Vấn đề duy nhất của lũy thừa một hàm phân hình với đa thức đạo hàm của chúng

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

▲➊ ❱❿◆ ❈❍×❒◆●

❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ❈Õ❆
▲Ô❨ ❚❍Ø❆ ▼❐❚ ❍⑨▼ P❍❹◆ ❍➐◆❍
❱❰■ ✣❆ ❚❍Ù❈ ✣❸❖ ❍⑨▼ ❈Õ❆ ❈❍Ó◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

▲➊ ❱❿◆ ❈❍×❒◆●

❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ❈Õ❆
▲Ô❨ ❚❍Ø❆ ▼❐❚ ❍⑨▼ P❍❹◆ ❍➐◆❍
❱❰■ ✣❆ ❚❍Ù❈ ✣❸❖ ❍⑨▼ ❈Õ❆ ❈❍Ó◆●
❈❤✉②➯♥ ♥❣➔♥❤ ✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍
▼➣ số


ữớ ữợ ❦❤♦❛ ❤å❝✿
P●❙✳ ❚❙ ❍⑨ ❚❘❺◆ P❍×❒◆●
❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵

▲í✐ ❝❛♠ ✤♦❛♥
❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣
t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❝→❝
❦➳t q✉↔ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥✱ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✈➔ ♥ë✐ ❞✉♥❣ tr➼❝❤ ❞➝♥ ✤↔♠
❜↔♦ t➼♥❤ tr✉♥❣ t❤ü❝ ❝❤➼♥❤ ①→❝✳
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✷✵

◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥
▲➯ ữỡ


ừ trữ


ừ ữớ ữợ

P ❚❙ ❍⑨ ❚❘❺◆ P❍×❒◆●
✐


▲í✐ ❝↔♠ ì♥
❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t tợ P r Pữỡ
ữớ t t ❝❤➾ ❜↔♦✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ✈➔ ❣✐ó♣ ✤ï tỉ✐ ❝â t❤➯♠ ♥❤✐➲✉ ❦✐➳♥
t❤ù❝✱ ❦❤↔ ♥➠♥❣ ♥❣❤✐➯♥ ❝ù✉✱ tê♥❣ ❤ñ♣ t➔✐ ❧✐➺✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♠ët
❝→❝❤ ❤♦➔♥ ❝❤➾♥❤✳
❚ỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ỗ
ở ú ù tổ q tr➻♥❤ ❤å❝ t➟♣ ❝õ❛ ♠➻♥❤✳
❉♦ t❤í✐ ❣✐❛♥ ✈➔ tr➻♥❤ ✤ë ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐

♥❤ú♥❣ t❤✐➳✉ sõt ú tổ rt ữủ sỹ õ ỵ ❝õ❛ ❝→❝ t❤➛②
❝ỉ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳
❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✷✵

◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥
▲➯ ❱➠♥ ❈❤÷ì♥❣

✐✐


▼ư❝ ❧ư❝
▲í✐ ❝❛♠ ✤♦❛♥
▲í✐ ❝↔♠ ì♥
▼ư❝ ❧ư❝
▼ð ✤➛✉
✶ ❑✐➳♥ t❤ù❝




ỵ ỡ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✶✳ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✈➔ t➼♥❤ t
ỵ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✸✳ ◗✉❛♥ ❤➺ sè ❦❤✉②➳t ✈➔ ✤✐➸♠ ❜ä ✤÷đ❝ P✐❝❛r❞ ✳
❍➔♠ ✤➳♠ ♠ð rë♥❣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤➳♠ ♠ð rë♥❣ ✳ ✳

✳

✳
✳
✳
✳
✳
✳

✳
✳
✳
✳
✳
✳
✳

✳
✳
✳
✳
✳
✳
✳

✐
✐✐
✐✐✐
✶
✸

✳ ✸

✳ ✸
✳ ✺
✳ ✻
✳ ✼
✳ ✼
✳ ✶✵

✷ ❱➜♥ ✤➲ ❞✉② ♥❤➜t

✶✼

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✵
✹✷

✷✳✶
✷✳✷

▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳
ỵ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✐✐✐


▼ð ✤➛✉
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱ a ∈ C ∪ {∞}✳
❚❛ ❦➼ ❤✐➺✉

E f (a) = f −1 (a) = {z ∈ C : f (z) = a}
Ef (a) = {(z, n) ∈ C × N : f (z) = a, ordf −a (z) = n}
❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C ✈➔ a ❧➔ ♠ët
❣✐→ trà ♣❤ù❝ ❤ú✉ ❤↕♥ ❤♦➦❝ ∞✳ ❚❛ ♥â✐ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ a ❦➸ ❝↔ ❜ë✐ ✭✈✐➳t
♥❣➢♥ ❣å♥ ❧➔ ❈▼✮ ♥➳✉ Ef (a) = Eg (a)✳ ❚❛ ♥â✐ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ a ❦❤æ♥❣
❦➸ ❜ë✐ ✭✈✐➳t ♥❣➢♥ ❣å♥ ❧➔ ■▼✮ ♥➳✉ E f (a) = E g (a) .
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ a(z) ✤÷đ❝ ❣å✐ ❧➔
❤➔♠ ♥❤ä ❝õ❛ f ♥➳✉ T (r, a) = o(T (r, f )). ❱ỵ✐ ❤➔♠ ♥❤ä a(z), t❛ ♥â✐ f, g
❝❤✉♥❣ ♥❤❛✉ ❤➔♠ a(z) ❈▼ ✭❤♦➦❝ ■▼✮ ♥➳✉ ❤➔♠ f − a ✈➔ g − a ❝❤✉♥❣ ♥❤❛✉
❣✐→ trà 0 ❈▼ ✭■▼ t÷ì♥❣ ù♥❣✮✳
◆➠♠ ✶✾✼✼✱ ❘✉❜❡❧ ✈➔ ❨❛♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤✿ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♥❣✉②➯♥
❦❤→❝ ❤➡♥❣✱ ♥➳✉ f ✈➔ f ❝❤✉♥❣ ♥❤❛✉ ❤❛✐ ❣✐→ trà ❤ú✉ ❤↕♥ ♣❤➙♥ ❜✐➺t a ✈➔ b
❦➸ ❝↔ ❜ë✐ t❤➻ f = f ✳ ◆➠♠ ✶✾✼✾✱ ▼✉❡s ✈➔ ❙t❡✐♥♠❡t③ ✭❬✶✹❪✮ ✤➣ ❝❤ù♥❣ ♠✐♥❤
❦➳t q✉↔ t÷ì♥❣ tü ❦❤✐ t❤❛② ✤✐➲✉ ❦✐➺♥ ❈▼ ❜ð✐ ■▼✳ ❚ø ♥❤ú♥❣ ❝æ♥❣ tr➻♥❤
♥➔② ❝õ❛ ❝→❝ t→❝ ❣✐↔ ✤➣ ♥↔② s✐♥❤ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤
✈ỵ✐ ✤↕♦ ❤➔♠ ❝õ❛ ❝❤ó♥❣✳
◆➠♠ ✷✵✵✽✱ ❚✳ ❩❤❛♥❣ ✈➔ ❲✳ ▲☎
✉ ✭❬✶✻❪✮ ✤➣ ①❡♠ ①➨t ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦
❧ô② t❤ø❛ ❜➟❝ n ❝õ❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä ✈ỵ✐ ✤↕♦
❤➔♠ ❝➜♣ k ❝õ❛ ♥â ✈➔ t❤✉ ✤÷đ❝ ♠ët sè ❦➳t q✉↔ ✈➲ ✈➜♥ ✤➲ ♥➔②✳ ❈ư t❤➸✱ ❝→❝
t→❝ ❣✐↔ ✤➣ ✤÷❛ r❛ ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ✤➸ ❝→❝ ❤➔♠ f n − a ✈➔ f (k) − a
✶


