✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖
❱❆◆❍◆❆❙❖◆❊ ❚❍❊PP❍❆❱❖◆●
❚➑◆❍ ❙■➊❯ ▲➬■✱ ❚➑◆❍ ❚❆❯❚
❱⑨ ❚➑◆❍ ❑ ✲ ✣❺❨ ❈Õ❆ ❈⑩❈ ❚❾P ▼Ð
❑❍➷◆● ❇➚ ❈❍➄◆ ❚❘❖◆● Cn
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼
✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖
❱❆◆❍◆❆❙❖◆❊ ❚❍❊PP❍❆❱❖◆●
❚➑◆❍ ❙■➊❯ ▲➬■✱ ❚➑◆❍ ❚❆❯❚
❱⑨ ❚➑◆❍ ❑ ✲ ✣❺❨ ❈Õ❆ ❈⑩❈ ❚❾P ▼Ð
❑❍➷◆● ❇➚ ❈❍➄◆ ❚❘❖◆● Cn
❈❤✉②➯♥ ♥❣➔♥❤✿ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿ ❚❙✳ ❚❘❺◆ ❍❯➏ ▼■◆❍
❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼
▲í✐ ❝❛♠ ✤♦❛♥
❚æ✐ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐✱ ❞÷î✐ sü ❤÷î♥❣
❞➝♥ t➟♥ t➻♥❤ ✈➔ ❝❤✉ ✤→♦ ❝õ❛ ❚❙✳ ❚r➛♥ ❍✉➺ ▼✐♥❤✳
❚r♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❧✉➟♥ ✈➠♥ tæ✐ ✤➣ ❦➳ t❤ø❛ t❤➔♥❤ q✉↔ ❦❤♦❛ ❤å❝ ❝õ❛
❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈➔ ✤ç♥❣ ♥❣❤✐➺♣ ✈î✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤✳
❍å❝ ✈✐➯♥
❱❛♥❤♥❛s♦♥❡ ❚❍❊PP❍❆❱❖◆●
✐
▲í✐ ❝↔♠ ì♥
▲✉➟♥ ✈➠♥ ♥➔② ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ t➟♥ t➻♥❤ ✈➔ sü ❝❤➾
❜↔♦ ♥❣❤✐➯♠ ❦❤➢❝ ❝õ❛ ❚❙✳ ❚r➛♥ ❍✉➺ ▼✐♥❤✱ tæ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥
t❤➔♥❤ ✈➔ s➙✉ s➢❝ ✤➳♥ ❝æ ❣✐→♦✳
❚æ✐ ❝ô♥❣ ①✐♥ ❦➼♥❤ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦
tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ❝ô♥❣ ♥❤÷ ❝→❝ t❤➛② ❝æ
❣✐→♦ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ❦❤â❛ ❤å❝ ✷✵✶✺✲✷✵✶✼✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ✤❡♠ ❤➳t
t➙♠ ❤✉②➳t ✈➔ sü ♥❤✐➺t t➻♥❤ ✤➸ ❣✐↔♥❣ ❞↕② ✈➔ tr❛♥❣ ❜à ❝❤♦ ❝❤ó♥❣ tæ✐ ♥❤✐➲✉
❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠✳
❱➔ ❝✉è✐ ❝ò♥❣✱ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❝↔♠ ì♥ ❝→❝ ✤ç♥❣ ♥❣❤✐➺♣✱ ❜↕♥
❜➧ ✤➣ ❧✉æ♥ ✤ç♥❣ ❤➔♥❤ ❣✐ó♣ ✤ï tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥
❝ù✉ ❝ô♥❣ ♥❤÷ tr♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✶✼
◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥
❱❛♥❤♥❛s♦♥❡ ❚❍❊PP❍❆❱❖◆●
✐✐
▼ö❝ ❧ö❝
▲í✐ ❝❛♠ ✤♦❛♥
✐
▲í✐ ❝↔♠ ì♥
✐✐
▼ö❝ ❧ö❝
✐✐✐
▼ð ✤➛✉
✶
✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✸
✶✳✶
⑩♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸
✶✳✷
✣à♥❤ ❧þ ❆s❝♦❧✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✹
✶✳✸
❍➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✺
✶✳✹
❍➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✺
✶✳✺
❍➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ♣❡❛❦ ✈➔ ❛♥t✐♣❡❛❦
