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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 TỐN HỌC

THÁI NGUYÊN – 2008


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

Chun 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


ử ử




ỵ tt





♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ỵ tt

✳ ✳ ✳ ✳

✽

❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳

✽

✶✳✷✳✶
✶✳✷✳✷

▼ët sè ✈➼ ❞ö ✈➲ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛

✳ ✳ ✳ ✳

✶✵

✶✳✷✳✸

▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛





ỵ ỡ tự t ừ





ỵ ỡ tự ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✷ ỵ ỡ tự rt
ữớ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤

✷✸
✷✳✶

❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤



ỵ ỡ tự ữớ ❤➻♥❤ ❝➢t
❝→❝ s✐➯✉ ♠➦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✷✳✷✳✶

▼ët sè ❜ê ✤➲ q✉❛♥ trå♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ỵ ỡ tự ❝→❝ ✤÷í♥❣ ❝♦♥❣
❝❤➾♥❤ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

✷✾



ỵ tt ố tr ừ ữủ ✤→♥❤ ❣✐→ ❧➔ ♠ët

tr♦♥❣ ♥❤ú♥❣ t❤➔♥❤ tü✉ ✤➭♣ ✤➩ ✈➔ s➙✉ s➢❝ ❝õ❛ t♦→♥ ❤å❝ tr♦♥❣ t❤➳
❦✛ ❤❛✐ ♠÷ì✐✳ ✣÷đ❝ ❤➻♥❤ t❤➔♥❤ tø ♥❤ú♥❣ ♥➠♠ ✤➛✉ ❝õ❛ ❝õ❛ t❤➳ ❦✛✱ ỵ
tt õ ỗ ố tứ ỳ ổ tr ừ ❍❛❞❛♠❛r❞✱
❇♦r❡❧ ✈➔ ♥❣➔② ❝➔♥❣ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ỹ
ừ t ồ ỵ tt ố tr ờ sỹ tờ qt õ
ỵ ỡ ừ số ỡ ỵ tt ♥❣❤✐➯♥ ❝ù✉ sü
♣❤➙♥ ❜è ❣✐→ trà ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ tứ

C C{} r t

ừ ỵ tt ỗ ỵ ỡ ừ ỵ ỡ
t❤ù ♥❤➜t ❧➔ ♠ët ❝→❝❤ ✈✐➳t ❦❤→❝ ❝õ❛ ❝æ♥❣ t❤ù❝ Pss s
ỵ õ r trữ

T (r, a, f )

❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦

a ♥➳✉ t➼♥❤ s❛✐ ❦❤→❝ ♠ët ✤↕✐ ❧÷đ♥❣ ❜à ❝❤➦♥✱ tr♦♥❣ ✤â a ❧➔ ởt số ự
tũ ỵ ỵ ỡ tự t❤➸ ❤✐➺♥ ♥❤ú♥❣ ❦➳t q✉↔ ✤➭♣ ♥❤➜t✱ s➙✉
s➢❝ ♥❤➜t ❝õ❛ ỵ tt ố tr ỵ ữ r❛ ♠è✐ q✉❛♥
❤➺ ❣✐ú❛ ❤➔♠ ✤➦❝ tr÷♥❣ ✈➔ ❤➔♠ ①➜♣
rt ự ỵ s❛✉ ✤➙②✿

❈❤♦ f : C −→ Pn(C) ❧➔ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥
t✉②➳♥ t➼♥❤✱ Hi✱ i = 1, ..., q✱ ❧➔ ❝→❝ s✐➯✉ ♣❤➥♥❣ ð ✈à tr➼ tê♥❣ q✉→t✳ ❱ỵ✐
✷


