Luận văn thạc sỹ toán học Định lý nevanlinna cartan padic và áp dụng

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

P❍Ò◆● ❚❍➚ ❍×❒◆●

✣➚◆❍ ▲Þ ◆❊❱❆◆▲■◆◆❆✲❈❆❘❚❆◆ P✲❆❉■❈
❱⑨ ⑩P ❉Ö◆●
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


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

P❍Ò◆● ❚❍➚ ❍×❒◆●

✣➚◆❍ ▲Þ ◆❊❱❆◆▲■◆◆❆✲❈❆❘❚❆◆ P✲❆❉■❈
❱⑨ ⑩P ❉Ö◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿
❚❙✳ ❱Ô ❍❖⑨■ ❆◆

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✐

▲í✐ ❝❛♠ ✤♦❛♥
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐✱ ❞÷î✐ sü
❤÷î♥❣ ❞➝♥ ❝õ❛ ❚❙✳ ❱ô ❍♦➔✐ ❆♥✳ ▲✉➟♥ ✈➠♥ ❝❤÷❛ tø♥❣ ✤÷ñ❝ ❝æ♥❣ ❜è tr♦♥❣
❜➜t ❦➻ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ♥➔♦ ✈➔ ♠å✐ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❧✉➟♥
✈➠♥ ❧➔ tr✉♥❣ t❤ü❝✳

❍å❝ ✈✐➯♥

P❤ò♥❣ ❚❤à ❍÷ì♥❣




ớ ỡ
ữủ t t s ồ ồ ữ
ồ ữợ sỹ ữợ ừ s ụ
ổ ỷ ớ ỡ t s s t s
ụ ữớ tớ t t ữợ ú
ù t tổ t tốt
ởt ỳ t tọ ỏ t ỡ t ồ ừ
ồ ữ ồ ồ
t
t ỡ t tr ợ
ồ ổ ừ ở ú ù tổ tr sốt q tr ồ
t tỹ
õ ố s tớ ỹ ừ t õ
õ tr ọ ỳ t sõt t ữủ
ỳ ỵ õ õ ừ ồ ồ ổ
t ỡ

t

Pũ ữỡ


✐✐✐

▼ö❝ ❧ö❝
▼ð ✤➛✉
✶ ▲þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝
✶✳✶

✶✳✷

▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥

✶
✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

♣ ✲❛❞✐❝

✶✳✶✳✶

❚r÷í♥❣ ❝→❝ sè

✶✳✶✳✷

❍➔♠ s✐♥❤ ❜ð✐ ❝❤✉é✐ ❧ô② t❤ø❛

✶✳✶✳✸

❍➔♠ ♣❤➙♥ ❤➻♥❤

♣ ✲❛❞✐❝

♣ ✲❛❞✐❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

❍➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤

♣ ✲❛❞✐❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✷✳✶

❍➔♠ ✤➦❝ tr÷♥❣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✷✳✷

❍❛✐ ✣à♥❤ ❧þ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✷✳✸

❇ê ✤➲ q✉❛♥ ❤➺ sè ❦❤✉②➳t

✷✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝ ✈➔ →♣ ❞ö♥❣✳
♣ ✲❛❞✐❝

✷✳✶

✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥

✷✳✷

❍❛✐ →♣ ❞ö♥❣ ❝õ❛ ✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✷✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

♣ ✲❛❞✐❝

✳ ✳ ✳

✷✹
✷✾

✹✼
✹✽


✐✈

❈→❝ ❦➼ ❤✐➺✉
• Cp

✿ ❚r÷í♥❣ sè ♣❤ù❝

♣ ✲❛❞✐❝

❢ ✿ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝

• Nf (a, r)✿ ❍➔♠ ✤➳♠ ❝õ❛ ❢ t↕✐ ❛
• mf (∞, r) ✿ ❍➔♠ ①➜♣ ①➾ ❝õ❛ ❢
• Tf (r)✿ ❍➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❢
•

• O(1)✿

✣↕✐ ❧÷ñ♥❣ ❣✐î✐ ♥ë✐

• Nf (r), Nk (f, r)✿
• W (f )
• Hj ✿

❍➔♠ ✤➳♠✱ ❤➔♠ ✤➳♠ ♠ù❝

❲r♦♥s❦✐❛♥ ❝õ❛ ❤➔♠

f

❙✐➯✉ ♣❤➥♥❣

• Fj (z) = 0✿

P❤÷ì♥❣ tr➻♥❤ ❝õ❛ s✐➯✉ ♣❤➥♥❣

k


✶

▼ð ✤➛✉
❚♦→♥ ❤å❝ ✤÷ñ❝ ❝♦✐ ❧➔ ✤➾♥❤ ❝❛♦ tr➼ t✉➺ ❝õ❛ ❝♦♥ ♥❣÷í✐✱ ♥â ①➙♠ ♥❤➟♣ ✈➔♦
❤➛✉ ❤➳t ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❧➔ ♥➲♥ t↔♥❣ ❝õ❛ ♥❤✐➲✉ ❧þ t❤✉②➳t ❦❤♦❛ ❤å❝
q✉❛♥ trå♥❣✳ ❚♦→♥ ❤å❝ ♥❣➔② ❝➔♥❣ ♣❤→t tr✐➸♥ ♠↕♥❤ ♠➩ q✉❛ tø♥❣ t❤í✐ ❦➻✳
✣➦❝ ❜✐➺t tr♦♥❣ ✤➛✉ t❤➳ ❦✛ ❳❳ ❧➔ sü r❛ ✤í✐ ❝õ❛ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✱
✤÷ñ❝ ❝♦✐ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ t❤➔♥❤ tü✉ ♥ê✐ ❜➟t ✈➔ s➙✉ s➢❝ ♥❤➜t✳ ❚rå♥❣
t➙♠ ❝õ❛ ❧þ t❤✉②➳t ♥➔② ❧➔ ❤❛✐ ✣à♥❤ ❧þ ❝❤➼♥❤ ❝õ❛ ◆❡✈❛♥❧✐♥♥❛✳
◆➠♠ ✶✾✸✸✱ ❍✳❈❛rt❛♥ ✤➣ ♠ð rë♥❣ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ tr÷í♥❣ ❤ñ♣
✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ✈➔ ✤÷❛ r❛ ♥❤✐➲✉ ù♥❣ ❞ö♥❣ q✉❛♥ trå♥❣✳ ❱➻ ✈➟② ❧þ
t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✤è✐ ✈î✐ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ✤÷ñ❝ ♠❛♥❣ t➯♥
❤❛✐ ♥❤➔ t♦→♥ ❤å❝ ①✉➜t s➢❝ ❝õ❛ t❤➳ ❦✛ ❳❳ ✤â ❧➔ ✧❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✲
❈❛rt❛♥✧✳ ❚❤æ♥❣ q✉❛ ❤÷î♥❣ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ♥❤✐➲✉ ❦➳t q✉↔ ✤➭♣ ✤➩ tr♦♥❣
❣✐↔✐ t➼❝❤ ❤➔♠✱ tr♦♥❣ ✤↕✐ sè ❝ô♥❣ ♥❤÷ tr♦♥❣ ❧þ t❤✉②➳t sè ✤÷ñ❝ r❛ ✤í✐✱
❣➢♥ ❧✐➲♥ ✈î✐ ♥❤✐➲✉ t➯♥ t✉ê✐ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ tr➯♥ t❤➳ ❣✐î✐ ♥❤÷ P❤✳
●r✐❢❢✐t❤s✱ ❍✳❲❡②❧✱ P✳❱♦❥t❛✱ ●✳❋❛❧t✐♥❣s✱✳✳✳

