣
S(r)
q
+m r,
ν=1
f
3q
1
+q log+ +log 2+log
✳
f − aν
δ
|f (0)|
tr♦♥❣ tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t s➩ ✤â♥❣ ✈❛✐ trá ❧➔ s❛✐ sè
❦❤æ♥❣ ✤→♥❣ ❦➸✳ ❙ü tê♥❣ ❤ñ♣ ❝→❝ ✈➜♥ ✤➲ ✤â tr♦♥❣ ✤à♥❤ ❧þ tr➯♥ s➩ ♠❛♥❣
❧↕✐ ✤à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐✳ ✣✐➲✉ ✤â ❝❤♦ t❤➜② r➡♥❣✱ tr♦♥❣ tr÷í♥❣ ❤ñ♣
tê♥❣ q✉→t tê♥❣ ❝õ❛ ❝→❝ sè ❤↕♥❣
m(r, aν )
✶✺
Footer Page 17 of 166.
t↕✐ ♠é✐ sè ❦❤æ♥❣ t❤➸ ❧î♥
Header Page 18 of 166.
❤ì♥
2T (r)✳
❇➙② ❣✐í ❝❤ó♥❣ t❛ ❜➢t ✤➛✉ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ tr÷í♥❣ ❤ñ♣ t÷ì♥❣ ✤è✐
✤ì♥ ❣✐↔♥ ❝õ❛ ✤à♥❤ ❧þ tr÷î❝ ❦❤✐ ①û ❧þ ✈î✐ ÷î❝ ❧÷ñ♥❣ ♣❤ù❝ t↕♣ ❤ì♥ ❝õ❛
S(r).
❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ❝→❝ sè ♣❤➙♥ ❜✐➺t aν , (1 ≤ ν ≤ q)✱ t❛ ①➨t ❤➔♠
q
F (z) =
ν=1
✶✳ ●✐↔ sû r➡♥❣ ✈î✐ ♠ët ✈➔✐
µ = ν✱
ν
1
.
f (z) − aν
♥➔♦ ✤â✱
|f (z) − aν | <
δ
✳
3q
❑❤✐ ✤â ✈î✐
t❛ ❝â
|f (z) − aµ | ≥ |aµ − aν | − |f (z) − aν | ≥ δ −
❜ð✐ ✈➟②✱ ✈î✐
µ=ν
2
δ
≥ δ,
3q
3
t❤➻
1
3
1
≤
≤
.
|f (z) − aµ | 2δ
|f (z) − aν |
◆❤÷ ✈➟②
|F (z)| ≥
1
−
|f (z) − aν |
µ=ν
1
|f (z) − aµ |
1
q−1
1−
|f (z) − aν |
2q
1
≥
.
2 |f (z) − aν |
≥
❚ø ✤â t❛ ❝â
q
+
log+
log |F (z)| ≥
µ=1
q
log+
≥
µ=1
1
2
− q log+ − log 2
|f (z) − aµ |
δ
1
3q
− q log+
− log 2.
|f (z) − aµ |
δ
✶✻
Footer Page 18 of 166.
✭✶✳✻✮
Header Page 19 of 166.
ợ
à=
log+
1
3
2
log+
log+
|f (z) aà |
2
t
õ
q
log+
à=1
1
1
= log+
+
|f (z) aà |
|f (z) a |
à=
1
|f (z) aà |
2
1
+ (q 1) log+ .
|f (z) a |
log+
r
log+
à=
1
2
(q 1) log+ .
|f (z) aà |
õ
q
log+
+
log |F (z)|
à=1
1
3q
q log+
log 2.
|f (z) aà |
ữủ ự
ữ tỗ t ởt tr
q
|f (z) a | <
3q
t ú
ữủ sỷ
|f (z) a |
3q
ợ ồ
õ t õ ởt
q
+
log+
log |F (z)|
=1
|f (z) a |
3q
q
log+
=1
1
3q
q log+
log 2.
|f (z) a |
ợ ồ
1
3q
|f (z) a |
ợ ồ
3q
1
q log+
+ log 2.
|f (z) a |
ứ õ
q
+
log+
log |F (z)| 0
=1
1
3q
q log+
log 2.
|f (z) a |
Footer Page 19 of 166.
Header Page 20 of 166.
◆❤÷ ✈➟②✱ tr♦♥❣ ♠å✐ tr÷í♥❣ ❤ñ♣ t❛ ✤➲✉ ❝â ✤÷ñ❝
q
+
log+
log |F (z)| ≥
ν=1
❱î✐
z = reiθ ✱
1
3q
− q log+
− log 2.
|f (z) − aν |
δ
❧➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ ❝❤ó♥❣ t❛ s✉② r❛
2π
log+ F (reiθ ) dθ
0
2π
q
log+
≥
0
ν=1
♥➯♥
3q
1
− q log+
− log 2 dθ.
|f (z) − aν |
δ
q
m(r, aν ) − q log+
m(r, F ) ≤
ν=1
3q
− log 2.