♠é✐ ε > 0 t❛ ❝â

q

m(r, Hj , f ) ≤ (n + 1 + ε)T (r, f ),
j=1

tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✤ó♥❣ ✈ỵ✐ ♠å✐ r > 0 ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❝â ✤ë
✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳
❑➳t q✉↔ tr➯♥ ❝õ❛ ❍✳ ❈❛rt❛♥ ❧➔ ❝æ♥❣ tr➻♥❤ ✤➛✉ t✐➯♥ ✈➲ ♠ð rở ỵ
tt ữớ ỷ ử t q õ
ổ ữ r ữợ ữủ số t ữớ
ợ s ♣❤➥♥❣ ð ✈à tr➼ tê♥❣ q✉→t✳ ❈æ♥❣ tr➻♥❤ ♥➔② ❝õ❛
æ♥❣ ✤➣ ✤÷đ❝ ✤→♥❤ ❣✐→ ❧➔ ❤➳t sù❝ q✉❛♥ trå♥❣ ✈➔ r ởt ữợ
ự ợ t tr ỵ tt
ỵ tt ữớ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ s❛✉ ♥➔② ✤÷đ❝
♠❛♥❣ t➯♥ ❤❛✐ ♥❤➔ t♦→♥ ồ ờ t ừ t õ ỵ t❤✉②➳t
◆❡✈❛♥❧✐♥♥❛ ✲ ❈❛rt❛♥✧✳
◆❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙②✱ ✈✐➺❝ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❈❛rt❛♥ ❝❤♦ tr÷í♥❣
❤đ♣ ❝→❝ s✐➯✉ ♠➦t t❤✉ ❤ót ữủ sỹ ú ỵ ừ t ồ
▼✳ ❘✉ ❬✶✷❪ ✤➣ ❝❤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♥❣ ✈ỵ✐ ♠å✐ r ✤õ ❧ỵ♥ ♥➡♠ ♥❣♦➔✐ ♠ët
t➟♣ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳ ❑➳t q✉↔ tr➯♥ ✤➣ ✤÷đ❝ ◗✳ ❨❛♥ ✈➔
✸


❩✳ ❈❤❡♥ ❬✹❪ ♠ð rë♥❣ ❝❤♦ tr÷í♥❣ ❤đ♣ ❤➔♠ ✤➳♠ t➼♥❤ ✤➳♥ ❜ë✐ ❝❤➦♥
✭❤❛② ❝á♥ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❝öt✮✳ ❑➳t q✉↔ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿

●✐↔ sû f : C → Pn(C) ❧➔ ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥
✤↕✐ sè ✈➔ Dj ✱ 1 ≤ j ≤ q ❧➔ q s✐➯✉ ♠➦t tr♦♥❣ Pn(C) ❝â ❜➟❝ dj tữỡ
ự tr tờ qt õ ợ ộ > 0 tỗ t ởt số
ữỡ M s ❝❤♦
q
M
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♥❣ ✈ỵ✐ ♠å✐ r ✤õ ❧ỵ♥ ♥➡♠ ♥❣♦➔✐ ♠ët
t➟♣ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳
❈❤♦ ✤➳♥ ♥❛②✱ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ sü tỗ t ừ
tổ q ♥❣÷đ❝ ❝õ❛ ❝→❝ s✐➯✉ ♠➦t✱ ♥❣÷í✐ t❛ t❤÷í♥❣ sû ❞ư♥❣
ỵ ỡ tự rt tổ q t
ở r ỵ ◆❡✈❛♥❧✐♥♥❛ ✲ ❈❛rt❛♥ ❝á♥ ❝❤♦ t❛ ❤✐➸✉

t❤➯♠ ✈➲ t➼♥❤ s✉② ❜✐➳♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤✳
▼ư❝ t✐➯✉ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ❝→❝ ❦➳t q✉↔ ✤➣ ✤÷đ❝
✤÷❛ r❛ ❝õ❛ ◗✳ ❨❛♥ ✈➔ ❩✳ ❈❤❡♥ ✈ỵ✐ ❝ỉ♥❣ ❝ư ự ừ
ỵ tt rt ❝→❝ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ tø