♣ ✲❛❞✐❝✮✱ ❧➛♥ ✤➛✉ t✐➯♥
❍❛ ❍✉② ❑❤♦❛✐ ✈➔ ▼② ❱✐♥❤ ◗✉❛♥❣ ✤➣ ①➙② ❞ü♥❣ t÷ì♥❣ tü ♣ ✲❛❞✐❝ ❝õ❛
❚r➯♥ tr÷í♥❣ ❝ì sð ❦❤æ♥❣ ❆❝s✐♠❡t ✭tr÷í♥❣ ❝→❝ sè

❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ t❤æ♥❣ q✉❛ ✷ ✣à♥❤ ❧þ ❝❤➼♥❤✳ ◆❤✐➲✉ ❦➳t q✉↔ ❤❛② ✈➔
♥❤ú♥❣ ♣❤→t tr✐➸♥ t✐➳♣ t❤❡♦ ❝õ❛ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛

♣ ✲❛❞✐❝ ❝â t❤➸ t➻♠

t❤➜② tr♦♥❣ ♥❤ú♥❣ ❝æ♥❣ tr➻♥❤ ❬✷❪✱ ❬✸❪✱ ❬✹❪✱ ❬✺❪✱✳✳✳
❈❤♦ ✤➳♥ ♥❛②✱ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❞÷í♥❣ ♥❤÷ ✤➣ ✤÷ñ❝ ❤♦➔♥
t❤✐➺♥ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♣❤ù❝✳ ❚✉② ♥❤✐➯♥ sü t❤➸ ❤✐➺♥ ❝õ❛ ❧þ t❤✉②➳t ♥➔②
tr➯♥ tr÷í♥❣ ❝ì sð ❦❤æ♥❣ ❆❝s✐♠❡t ♠î✐ ❝❤➾ ❜➢t ✤➛✉ ✈➔ ❝á♥ ❧➙✉ ♠î✐ ✤÷ñ❝
❤♦➔♥ t❤✐➺♥✳ ◆➠♠ ✶✾✽✸✱ ❍❛ ❍✉② ❑❤♦❛✐ ✈➔ ▼② ❱✐♥❤ ◗✉❛♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤

✤÷ñ❝ ❝→❝ ✣à♥❤ ❧þ ❝❤➼♥❤ ❝õ❛ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛

♣ ✲❛❞✐❝ tr♦♥❣ tr÷í♥❣ ❤ñ♣


✷

♠ët ❝❤✐➲✉✳ ◆➠♠ ✶✾✾✸✱ ❲✳❈❤❡rr② ✤➣ ①➙② ❞ü♥❣ ♠ët ❜↔♥ s❛♦

♣ ✲❛❞✐❝ ❤➛✉

❤➳t ❝→❝ ❦➳t q✉↔ ❝õ❛ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✤è✐ ✈î✐ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ①→❝
✤à♥❤ tr➯♥ ✤➽❛ t❤õ♥❣ ❝õ❛ ♠➦t ♣❤➥♥❣

♣ ✲❛❞✐❝ Cp✳ ✣➸ ❣â♣ ♣❤➛♥ ❧➔♠ ♣❤♦♥❣

♣❤ó t❤➯♠ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ✈î✐ ❝❤✐➲✉ ❝❛♦ tr♦♥❣ tr÷í♥❣ ❤ñ♣

♣ ✲❛❞✐❝✱ ✈➔♦ ♥➠♠ ✶✾✾✺ ❍❛ ❍✉② ❑❤♦❛✐ ✈➔ ▼❛✐ ❱❛♥ ❚✉ ❬✺❪ ✤➣ ♣❤→t ❜✐➸✉ ✈➔
❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝✳
❚❤❡♦ ❤÷î♥❣ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ tæ✐ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ✿

✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝ ✈➔ →♣ ❞ö♥❣✳
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② tæ✐ s➩ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❧↕✐ ✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲

♣ ✲❛❞✐❝ ❬✺❪✳ ❙❛✉ ✤â ❝❤➾ r❛ ♠ët sè ù♥❣ ❞ö♥❣ q✉❛♥ trå♥❣ ❝õ❛ ✣à♥❤ ❧þ
◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝ ✈➔♦ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ sü s✉② ❜✐➳♥ ❝õ❛ ✤÷í♥❣
❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ♣ ✲❛❞✐❝✳ ▼➦t ❦❤→❝✱ ✣à♥❤ ❧þ ▼❛s♦♥ ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥
❈❛rt❛♥

q✉❛♥ ✭①❡♠ ❬✶✲✷✲✸❪✮ ❧➔ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ sæ✐ ✤ë♥❣ ✈➔ t❤í✐ sü✳ ❱➻ ✈➟②
❝❤ó♥❣ tæ✐ ❝ô♥❣ tr➻♥❤ ❜➔② ❧↕✐ t÷ì♥❣ tü ❝õ❛ ✣à♥❤ ❧þ ▼❛s♦♥ ❝❤♦ ❝→❝ ❤➔♠