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✻
✶✳✻
●✐↔ ♠➯tr✐❝ ✈✐ ♣❤➙♥ ❘♦②❞❡♥ ✕ ❑♦❜❛②❛s❤✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✻
✶✳✼
●✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✼
✶✳✽
❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♠ët ♠✐➲♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✽
✶✳✾
▼✐➲♥ t❛✉t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✾
✷ ❚➼♥❤ s✐➯✉ ❧ç✐✱ t➼♥❤ t❛✉t ✈➔ t➼♥❤ k✲✤➛② ❝õ❛ ❝→❝ t➟♣ ♠ð tr♦♥❣
✶✵
Cn
✐✐✐
✷✳✶
❚➼♥❤ s✐➯✉ ❧ç✐ ❝õ❛ ♠ët t➟♣ ♠ð ❦❤æ♥❣ ❜à ❝❤➦♥ tr♦♥❣ Cn ✳ ✳ ✳
✷✳✷
❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ✈➔ t➼♥❤ t❛✉t ❝õ❛ ♠ët t➟♣ ♠ð ❦❤æ♥❣ ❜à ❝❤➦♥
✶✵
tr♦♥❣ Cn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✺
✷✳✸
❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② ❝õ❛ ♠ët ♠✐➲♥ tr♦♥❣ Cn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✵
✷✳✹
❚➼♥❤ k ✲ ✤➛② ❝õ❛ ♠ët t➟♣ ♠ð ❦❤æ♥❣ ❜à ❝❤➦♥
tr♦♥❣ Cn ✳
✷✳✺
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✼
❈→❝ ♠✐➲♥ ❍❛rt♦❣s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✵
❑➳t ❧✉➟♥
✸✽
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
✸✾
✐✈
▼ð ✤➛✉
◆❤÷ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t ❧þ t❤✉②➳t ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ❤②♣❡r❜♦❧✐❝ r❛ ✤í✐
✈➔♦ ❝✉è✐ ♥❤ú♥❣ ♥➠♠ ✻✵ ❝õ❛ t❤➳ ❦✛ tr÷î❝✱ s❛✉ ♥❤ú♥❣ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥
❝ù✉ ❝õ❛ ♥❤➔ t♦→♥ ❤å❝ ◆❤➟t ❇↔♥ ❙✳ ❑♦❜❛②❛s❤✐✳ ❈❤♦ ✤➳♥ ♥❛②✱ ❧þ t❤✉②➳t
♥➔② ✤➣ trð t❤➔♥❤ ♠ët ♥❣➔♥❤ ♥❣❤✐➯♥ ❝ù✉ q✉❛♥ trå♥❣ ❝õ❛ ❣✐↔✐ t➼❝❤ ♣❤ù❝
❤②♣❡r❜♦❧✐❝✳ ◆❤✐➲✉ ❦➳t q✉↔ s➙✉ s➢❝ ✈➔ ✤➭♣ ✤➩ ✤➣ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐
♥❤ú♥❣ ♥❤➔ t♦→♥ ❤å❝ ❧î♥ tr➯♥ t❤➳ ❣✐î✐ ♥❤÷ ❙✳ ❑♦❜❛②❛s❤✐✱ ▼✳ ●r❡❡♥❡✱ ❏✳
◆♦❣✉❝❤✐✱✳✳✳✳ ▲þ t❤✉②➳t ♥➔② ✤÷ñ❝ ù♥❣ ❞ö♥❣ rë♥❣ r➣✐ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝
❦❤→❝ ♥❤❛✉ ♥❤÷ ❍➺ ✤ë♥❣ ❧ü❝ ♣❤ù❝✱ ▲þ t❤✉②➳t ♣❤➙♥ ❜è ❣✐→ trà ✈➔ ①➜♣ ①➾
❉✐♦♣❤❛♥t✐♥❡✳ ❚✉② ♥❤✐➯♥ ✤❛ sè ❝→❝ ❦➳t q✉↔ ❝❤➾ ✤↕t ✤÷ñ❝ tr♦♥❣ ✤✐➲✉ ❦✐➺♥
❝â t➼♥❤ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ❝õ❛ ❝→❝ ♠✐➲♥✳ ❱î✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ✈➔
♥❣❤✐➯♥ ❝ù✉ ✈➲ ❤➻♥❤ ❤å❝ ❝õ❛ ❝→❝ ♠✐➲♥ ❦❤æ♥❣ ❜à ❝❤➦♥✱ ❡♠ ✤➣ ❧ü❛ ❝❤å♥ ✤➲
t➔✐ ✧❚➼♥❤ s✐➯✉ ❧ç✐✱ t➼♥❤ t❛✉t ✈➔ t➼♥❤ ❦✲ ✤➛② ❝õ❛ ❝→❝ t➟♣ ♠ð ❦❤æ♥❣
❜à ❝❤➦♥ tr♦♥❣ Cn ✧ ♥❤➡♠ t➻♠ ❤✐➸✉ ♠ët sè ❝→❝ ❦➳t q✉↔ ✤à❛ ♣❤÷ì♥❣ ✈➲
t➼♥❤ ❤②♣❡r❜♦❧✐❝✱ t➼♥❤ t❛✉t ✈➔ t➼♥❤ ❦✲ ✤➛② ❝õ❛ ❝→❝ t➟♣ ♠ð ❦❤æ♥❣ ❜à ❝❤➦♥
tr♦♥❣ Cn .