✈➔ ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ✵ ❦❤æ♥❣ ❦➸ ❜ë✐ ❤♦➦❝ ❦➸ ❝↔ ❜ë✐ t❤➻ f n = f (k) ✱ tr♦♥❣
✤â a(z) ❧➔ ♠ët ❤➔♠ ♥❤ä✳
❑➼ ❤✐➺✉

Mj (f ) = (f )n0i f (1)
✈➔

n1i

... f (k)

nki

t

P [f ] =

Mj (f ) .
j=1

●➛♥ ✤➙② ❝â ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ♠ð rë♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❚✳ ❩❤❛♥❣ ✈➔ ❲✳ ▲☎
✉
❝❤♦ ❝→❝ tr÷í♥❣ ❤đ♣✿ t❤❛② t❤➳ ❧ơ② t❤ø❛ ❜➟❝ n ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f tr♦♥❣
❦➳t q✉↔ ❚✳ ❩❤❛♥❣ ✈➔ ❲✳ ▲☎
✉ ❝õ❛ ❜ð✐ ✤❛ t❤ù❝ ❜➟❝ n ❝õ❛ ❤➔♠ ✤â❀ t❤❛② t❤➳
✤↕♦ ❤➔♠ ❝➜♣ k ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ❜ð✐ ♠ët ✤ì♥ t❤ù❝ ❝❤ù❛ ❝→❝ ✤↕♦ ❤➔♠
❝→❝ ❝➜♣ Mj [f ] ❤♦➦❝ ✤❛ t❤ù❝ ❝❤ù❛ ❝→❝ ✤↕♦ ❤➔♠ P [f ] ❝õ❛ ❤➔♠ ✤â✳
▼ư❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ❣✐ỵ✐ t❤✐➺✉ ♠ët sè ♥❣❤✐➯♥ ❝ù✉ ❣➛♥ ✤➙②
❝õ❛ ❚✳ ❩❤❛♥❣✱ ❲✳ ▲☎
✉✱ ❆✳ ❇❛♥❡r❥❡❡✱ ❇✳ ❈❤❛❦r❛❜♦rt② ✈➔ ♠ët sè t→❝ ❣✐↔
❦❤→❝ t ữợ ự õ tr
❝❤÷ì♥❣✱ tr♦♥❣ ❈❤÷ì♥❣ ✶ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔②
♠ët sè ❦✐➳♥ t❤ù❝ ❝➛♥ ❝❤✉➞♥ ❜à✱ ❝➛♥ t❤✐➳t ❝❤♦ ❝→❝ ♥ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥✳
❈❤÷ì♥❣ ✷ ❧➔ ❝❤÷ì♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦➳t
q✉↔ ✈➲ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤✐ ❧ô② t❤ú❛ ❝õ❛ ♠ët
❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝â ❝❤✉♥❣ ♠ët ❣✐→ trà ❤❛② ❤➔♠ ♥❤ä ✈ỵ✐ ✤ì♥ t❤ù❝ ❤♦➦❝ ✤❛

t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ♥â✳
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✷✵

◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥
▲➯ ❱➠♥ ❈❤÷ì♥❣

✷


❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✈➔ ❤❛✐ ỵ ỡ
r ỵ tt ố tr ◆❡✈❛♥❧✐♥♥❛✱ ❝→❝ ❤➔♠ ①➜♣ ①➾✱ ❤➔♠ ✤➳♠✱
❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✱
✤â♥❣ ởt trỏ q trồ sốt ỵ tt r ♣❤➛♥ ♥➔② ❝❤ó♥❣
tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ♥➔② ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳

✶✳✶✳✶✳ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✈➔ t➼♥❤ ❝❤➜t
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤ù❝ C ✈➔ r > 0 ❧➔ ♠ët sè t❤ü❝
❞÷ì♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❍➔♠
2π

m(r, f ) =

1
2π

log+ f (reiϕ ) dϕ

0

✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ①➜♣ ①➾ ❝õ❛ ❤➔♠ f ✳
❇➙② ❣✐í t❛ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❤➔♠ ✤➳♠✳ ❑➼ ❤✐➺✉ 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❛ ❢ tr♦♥❣

Dr = {z ∈ C : |z| ≤ |r|}✳
✸


✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❍➔♠
r

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

N (r, ∞; f ) = N (r, f ) =
0

✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ ❝õ❛ ❢ ✭❝á♥ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ t↕✐ ❝→❝ ❝ü❝
✤✐➸♠✮✳ ❍➔♠
r

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

N (r, ∞; f ) = N (r, f ) =

0

✤÷đ❝ ❣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 )
ỡ tr ỵ tt ❜è ❣✐→ trà✱ ♥â ❝á♥ ❣å✐ ❧➔ ❝→❝ ❤➔♠
◆❡✈❛♥❧✐♥♥❛✳

▼➺♥❤ ✤➲ ✶✳✶✳✹ ✭▼ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✮✳ ❈❤♦
f1 , f2 , . . . , fp ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C, ❦❤✐ ✤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,
=1

T (r, f ).
=1

ỵ ỡ
ỵ ỵ ỡ tự t f ≡ 0 ❧➔ ♠ët ❤➔♠ ♣❤➙♥
❤➻♥❤ tr➯♥ C✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ r > 0✱ t❛ ❝â
1
1
+ N r,
+ log |cj |
✭✶✮ T (r, f ) = m r,
f
f

✭✷✮ ❱ỵ✐ ♠é✐ sè ♣❤ù❝ a ∈ C,

T (r, f ) − m r,

1
1
+ N r,
f −a

f −a

≤ log

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♥❣ ❞ị♥❣ ừ ỵ ỡ tự t ữợ


1
= T (r, f ) + O(1),
f −a
tr♦♥❣ ✤â O(1) ❧➔ ✤↕✐ ❧÷đ♥❣ ❜à ❝❤➦♥ ❦❤✐ r → ∞.
T r,

❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ r > 0✳ ❑➼ ❤✐➺✉
1
Nram (r, f ) = N r,
+ 2N (r, f ) − N (r, f )
f
✈➔ ❣å✐ ❧➔ ❤➔♠ ❣✐→ trà ♣❤➙♥ ♥❤→♥❤ ❝õ❛ ❤➔♠ f ✳ ❍✐➸♥ ♥❤✐➯♥ Nram (r, f ) 0.