♣ ✲❛❞✐❝✳ ✣➙② ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✈➜♥ ✤➲ ♠❛♥❣ t➼♥❤ t❤í✐ sü ✈➔
❝➜♣ t❤✐➳t ❝õ❛ ❣✐↔✐ t➼❝❤ ♣ ✲❛❞✐❝✱ ✤÷ñ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥

♥❣✉②➯♥

❝ù✉✳
◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ✤÷ñ❝ ❝❤✐❛ t❤➔♥❤ ✷
❝❤÷ì♥❣✿

❈❤÷ì♥❣ ✶✳ ❚r➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦
❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝✳
❈❤÷ì♥❣ ✷✳ ❚r➻♥❤ ❜➔② ❧↕✐ ✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝✱ ù♥❣ ❞ö♥❣
✤à♥❤ ❧þ ✈➔♦ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ sü s✉② ❜✐➳♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤

♣ ✲❛❞✐❝ ✈➔ t÷ì♥❣ tü ❝õ❛ ✣à♥❤ ❧þ ▼❛s♦♥ ❝❤♦ ❝→❝ ❤➔♠ ♥❣✉②➯♥ ♣ ✲❛❞✐❝✳


✸

❈❤÷ì♥❣ ✶

▲þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠
♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝
❍✐➺♥ ♥❛② tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❝❛♦ ❤å❝ ❝❤✉②➯♥ ♥❣➔♥❤ t♦→♥ ❣✐↔✐ t➼❝❤ t↕✐
❑❤♦❛ ❚♦→♥ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❣✐→♦ tr➻♥❤
❣✐↔✐ t➼❝❤

♣ ✲❛❞✐❝ ❬✶❪ ✤➣ ✤÷ñ❝ ✤÷❛ ✈➔♦ ❣✐↔♥❣ ❞↕②✳ ◆❣♦➔✐ r❛✱ ❝ô♥❣ ❝â ♠ët

sè t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❜➡♥❣ t✐➳♥❣ ❆♥❤ ❬✷❪✱ ❬✸✲✹❪ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❦✐➳♥
t❤ù❝ ❝ì ❜↔♥ ♥➔②✳ ❚ø ✤â ❝→❝ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝✱ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤ ✈➔ ♥❤ú♥❣
♥❣÷í✐ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✱ ❝â t❤➸ t❤❛♠ ❦❤↔♦ ❜ê s✉♥❣ ✈➔ ♠ð rë♥❣ t❤➯♠
❦✐➳♥ t❤ù❝ ✈➲ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛

♣ ✲❛❞✐❝✳ ❚❤æ♥❣ q✉❛ ❝→❝ t➔✐ ❧✐➺✉ ♥➔②✱

tr➯♥ ❝ì sð ❝→❝ ❦✐➳♥ t❤ù❝ ✤➣ ❜✐➳t✱ tr♦♥❣ ❈❤÷ì♥❣ ✶ tæ✐ ①✐♥ tr➻♥❤ ❜➔② ♠ët
sè ❦✐➳♥ t❤ù❝ ✈➲ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤

♣ ✲❛❞✐❝ ✤➸ ❞ò♥❣

❝❤♦ ❈❤÷ì♥❣ ✷✳

✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥
✶✳✶✳✶ ❚r÷í♥❣ ❝→❝ sè ♣ ✲❛❞✐❝
❱î✐

p

❧➔ ♠ët sè ♥❣✉②➯♥ tè ❝è ✤à♥❤✱ ❖str♦✇s❦✐ ✤➣ ❦❤➥♥❣ ✤à♥❤✿

❈❤➾ ❝â ❤❛✐ ❝→❝❤ tr❛♥❣ ❜à ❝❤✉➞♥ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝❤♦ tr÷í♥❣ ❤ú✉ t➾
▼ð rë♥❣ t❤❡♦ ❝❤✉➞♥ t❤æ♥❣ t❤÷í♥❣ t❛ ❝â tr÷í♥❣ sè t❤ü❝
❝❤✉➞♥

♣ ✲❛❞✐❝ t❛ ❝â tr÷í♥❣ sè Qp✳

❑➼ ❤✐➺✉

Cp = Qp

❧➔ ❜ê s✉♥❣ ❝õ❛ ❜❛♦ ✤â♥❣ ✤↕✐ sè ❝õ❛

R✱
Qp ✳

Q✳

♠ð rë♥❣ t❤❡♦

❚❛ ❣å✐

Cp

❧➔




trữớ số ự



tr

ữủ rở tỹ ừ

Cp

tr Qp



Dr = {z Cp : |z| r} , D<r> = {z Cp : |z| = r} .
sỷ

f (z)

Dr

tr

ữủ

f (z) =

an z n
n0


lim |an | |z n | = 0

n

tỗ t

n N

|an | |z n |

t tr ợ t

õ t t

|f |r = max {|an | |z n |} .
n0

r sốt t q ữợ

log



logp

s ộ ụ tứ
s ộ ụ tứ õ


an z n ,

f (z) =

(an Cp ).

n=0

õ t

f (z)

tr ừ tờ ộ

an z n

ợ ộ

z Cp

n=0




|an z n | 0



n

õ ộ ở tử ở tử

ừ ộ ữủ t ổ tự

1
1
= lim sup |an | n .
n


sỷ ộ ụ tứ

0 < +

f (z) =

ợ ộ

an z n

õ ở tử

n=0
+

r R : 0 < r <

số ợ

t

à(r, f ) = max |an |rn
n0

số tr t

(r, f ) = max {n : |an |rn = à(r, f )} .
n0







t ộ

an z n

ở tử t

z Cp : |z| = r <

t

n=0

lim |an |rn = 0

n

t

{|an |rn }

tr

R+

õ t

õ t t s

[1]

ợ ộ s ộ ụ tứ t õ
ợ ộ r : 0 < r < , à(r, f ) ổ tỗ t ỳ
à(r, f ) tử t r
ợ ộ r số tr t (r, f ) ổ tỗ t ỳ ởt
số ổ t õ
à(r, f ) = |a(r,f ) |r(r,f ) .

z Cp |z| r t
|f (z)| max |an ||z n | max |an |rn = à(r, f ).
n0

n0

[1]

số tr t (r, f ) t t r r tọ
r

log à(r, f ) = log |a(0,f ) |+

0

(t, f ) (0, f )
dt+(0, f ) log r, (0 < r < ),
t

tr õ log rt tỹ ỡ số

ỵ [1]