▲✉➟♥ ✈➠♥ ❣ç♠ ✸✾ tr❛♥❣✱ tr♦♥❣ ✤â ❝â ♣❤➛♥ ♠ð ✤➛✉✱ ❤❛✐ ❝❤÷ì♥❣ ♥ë✐ ❞✉♥❣✱
♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ ❞❛♥❤ ♠ö❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳
❈❤÷ì♥❣ ✶✿ ❍➺ t❤è♥❣ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ ❦➳t q✉↔ ❝➛♥ t❤✐➳t ❝❤♦
❝❤÷ì♥❣ s❛✉✳
❈❤÷ì♥❣ ✷✿ ❚r➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ t➼♥❤ ❤②♣❡r❜♦❧✐❝✱ t➼♥❤ t❛✉t✱
✶
t s ỗ ừ ởt t ổ tr Cn ự t
r ừ ởt ổ tr Cn q sỹ tỗ t ừ
ởt ữỡ t ộ t
ừ ỗ tớ t ố ỳ t tt ữỡ
t tt t ử ừ ởt tr Cn P ố ừ ữỡ tr
ự ử ừ t q tr ự t r ừ
rts r ừ ởt rts tt
s ỗ
ổ tr ọ ỳ t rt
ữủ ỳ ỵ õ õ ừ t ổ ồ
ữủ t ỡ
❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ sû
❞ö♥❣ ❝❤♦ ❝❤÷ì♥❣ s❛✉ ♥❤÷✿ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤✱ ✤à♥❤ ❧þ ❆s❝♦❧✐✱ ❤➔♠ ✤✐➲✉
❤á❛ ❞÷î✐✱ ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐✱ ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ♣❡❛❦ ✈➔ ❛♥t✐♣❡❛❦✱
❣✐↔ ♠➯tr✐ ✈✐ ♣❤➙♥✱ ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐✱ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♠ët
♠✐➲♥✱ ♠✐➲♥ t❛✉t✳ ❈→❝ ♥ë✐ ❞✉♥❣ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ ✈✐➺t t❤❡♦ ❝→❝ t➔✐
❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✺❪✳
✶✳✶
⑩♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤
●✐↔ sû X ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ Cn ✈➔ f : X → C ❧➔ ♠ët ❤➔♠ sè✳
❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ♣❤ù❝ t↕✐ x0 ∈ X ♥➳✉ tç♥ t↕✐ →♥❤ ①↕ t✉②➳♥
t➼♥❤ λ : Cn → C s❛♦ ❝❤♦
|f (x0 + h) − f (x0 ) − λ (h)|
= 0,
|h|→0
|h|
lim
tr♦♥❣ ✤â h = (h1 , ..., hn ) ∈ C ✈➔ |h| =
n
n
2
|hi |
1/2
✳
i=1
❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ t↕✐ x0 ∈ X ♥➳✉ f ❦❤↔ ✈✐ ♣❤ù❝ tr♦♥❣ ♠ët
❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛ x0 ✈➔ ✤÷ñ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ X ♥➳✉ f ❝❤➾♥❤ ❤➻♥❤
t↕✐ ♠å✐ ✤✐➸♠ t❤✉ë❝ X ✳
✸
ởt f : X Cm õ t t ữợ f = (f1 , ..., fm ) tr
õ fi = i f : X C, i = 1, ..., m tồ ở õ f ữủ ồ
tr X fi tr X ợ ồ i = 1, ..., m
f : X f (X) Cn ữủ ồ s f s
f 1 ụ
ỵ s
sỷ F ởt ồ õ tứ ổ
tổ ổ X ổ tổ ổ Y ồ F ữủ ồ tử ỗ
ts tứ x X tợ y Y ợ ộ U ừ y
t ữủ ởt V ừ x W ừ y s
f (x) t f (V ) U ợ ồ f F
F tử ỗ ợ ồ x X ồ y Y t F ữủ
ồ tử ỗ tứ X Y
ỵ ỵ s ố ợ ồ tử ỗ
sỷ F t ừ t tử C(X, Y ) tứ ổ
q t ữỡ X ổ sr Y C(X, Y )
õ tổ ổ t õ F t tữỡ ố tr C(X, Y )
s ữủ tọ
F ồ tử ỗ
ợ ộ x X t ủ Fx = {f (x)|f F } t tữỡ ố
tr Y
ỏ ữợ
sỷ G ởt tr Cn u : G [ , ) ữủ ồ
ỏ ữợ tr G Cn u tọ s
u ỷ tử tr tr G tự t {z G|u(z) < s} t
ợ ộ số tỹ s
ợ ộ t t tữỡ ố ừ G ồ h : R
t õ u h tr
ỏ tr tử tr
t u h tr
õ t ỏ ữợ s
ỷ tử tr tr G ỏ ữợ tr G
ừ ợ ộ z G tỗ t r0 (z) > 0 s
u(z)
1
2
0
2
u(z + reit )dt ợ ồ r < r0 (z).
ỏ ữợ
sỷ G ởt tr Cn
: G [ , ) ữủ ồ ỏ ữợ tr
G Cn ỵ P SH(G)
ỷ tử tr tr G s = tr ộ t
tổ ừ G
ợ ộ a G ợ ồ b Cn , b = 0 (a + b)
ỏ ữợ ỗ t tr ộ t tổ
ừ t C : a + b G .