ỵ ỵ ỡ tự ❤❛✐✮✳ ●✐↔ sû f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝

❤➡♥❣ tr➯♥ C✱ a1 , . . . , aq ∈ C, (q > 2) ❧➔ ❝→❝ ❤➡♥❣ sè ♣❤➙♥ ❜✐➺t✱ ❦❤✐ ✤â ✈ỵ✐
♠é✐ ε > 0✱ ❜➜t ✤➥♥❣ t❤ù❝
q

(q − 1)T (r, f ) ≤

N r,
j=1

1
+ N (r, f ) − Nram (r, f ) + log T (r, f )
f − aj
✺


+ (1 + ε) log+ log T (r, f ) + O(1)
q

≤

N r,
j=1

1
+ N (r, f ) + log T (r, f )
f − aj

+ (1 + ε) log+ log T (r, f ) + O(1)
✤ó♥❣ ✈ỵ✐ ♠å✐ r ≥ r0 ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ E ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳

✶✳✶✳✸✳ ◗✉❛♥ ❤➺ sè ❦❤✉②➳t ✈➔ ✤✐➸♠ ❜ä ✤÷đ❝ P✐❝❛r❞
●✐↔ sû f (z) ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C✳ ❚❛ ❦➼ ❤✐➺✉

1
1
N (r,
)
)
f −a
f −a
= 1 − lim sup
;
δf (a) = lim inf
r→∞
T (r, f )
T (r, f )
r→∞
1
N (r,
)
f −a
Θf (a) = 1 − lim sup
;
T (r, f )
r→∞
1
1
N (r,
) − N (r,
)

f −a
f −a
θf (a) = lim inf
.
r→∞
T (r, f )
m(r,

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✽✳ δf (a) ✤÷đ❝ ❣å✐ ❧➔ sè ❦❤✉②➳t✱ Θf (a) ❣å✐ ❧➔ sè ❦❤✉②➳t
❦❤æ♥❣ ❦➸ ❜ë✐✱ θf (a) ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ❜ë✐ ❝õ❛ sè ❦❤✉②➳t✳

◆❤➟♥ ①➨t✳ ✶✳ ◆➳✉ f (z) = a ✈æ ♥❣❤✐➺♠ t❤➻ N (r, f −1 a ) = 0 ✈ỵ✐ ♠å✐ r s✉②
r❛ δf (a) = 1✳ ❈❤➥♥❣ ❤↕♥ f (z) = ez t❤➻ δf (0) = 1.
1
✷✳ ◆➳✉ N (r,
) = o(T (r, f )) ❦❤✐ ✤â δf (a) = 1✳ ◆❤÷ ✈➟② sè ❦❤✉②➳t
f −a
❜➡♥❣ 1 ❦❤✐ sè ừ ữỡ tr q t s ợ t ❝õ❛ ♥â✳
✸✳ ❱ỵ✐ ♠é✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ✈➔ a C t ổ õ

0

f (a)

f (a)

1.

ỵ s t❛ ♠ët t➼♥❤ ❝❤➜t ❝õ❛ sè ❦❤✉②➳t✱ t❤÷í♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❜ê
✤➲ q✉❛♥ ❤➺ sè ❦❤✉②➳t✳

✻


ỵ f ❤➡♥❣ tr➯♥ C✳ ❑❤✐ ✤â t➟♣
❤ñ♣ ❝→❝ ❣✐→ trà ❝õ❛ a ♠➔ Θf (a) > 0 ❝ò♥❣ ❧➢♠ ❧➔ ✤➳♠ ữủ ỗ tớ t
õ

f (a) + f (a)
aC

f (a)

2.

aC

q ỵ Pr sỷ f ❤➻♥❤ tr➯♥ C✱ ♥➳✉
f ❦❤æ♥❣ ♥❤➟♥ ✸ ❣✐→ trà a1 , a2 , a3 ∈ C ∪ {∞} t❤➻ f ❧➔ ❤➔♠ ❤➡♥❣✳
❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C✳ ▼ët ♣❤➛♥ tû a ∈ C ∪ {∞} ✤÷đ❝
❣å✐ ❧➔ ❣✐→ trà ❜ä ✤÷đ❝ P✐❝❛r❞ ❝õ❛ f ♥➳✉ a ∈ f (C), tù❝ ❧➔ ♣❤÷ì♥❣ tr➻♥❤

f (z) = a ổ õ tr C ỵ r a ❧➔ ♠ët ✤✐➸♠ ❜ä ✤÷đ❝
1
P✐❝❛r❞ t❤➻ N r,
= 0✱ ❞♦ õ f (a) = 1 ứ ỵ Pr t ❞➵
f −a
❞➔♥❣ s✉② r❛✿

▼➺♥❤ ✤➲ ✶✳✶✳✶✶✳ ▼é✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ C ❝â ♥❤✐➲✉ ♥❤➜t

❧➔ ❤❛✐ ✤✐➸♠ ✤✐➸♠ ❜ä ✤÷đ❝ P✐❝❛r❞✱ ♠é✐ ❤➔♠ ♥❣✉②➯♥ tr➯♥ C ❝â ♥❤✐➲✉ ♥❤➜t
♠ët ✤✐➸♠ ❜ä ✤÷đ❝ P✐❝❛r❞✳

✶✳✷ ❍➔♠ ✤➳♠ ♠ð rë♥❣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t

✶✳✷✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠
❱ỵ✐ ❤➔♠ ♣❤➙♥ f s ừ f ỵ ❜ð✐ ρ2 (f ), ✤÷đ❝ ①→❝ ✤à♥❤
❜ð✐

ρ2 (f ) = lim sup
r→∞

log log T (r, f )
.
log r

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❈❤♦ p ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ a ∈ C ∪ {∞}✳
✭✐✮ ❑➼ ❤✐➺✉ N (r, a; f | ≥ p) ❧➔ ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ t↕✐ ❝→❝ a✲✤✐➸♠ ❝õ❛ f
♠➔ ❝â ❜ë✐ ❦❤ỉ♥❣ ♥❤ä ❤ì♥ p✳
✭✐✐✮ ❑➼ ❤✐➺✉ N (r, a; f ≥ p) ❧➔ ❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐ t↕✐ ❝→❝ a✲✤✐➸♠ ❝õ❛

f ♠➔ ❝â ❜ë✐ ❦❤æ♥❣ ♥❤ä ❤ì♥ p✳
✼


✭✐✐✐✮ ❑➼ ❤✐➺✉ N (r, a; f | ≤ p) ❧➔ ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ t↕✐ ❝→❝ a✲✤✐➸♠ ❝õ❛ f
♠➔ ❝â ❜ë✐ ❦❤ỉ♥❣ ❧ỵ♥ ❤ì♥ p✳
✭✐✈✮ ❑➼ ❤✐➺✉ N (r, a; f ≤ p) ❧➔ ❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐ t↕✐ ❝→❝ a✲✤✐➸♠ ❝õ❛

f ♠➔ ❝â ❜ë✐ ❦❤ỉ♥❣ ❧ỵ♥ ❤ì♥ p✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳ ❱ỵ✐ a ∈ C ∪ {∞} ✈➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ p✱ ❦➼ ❤✐➺✉
Np (r, a; f ) = N (r, a; f ) + N (r, a; f | ≥ 2) + · · · + N (r, a; f | ≥ p).
❘ã r➔♥❣ N1 (r, a; f ) = N (r, a; f )✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳ ❱ỵ✐ a ∈ C ∪ {∞} ✈➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ p✱ ✤➦t
δp (a, f ) = 1 − lim sup
r→∞