ợ r > 0 à(r, .) : Ar (Cp) R+ tọ t t s
à(r, f ) = 0 f 0
à(r, f + g) max {à(r, f ), à(r, g)}
à(r, f g) = à(r, f )à(r, g)


✻

✶✳✶✳✸ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝
❚r÷í♥❣ ❝→❝ ❤➔♠ ♣❤➙♥ t❤ù❝ ❝õ❛ ❝→❝ ❤➔♠ tr♦♥❣

H(D)

M(D)✳
❉✳ ◆➳✉ f

❦➼ ❤✐➺✉ ❧➔

f ∈ M(D) ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥
❦❤æ♥❣ ❝â ❝ü❝ ✤✐➸♠ tr➯♥ D t❤➻ t❛ ❣å✐ f ❧➔ ❝❤➾♥❤ ❤➻♥❤✳

❱î✐ ρ > 0✱ ♥➳✉ f ∈ H(Cp (0; ρ)) t❤➻ ✤➽❛ ❣✐↔✐ t➼❝❤ ❧î♥ ♥❤➜t ❝õ❛ f t↕✐ ♠é✐
✤✐➸♠ a ∈ Cp (0; ρ) ❝❤➼♥❤ ❧➔ Cp (0; ρ)✱ ❞♦ ✤â f ∈ A(ρ (Cp )✳ ◆❤÷ ✈➟②

▼ët ♣❤➛♥ tû

H(Cp (0; ρ)) = A(ρ (Cp ).
❙✉② r❛

M(Cp (0; ρ)) =

g
|g, h ∈ A(ρ (Cp ), h ≡ 0 .
h

❚❛ ❝ô♥❣ ✈✐➳t

M(ρ (Cp ) = M(Cp (0; ρ)).
✣➦❝ ❜✐➺t✱ ♠ët ♣❤➛♥ tû ❝õ❛ t➟♣ ❤ñ♣

M(∞ (Cp ) = M(Cp (0; ∞)) = M(Cp )
✤÷ñ❝ ❣å✐ ❧➔ ♠ët
t➾ tr➯♥

❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ Cp✳ ❑➼ ❤✐➺✉ Cp(z) ❧➔ t➟♣ ❝→❝ ❤➔♠ ❤ú✉

Cp ✱ ❦❤✐ ✤â Cp (z) ⊂ M(Cp )✳ ▼é✐ ♣❤➛♥ tû tr♦♥❣ t➟♣ M(Cp )− Cp (z)

✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ s✐➯✉ ✈✐➺t✳

✶✳✷ ❍➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝

✶✳✷✳✶ ❍➔♠ ✤➦❝ tr÷♥❣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t
●✐↔ sû

❢=

f ❧➔ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ Cp ✳ ❱î✐ ♠é✐ a ∈ Cp ✱ ✈✐➳t
Pi (z − a) ✈î✐ Pi ❝→❝ ✤❛ t❤ù❝ ❜➟❝ i✳

✣à♥❤ ♥❣❤➽❛

vf (a) = min {i : Pi = 0} .
d ∈ Cp ✱ ✤à♥❤ ♥❣❤➽❛
vfd (a) = vf −d (a)✳ ❈è ✤à♥❤
❈❤♦

vfd : a ∈ Cp −→ N
ρ0 ✈î✐ 0 < ρ0 ≤ r✳

♠ët ❤➔♠
sè t❤ü❝

①→❝ ✤à♥❤ ❜ð✐


✼

✣à♥❤ ♥❣❤➽❛

r

1
Nf (a, r) =
log p

nf (a, x)
dx,
x
ρ0

nf (a, x) ❧➔ sè
✤➽❛ |z| ≤ x✳
◆➳✉ a = 0 t❤➻ ✤➦t
ð ✤â

♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

f (z) = a

t➼♥❤ ❝↔ ❜ë✐ tr➯♥

Nf (r) = Nf (0, r).
❈❤♦

l

❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✣➦t

r

nl,f (a, x)

dx,
x

1
Nl,f (a, r) =
log p
ρ0
ð ✤â

min {vf −a (z), l} .

nl,f (a, x) =
|z|≤r
❈❤♦

k

❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠

✤à♥❤ ❜ð✐✿


0
≤k
vf (z) =
v (z)
f

vf≤k

♥➳✉

vf (z) > k

♥➳✉

vf (z) ≤ k

tø

✈➔

n≤k
f (r) =

vf≤k (z),

≤k
n≤k
f (a, r) = nf −a (r).

|z|≤r
✣à♥❤ ♥❣❤➽❛

Nf≤k (a, r)
◆➳✉

a=0

1

=
log p

r
ρ0

n≤k
f (a, x)
x

dx.

t❤➻ ✤➦t

Nf≤k (r) = Nf≤k (0, r).
❚❛ ✤➦t

Nf≤k (a, r)

1
=
log p

r
ρ0

n≤k
f (a, x)
x

dx,

Cp

✈➔♦

N

①→❝


✽

ð ✤â

min vf≤k
−a (z), l .

n≤k
l,f (a, x) =
|z|≤r
❚÷ì♥❣ tü t❛ ✤à♥❤ ♥❣❤➽❛

≥k
>k
Nf(a, r), Nf>k (a, r), Nf≥k (a, r), Nl,f
(a, r), Nl,f
(a, r)✳

●✐↔ sû f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ Cp ✱ ❦❤✐ ✤â tç♥ t↕✐ ❤❛✐ ❤➔♠ f2 , f1
f1
s❛♦ ❝❤♦ f1 , f2 ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ✈➔ f =
✳ ❱î✐ a ∈ Cp ∪ {∞}✱
f2
t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➳♠ sè ❦❤æ♥❣ ✤✐➸♠ nf (a, r) ❝õ❛ ❢ t↕✐ ❛ ❤❛② ❝á♥ ❣å✐
❤➔♠ ✤➳♠ sè

❛ ✲ ✤✐➸♠ ❝õ❛ ❢ ❜ð✐✿


n (∞, r) = n (0, r)
f
f2
nf (a, r) =
n
(0, r).
f1 −af2

Nf (a, r) ❝õ❛ ❢ t↕✐ ❛ ❜ð✐✿

N (∞, r) = N (0, r)
f
f2
Nf (a, r) =
N
f −af (0, r).