ỵ : G [ , ) ởt ỷ tử tr
= tr t ự t tổ ừ G õ P SH(G)
ợ ộ a G b Cn a + b : G, ||
1 G
t õ
(a)
tr õ l(; a, b) =
1
2
2
0
l(; a, b),
(a + eit b)dt.
ỏ ữợ t
D ởt tr Cn
ởt ữủ ồ ởt ỏ ữợ ữỡ t
ởt p tở D tỗ t ởt U ừ p s
U tọ
ỏ ữợ tr D U tử tr tr D
(p) = 0
(z) < 0, ợ ồ z D
U.
ởt ữủ ồ ởt ỏ ữợ t t ởt
p tở D {} tỗ t ởt U ừ p s
U tọ
ỏ ữợ tr D U tử tr tr D
(p) =
(z) > , ợ ồ z (D
U )\{p}.
tr s
D ởt tr Cn FG : G ì Cn [0, )
FG (z; X) := inf{()|| : Hol(, D), : () = z,
. () = X}
✤÷ñ❝ ❣å✐ ❧➔ ❣✐↔ ♠➯tr✐❝ ✈✐ ♣❤➙♥ ❘♦②❞❡♥ ✕ ❑♦❜❛②❛s❤✐ tr➯♥ D✱ ð ✤â
γ(λ) :=
1
, λ ∈ ∆.
1 − |λ|2
❘ã r➔♥❣
❛✮ FD (z, X) = inf{α > 0 : ∃ϕ ∈ Hol(∆, D) : ϕ(0) = z, αϕ (0) = X}
¯ D) : ϕ(0) = z, αϕ (0) = X}.
= inf{α > 0 : ∃ϕ ∈ Hol(∆,
❜✮ FD (z, λX) = |λ|.FD (z, X), λ ∈ C, z ∈ D ⊂ Cn , X ∈ Cn .
❝✮ FG (F (z); F (z)X)
FD (z; X), F ∈ Hol(∆, G), z ∈ D ⊂ Cn , X ∈ Cn .
✣➦❝ ❜✐➺t✱ ♥➳✉ F : D → G ❧➔ →♥❤ ①↕ s♦♥❣ ❝❤➾♥❤ ❤➻♥❤✱ t❤➻
❞✮ FG (F (z); F (z)X) = FD (z; X), z ∈ D; X ∈ Cn .
❡✮ F∆ (λ, X)
✶✳✼
γ(λ).|X|, λ ∈ ∆, X ∈ C.
●✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐
❈❤♦ D ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ Cn ✱ ❝è ✤à♥❤ ❤❛✐ ✤✐➸♠ z0 , z0 t❤✉ë❝ D✳ ❚ç♥ t↕✐
♠ët ✤÷í♥❣ ❝♦♥❣ α : ❬0, 1] → D ♥è✐ ❤❛✐ ✤✐➸♠ z0 , z0 ✳ ⑩♣ ❞ö♥❣ ✤à♥❤ ❧þ ①➜♣
①➾ ❲❡✐❡rstr❛ss✱ t❛ t➻♠ ✤÷ñ❝ ♠ët →♥❤ ①↕ ✤❛ t❤ù❝ P : ❬0, 1] → D ♠➔
P (0) = z0 , P (1) = z0 .
❉➵ ❝❤å♥ ✤÷ñ❝ ♠ët ♠✐➲♥ ❧✐➯♥ t❤æ♥❣ G ⊂ C, ❬0, 1] ⊂ G s❛♦ ❝❤♦ P (λ) ∈ D
✈î✐ λ ∈ G✳ ❚❤❡♦ ✤à♥❤ ❧þ →♥❤ ①↕ ❘✐❡♠❛♥♥✱ t❛ ❝â t❤➸ ❦➳t ❧✉➟♥ r➡♥❣ z0 , z0
♥➡♠ tr➯♥ ♠ët ✤➽❛ ❣✐↔✐ t➼❝❤ ϕ : ∆ → G ♠➔
ϕ(0) = z0 ✈➔ ϕ(σ) = z0 , (0
σ < 1).
▲➜② z0 , z0 ∈ D✳ ❚❛ ✤➦t
¯ D) : ϕ(λ ) =
lD (z , z ) := inf{p(λ , λ ) : λ , λ ∈ ∆, ∃ϕ ∈ Hol(∆,
z , ϕ(λ ) = z ⑥
✼
¯ D) : ϕ(0) = z , ϕ(λ ) = z ⑥ ✱
= inf{p(0, λ ) : λ ∈ ∆, ∃ϕ ∈ Hol(∆,
ð ✤➙②
p(z , z ) := inf{Lγ (α)| α : ❬0; 1] → ∆ ❧➔ ✤÷í♥❣ ❝♦♥❣ ❧î♣ C 1 ✱ λ = α(0),
λ = α(1)⑥ , Lγ (α) :=
1
0 γ(α(t)|α
(t)|dt.