C

✈➔♦

Pn (C).
▲✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t ữỡ ũ ợ t
❞❛♥❤ ♠ư❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳
❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ❝→❝
✤à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✳ ❚r➻♥❤ ❜➔② ự
ỵ ỡ tự ừ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳
❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ ♠ët ❞↕♥❣ ỵ ỡ tự




❝❤♦ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❝➢t ❝→❝ s✐➯✉ ♠➦t ð ✈à tr➼ tê♥❣ q✉→t✳ ❈❤÷ì♥❣
♥➔② ✤÷đ❝ ✈✐➳t ❞ü❛ tr➯♥ ❝ỉ♥❣ tr➻♥❤ ❝õ❛ ◗✳ ❨❛♥✱ ❩✳ ❈❤❡♥ ❬✹❪✳

▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t ữợ sỹ ữợ ồ ừ

❍♦➔✐ ❆♥

✳ ❚→❝ ❣✐↔ 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 ❝ỉ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝
❙÷ ♣❤↕♠ t❤✉ë❝ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ✤➦❝ ❜✐➺t ❧➔ ❚❤➔②

P❤÷ì♥❣

❍➔ ❚r➛♥

✈➔ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✈➔ ❝→❝

t❤➛② ❝ỉ ❣✐→♦ ❱✐➺♥ ❚♦→♥ ❤å❝ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ ❤♦➔♥
t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥✳
❚→❝ ❣✐↔ ❝ô♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ❈❛♦
✤➥♥❣ ❈ỉ♥❣ ♥❣❤➺ ✈➔ ❑✐♥❤ t➳ ❈æ♥❣ ♥❣❤✐➺♣✱ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ t↕♦
♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ♥❤➜t ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣✳

✺


ữỡ
ỵ tt

r ữỡ ú tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ s➩
✤÷đ❝ sû ❞ö♥❣ tr♦♥❣ ❝→❝ ♣❤➛♥ s❛✉✳ ❈→❝ ❦✐➳♥ t❤ù❝ ❝õ❛ ❝❤÷ì♥❣ ♥➔②
✤÷đ❝ tr➼❝❤ ❞➝♥ tø ❬✶❪✱ ❬✺❪✱ ❬✼❪✱ ❬✾❪✱ ✳✳✳

✶✳✶ ❍➔♠ ♣❤➙♥ ❤➻♥❤
✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳

❈❤♦

D

❧➔ ♠ët ♠✐➲♥ tr♦♥❣ ♠➦t ♣❤➥♥❣ ♣❤ù❝

C✱

f (z) = u(x, y) + iv(x, y) ✤÷đ❝ ❣å✐ C t z0 C tỗ
f (z0 + h) − f (z0 )
t↕✐ ❣✐ỵ✐ ❤↕♥ ❤ú✉ ❤↕♥ lim
✳
h→0
h
●✐→ trà ✤â ✤÷đ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ ♣❤ù❝ ❝õ❛ ❤➔♠ f (z) t↕✐ z0 ✳
❤➔♠

❍➔♠

f (z)

✤÷đ❝ ❣å✐ ❧➔

C✲❦❤↔

✈✐ tr♦♥❣ D ♥➳✉ ♥â C ✲ ❦❤↔ ✈✐ t↕✐ ♠å✐

z0 ∈ D.

✶✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳
♥â

C

❍➔♠

❍➔♠

f (z)

✤÷đ❝ ❣å✐ ❧➔

❝❤➾♥❤ ❤➻♥❤ t↕✐ z0 ∈ C ♥➳✉

✲ ❦❤↔ ✈✐ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛

f (z)

✤÷đ❝ ❣å✐ ❧➔

z0 ✳

❝❤➾♥❤ ❤➻♥❤ tr➯♥ D ♥➳✉ ♥â ❝❤➾♥❤ ❤➻♥❤ t↕✐ ♠å✐
✻


✤✐➸♠

z

t❤✉ë❝

D✳

❚➟♣ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ♠✐➲♥

✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳
♣❤ù❝

C

✤÷đ❝ ❣å✐ ❧➔

f (z)