✣à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➳♠

1

❚÷ì♥❣ tü t❛ ❝ô♥❣ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❤➔♠

2

nf (∞, r), N f (∞, r), nf (a, r), N f (a, r).

❚❛ ❝â

Nf (a, r) = Nf1 −af2 (r), Nf (∞, r) = Nf2 (r).
●✐↔ sû

∞

∞
n

f1 =
n=m1
tr♦♥❣ ✤â

m2 , m1 ∈ N

✈➔

bn z n ,

an z , f2 =
n=m2

am1 = 0✱ bm2 = 0✳

❚❛ ❝â

Nf (0, r) = Nf1 (0, r) = log |f1 |r − log |am1 | ,
Nf (∞, r) = Nf2 (0, r) = log |f2 |r − log |bm2 | .
❑➨♦ t❤❡♦

Nf (0, r) − Nf (∞, r) = log |f |r − log

|am1 |
= log |f |r − log |f ∗ ( 0)|,
|bm2 |


✾

❚r♦♥❣ ✤â

am1
✳
bm2

f ∗ (0) =

❚❛ ❝â

f ∗ (0) = lim z m2 −m1 f (z) ∈ Cp∗ .
z−→0

❍ì♥ ♥ú❛ t❛ ❝â

Nf (0, r) − Nf (∞, r) = Nf1 (0, r) − Nf2 (0, r)
= log |f1 |r − log |f1 |ρ0 − log |f2 |r + log |f2 |ρ0
|f1 |ρ0
|f1 |r
− log
= log
|f2 |r
|f2 |ρ0
= log |f |r − log |f |ρ0 .
❚✐➳♣ t❤❡♦ t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ①➜♣ ①➾ ❝õ❛ ❤➔♠

f

❜ð✐ ❝æ♥❣ t❤ù❝

mf (∞, r) = max {0, log |f |r } .
❱î✐ ♠é✐

a ∈ Cp ✱

✤➦t

mf (a, r) = m

1
(∞, r).
f −a

❚❛ ❝â

mf (0, r) = log+ µf (0, r) = max {0, − log |f |r } .
❙❛✉ ✤➙② t❛ ❝â ♠ët sè t➼♥❤ ❝❤➜t ✤ì♥ ❣✐↔♥ ❝õ❛ ❤➔♠ ✤➳♠ ✈➔ ❤➔♠ ①➜♣ ①➾✳

▼➺♥❤ ✤➲ ✶✳✹✳ [1]

●✐↔ sû ❢i ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ✵ tr➯♥ Cp✱ i = 1, 2, ..., k✳
❑❤✐ ✤â ✈î✐ ♠é✐ r > 0✱ t❛ ❝â
k

k

(∞, r) ≤

Nk
fi

Nfi (∞, r) + O(1); N k
i=1

i=1

(∞, r) ≤
fi

Nfi (∞, r) + O(1);
i=1

i=1

k

mk
fi

(∞, r) ≤ max mfi (∞, r)+O(1); m k
i∈{1,....k}

i=1

❈❤ù♥❣ ♠✐♥❤✳

(∞, r) ≤
fi

i=1

i=1

❱î✐ ♠é✐ ❦➼ ❤✐➺✉

fi =

fi1
✱
fi2

tr♦♥❣ ✤â

fi1 , fi2 ∈ A (Cp )✳

✤â✱ ✈✐➳t

k

i=1

F
fi =
;
f12 ...fk2

k

fi =
i=1

mfi (∞, r)+O(1).

G
,
f12 ...fk2

❑❤✐


✶✵

tr♦♥❣ ✤â

F, G ∈ Ar (Cp )✳
k

k

fi

❉♦ ✤â✱ ♠é✐ ❝ü❝ ✤✐➸♠ ❝õ❛ ❤➔♠

fi

❤♦➦❝

❝õ❛ ❤➔♠

f12 ....fk2 ✱

❝❤➾ ❝â t❤➸ ❧➔ ❦❤æ♥❣ ✤✐➸♠

i=1

i=1

♥➯♥ ♥â ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ ♠ët tr♦♥❣ ❝→❝ ❤➔♠

k
fi

nfi (∞, r);

(∞, r) ≤

nk
fi

i=1

i=1

❙✉② r❛

k

(∞, r) ≤

nk

fi ✳

nfi (∞, r).
i=1

i=1

✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦

k

k

(∞, r) ≤

Nk
fi

Nfi (∞, r)+O(1);

fi

i=1

i=1

(∞, r) ≤

Nk

Nfi (∞, r)+O(1).
i=1

i=1

◆❣♦➔✐ r❛ t❛ ❝â

k

log |

fi |r ≤ log max |fi |r = max log |fi |r ,
i∈{1,...,k}

i=1

i∈{1,...,k}

♥➯♥

mk
fi

(∞, r) ≤ max mfi (∞, r).
i∈{1,...,k}

i=1

❱➔

k

k

fi |r =

log|
i=1
❉♦ ✤â

log|fi |r .

i=1

k

(∞, r) ≤

mk
fi

mfi (∞, r) + O(1).
i=1

i=1

▼➺♥❤ ✤➲ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❚✐➳♣ t❤❡♦ t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➦❝ tr÷♥❣ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝

Tf (∞, r) = mf (∞, r) + Nf (∞, r).
❚❛ ❝â

Tf (r) = max log|fi |r + O(1).
1≤i≤2

f

✤÷ñ❝ ❣å✐ ❧➔ s✐➯✉ ✈✐➺t ♥➳✉

Tf (r)
= ∞✳
r−→∞ log r

lim


✶✶

▼➺♥❤ ✤➲ ✶✳✺✳ [1]

●✐↔ sû ❢i ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ✵ tr➯♥ Cp✱ i = 1, 2, ..., k✳
❑❤✐ ✤â ✈î✐ ♠é✐ ρ0 < r✱ t❛ ❝â
k
k
Tf (r) + O(1)✳
(r) ≤
Tf (r) + O(1)❀ T
(r) ≤
T
f
f
k

k

i

i=1

i

i=1

i

i=1

i

i=1

❍ì♥ ♥ú❛ Tf (r) ❧➔ ♠ët ❤➔♠ t➠♥❣ t❤❡♦ r✳

❚r♦♥❣ ❧þ t❤✉②➳t ♣❤➙♥ ❜è ❣✐→ trà✱ ❝æ♥❣ t❤ù❝ P♦✐ss♦♥✲❏❡♥s❡♥ s❛✉ ✤➙② ❧➔ ❦➳t
q✉↔ q✉❛♥ trå♥❣✳
●✐↔ sû