❚❛ ❣å✐ lD ❧➔ ❤➔♠ ▲❡♠♣❡rt ❝õ❛ D✳
✲ ❱î✐ z , z ∈ D, t❛ ✤➦t
N
lD (zj−1 , zj ) : N ∈ N, z0 = z , z1 , ..., zN −1 ∈ D,
kD (z , z ) := inf{
j=1
zN = z }. ❍➔♠ kD ✤÷ñ❝ ❣å✐ ❧➔ ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐ tr➯♥ D✳
◆❤➟♥ ①➨t ✶✳✼✳✶✳ ●✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐ kD (z , z ) ❝á♥ ✤÷ñ❝ ✤à♥❤
♥❣❤➽❛ ❜ð✐
1
kD (z , z ) = inf
FD (γ(t), γ (t))dt,
0
tr♦♥❣ ✤â inf ❧➜② t❤❡♦ t➜t ❝↔ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ ✈✐ ♥è✐ z ✈➔ z .
✶✳✽
❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♠ët ♠✐➲♥
✲ ▼ët ♠✐➲♥ D ⊂ Cn ✤÷ñ❝ ❣å✐ ❧➔ k ✲ ❤②♣❡r❜♦❧✐❝ ♥➳✉ kD ❧➔ ❦❤♦↔♥❣ ❝→❝❤
tr➯♥ D✳
✲ ▼ët ♠✐➲♥ k ✲ ❤②♣❡r❜♦❧✐❝ D ✤÷ñ❝ ❣å✐ ❧➔ k ✲ ❤②♣❡r❜♦❧✐❝ ✤➛②
✭❤❛② k ✲ ✤➛②✮ ♥➳✉ ♥â ✤➛② ✤è✐ ✈î✐ ❦❤♦↔♥❣ ❝→❝❤ kD ✳ ▼✳▲✳❘♦②❞❡♥ [Ro] ✤➣
❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♠ët ♠✐➲♥ D ❧➔ ❤②♣❡r❜♦❧✐❝ ♥➳✉ ✈î✐ ♠å✐ ✤✐➸♠ p ∈ D✱ tç♥
t↕✐ ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ p ✈➔ ♠ët ❤➡♥❣ sè ❞÷ì♥❣ c s❛♦ ❝❤♦ FD (y, x)
c||x||
✈î✐ ♠å✐ y ∈ U ✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ❤å ❝→❝ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ tø ∆ ✈➔♦
D ❧➔ ✤ç♥❣ ❧✐➯♥ tö❝ ✤è✐ ✈î✐ ❦❤♦↔♥❣ ❝→❝❤ dD ✱ t❤➻ ♠✐➲♥ ❤②♣❡r❜♦❧✐❝ D ❧➔
❤②♣❡r❜♦❧✐❝ ✤➛② ♥➳✉ ✈î✐ ♠é✐ ✤✐➸♠ z ∈ D ✈➔ ♠é✐ sè t❤ü❝ ❞÷ì♥❣ r✱ ❤➻♥❤ ❝➛✉
④ y ∈ D : dD (z, y)
r⑥ ❧➔ ❝♦♠♣❛❝t tr♦♥❣ D✳
✽
✶✳✾
▼✐➲♥ t❛✉t
❈❤♦ D ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ Cn ✱ tr➯♥ Hol(∆, D) t❛ tr❛♥❣ ❜à tæ ♣æ ❝♦♠♣❛❝t
♠ð✳
✲ ❉➣② ④ fj ⑥
∞
j=1
⊂ Hol(∆, D) ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➙♥ ❦ý ❝♦♠♣❛❝t ♥➳✉ ✈î✐ ♠é✐
t➟♣ ❝♦♥ ❝♦♠♣❛❝t K ❝õ❛ ∆✱ ♠é✐ t➟♣ ❝♦♥ ❝♦♠♣❛❝t L ❝õ❛ D✱ tç♥ t↕✐ f0 ∈ N
s❛♦ ❝❤♦ fj (K) ∩ L = ∅ ✈î✐ ♠å✐ j
t➢❝ ♥➳✉ ♠é✐ ❞➣② ④ fj ⑥
∞
j=1
j0 ✳ ❍å Hol(∆, D) ✤÷ñ❝ ❣å✐ ❧➔ ❤å ❝❤✉➞♥
tr♦♥❣ Hol(∆, D) ❝❤ù❛ ♠ët ❞➣② ❝♦♥ ④ fjν ⑥ ❤♦➦❝
❧➔ ❤ë✐ tö ✤➲✉ tr➯♥ ♠é✐ t➟♣ ❝♦♥ ❝♦♠♣❛❝t tî✐ →♥❤ ①↕ f ∈ Hol(∆, D) ✭❦þ
K
❤✐➺✉ fjν →
→
− f ✮ ❤♦➦❝ ❧➔ ♣❤➙♥ ❦ý ❝♦♠♣❛❝t✳
✲ ▼✐➲♥ D ✤÷ñ❝ ❣å✐ ❧➔ ♠✐➲♥ t❛✉t ♥➳✉ ❤å Hol(∆, D) ❧➔ ♠ët ❤å ❝❤✉➞♥ t➢❝✳
✲ ▼✐➲✉ D ✤÷ñ❝ ❣å✐ ❧➔ t❛✉t ✤à❛ ♣❤÷ì♥❣ t↕✐ ♠ët ✤✐➸♠ p ∈ ∂D ♥➳✉ tç♥ t↕✐
♠ët ❧➙♥ ❝➟♥ U ❝õ❛ p s❛♦ ❝❤♦ D ∩ U ❧➔ ♠ët ♠✐➲♥ t❛✉t✳
✾
❈❤÷ì♥❣ ✷