❍➔♠

D✱

❦➼ ❤✐➺✉ ❧➔

H(D)✳

❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ t♦➔♥ ♠➦t ♣❤➥♥❣

❤➔♠ ♥❣✉②➯♥✳

✶✳✶✳✹ ỵ 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

ỵ sû f (z) ❧➔ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ ♠✐➲♥ ❤ú✉

❤↕♥ D ⊂ C✳ ❑❤✐ ✤â tr♦♥❣ ♠é✐ ❧➙♥ ❝➟♥ ❝õ❛ ♠é✐ ✤✐➸♠ z ∈ D✱ ❤➔♠
f (z) ✤÷đ❝ ❦❤❛✐ tr✐➸♥ t❤➔♥❤ ❝❤✉é✐
(z − z0 )
(z − z0 )2
f (z) = f (z0 ) +
f (z0 ) +
f (z0 ) + . . .
1!
2!

❍ì♥ ♥ú❛✱ ❝❤✉é✐ tr➯♥ ❤ë✐ tư ✤➲✉
|z z0 | tũ ỵ tr♦♥❣ D.
❈❤✉é✐ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔
❝õ❛ ✤✐➸♠

✣✐➸♠

✭❤❛② ❦❤ỉ♥❣✲✤✐➸♠ ❝➜♣

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)

✭✶✳✶✮

✤÷đ❝ ❣å✐ ❧➔

❝õ❛ ❤➔♠

f (z)

❦❤ỉ♥❣ ✤✐➸♠ ❜➟❝ m > 0
♥➳✉

f (n) (z0 ) = 0,

❝❤♦

f (m) (z0 ) = 0.

❍➔♠

tr♦♥❣ ✤â

f (z)
g, h

✤÷đ❝ ❣å✐ ❧➔

❤➔♠ ♣❤➙♥ ❤➻♥❤

❧➔ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣

✼

tr♦♥❣

D.


◆➳✉
❧➔

D = C t❤➻ t❛ ♥â✐ f (z) ♣❤➙♥ ❤➻♥❤ tr➯♥ C ❤❛② ✤ì♥ ❣✐↔♥ ❧➔ f (z)

❤➔♠ ♣❤➙♥ ❤➻♥❤✳

✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳

✣✐➸♠

z0

✤÷đ❝

❣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 ừ

ỵ ổ tự P♦✐✐s♦♥ ✲ ❏❡♥s❡♥✮✳ ●✐↔ sû f (z) ≡ 0 ❧➔

♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr♦♥❣ ❤➻♥❤ trá♥ {|z| ≤ R} ✈ỵ✐ 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
.
2b z
R 2 aà z
R

=1

ỵ tt ❤➔♠ ♣❤➙♥ ❤➻♥❤
✶✳✷✳✶ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤
●✐↔ sỷ

f

ỵ

tr

r.

r < R✳

n(r, ∞, f ) ✭t÷ì♥❣ ù♥❣✱ n(r, ∞, f ), ❧➔ sè ❝→❝ ❝ü❝ ✤✐➸♠ t➼♥❤

❝↔ ❜ë✐✱ ✭t÷ì♥❣ ù♥❣✱ ❦❤ỉ♥❣ t➼♥❤ ❜ë✐✮✮✱ ❝õ❛ ❤➔♠
❦➼♥❤

R

●✐↔ sû

a ∈ C✱

f

t❛ ✤à♥❤ ♥❣❤➽❛

n(r, a, f ) = n r, ∞,

✽

1
,
f −a

tr♦♥❣ ✤➽❛ ✤â♥❣ ❜→♥


n(r, a, f ) = n r, ∞,

1
.
f −a

✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ✤➳♠ t➼♥❤ ❝↔ ❜ë✐ N (r, a, f ),
❦❤ỉ♥❣ t➼♥❤ ❜ë✐