∞

an z n

f (z) =
n=0

❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤

♣ ✲❛❞✐❝ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ❦❤æ♥❣ tr➯♥ Dr ✳ ✣➦t
T = − log r,

nf (0, r) = n−
f (T ),

c = − log ρ,

+
Γf (T ) = {t ∈ R : (n−
f (t) − nf (t)) = 0, t ≥ T },
ð ✤â

T = − log r

❱î✐ ❝→❝ ❦➼ ❤✐➺✉ ✤➣ ✤÷ñ❝ ①→❝ ✤à♥❤ ♥➔② ✈➔ ❝❤ó þ r➡♥❣ sè ❝→❝ ♣❤➛♥ tû ❝õ❛

Γf (T )

❧➔ ❤ú✉ ❤↕♥✱ ❝❤ó♥❣ t❛ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ s❛✉ ✤➙②

✣à♥❤ ❧þ ✶✳✻

✳

✭❈æ♥❣ t❤ù❝ P♦✐ss♦♥✲❏❡♥s❡♥✮

●✐↔ sû ❢ ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ♣✲❛❞✐❝ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ❦❤æ♥❣ tr➯♥ Dr ✳ ❑❤✐
✤â
+
−
(n−
f (t) − nf (t))(t − T ) + nf (c)(c − T ).

Tf (r) − Tf (ρ) = Nf (r) =
c>t≥T

❈❤ù♥❣ ♠✐♥❤✳

❱✐➳t

∞

an z n .

f (z) =
n=0
✣➸ ✤ì♥ ❣✐↔♥ t❛ ✤➦t

= nf (0, 0),

a = log |a |,


✶✷

r

1
M=
log p

nf (0, x) −
dx + log r,
x

0
ρ

nf (0, x) −
dx + log ρ,
x

1
M1 =
log p
0
r

1
M2 =
log p

nf (0, x) −
r
dx + log ,
x
ρ

ρ
r

1
M3 =
log p

nf (0, x) −
dx,

x

ρ
+
−
((n−
f (t) − nf (t))(t − T ) + nf (c)(c − T ),

M4 =
c>t≥T

+
((n−
f (t) − nf (t))(t − T ) + log r,

M5 =
t≥T

+
((n−
f (t) − nf (t))(t − T ),

M6 =
t≥T

Γ = Γf (T ).
❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤

Tf (r) = Tf (ρ) = M3 = M4 .

✭✶✳✷✮

✣➸ ❝❤ù♥❣ ♠✐♥❤ ✭✶✳✷✮✱ tr÷î❝ t✐➯♥ t❛ ❝❤ù♥❣ ♠✐♥❤

Tf (r) − a = M = M5 .

❚r÷í♥❣ ❤ñ♣ ✶✳

= 0.

❑❤✐ ✤â

r

nf (0, x)
dx,
x

1
M=
log p

M5 = M6 .

0
◆➳✉

Γ=∅

t❤➻

Tf (r) = a

✈➔

M = 0,

M6 = 0.

❉♦ ✤â

Tf (r) − a = M = M6 .

✭✶✳✸✮




= t số tỷ ừ ỳ sỷ
t(1) , t(2) , ...., t(n) õ T t(1) < t(2) < .... < t(n) . t



(i)

bi = pt ,

ỗ

n

tỷ

i = 1, 2, ..., n,

s = nf (0, r),

s1 = nf (0, b2 ),

c1 = |as |,

c2 = log |f |b1 ,

c3 = log |f |b2 ,

c4 = |as1 |,

+
(1)
(n
f (t) nf (t))(t t ),

M7 =
tt(1)

+
(2)
(n
f (t) nf (t))(t t ).

M8 =
tt(2)

(i+1)
(i)
n+
), i = 1, ..., n 1
f (t ) = nf (t
t = t(i) , i = 1, ..., n

ỵ r
ợ ồ

M7 = M6 + s(T t(1) ),
ỡ ỳ t õ

+
n
f (t) nf (t) = 0

M8 = M7 + s1 (t(1) t(2) ).

0 < bn < bn1 < ... < b1 r.

ự tự q t

n

ợ n = 1

b1 = r õ nf (0, x) = 0,
ừ Tf (r) t ữủ
t b1 < r õ
t

0 < x < r

ứ t tử

M = s(log r log b1 ) = log c1 rs log c1 bs1 ,
Tf (b1 ) = log c1 bs1 .


b1 < r



n=1



(1)
s = n
f (t ),
tỷ

t

ợ

t > t(1)

Tf (r) = log c1 rs .

ổ ử tở





Tf (r)



(1)
n+
f (t ) = 0,

Tf (b1 ) = a.

õ

(1)
+ (1)
(1)
M = (n

t) = Tf (r) a.
f (t ) nf (t ))(t

tử


✶✹

❱➟②

M = M6 = Tf (r) − a.
❉♦ ✤â ✭✶✳✷✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✳

•

❱î✐ n ≥ 2.

●✐↔ sû ❤➺ t❤ù❝ ✭✶✳✷✮ ✤ó♥❣ ✈î✐ ♠å✐
❤➺ t❤ù❝ ✭✶✳✷✮ ✤ó♥❣ ✈î✐ ♠å✐

(1 ≤ v ≤ n − 1).

v

❚❛ ❝❤ù♥❣ ♠✐♥❤

n✳

b1 < r ✳
❑❤✐ ✤â 0 < bn < bn−1 < .... < b1 < r ✈➔

T < t(1) < .... < t(n) . ⑩♣ ❞ö♥❣ ❣✐↔ t❤✐➳t
❳➨t

1
log p

b1

✈➻ ✈➟②
q✉② ♥↕♣ t❛ ❝â

nf (0, x)
dx = c2 − a = M7 .
x

0
❱➟②

r

1
M = c2 − a +
log p

nf (0, x)
dx
x
b1

r

= M7 +

1
log p

nf (0, x)
dx
x
b1
r

1
= M6 + s(T − t(1) ) +
log p

nf (0, x)
dx.
x

✭✶✳✹✮

b1
▼➦t ❦❤→❝

r

1
log p

nf (0, x)
dx = s(log r−log b1 ) = s(t(1) −T ) = log(a1 rs )−log(a1 bs1 ),
x
b1

c2 = log(c1 bs1 ).
❉♦ ❝→❝ ♣❤➛♥ tû

t

✈î✐

T ≤ t < t(1)

❦❤æ♥❣ ♣❤ö t❤✉ë❝

Tf (r) = log c1 rs .