❚➼♥❤ s✐➯✉ ❧ç✐✱ t➼♥❤ t❛✉t ✈➔ t➼♥❤ k✲✤➛②
❝õ❛ ❝→❝ t➟♣ ♠ð tr♦♥❣ Cn
✷✳✶
❚➼♥❤ s✐➯✉ ❧ç✐ ❝õ❛ ♠ët t➟♣ ♠ð ❦❤æ♥❣ ❜à ❝❤➦♥ tr♦♥❣ Cn
❚r÷î❝ t✐➯♥ t❛ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ s❛✉✿
✲ ▼ët ✤✐➸♠ ❜✐➯♥ a ❝õ❛ ♠ët t➟♣ ♠ð D tr♦♥❣ Cn ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ✤✐➸♠ ❝❤➢♥
❝õ❛ D ♥➳✉ tç♥ t↕✐ ♠ët ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ➙♠ ϕ tr➯♥ D ♠➔ lim u(z) = 0✱
z→a
ϕ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ❤➔♠ ❝❤➢♥ ❝õ❛ D t↕✐ a✳
✣✐➸♠ a ❣å✐ ❧➔ ✤✐➸♠ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ D ♥➳✉ tç♥ t↕✐ ♠ët ❧➙♥ ❝➟♥ ♠ð
U ❝õ❛ a s❛♦ ❝❤♦ a ❧➔ ♠ët ✤✐➸♠ ❝❤➢♥ ❝õ❛ D ∩ U ✳
✲ ▼ët ✤✐➸♠ ❜✐➯♥ a ❝õ❛ ♠ët t➟♣ ♠ð D tr♦♥❣ Cn ❣å✐ ❧➔ ♠ët ✤✐➸♠ ♣❡❛❦ ✤❛
✤✐➲✉ ❤á❛ ❞÷î✐ ❝õ❛ D ♥➳✉ tç♥ t↕✐ ♠ët ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ϕ tr➯♥ D ♠➔
lim ϕ(z) = 0 ✈➔ lim sup ϕ(z) < 0 ✈î✐ ❜➜t ❦ý b ∈ ∂D\{a}✳ ❍➔♠ ϕ ✤÷ñ❝ ❣å✐
z→a
z→b
❧➔ ❤➔♠ ♣❡❛❦ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ❝õ❛ D t↕✐ a✳
✲ ▼ët t➟♣ ♠ð D tr♦♥❣ Cn ✤÷ñ❝ ❣å✐ ❧➔ s✐➯✉ ❧ç✐ ♥➳✉ tç♥ t↕✐ ♠ët ❤➔♠ ✈➨t
❝↕♥ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ❧✐➯♥ tö❝✱ ➙♠ tr➯♥ D✱ tù❝ ❧➔ lim ϕ(z) = 0✳
z→∂D
◆❤➟♥ ①➨t ✷✳✶✳✶✳ ❬✻❪ D ❧➔ s✐➯✉ ❧ç✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♠å✐ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥
t❤æ♥❣ ❝õ❛ ♥â ❧➔ s✐➯✉ ❧ç✐✳
✶✵
t t t trữớ ủ D õ ổ số t tổ
D1 , D2 , ... sỷ ồ Dj s ỗ j t ỏ
ữợ tữỡ ự ừ Dj t j j N := max{j , j 1 }
t D s ỗ ợ ỏ ữợ tr D õ |
Dj := j
õ s
ởt ừ ởt t ổ
D tr Cn t tr D õ ởt ỏ ữợ t
t t t ý t tổ ừ D
r
ự sỷ ởt ừ D t õ t tt
r 1 < < 0 {Dj } ởt t s Dj Dj+1
Dj = D õ tỗ t
j=1
B(0, rj ) = {z Cn : z rj }, j N,
s j :=
inf > j := sup .
Dj
(z), z D\B(0, rj )
D\B(0,rj )
t j (z) :=
max{(z), r2 (j j )||z||2 + j , z D B(0, rj ).
j
tr ữủ :=
j=1
i
2j
t
D ởt t ừ Cn a ởt
tở D
gD (a, ã ) := sup{u( ã ) : u La , u 0},
tr õ La = {u P SH(D) : u( ã ) log
ã a o(1) ã a}
gD ữủ ồ r ự ợ ỹ t a ó r gD (a, ã)
ởt ỏ ữợ
D ởt tr Cn a D t t õ t q s
D ởt tr Cn a D t
D s ỗ
lim gD (a, z) = 0 , b D.
zb
r t t t s tr ởt t q t s ỗ ừ
ởt t ổ tr Cn . rữợ t t õ ờ s
ờ D2 D1 D t tr Cn ợ D1 = D
D2 D D1 sỷ ởt ỏ ữợ tr D s
:= inf > := sup .