Np (r, a; f )
.
T (r, f )

❉➵ t❤➜②

0 ≤ δ(a, f ) ≤ δp (a, f ) ≤ δp−1 (a, f )
≤ . . . ≤ δ2 (a, f ) ≤ δ1 (a, f ) = Θ(a, f ) ≤ 1.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✹✳ ❈❤♦ ❤❛✐ sè ♥❣✉②➯♥ n, p✱ ✤à♥❤ ♥❣❤➽❛
µp = min{n, p} ✈➔ µ∗p = p + 1 − µp .
❑❤✐ ✤â t❛ ❞➵ r➔♥❣ ❦✐➸♠ ❝❤ù♥❣

Np (r, 0; f n ) ≤ µp Nµ∗p (r, 0; f ).

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✺✳ ❈❤♦ f

✈➔ g ❧➔ ❤❛✐ ❤➔♠ tr C a

C {}

ỵ ❤✐➺✉ N L (r, a; f ) ❧➔ ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ t↕✐ ❝→❝ a✲✤✐➸♠ ❝❤✉♥❣ ✭❝â
❜ë✐

1✮ ❝õ❛ f ✈➔ g t❤ä❛ ♠➣♥ ❜ë✐ ❝õ❛ a−✤✐➸♠ ✤â ✤è✐ ✈ỵ✐ f ợ ỡ

ố ợ g
1)

ỵ NE (r, a; f ) ❧➔ ❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐ t↕✐ ❝→❝ a✲✤✐➸♠ ❝❤✉♥❣
❝õ❛ f ✈➔ g t❤ä❛ ♠➣♥ ❜ë✐ ❝õ❛ a−✤✐➸♠ ✤â ✤è✐ ✈ỵ✐ f ❜➡♥❣ ✤è✐ ✈ỵ✐ g ✈➔
❜➡♥❣ ✶✳
✽


(2

ỵ N E (r, a; f ) ❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐ t↕✐ ❝→❝ a✲✤✐➸♠ ❝❤✉♥❣
❝õ❛ f ✈➔ g t❤ä❛ ♠➣♥ ❜ë✐ ❝õ❛ a−✤✐➸♠ ✤â ✤è✐ ✈ỵ✐ f ố ợ g
ợ ỡ
(2

1)

ữỡ tỹ t ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❤➔♠ N L (r, a; g), NE (r, a; g), N E (r, a; g).

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✻✳ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ n0i, n1i, . . . , nki ❧➔ ❝→❝
sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✳ ❇✐➸✉ ❞✐➵♥

Mj (f ) = (f )n0i f (1)

n1i

... f (k)

nki

✤÷đ❝ ❣å✐ ❧➔ ✤ì♥ t❤ù❝ ✈✐ ♣❤➙♥ s✐♥❤ ❜ð✐ f ✈ỵ✐ ❜➟❝
k

dMj = d(Mj ) =

nij
i=0

✈➔ ✤ë ❝❛♦

k

ΓMj =

(i + 1)nij .
i=0

❚ê♥❣

t

P [f ] =

Mj (f )

j=1

✤÷đ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ s✐♥❤ ❜ð✐ f ✈ỵ✐ ❜➟❝

d(P ) = max{d(Mj ) : 1

j

t}

✈➔ ✤ë ❝❛♦

ΓP = max{ΓMj : 1
❙è d(P ) = max{d(Mj ) : 1

j

t}.

t} ✈➔ k ✭❜➟❝ ❝❛♦ ♥❤➜t ❝õ❛ ✤↕♦ ❤➔♠

j

❝õ❛ f tr♦♥❣ P [f ]✮ t÷ì♥❣ ù♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ t❤➜♣ ✈➔ ❜➟❝ ❝õ❛ P [f ].

P [f ] ✤÷đ❝ ❣å✐ ❧➔ t❤✉➛♥ ♥❤➜t ♥➳✉ d(P ) = d(P )✳ P [f ] ✤÷đ❝ ❣å✐ ❧➔ ✤❛
t❤ù❝ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ s✐♥❤ ❜ð✐ f ♥➳✉ d(P ) = 1✱ ♥❣÷đ❝ ❧↕✐ P [f ] ✤÷đ❝ ❣å✐
❧➔ ❦❤æ♥❣ t✉②➳♥ t➼♥❤✳
✾

❑➼ ❤✐➺✉

Q = max{ΓMj − d(Mj ) : 1

j

t}

= max{n1j + 2n2j + · · · + knkj : 1

j

t}.

❚÷ì♥❣ tü✱ t❛ ❝ơ♥❣ ❝â ✤à♥❤ ♥❣❤➽❛ ✈ỵ✐ ✤ì♥ t❤ù❝ ✈✐ ♣❤➙♥ M [f ] ✿

λ = ΓM − dM .

✶✳✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤➳♠ ♠ð rë♥❣
❚r♦♥❣ ♣❤➛♥ ♥➔②✱ t❛ tr➻♥❤ ❜➔② ♠ët sè ❜ê ✤➲ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤➳♠
♠ð rë♥❣✳ ❱ỵ✐ F, G ❧➔ ❤❛✐ ỵ H
s

H=

2F
F

F

F −1

−

G
2G
−
.
G
G−1

✭✶✳✶✮

❇ê ✤➲ ✶✳✷✳✼ ✭❬✷❪✮✳ ❱ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱ t❛ ❝â
1 + δ2 (0, f ) ≥ 2Θ(0, f ).
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â

2N (r, 0; f )
N2 (r, 0; f )
− lim sup
T (r, f )
T (r, f )
r→∞
r→∞
N2 (r, 0; f ) − 2N (r, 0; f )
≤ lim sup
T (r, f )
r→∞
≤ 0,

2Θ(0, f ) − δ2 (0, f ) − 1 = lim sup

❜ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

❇ê ✤➲ ✶✳✷✳✽ ✭❬✷❪✮✳ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ ✈➔ M [f ] ❧➔ ✤ì♥
t❤ù❝ ✈✐ ♣❤➙♥ ❝â ❜➟❝ dM ✈➔ ✤ë ❝❛♦ ΓM ✳ ❑❤✐ ✤â

T (r, M ) ≤ dM T (r, f ) + λN (r, ∞; f ) + S(r, f ).

❇ê ✤➲ ✶✳✷✳✾ ✭❬✷❪✮✳ ❚❛ ❝â
N (r, 0; M ) ≤ T (r, M ) − dM T (r, f ) + dM N (r, 0; f ) + S(r, f ).
✶✵


❇ê ✤➲ ✶✳✷✳✶✵ ✭❬✷❪✮✳ ❚❛ ❝â
N (r, 0; M ) ≤ dM N (r, 0; f ) + λN (r, ∞; f ) + S(r, f ).