δ
✭✶✳✼✮
▼➦t ❦❤→❝✱ t❛ ①➨t
1 f
m(r, F ) = m r, . .f F
f f
≤
r,
1
f
+m r,
f
f
+m (r, f F ) .
✭✶✳✽✮
❚❤❡♦ ❝æ♥❣ t❤ù❝ ❏❡♥s❡♥ ✭✶✳✹✮✱ t❛ ❝â
T (r, f ) = T
T
r,
1
f
+ log |f (0)| ,
r,
f
f
=T
r,
f
f
+ log
f (0)
.
f (0)
r,
f
f
= m r,
f
f
+N
r,
+N
r,
◆â✐ ❝→❝❤ ❦❤→❝
m r,
f
f
+N
f
f
+ log
f (0)
.
f (0)
❙✉② r❛
m r,
f
f
= m r,
f
f
f
f
−N
✶✽
Footer Page 20 of 166.
−
r,
f
f
+ log
f (0)
.
f (0)
✭✶✳✾✮
Header Page 21 of 166.
✈➔ ♥❣♦➔✐ r❛ t❛ ❝â
T (r, f ) = m r,
1
f
+N
1
f
r,
+ log |f (0)| .
❞♦ ✤â
m r,
1
f
= T (r, f ) − N
r,
1
f
+ log
1
.
|f (0)|
✭✶✳✶✵✮
❑➳t ❤ñ♣ ✭✶✳✾✮ ✈➔ ✭✶✳✶✵✮ ✈➔ t❤❛② ✈➔♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✽✮✱ t❛ ❝â
m(r, F ) ≤ T (r, f ) − N
+N
r,
f
f
−N
r,
r,
1
f
f
f
1
f
+ m r,
+
|f (0)|
f
f (0)
+m(r, f F ). ✭✶✳✶✶✮
+ log
f (0)
+ log
❑➳t ❤ñ♣ ✭✶✳✼✮ ✈î✐ ✭✶✳✶✶✮ t❛ ❝â
q
3q
+ log 2.
δ
ν=1
1
f
f
f
≤ T (r, f ) − N r,
+ N r,
− N r,
+ m r,
f
f
f
f
1
3q
+m(r, f F ) + log
+ T (r, f ) − N (r, f ) + q log+
+ log 2✳
|f (0)|
δ
f
❙û ❞ö♥❣ ❝æ♥❣ t❤ù❝ ❏❡♥s❡♥ ❝❤♦ ❤➔♠
✱ t❛ ❝â
f
m(r, aν ) + m(r, ∞) ≤ m(r, F ) + m(r, f ) + q log+
2π
1
f (0)
log
=
f (0)
2π
f (reiφ )
log
dφ + N
f (reiφ )
r,
f
f
−N
r,
f
f
.
0
❙✉② r❛
2π
N
f
r,
f
−N
f
r,
f
1
=
2π
log
f (reiφ )
f (0)
dφ
−
log
f (reiφ )
f (0)
0
2π
1
=
2π
2π
1
log f (reiφ ) dφ−log |f (0)|−
2π
0
0
✶✾
Footer Page 21 of 166.
log f (reiφ ) dφ−log |f (0)|
Header Page 22 of 166.
=N
r,
1
f
− N (r, f ) − N
r,
1
f
+ N (r, f ) .
❈✉è✐ ❝ò♥❣ t❛ ♥❤➟♥ ✤÷ñ❝
q
m(r, aν ) + m(r, ∞)
ν=1
≤ 2T (r, f ) − 2N (r, f ) − N (r, f ) + N
+ m r,
f
f
+ m (r, f F ) + log
r,
1
f
+
1
3q
+ q log+
+ log 2.
|f (0)|
δ
❈❤ó þ r➡♥❣
q
m (r, f F ) = m r,
ν=1
f
f − aν
✈➔ ✤➦t
N1 (r) = N
r,
1
f
f
S(r) = m r,
f
+ 2N (r, f ) − N (r, f ) ,
q
+ m r,
ν=1
f
f − aν
3q
1
+ log 2 + log
.
+ q log+
δ
|f (0)|
❑❤✐ ✤â✱ t❛ ❝â
q
m(r, ∞) +
m(r, aν ) ≤ 2T (r, f ) − N1 (r) + S(r).
ν=1
✣â ❧➔ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳
q
◆❤➟♥ ①➨t✳
log
ν=1
R
bν
N1 (r)
tr♦♥❣ ✤à♥❤ ❧þ tr➯♥ ❧➔ ❞÷ì♥❣ ✈➻
✱ tr♦♥❣ tê♥❣ tr➯♥ ♥➳✉
bv
✷✵
Footer Page 22 of 166.