✭✶✳✺✮

Γ

♥➯♥
✭✶✳✻✮


✶✺

❚ø ✭✶✳✹✮ ✱ ✭✶✳✺✮ ✱ ✭✶✳✻✮ t❛ ❝â

M = Tf (r) − a = M6 + s(T − t(1) ) + s(t(1) − T ) = M6 .
❳➨t

b1 = r ✳
0 < bn < ..... < b2 < b1 = r ✈➔ ❞♦ ✤â
< t(2) < .... < t(n) . ⑩♣ ❞ö♥❣ ❣✐↔ t❤✐➳t q✉②

❑❤✐ ✤â t❛ ❝â

T = t(1)

1
log p

b2

♥↕♣ t❛ ❝â

nf (0, x)
dx = c3 − a = M8 = M7 + s1 (t(1) − t(2) ).
x

0
❱➟②

1
M = c3 −a+
log p

b1

nf (0, x)
1
dx = M7 +s1 (t(1) −t(2) )+
x
log p

b2

b1

nf (0, x)
dx.
x

b2
✭✶✳✼✮

❍ì♥ ♥ú❛✱ t❛ ❝â

b1

1
log p

nf (0, x) = s1

❦❤✐

b2 ≤ x ≤ b1

✈➔

nf (0, x)
dx = s1 (log b1 −log b2 ) = s1 (t(2) −t(1) ) = log c4 bs11 −log c4 bs21 ,
x

b2

c3 = log c4 bs21 .
❱➻

t∈
/Γ

❦❤✐

t(1) < t < t(2) ✱

✈➔

Tf (r)

❧✐➯♥ tö❝ ♥➯♥

Tf (r) = log c4 bs11 .
❚ø ✭✶✳✼✮✱ ✭✶✳✽✮ ✈➔ ✭✶✳✾✮✱ t❛ ♥❤➟♥ ✤÷ñ❝

M = Tf (r) − a = M7 + s1 (t(1) − t(2) ) + s1 (t(2) − t(1) ) = M7 .
❱➻

T = t(1)

♥➯♥

M7 = M6 .
❉♦ ✤â

M = Tf (r) − a = M6 .

❚r÷í♥❣ ❤ñ♣ ✷✳
❑❤✐ ✤â

= 0.
f = f1 f2 ✈î✐ f1 = z .

✭✶✳✽✮

✭✶✳✾✮


✶✻

❚❛ ❝â

nf2 (0, 0) = 0;

nf (0, 0) = ;

nf (0, x) = nf2 (0, x) + .

❑➨♦ t❤❡♦

Tf (r) = Tf1 (r) + Tf2 (r) = log r + Tf2 (r).
❍➔♠

f2

✤â♥❣ ✈❛✐ trá ♥❤÷ ❤➔♠

f

tr♦♥❣ tr÷í♥❣ ❤ñ♣ ✶✱ ❞♦ ✤â

r

nf2 (0, x)
dx = Tf2 (r) − a = M6 .
x

1
log p
0
❱➟②

M = Tf2 (r) − a + log r = Tf (r) − a = M5 .
❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ♥❤➟♥ ✤÷ñ❝

M1 = Tf (ρ) − a = M9 ,

✭✶✳✶✵✮

ð ✤â

+
(n−
f (t) − nf (t))(t − c) + log ρ.

M9 =
t≥c

❚✐➳♣ tö❝ ❝❤ù♥❣ ♠✐♥❤ ✭✶✳✷✮✳
❚❛ ❝â

M = M1 + M2 = M5 ,

M3 = M2

⑩♣ ❞ö♥❣ ✭✶✳✸✮ ✈➔ ✭✶✳✶✵✮ ♥❤➟♥ ✤÷ñ❝

M3 = M − M1 = Tf (r) − Tf (ρ) = M5 − M9 = M4 .
✣à♥❤ ❧þ ✶✳✻ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤

❍➺ q✉↔ ✶✳✼✳

●✐↔ sû ❢ ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣✲❛❞✐❝ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ❦❤æ♥❣ tr➯♥ Dr ✳ ❑❤✐
✤â
Tf (r) = Nf (r) − N f1 (r) + O(1),

ð ✤â O(1) ❧➔ ✤↕✐ ❧÷ñ♥❣ ❜à ❝❤➦♥ ❦❤✐ r −→ +∞✳

✶✼

✶✳✷✳✷ ❍❛✐ ✣à♥❤ ❧þ ❝❤➼♥❤
❚r♦♥❣ ♠ö❝ ♥➔② tæ✐ s➩ ❞✐➵♥ ✤↕t ❤❛✐ ✣à♥❤ ❧þ ❝❤➼♥❤ tr♦♥❣ ❧þ t❤✉②➳t ♣❤➙♥
❤➻♥❤ ◆❡✈❛♥❧✐♥♥❛

♣ ✲❛❞✐❝✳ ❚❛ ✈➝♥ ❞ò♥❣ ❦➼ ❤✐➺✉ |.| t❤❛② ❝❤♦ |.|p tr➯♥ Cp✳ ❚❛

❝è ✤à♥❤ ❤❛✐ sè t❤ü❝

ρ

✈➔

ρ0

s❛♦ ❝❤♦

0 < ρ0 < ρ < ∞✳

✣➛✉ t✐➯♥ t❛ s➩ ✤✐

❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ❧þ ❝❤➼♥❤ t❤ù ♥❤➜t✳

✣à♥❤ ❧þ ✶✳✽✳ ✭✣à♥❤ ❧þ ❝❤➼♥❤ t❤ù ♥❤➜t✮ [1]

◆➳✉ f ❧➔ ♠ët ❤➔♠ ❦❤→❝ ❤➡♥❣ tr➯♥ Cp(0, ρ) t❤➻ ✈î✐ ♠å✐ a ∈ Cp t❛ ❝â
mf (a, r) + Nf (a, r) = Tf (r) + O(1).