DD1
DD2
ợ a D2 t d(a) := inf gD1 (a, ã) õ
DD2
gD (a, z) gD1 (a, z) +
d(a) z D1 ,
gD (a, z)
(z)
d(a) z D\D1 .
ự õ t tt r d(a) > õ
u(a, z) :=
(z)
d(a), z D
tọ
u(a, z) gD1 (a, z), z D D2 ,
u(a, z) 0 lim sup gD1 (a, ), z D D1 .
D1 z
õ t õ
g (a, z)
, z D2
D1
v(a, z) := max{gD1 (a, z), u(a, z)}, z D1 \D2
u(a, z)
, z D\D1
ởt ỏ ữợ ố ợ tự ợ ỹ rt t a
ỡ ỳ
v(a, z)
d(a), z D.
tứ ừ gD t õ
gD (a, z) v(a, z) +
d(a).
ứ ờ tr t ự ữủ s t s ỗ ừ
ởt t ổ tr Cn õ
D ởt t ổ tr Cn
D s ỗ ữỡ t t ý ỳ tự ợ t
ý ỳ a D tỗ t ởt U ừ a s ồ t
tổ ừ D U s ỗ ởt
ỏ ữợ t D s ỗ
ự ự tọ
lim gD (z, w) = 0,
D wa
ợ ồ a D z D
rữợ t a = s ự ữợ ỡ
r ởt tỗ t ởt
ỏ ữợ t t ồ ởt trỡ s = 1 z
sup p D õ tỗ t ởt số c > 0 s
uz (w) := c(w) + (w) log w z ,
wD
ởt ỏ ữợ tr D ợ ỹ rt t z
gD (z, w) uz (w), w D,
õ t õ
ớ sỷ a D Cn r > 0 s a, z B(0, r)
ởt ỏ ữợ ừ D t t
< 0
sup
D B(0,r)
õ tỗ t r > r s
2
inf
>
D B(0,r )
sup
.
D B(0,r)
:= D B(0, r ) õ ờ t õ
t D
gD (z, )
gD (z, ) +
inf
D B(0,r)
gD (z, ã)
gD (z, ) +
inf
D B(0,r)
gD (z, ã).
t tổ ừ D
ự z D
s
D
ỗ tr t õ
lim g (z, w)
wa D
D
= 0.
õ
t gD (z, w) = 0 w
/D
lim inf gD (z, w)
wa
inf
D B(0,r)
gD (z, ã).
r tũ ỵ t trữớ ủ tự t t ự ữủ
inf
DB(0,r)
g(z, ã) 0 r .
õ ữủ ự
r t tt ừ ởt t ổ
tr Cn
r t s tr ởt số t q t r
t tt ừ ởt t ổ tr Cn
rữợ t t ởt số s
ởt t D tr Cn ữủ ồ tt ồ Hol(, D)
tứ ỡ = z C | |z| < 1} D ởt ồ
t tự t ý {fj } Hol(, D) ự ởt fjk
ở tử tr t t ởt f Hol(, D)
ự ởt ý t
ú ỵ r t D tr Cn tt ồ t
tổ ừ õ tt t sỷ D õ ổ t
tổ D1 , D2 , ... {fj } Hol(, D) s #{j : fj () Dk } < ,
k N t fj ý t
ỵ ừ ừ s t ý t K D õ ỳ
ợ t tổ ừ D
D ởt t tr Cn ởt a D ữủ ồ tt
ữỡ ừ D tỗ t ởt U ừ a s t D U tt
D ữủ ồ tt ữỡ t ý ỳ ừ D
ữủ t ý ừ ởt t tt tt
ữỡ
ởt lD z, w D,
lD (z, | : Hol(, D) (0) = z, () = w}
ữủ ồ rt ừ D
˜ ❝õ❛ D✱ t❤➻ t❛ ✤➦t
◆➳✉ z, ω ❝ò♥❣ t❤✉ë❝ ♠ët t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ D
lD (z, w) := lD˜ (z, w),
♥➳✉ ❦❤æ♥❣ t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ lD (z, w) := 1.