❇ê ✤➲ ✶✳✷✳✶✶ ✭❬✶✸❪✮✳ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ ✈➔ ✤➦t
n
i
i=0 ai f
m
j
j=0 bj f

R(f ) =

❧➔ ❤➔♠ ❤ú✉ t✛ ❜➜t ❦❤↔ q✉② t❤❡♦ ❜✐➳♥ f ✈ỵ✐ ❤➺ sè ❤➡♥❣ sè {ai } ✈➔ {bj }✱
tr♦♥❣ ✤â an = 0 ✈➔ bm = 0✳ ❑❤✐ ✤â t❛ ❝â

T (r, R(f )) = pT (r, f ) + S(r, f ),
tr♦♥❣ ✤â p = max{n, m}✳

❇ê ✤➲ ✶✳✷✳✶✷ ✭❬✷❪✮✳ ❚❛ ❝â
N (r, ∞;

M
) ≤ dM N (r, 0; f ) + λN (r, ∞; f ) + S(r, f ).
f dM

❈❤ù♥❣ ♠✐♥❤✳ ●å✐ z0 ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ f ❝â ❜➟❝ t✳ ❑❤✐ ✤â ♥â ❧➔ ♠ët ❝ü❝

✤✐➸♠ ❝õ❛

d
f dM

✈ỵ✐ ❜➟❝ n1 + 2n2 + · · · + knk = λ.

●å✐ z0 ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f ❝â ❜➟❝ s✳ ❑❤✐ ✤â ❧➔ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛

d
f dM

✈ỵ✐ ❜➟❝ ♥❤✐➲✉ ♥❤➜t ❧➔ sdM ✳ ❉♦ ✤â

N (r, ∞;

M
) ≤ dM N (r, 0; f ) + λN (r, ∞; f ) + S(r, f ).

f dM

❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

❇ê ✤➲ ✶✳✷✳✶✸ ✭❬✷❪✮✳ ❱ỵ✐ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f1 ✈➔ f2 ❜➜t ❦ý t❛
❝â

Np (r, ∞; f1 f2 ) ≤ Np (r, ∞; f1 ) + Np (r, ∞; f2 ).
❈❤ù♥❣ ♠✐♥❤✳ ●å✐ z0 ❧➔ ♠ët ❝ü❝ ✤✐➸♠ ❝õ❛ fi ✈ỵ✐ ❜➟❝ ti ✈ỵ✐ i = 1, 2✳ ❑❤✐

✤â✱ z0 ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ f1 f2 ✈ỵ✐ ❜➟❝ ♥❤✐➲✉ ♥❤➜t t1 + t2 ✳
❚r÷í♥❣ ❤đ♣ ✶✳ ●✐↔ sû t1 ≥ p ✈➔ t2 ≥ p✳ ❑❤✐ ✤â t1 + t2 ≥ p✳ ❉♦ ✤â z0

✤÷đ❝ t➼♥❤ ♥❤✐➲✉ ♥❤➜t p ❧➛♥ ð ✈➳ tr→✐ ❝õ❛ ❤➔♠ ✤➳♠ ❜➯♥ tr➯♥✱ tr♦♥❣ ❦❤✐ z0
✤÷đ❝ ✤✐➸♠ p + q ❧➛♥ ð ✈➳ ♣❤↔✐ ❝õ❛ ❤➔♠ ✤➳♠ ❜➯♥ tr➯♥✳
✶✶


❚r÷í♥❣ ❤đ♣ ✷✳ ●✐↔ sû t1 ≥ p ✈➔ t2 < p✳
❚r÷í♥❣ ❤đ♣ ✷✳✶✳ ●✐↔ sû t1 + t2 ≥ p✳ ❑❤✐ ✤â z0 ✤÷đ❝ ✤➳♠ ♥❤✐➲✉ ♥❤➜t p ð

✈➳ tr→✐ ❝õ❛ ❤➔♠ ✤➳♥ ❜➯♥ tr➯♥ tr♦♥❣ ❦❤✐ ♥â ✤÷đ❝ ✤➳♠ p + q ❧➛♥ ð ✈➳ ♣❤↔✐
❝õ❛ ❤➔♠ ✤➳♠✳
❚r÷í♥❣ ❤đ♣ ✷✳✷✳ ●✐↔ sû t1 + t2 < p✳ ❚r÷í♥❣ ❤ñ♣ ♥➔② ①✉➜t ❤✐➺♥ ❦❤✐ t2

➙♠ tù❝ ❧➔ ❦❤✐ z0 ❧➔ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ f2 ✳ ❑❤✐ ✤â z0 ✤÷đ❝ ✤➳♠ ♥❤✐➲✉ ♥❤➜t

max{0, t1 + t2 } ð ✈➳ tr→✐ ❝õ❛ ❤➔♠ ✤➳♥ ❜➯♥ tr➯♥ tr♦♥❣ ❦❤✐ ♥â ✤÷đ❝ ✤➳♠ p
❧➛♥ ð ✈➳ ♣❤↔✐ ❝õ❛ ❤➔♠ ✤➳♠✳
❚r÷í♥❣ ❤đ♣ ✸✳ ●✐↔ sû t1 < p ✈➔ t2 ≥ p✳ ❑❤✐ ✤â t1 + t2 ≥ p✳ ❚r÷í♥❣ ❤đ♣

♥➔② ❝â ✤÷đ❝ ♣❤➙♥ t➼❝❤ ♥❤÷ ✤➣ ❧➔♠ ð ❚r÷í♥❣ ❤đ♣ ✷✳
❚r÷í♥❣ ❤đ♣ ✹✳ ●✐↔ sû t1 < p ✈➔ t2 < p✳
❚r÷í♥❣ ❤đ♣ ✹✳✶✳ ●✐↔ sû t1 + t2 ≥ p. ❑❤✐ ✤â z0 ✤÷đ❝ ✤➳♠ ♥❤✐➲✉ ♥❤➜t p ❧➛♥

ð ✈➳ tr→✐ tr♦♥❣ ❦❤✐ ♥â ✤÷đ❝ ✤➳♠ max{0, t1 } + max{0, t2 } ❧➛♥ ð ✈➳ ♣❤↔✐
❝õ❛ ❜✐➸✉ t❤ù❝ ❜➯♥ tr➯♥✳
❚r÷í♥❣ ❤đ♣ ✹✳✷✳ ●✐↔ sû t1 + t2 < p✳ ❑❤✐ ✤â z0 ✤÷đ❝ ✤➳♠ ♥❤✐➲✉ ♥❤➜t

max{0, t1 + t2 } ❧➛♥ tr♦♥❣ ❦❤✐ ♥â ✤÷đ❝ ✤➳♠ max{0, t1 } + max{0, t2 } ð ✈➳
♣❤↔✐ ❝õ❛ ❤➔♠ ✤➳♠ ❜➯♥ tr➯♥✳
❑➳t ❤ñ♣ t➜t ❝↔ ❝→❝ tr÷í♥❣ ❤đ♣ t❛ s✉② r❛ ❦➳t ❧✉➟♥ ❝õ❛ ❜ê ✤➲✳

❇ê ✤➲ ✶✳✷✳✶✹ ✭❬✶✵❪✮✳ ❱ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f t❛ ❝â
Np (r, 0; f (k) ) ≤ Np+k (r, 0; f ) + kN (r, ∞; f ) + S(r, f ).