N (r, a, f )✮✱

❝õ❛ ❤➔♠

f

t↕✐ ❣✐→ trà

a

✭t÷ì♥❣ ù♥❣✱

✤÷đ❝ ✤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

✤÷đ❝ ✤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 ❧ỵ♥ tr✉♥❣ ❜➻♥❤ ❝õ❛

|z| = r✳
✾

log |f |

tr➯♥ ✤÷í♥❣ trá♥


✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ✤➦❝ tr÷♥❣ T (r, a, f )
a∈C

❝õ❛ ❤➔♠

f

t↕✐ ❣✐→ trà

✤÷đ❝ ✤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 ♥➔♦ õ trữ ố ợ ỵ tt
❤➻♥❤ ❝â ✈❛✐ trá t÷ì♥❣ tü ♥❤÷ ❜➟❝ ❝õ❛ ✤❛ tự tr ỵ
tt tự ứ tr÷♥❣ t❛ ❝â

T (r, a, f ) ≥ N (r, a, f ) + O(1),
tr♦♥❣ ✤â

O(1)

❧➔ ✤↕✐ ❧÷đ♥❣ ❜à ❝❤➦♥ ❦❤✐

r→∞

✳

❱ỵ✐ ❝→❝❤ ✤à♥❤ ♥❣❤➽❛ ♥➔② t❤➻ ❝ỉ♥❣ t❤ù❝ P♦✐✐s♦♥✲❏❡♥s❡♥ ✭✣à♥❤ ỵ
ữủ t ữ 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

✶✵

dt
= p log r + O(1)
t


❦❤✐

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

✈ỵ✐

a = ∞.

P❤÷ì♥❣

♥❣❤✐➺♠✱ ✈➻ t❤➳

N (r, ∞, f ) = q log r + O(1),
ỵ ỡ tự ♥❤➜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),
❑❤✐

✈ỵ✐

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



ợ

a = c.

ỡ ỳ ỵ



õ

m(r, c, f ) = k log r + O(1),

N (r, c, f ) = (q − k) log r + O(1).

❱➟② tr♦♥❣ ♠å✐ tr÷í♥❣ ❤đ♣

T (r, f ) = d log r + O(1),
tr♦♥❣ ✤â

k

d = max(p, q).
✶✶

✶✳✷✳✺ ❱➼ ❞ư✳

❳➨t ❤➔♠

f (z) = ez .

❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱
π
2

2π
iθ

log+ ere

m(r, f ) =

dθ
=
2π

f

❧➔ ❤➔♠ ♥❣✉②➯♥ ♥➯♥

❱ỵ✐

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 ♥❣❤✐➺♠ ✈ỵ✐ ❝❤✉ ❦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✱

✶✳✷✳✻ ❱➼ ❞ư✳
❱ỵ✐ ♠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

✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜➡♥❣ tê ❤đ♣ 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).
✶✷

eiz

✈➔


✶✳✷✳✸ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛
❈❤ó♥❣ t❛ t✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ✤ì♥ ❣✐↔♥ ừ
ú ỵ r

p

p

log+

=1
p

log

log+ |a |

a

+

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❤✉ ✤÷đ❝
❝→❝ ❜➜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 ❞ư♥❣ ✭✶✳✸✮ t❛ t❤✉ ✤÷đ❝

p
✺✳

✻✳

T

T

r,

p

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✳
✶✸

❧➔

❱➔ tø ✤â


❝❤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.



ỵ ỡ tự t ừ
ỵ sỷ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ a ❧➔ ♠ët sè ♣❤ù❝ tũ ỵ
õ t õ
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.
❚❛ tữớ ũ ỵ ỡ tự t ữợ

m r,

1
f −a

+N

r,

1
f −a

= T (r, f ) + O(1),

tr♦♥❣ ✤â ởt ữủ ợ ở

ị
tr tr ổ tự ừ ỵ số
❤➔♠

T (r, f )

❦❤ỉ♥❣ ♣❤ư t❤✉ë❝

a✱

f −a

✈➔

f

❣➛♥

s❛✐ ❦❤→❝ ♠ët ✤↕✐ ữủ

ợ ở

ự t õ
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),
✶✹

a✳


✈ỵ✐

|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 õ

ỵ ữủ ự

ỵ ỡ tự
ỡ ú t s➩ ✈✐➳t

m(r, ∞)

t❤❛② ❝❤♦

m(r, a)

t❤❛② ❝❤♦

m r,

1
f −a

✈➔

m(r, f ).