❈❤ù♥❣ ♠✐♥❤✳

❚❛ ❝â

mf (a, r) + Nf (a, r) = Tf (a, r) = Tf −a (r) − log|f − a|ρ0 .
❚❛ ❧↕✐ ❝â

Tf −a (r) ≤ Tf (r) + log+ |a|,
Tf (r) ≤ Tf −a (r) + log+ |a|.
❚ø ✤â t❛ ❝â ❦➳t ❧✉➟♥ ❝õ❛ ✤à♥❤ ❧þ✳
▼➺♥❤ ✤➲ s❛✉ ❧➔ ❇ê ✤➲ ✤↕♦ ❤➔♠ ❧♦❣❛r✐t✳

▼➺♥❤ ✤➲ ✶✳✾✳ [1]

◆➳✉ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ Cp(0, ρ) t❤➻ ✈î✐ ♠é✐ sè ♥❣✉②➯♥
f (k)
1
f (k)
1
k > 0 t❛ ❝â
≤ k ✱ ✤➦❝ ❜✐➺t
≤ k log+ ✳
f
r
f
r
r

r

❈❤ù♥❣ ♠✐♥❤✳ f ∈ A(ρ (Cp)

t❛ ❝â

f
f
❉♦ ✤â

f (k)
f

k

=
r

i=1

= |f |r ≤
r

f (i)
f (i−1)

k

=
r

i=1

1
.
r
f (i)
f (i−1)

≤
r

1
,
rk


✶✽

f (0) = f ✳
g
①➨t f =
∈ M(ρ (Cp )✳
h

tr♦♥❣ ✤â
❇➙② ❣✐í

f
f

❑❤✐ ✤â

h
hg − gh h
g
.
−
=
2
h
g r
g
h
1
g
h
≤ max
,
≤ .
g r h r
r

=
r

r

❚÷ì♥❣ tü t❛ ❝ô♥❣ t❤✉ ✤÷ñ❝

f (k)

f

≤
r

1
.
rk

▼➺♥❤ ✤➲ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❱î✐ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣

f

tr♦♥❣

Cp (0, ρ)✱

t❛ ✤à♥❤ ♥❣❤➽❛

NRamf (∞, r) = 2Nf (∞, r) − Nf (∞, r) + Nf (0, r).
❚✐➳♣ t❤❡♦ t❛ ❣✐î✐ t❤✐➺✉ ✣à♥❤ ❧þ ❝❤➼♥❤ t❤ù ❤❛✐✳

✣à♥❤ ❧þ ✶✳✶✵✳ ✭✣à♥❤ ❧þ ❝❤➼♥❤ t❤ù ❤❛✐✮ [1]

◆➳✉ ❢ ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ Cp(0, ρ) ✈➔ a1, ...., aq ∈ Cp ❧➔ ❝→❝
sè ♣❤➙♥ ❜✐➺t✳ ✣➦t
δ = min {1, |ai − aj |} , A = max {1, |ai |} .
i

i=j

❑❤✐ ✤â ✈î✐ 0 < r < ρ✱
q

(q − 1)Tf (r) ≤ Nf (r) +

Nf (aj , r) − NRamf (∞, r) − log r + Sf
j=1
q

≤ N f (r) +

N f (aj , r) − log r + Sf ,
j=1

tr♦♥❣ ✤â
q

log |f − aj |ρ0 − log |f |ρ0 + (q − 1) log

Sf =
j=1

A
.
δ


✶✾

❈❤ù♥❣ ♠✐♥❤✳
▲➜②

r ∈ |Cp |

Ar (Cp )

❚r♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❦❤✐ ✈✐➳t

s❛♦ ❝❤♦

ρ0 < r < ρ ✳

| |

❚❛ ✈✐➳t

t❛ ❤✐➸✉ ❧➔

f1
f =
f0

| |p ✳

tr♦♥❣ ✤â

f1 , f0 ∈

❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ✈➔ ✤➦t

F0 = f0 , Fi = f1 − ai f0

(i = 1, 2, ..., q).

❑❤✐ ✤â

|fk (z)| ≤ A max {|F0 (z)|, |Fi (z)|} ,
i

(k = 0, 1).

❚❛ ❧✉æ♥ sû ❞ö♥❣

W = W (f0 , f1 ) =

f0 f1
f0 f1

❧➔ ❦➼ ❤✐➺✉ ❲r♦♥s❦✐❛♥ ❝õ❛

f0

✈➔

f1 ✳

✣➦t

Wi = W (F0 , Fi ) = W.
❇➙② ❣✐í t❛ ❝è ✤à♥❤

z ∈ Cp [0, ρ0 ] − Cp [0, ρ]

W (z), f1 (z), Fi (z) = 0,
❑❤✐ ✤â tç♥ t↕✐ ♠ët ❝❤➾ sè

j ∈ {1, 2, ...., q}

s❛♦ ❝❤♦

i = 0, 1, ...., q.
s❛♦ ❝❤♦

|Fj (z)| = min |Fi (z)|.
1≤j≤q

❈❤ó þ r➡♥❣

|Fi (z) − Fj (z)| 1
≤ |Fi (z)| (i = j).
|aj − ai |
δ

|f0 (z)| =

◆❤÷ ✈➟② ❝❤ó♥❣ t❛ ❝â t❤➸ ❧➜② ❝→❝ ❝❤➾ sè ♣❤➙♥ ❜✐➺t

j

(l = 1, 2, ..., q − 1)

β1 , ...., βq−1

✈î✐

s❛♦ ❝❤♦

0 < max {δ|f0 (z)|, |Fj (z)|} ≤ |Fβ1 (z)| ≤ .... ≤ |Fβq−1 (z)| < ∞
❑❤✐ ✤â t❛ ❝â

|fk (z)| ≤
✈î✐ ♠é✐

k = 0, 1;

A
A
max {δ|f0 (z)|, |Fj (z)|} ≤ |Fβl (z)|,
δ
δ

l = 1, 2, ....q − 1✳

|f (z)| = max |fk (z)| ≤
k

◆❤÷ ✈➟② t❛ t❤✉ ✤÷ñ❝

A
|Fβl (z) |,
δ

l = 0, ..., q − 1,

βl =