❇ê ✤➲ ✶✳✷✳✶✺ ✭❬✷❪✮✳ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ M [f ] ❧➔ ♠ët ✤ì♥
t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ f ✱ ❦❤✐ ✤â

Np (r, 0; M [f ]) ≤ dM Np+k (r, 0; f ) + λN (r, ∞; f ) + S(r, f ).
❈❤ù♥❣ ♠✐♥❤✳ ❘ã r➔♥❣ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ❦❤→❝ ❤➡♥❣ ❜➜t ❦ý✱ ♥➳✉ p ≤ q

t❤➻

Np (r, f ) ≤ Nq (r, f ).

✶✷


❚✐➳♣ t❤❡♦✱ ❦➳t ❤đ♣ ✈ỵ✐ ❇ê ✤➲ ✶✳✷✳✶✸ ✈➔ ❇ê ✤➲ ✶✳✷✳✶✹✱ t❛ t❤✉ ✤÷đ❝

k

ni Np (r, 0; f (i) ) + S(r, f )

Np (r, 0; M [f ]) ≤
i=0
k

≤

ni {Np+i (r, 0; f ) + iN (r, ∞; f )} + S(r, f )
i=0
k

≤

ni Np+i (r, 0; f ) + λN (r, ∞; f )} + S(r, f )
i=0
k

≤

ni Np+k (r, 0; f ) + λN (r, ∞; f )} + S(r, f )
i=0

≤ dM Np+k (r, 0; f ) + λN (r, ∞; f )} + S(r, f ),
✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ ❦➳t ❧✉➟♥ ❝õ❛ ❜ê ✤➲✳

❇ê ✤➲ ✶✳✷✳✶✻ ✭❬✼❪✮✳ ❈❤♦ p, n ❧➔ ❤❛✐ sè ♥❣✉②➯♥ ữỡ õ ợ > 0
t õ

Np (r, 0; f n ) ≤ (n − nδp (0, f ) + ε)T (r, f ).
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ t❤➜② r➡♥❣

Np (r, 0; f n ) ≤ nNp (r, 0; f ).
❚ø ✤à♥❤ ♥❣❤➽❛ ❤➔♠ δp (0, f )✱ ✈ỵ✐ ♠å✐ n t❛ ❝â

δp (0, f )

1−

Np (r, 0; f ) ε
+ ,
T (r, f )
n

t❤❛② t❤➳ ✈➔♦ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â ❦➳t ❧✉➟♥ ❝õ❛ ❇ê ✤➲✳

❇ê ✤➲ ✶✳✷✳✶✼ ✭❬✶✷❪✮✳ ❱ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ P [f ] s✐♥❤
❜ð✐ f t❛ ❝â

N (r, ∞; P ) ≤ d(P )N (r, ∞; f ) + (ΓP − d(P ))N (r, ∞; f ).

❇ê ✤➲ ✶✳✷✳✶✽ ✭❬✻❪✮✳ ❱ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ P [f ] s✐♥❤
❜ð✐ f t❛ ❝â

m r,

P [f ]
1

≤ (d(P ) − d(P ))m r,
+ S(r, f ).
f
f d(P )
✶✸


❇ê ✤➲ ✶✳✷✳✶✾ ✭❬✸❪✮✳ ●✐↔ sû P [f ] ❧➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ s✐♥❤ ❜ð✐ ❤➔♠ ♣❤➙♥
❤➻♥❤ ❦❤→❝ ❤➡♥❣ f ✳ ❑❤✐ ✤â

N r, ∞;

P [f ]
f d(P )
≤ (ΓP − d(P ))N (r, ∞; f ) + (d(P ) − d(P ))N (r, 0; f | ≥ k + 1)
+ QN (r, 0; f | ≥ k + 1) + d(P )N (r, 0; f | ≤ k) + S(r, f ).

❇ê ✤➲ ✶✳✷✳✷✵ ✭❬✼❪✮✳ ❈❤♦ P [f ] ❧➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ✱
❦❤✐ ✤â

N (r, 0; P [f ]) ≤ (ΓP − d(P ))N (r, ∞; f ) + d(P )N (r, 0; f )
1
+ (d(P ) − d(P )) m(r, ) + T (r, f ) + S(r, f ).
f
❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❇ê ✤➲ ✶✳✷✳✶✽✱ t❛ ❝â

d(P )m r,

1
1

≤ m r,
+ S(r, f ).
f
P [f ]

✭✶✳✷✮

❙û ❞ö♥❣ ❜ê ✤➲ ✶✳✷✳✶✼✱ ✶✳✷✳✶✽ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✷✮ t❛ ❝â

1
+ O(1)
p
1
≤ T (r, P [f ]) − d(P )m r,
+ S(r, f )
f
1
≤ (d(P ) − d(P ))m r,
+ d(P )m(r, f ) + d(P )N (r, ∞; f )
f
1
+ S(r, f )
+ (ΓP − d(P ))N (r, ∞; f ) − d(P )m r,
f
≤ (ΓP − d(P ))N (r, ∞; f ) + d(P )N (r, 0; f )
1
+ (d(P ) − d(P )) m r,
+ T (r, f ) + S(r, f ).
f

N (r, 0; P [f ]) = T (r, P [f ]) − m r,

✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

❇ê ✤➲ ✶✳✷✳✷✶ ✭❬✼❪✮✳ ❈❤♦ j ✈➔ p ❧➔ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ ♠➣♥ j ≥ p+1✳
●✐↔ sû P [f ] ❧➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈ỵ✐ ΓP > (k + 1)d(P ) − (p + 1)✳ ❑❤✐ ✤â

N (j+ΓP −d(P ) r, 0; f d(P ) ≤ N (j (r, 0; P [f ]).
✶✹


❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû z0 ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f ✈ỵ✐ ❜➟❝ t✳ ◆➳✉

td(P ) < j + ΓP − d(P )
t❤➻ ❜ê ✤➲ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ❉♦ ✤â t❛ ❣✐↔ sû

td(P ) ≥ j + ΓP − d(P ).
❳➨t ❤❛✐ tr÷í♥❣ ❤đ♣✿
❚r÷í♥❣ ❤đ♣ ✶✳ ●✐↔ sû t ≥ k + 1✳ ❑❤✐ ✤â z0 ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ P [f ] ❝â