✶✳✷✳✽ ỵ sỷ ❤➡♥❣ sè tr♦♥❣
|z| ≤ r✳

●✐↔ sû a1, a2, ..., aq ✈ỵ✐ q > 2 ❧➔ ❝→❝ sè ♣❤ù❝ ❤ú✉ ❤↕♥✱ r✐➯♥❣
❜✐➺t✱ δ > 0 ✈➔ ❣✐↔ sû r➡♥❣ |aµ − a | ợ 1 à < ≤ q✳ ❑❤✐ ✤â
q

m(r, ∞) +

m(r, aν ) ≤ 2T (r, f ) − N1 (r) + S(r),
ν=1

tr♦♥❣ ✤â N1(r) ❧➔ ❞÷ì♥❣ ✈➔ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐
1
N1 (r) = N r,
+ 2N (r, f ) − N (r, f ) ✈➔
f
f
S(r) = m r,
f
▲÷đ♥❣

S(r)

q

+m r,
ν=1

f
3q
1
+q log+ +log 2+log
✳
f − aν
δ
|f (0)|

tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t s➩ ✤â♥❣ ✈❛✐ trá ❧➔ s❛✐ sè

❦❤ỉ♥❣ ✤→♥❣ ❦➸✳ ❙ü tê♥❣ ❤đ♣ ❝→❝ ✈➜♥ ✤➲ õ tr ỵ tr s
ỵ ỡ ❜↔♥ t❤ù ❤❛✐✳ ✣✐➲✉ ✤â ❝❤♦ t❤➜② r➡♥❣✱ tr♦♥❣ tr÷í♥❣ ❤ñ♣
tê♥❣ q✉→t tê♥❣ ❝õ❛ ❝→❝ sè ❤↕♥❣

m(r, aν )
✶✺

t↕✐ ♠é✐ sè ❦❤ỉ♥❣ t❤➸ ❧ỵ♥


❤ì♥

2T (r)✳

❇➙② ❣✐í ❝❤ó♥❣ t❛ ❜➢t ✤➛✉ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ trữớ ủ tữỡ ố
ỡ ừ ỵ trữợ ỷ ỵ ợ ữợ ữủ ự t ỡ ừ

S(r).