❧➔ ❝ü❝ ✤✐➸♠ ❜ë✐
k
N (r, f ) =
t❤➻ ✤÷ñ❝ t➼♥❤
Header Page 23 of 166.
k
❧➛♥✳
●✐↔ sû
b1 , b2 , ..., bN
❧➔ ❝→❝ ❝ü❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛
f (z)
✈î✐ ❝➜♣ ❧➛♥
k1 , k2 , ..., kN ✳ ❳➨t t↕✐ ✤✐➸♠ bv ✱ t❛ t❤➜② ❦❤❛✐ tr✐➸♥ ❝õ❛ f (z) s➩
ckν
❝â ❞↕♥❣ f (z) =
+ ...✳
(z − bν )kν
c−kν
❑❤✐ ✤â f (z) s➩ ❝â ❦❤❛✐ tr✐➸♥ ❧➔ f (z) =
+ ...✱ tù❝ ❧➔ bv s➩
(z − bν )kν +1
❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ kv + 1 ❝õ❛ ❤➔♠ f (z). ◆❤÷ ✈➟② b1 , b2 , ..., bN ❧➔ ❝→❝ ❝ü❝
❧÷ñt ❧➔
✤✐➸♠ ❝õ❛
f (z)
kν log |
N (r, f ) =
k1 + 1, k2 + 1, ..., kN + 1✳ ◆❤÷
N
r
N (r, f ) =
(kν + 1) log | | ♥➯♥
bν
ν=1
✈î✐ ❝➜♣ ❧➛♥ ❧÷ñt ❧➔
N
ν=1
r
|
bν
✈➔
N
r
2N (r, f ) − N (r, f ) =
2kν log | | −
bν
ν=1
N
(2kν − (kν + 1)) log |
ν=1
R
|=
bν
N
(kν + 1) log |
ν=1
N
(2kν − 1)) log |
ν=1
✈➟②
r
|=
bν
r
| ≥ 0.
bν
❚❛ ❝â ✤à♥❤ ❧þ s❛✉ ✤➙②✳
✶✳✷✳✾ ✣à♥❤ ❧þ ✭✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐✮✳ ◆➳✉ f ❧➔ ❤➔♠ ♣❤➙♥
❤➻♥❤✱ ❦❤→❝ ❤➡♥❣ sè tr➯♥ C ✈➔ a1, a2, ..., aq ❧➔ q > 2 ✤✐➸♠ ♣❤➙♥ ❜✐➺t✳
❑❤✐ ✤â
q
(q − 1)T (r, f ) ≤ N (r, f ) +
N
r,
1
f − aj
− N1 (r, f ) + S (r, f )
N
r,
1
f − aj
− N0 (r, f ) + S (r, f ) .
j=1
q
≤ N (r, f ) +
j=1
tr♦♥❣ ✤â S (r, f ) = o(T (r, f )) ❦❤✐ r → ∞ ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❝â ✤ë
✤♦ ❤ú✉ ❤↕♥✱
N1 (r, f ) = N
✈➔ N0
r,
1
f
r
1
f
+ 2N (r, f ) − N (r, f ) + S (r, f )
❧➔ ❤➔♠ ✤➳♠ t↕✐ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f ♠➔ ❦❤æ♥❣ ♣❤↔✐
✷✶
Footer Page 23 of 166.
Header Page 24 of 166.
❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f − aj , j = 1, ..., q✳
✷✷
Footer Page 24 of 166.
Header Page 25 of 166.
❈❤÷ì♥❣ ✷
✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❦✐➸✉
◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❝❤♦ ❝→❝ ✤÷í♥❣
❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
✷✳✶ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❝❤♦ ✤÷í♥❣ ❝♦♥❣
❝❤➾♥❤ ❤➻♥❤
❈❤ó♥❣ tæ✐ s➩ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠✱ ❦➼ ❤✐➺✉ ❝❤✉➞♥ ❝õ❛ ▲þ
t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✲ ❈❛rt❛♥ ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ tø
C ✈➔♦
Pn (C)✳
✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳
❣å✐ ❧➔
tr➯♥
⑩♥❤ ①↕
f := (f0 : ... : fn ) : C → Pn (C)
✤÷ñ❝
✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ C ♥➳✉ f0, ..., fn ❧➔ ❝→❝ ❤➔♠ ♥❣✉②➯♥
C✳
❚❛ ❝â t❤➸ ✈✐➳t
f = (f0 : f1 : · · · : fn )
❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ tr➯♥
C.
tr♦♥❣ ✤â
❑❤✐ ✤â
fi
❧➔ ❝→❝ ❤➔♠ ♥❣✉②➯♥
f = (f0 , f1 , . . . , fn )
❣å✐ ❧➔ ❜✐➸✉ ❞✐➵♥ rót ❣å♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
✷✸
Footer Page 25 of 166.
f✳
✤÷ñ❝