❜➟❝ ➼t ♥❤➜t ❧➔

min{n0j t + n1j (t − 1) + · · · + nkj (t − k)}
j

= min{tdMj − (ΓMj − dMj )}
j

= (t + 1)d(P ) − max{ΓMj }
j

≥ (j + ΓP − d(P )) + d(P ) − ΓP
≥ j,
✤✐➲✉ ♥➔② s✉② r❛ ❦➳t ❧✉➟♥ ❝õ❛ ❜ê ✤➲✳
❚r÷í♥❣ ❤đ♣ ✷✳ ❚✐➳♣ t❤❡♦ t❛ ❣✐↔ sû t ≤ k. ❑❤✐ ✤â

kd(P ) ≥ td(P ) ≥ j + ΓP − d(P )
≥ p + 1 + ΓP − d(P ),
♠➙✉ t❤✉➝♥ ✈➻ ΓP > (k + 1)d(P ) − (p + 1)✳

❇ê ✤➲ ✶✳✷✳✷✷ ✭❬✼❪✮✳ ❈❤♦ j ✈➔ p ❧➔ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ ♠➣♥ j ≥ p+1✳
●✐↔ sû P [f ] ❧➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ t❤✉➛♥ ♥❤➜t ✈ỵ✐

ΓP > (k + 1)d(P ) − (p + 1).
❑❤✐ ✤â

Np (r, 0; P [f ]) ≤ Np+ΓP −d(P ) (r, 0; f d(P ) ) + (ΓP − d(P ))N (r, ∞; f ) + S(r, f ).
✶✺


❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❇ê ✤➲ ✶✳✷✳✷✵ ✈➔ ✶✳✷✳✷✶✱ t❛ ❝â

Np (r, 0; P [f ]) ≤ (ΓP − d(P ))N (r, ∞; f ) + N (r, 0; f d(P ) )
∞

−

N (j (r, 0, P [f ]) + S(s, f )
j=p+1

≤ (ΓP − d(P ))N (r, ∞; f ) + Np+ΓP −d(P ) (r, 0; f d(P ) )
∞

N (j (r, 0; f d(P ) )

+
j=p+ΓP −d(P )+1
∞

−

N (j (r, 0, P [f ]) + S(s, f )
j=p+1

≤ (ΓP − d(P ))N (r, ∞; f ) + Np+ΓP −d(P ) (r, 0; f d(P ) ) + S(r, f ).
✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

✶✻


❈❤÷ì♥❣ ✷
❱➜♥ ✤➲ ❞✉② ♥❤➜t
✷✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝❤✉➞♥ ❜à
❈❤♦ f, g ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ①→❝ ✤à♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱

a ∈ C ∪ {∞}✳ ❱ỵ✐ a ∈ C✱ t❛ ♥â✐ f ✈➔ g ❝â ❝❤✉♥❣ t➟♣ a✲✤✐➸♠ ❦➸ ❝↔ ❜ë✐
❤❛② f ✈➔ g ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà a ❦➸ ở ỵ CM
f a ✈➔ g − a ❝â ❝❤✉♥❣ t➟♣ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✳ ❚❛ ♥â✐ f
✈➔ g ❝â ❝❤✉♥❣ t➟♣ a✲✤✐➸♠ ❦❤ỉ♥❣ ❦➸ ❜ë✐ ❤❛② f, g ✤÷đ❝ ❣å✐ ❧➔
tr a ổ ở ỵ IM ♥➳✉ ❝→❝ ❤➔♠ f − a ✈➔ g − a ❝â ❝❤✉♥❣

t➟♣ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ♣❤➙♥ ❜✐➺t✳ ◆➳✉ f ✈➔ g ❝â ❝ị♥❣ sè ❝ü❝ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐
✭❦❤ỉ♥❣ ❦➸ ❜ë✐✮ t❤➻ t❛ ♥â✐ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ a = ở tữỡ ự
ổ ở
ợ f t ý ỵ S(r, f ) ❧➔ ✤↕✐ ❧÷đ♥❣
t❤ä❛ ♠➣♥

S(r, f ) = o(T (r, f ))
❦❤✐ r → ∞ ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ E ❝â ✤ë ✤♦ t✉②➳♥ t➼♥❤
❤ú✉ ❤↕♥✳ ❍➔♠ ♣❤➙♥ ❤➻♥❤ a = a(z) (≡ ∞) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♥❤ä ✤è✐ ✈ỵ✐ f
♥➳✉ T (r, a) = S(r, f ) ❦❤✐ r → ∞, r ∈
/ E.
◆➳✉ a = a(z) ❧➔ ❤➔♠ ♥❤ä t❤➻ t❛ ♥â✐ r➡♥❣ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ a IM ❤♦➦❝
❝❤✉♥❣ ♥❤❛✉ a CM ♥➳✉ f − a ✈➔ g − a t÷ì♥❣ ù♥❣ ❝❤✉♥❣ ♥❤❛✉ 0 IM ❤♦➦❝
❝❤✉♥❣ ♥❤❛✉ ✵ CM✳
✶✼


❈❤♦ k ❧➔ sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠ ❤♦➦❝ ✈ỉ ❝ị♥❣✳ ợ a C {} ỵ

Ek (a; f ) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ a✲✤✐➸♠ ❝õ❛ f ✱ tr♦♥❣ ✤â ♠é✐ a✲✤✐➸♠ ❜ë✐ m ✤÷đ❝
✤➳♠ m ❧➛♥ ♥➳✉ m ≤ k ✈➔ ✤÷đ❝ ✤➳♠ k + 1 ❧➛♥ ♥➳✉ m > k ✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ✭❬✷❪✮ ◆➳✉
Ek (a; f ) = Ek (a; g)
t❤➻ t❛ ♥â✐ f, g ❝❤✉♥❣ ❣✐→ trà a ✈ỵ✐ trå♥❣ sè k.
✣à♥❤ ♥❣❤➽❛ ♥➔② ❦➨♦ t❤❡♦ r➡♥❣ ♥➳✉ f, g ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà a ✈ỵ✐ trå♥❣
sè k t❤➻ z0 ❧➔ a✲✤✐➸♠ ❝õ❛ f ✈ỵ✐ ❜ë✐ m ≤ k ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ a✲✤✐➸♠ ❝õ❛

g ✈ỵ✐ ❜ë✐ m ≤ k ✈➔ z0 ❧➔ ♠ët a✲✤✐➸♠ ❝õ❛ f ✈ỵ✐ ❜ë✐ m > k ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
♥â ❧➔ a✲✤✐➸♠ ❝õ❛ g ✈ỵ✐ ❜ë✐ n > k ✱ tr♦♥❣ ✤â m ❦❤æ♥❣ ♥❤➜t t❤✐➳t ❜➡♥❣ n✳

❚❛ ♥â✐ f, g ❝❤✉♥❣ ♥❤❛✉ (a, k) ♥➳✉ f, g ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà a ✈ỵ✐ trå♥❣
sè k ✳ ❘ã r➔♥❣ ♥➳✉ f, g ❝❤✉♥❣ ♥❤❛✉ (a, k) t❤➻ f, g ❝❤✉♥❣ ♥❤❛✉ (a, p) ✈ỵ✐
♠å✐ sè ♥❣✉②➯♥ p, 0 ≤ p < k ✳ ◆❣♦➔✐ r❛ t❛ ú ỵ r f, g a

IM CM ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f, g t÷ì♥❣ ù♥❣ ❝❤✉♥❣ ♥❤❛✉ (a, 0) ❤♦➦❝ ❝❤✉♥❣
♥❤❛✉ (a, ∞)✳

❇ê ✤➲ ✷✳✶✳✷ ✭❬✷❪✮✳ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ ✈➔ a(z) ❧➔ ❤➔♠
♥❤ä ✤è✐ ✈ỵ✐ f ✳ ✣➦t F =

fn
a ,G

=

M
a✳

❑❤✐ ✤â✱ F G ≡ 1.