ự ❱ỵ✐ ❝→❝ sè ♣❤➙♥ ❜✐➺t aν , (1 ≤ ν ≤ q)✱ t❛ ①➨t ❤➔♠
q

F (z) =
ν=1
✶✳ ●✐↔ sû r➡♥❣ ợ ởt

à =

1
.
f (z) a

õ

|f (z) a | <



3q

õ ợ

t õ

|f (z) aà | ≥ |aµ − aν | − |f (z) − a |
ợ

à=

2

,
3q
3

t

1
3
1


.
|f (z) aà | 2
|f (z) a |
ữ

|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
✶✻

✭✶✳✻✮


ợ

à=



log+

1
3
2
log+
log+
|f (z) aà |
2


t

õ

q

log+

à=1

1
1
= log+
+
|f (z) − aµ |
|f (z) − aν |

µ=ν

1
|f (z) − aµ |

2
1
+ (q − 1) log+ .
|f (z) − aν |
δ

≤ log+
❙✉② r❛✱

log+
µ=ν

1
2
≤ (q − 1) log+ .
|f (z) − aµ |

δ

❉♦ ✤â✱

q

log+

+

log |F (z)| ≥
µ=1

1
3q
− q log+
− log 2.
|f (z) aà |


ữủ ự
ữ tỗ t ởt tr

q



|f (z) a | <

δ

3q

t❤➻ ✭✶✳✻✮ ❧➔ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳

✷✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû

|f (z) − aν | ≥

δ
✱
3q

✈ỵ✐ ♠å✐

ν✳

❑❤✐ ✤â t❛ ❝â ♠ët

✤✐➲✉ ❤✐➸♥ ♥❤✐➯♥ ❧➔

q
+

log+

log |F (z)| ≥
ν=1
❉♦

ν✳

|f (z) − aν | ≥

❱➟②

δ
✱
3q

q

log+
ν=1

1
3q
− q log+
− log 2.
|f (z) − aν |
δ
ν

✈ỵ✐ ♠å✐

♥➯♥

1
3q
≤ ✱
|f (z) − aν |

δ

✈ỵ✐ ♠å✐

3q
1
≤ q log+
+ log 2.
|f (z) − aν |
δ

❚ø ✤â

q
+

log+

log |F (z)| ≥ 0 ≥
ν=1

1
3q
− q log+
− log 2.
|f (z) − aν |
δ

✶✼

◆❤÷ ✈➟②✱ tr♦♥❣ ♠å✐ tr÷í♥❣ ❤đ♣ t❛ ✤➲✉ ❝â ✤÷đ❝

q
+

log+

log |F (z)| ≥
ν=1
❱ỵ✐

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
✶✽

−
r,

f
f

+ log

f (0)
.
f (0)

✭✶✳✾✮


✈➔ ♥❣♦➔✐ 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♣ ✭✶✳✼✮ ✈ỵ✐ ✭✶✳✶✶✮ 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

log f (reiφ ) dφ−log |f (0)|
0

✶✾

=N

r,

1
f

− N (r, f ) − N

r,

1
f

+ N (r, f ) .

❈✉è✐ ❝ị♥❣ t❛ ♥❤➟♥ ✤÷đ❝

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)|


ú ỵ r

q

m (r, f F ) = m r,
=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 ❧➔ ✤✐➲✉ ❝➛♥ ự

q

t
log

=1

R
b

N1 (r)

tr ỵ tr ữỡ

tr♦♥❣ tê♥❣ tr➯♥ ♥➳✉

bv
✷✵

❧➔ ❝ü❝ ✤✐➸♠ ❜ë✐

k

N (r, f ) =

t❤➻ ✤÷đ❝ t➼♥❤


k

❧➛♥✳

●✐↔ sû

b1 , b2 , ..., bN

❧➔ ❝→❝ ❝ü❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛

f (z)

✈ỵ✐ ❝➜♣ ❧➛♥

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❝

❧÷đt ❧➔

✤✐➸♠ ❝õ❛

f (z)

kν log |

N (r, f ) =

k1 + 1, k2 + 1, ..., kN + 1✳ ◆❤÷
N
r
N (r, f ) =
(kν + 1) log | |
b
=1

ợ ữủt

N

=1

r
|
b



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 |
=1



r
|=
b

r
| 0.
b

õ ỵ s

ỵ ỵ ỡ tự 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♥❣ ♣❤↔✐
✷✶

❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f − aj , j = 1, ..., q✳

✷✷


ữỡ
ỵ ỡ tự
rt ✤÷í♥❣
❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
✷✳✶ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❝❤♦ ✤÷í♥❣ ❝♦♥❣
❝❤➾♥❤ ❤➻♥❤
❈❤ó♥❣ tæ✐ s➩ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠✱ ❦➼ ❤✐➺✉ ừ ỵ
tt rt ữớ ❝❤➾♥❤ ❤➻♥❤ tø

C ✈➔♦

Pn (C)✳

✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳
❣å✐ ❧➔
tr➯♥

⑩♥❤ ①↕

f := (f0 : ... : fn ) : C → Pn (C)

✤÷đ❝

✤÷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♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤

✷✸

f✳

✤÷đ❝