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♣❤↔♥ ❝❤ù♥❣ r➡♥❣ F G 1 ờ

ỵ ❝ì ❜↔♥ t❤ù ♥❤➜t✱ t❛ ❝â

M
+ S(r, f )
ddM
≤ dM N (r, 0; f ) + λN (r, ∞; f ) + S(r, f )

(n + dM )T (r, f ) = T r,

= S(r, f ),
✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

❇ê ✤➲ ✷✳✶✳✸ ✭❬✹❪✮✳ ●✐↔ sû F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ (1, l)✱ N (r, F ) = N (r, G)
✈➔ H ≡ 0, tr♦♥❣ ✤â F, G ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳✷ ✈➔

H=

F
2F
−
F
F −1

−
✶✽

G
2G
−
.
G
G−1

✭✷✳✶✮


❑❤✐ ✤â

N (r, ∞; H) ≤ N (r, ∞; F ) + N (r, 0; F | ≥ 2) + N (r, 0; G| ≥ 2)
+ N 0 (r, 0; F ) + N 0 (r, 0; G )
+ N L (r, 1; F ) + N L (r, 1; G) + S(r, f ).

❇ê ✤➲ ✷✳✶✳✹ ✭❬✷❪✮✳ ●✐↔ sû F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ (1, l)✳ ❑❤✐ ✤â✱ ♥➳✉ l ≥ 1
t❤➻

1
1
N L (r, 1; F ) ≤ N L (r, ∞; F ) + N L (r, 0; F ) + S(r, F )
2
2
✈➔ ♥➳✉ l = 0 t❤➻
N L (r, 1; F ) ≤ N L (r, ∞; F ) + N L (r, 0; F ) + S(r, F ).

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû l ≥ 1✳ ❑❤✐ ✤â sè ❜ë✐ ❝õ❛ 1✲✤✐➸♠ ❝õ❛ F ✤÷đ❝ ✤➳♠

tr♦♥❣ N L (r, 1; F ) ➼t ♥❤➜t ❧➔ ✸ ✈➻ l ≥ 1. ❉♦ ✤â

1
N (r, 1; F ) ≤ N (r, 0; F |F = 0)
2
1
1
≤ N (r, ∞; F ) + N (r, 0; F ) + S(r, F ).
2
2
●✐↔ sû l = 0. ❑❤✐ ✤â sè ❜ë✐ ❝õ❛ 1✲✤✐➸♠ ❝õ❛ F ✤÷đ❝ ✤➳♠ tr♦♥❣

N L (r, 1; F ) ➼t ♥❤➜t ❧➔ ✷ ❜ð✐ ✈➻ l = 0. ❉♦ ✤â

1
N (r, 1; F ) ≤ N (r, 0; F |F = 0)
2
≤ N (r, ∞; F ) + N (r, 0; F ) + S(r, F ).
❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

❇ê ✤➲ ✷✳✶✳✺ ✭❬✷❪✮✳ ●✐↔ sû F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ (1, l) ✈➔ H ≡ 0✳ ❑❤✐ ✤â
(2

N (r, 1; F ) + N (r, 1; G) ≤ N (r, ∞, H) + N E (r, 1; F ) + N L (r, 1; F )
+ N L (r, 1; G) + N (r, 1; G) + S(r, f ).
❈❤ù♥❣ ♠✐♥❤✳ ❘ã r➔♥❣
(2

N (r, 1; F ) = N (r, 1; F | = 1) + NE (r, 1; F ) + N L (r, 1; F ) + N L (r, 1; G)
✶✾


✈➔ t➼♥❤ t♦→♥ t❛ ❝â

N (r, 1; F | = 1) ≤ N (r, 0; H) + S(r, f ) ≤ N (r, ∞; H) + S(r, f ),
tø ✤â s✉② r❛ ❦➳t ❧✉➟♥ ❝õ❛ ❜ê ✤➲✳

❇ê ✤➲ ✷✳✶✳✻ ✭❬✷❪✮✳ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ ✈➔ a(z) ❧➔ ❤➔♠
♥❤ä ❝õ❛ f ✳ ▲➜② F =

fn
a

✈➔ G =

M
a

s❛♦ ❝❤♦ F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ (1, ∞)✳

❑❤✐ ✤â ♠ët tr♦♥❣ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤ó♥❣✿

✭✶✮ T (r) ≤ N2 (r, 0; F ) + N2 (r, 0; G) + N (r, ∞; F ) + N (r, ∞; G)

+N L (r, ∞; F ) + N L (r, ∞; G) + S(r),
✭✷✮ F ≡ G,
✭✸✮ F G ≡ 1,
tr♦♥❣ ✤â T (r) = max{T (r, F ), T (r, G)} ✈➔ S(r) = o(T (r)), r ∈ I, I ❧➔ t➟♣
❝â ✤ë ✤♦ t✉②➳♥ t➼♥❤ ✈æ ❤↕♥ ❝õ❛ r ∈ (0, ∞)✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû z0 ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ f ♠➔ ❦❤æ♥❣ ❧➔ ❝ü❝ ✤✐➸♠ ❤❛②

❦❤æ♥❣ ✤✐➸♠ ❝õ❛ a(z) õ z0 ỗ tớ ỹ ừ F ✈➔ G✳ ❉♦
✤â F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ ❝→❝ ❝ü❝ ✤✐➸♠ ❝õ❛ f ♠➔ ❦❤æ♥❣ ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❤❛②
❝ü❝ ✤✐➸♠ ❝õ❛ a(z)✳ ❘ã r➔♥❣

N (r, H) ≤ N (r, 0; F ≥ 2) + N (r, 0; G ≥ 2) + N L (r, ∞; F ) + N L (r, ∞; G)
+ N 0 (r, 0; F ) + N 0 (r, 0; G ) + S(r, f ).
P❤➛♥ ❝á♥ ❧↕✐ ❝õ❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ t✐➳♥ ❤➔♥❤ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❝❤ù♥❣
♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✷✳✶✸ tr♦♥❣ ❬✶❪✳

❇ê ✤➲ ✷✳✶✳✼ ✭❬✼❪✮✳ ●✐↔ sû f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ ✈➔ a(z) ❧➔ ❤➔♠
♥❤ä ❝õ❛ f ✳ ✣➦t F =

fn

a

✈➔ G =

P [f ]
a ✳

◆➳✉ F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ (1, ∞) t❤➻

♠ët tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉ t❤ä❛ ♠➣♥✿

✭✐✮ T (r) ≤ N2 (r, 0; F ) + N2 (r, 0; G) + N (r, ∞; F ) + N (r, ∞; G)

+N L (r, ∞; F ) + N L (r, ∞; G) + S(r),
✷✵