✲ ▼ët ✤✐➸♠ a ∈ ∂D ✤÷ñ❝ ❣å✐ ❧➔ ♠ët t ✲ ✤✐➸♠ ❝õ❛ D ♥➳✉
lim l
z→a D
w→b
(z, w) = 1,
✈î✐ ❜➜t ❦ý b ∈ D✳
✣✐➸♠ a ∈ ∂D ❣å✐ ❧➔ ♠ët t ✲ ✤✐➸♠ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ D ♥➳✉ tç♥ t↕✐ ♠ët
❧➙♥ ❝➟♥ U ❝õ❛ a s❛♦ ❝❤♦ a ❧➔ ♠ët t ✲ ✤✐➸♠ ❝õ❛ D ∩ U ✳ ❉♦ ✈➟② ♠ët ✤✐➸♠
t❛✉t ✤à❛ ♣❤÷ì♥❣ ❧➔ ♠ët t ✲ ✤✐➸♠ ✤à❛ ♣❤÷ì♥❣✳
✣✐➲✉ ♥❣÷ñ❝ ❧↕✐ ❦❤æ♥❣ ✤ó♥❣✱ t❤➟t ✈➟②✱ t❛ ❝â t❤➸ ❝❤➾ r❛ ♠ët ✈➼ ❞ö ❝❤ù♥❣
tä tç♥ t↕✐ ❝→❝ t ✲ ✤✐➸♠ ♠➔ ❦❤æ♥❣ ❧➔ ❝→❝ ✤✐➸♠ t❛✉t ✤à❛ ♣❤÷ì♥❣✳
❈❤➥♥❣ ❤↕♥✱ ✈î✐ ε > 0 ✤õ ♥❤ä✱ ♠✐➲♥
Dε := {(z, w) ∈ C2 : |z + 1|2 + |w|2 · sin |w−1 | < 1, |z| < 2, |w| < ε}
❧➔ ♠✐➲♥ ❦❤æ♥❣ ❣✐↔ ❧ç✐ t↕✐ (0, 0) ❞♦ ✤â ♥â ❦❤æ♥❣ ❧➔ t❛✉t t↕✐ (0, 0✮ ♥❤÷♥❣
|ez | − 1 ❧➔ ♠ët ❤➔♠ ❝❤➢♥ ❝õ❛ Dε t↕✐ ❣è❝ ❞♦ ✈➟② (0, 0) ❧➔ ♠ët t ✲ ✤✐➸♠ ❝õ❛
Dε ✳
◆❤➟♥ ①➨t ✷✳✷✳✶✳ ❬✻❪ ✣✐➸♠ a ❧➔ ♠ët t ✲ ✤✐➸♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈î✐ ❜➜t ❦ý
❞➣② {fj } ⊂ Hol(∆, D) ♠➔ fj (0) → a t❤➻ fj ♣❤➙♥ ❦ý ❝♦♠♣❛❝t✳
✲ ❚❛ ♥â✐ r➡♥❣ ♠ët t➟♣ ♠ð D ⊂ Cn ❧➔ ❤②♣❡r❜♦❧✐❝ ♥➳✉ ❜➜t ❦ý t❤➔♥❤ ♣❤➛♥
❧✐➯♥ t❤æ♥❣ ♥➔♦ ❝õ❛ D ❝ô♥❣ ❧➔ ❤②♣❡r❜♦❧✐❝✳
❑➳t q✉↔ s❛✉ ❝❤ù♥❣ tä t➼♥❤ t❛✉t ❝õ❛ D s✉② r❛ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♥â✳
❚❛ ❝â ♠➺♥❤ ✤➲ s❛✉✿
✶✻
▼➺♥❤ ✤➲ ✷✳✷✳✶✳ ❬✻❪ ❇➜t ❦ý t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ ♠ët t➟♣ ♠ð ❦❤æ♥❣
❜à ❝❤➦♥ D tr♦♥❣ Cn ❧➔ ❤②♣❡r❜♦❧✐❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
lim l (z, w)
z→∞ D
ω→b
> 0,
✈î✐ ❜➜t ❦ý b ∈ D✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✤➣ ❜✐➳t D ❧➔ ❤②♣❡r❜♦❧✐❝ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❤å Hol(∆, D) ❧➔
❧✐➯♥ tö❝ ✤ç♥❣ ✤➲✉ ✭①❡♠ ❬✹❪ ✮✱ tù❝ ❧➔ ✈î✐ ❜➜t ❦ý b ∈ D ✈➔ ❜➜t ❦ý ❧➙♥ ❝➟♥ U
❝õ❛ b✱ tç♥ t↕✐ ♠ët ❧➙♥ ❝➟♥ V ❝õ❛ b ✈➔ s ∈ (0, 1) s❛♦ ❝❤♦ ♥➳✉ f (0) ∈ V ✱ t❤➻
f (s∆) ⊂ U ✈î✐ ❜➜t ❦ý f ∈ Hol(∆, D)✳ ❈❤➼♥❤ ✈➻ ✈➟②✱ ♥➳✉ D ❧➔ ❤②♣❡r❜♦❧✐❝
t❤➻ t❛ ❧✉æ♥ ❝â
lim inf lD (z, ω) > 0, ✈î✐ b ∈ D.
z→∞
ω→b
✣➸ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐➲✉ ♥❣÷ñ❝ ❧↕✐✱ t❛ ❣✐↔ sû D ❦❤æ♥❣ ❧➔ ❤②♣❡r❜♦❧✐❝✱ ❦❤✐ ✤â
t❛ ❝â t❤➸ t➻♠ ✤÷ñ❝ ♠ët ✤✐➸♠ b ∈ D ✈➔ ❞➣② {sj } ⊂ ∆, {fj } ⊂ Hol(∆, D)
♠➔ sj → 0 ✈➔ fj (0) → b, fj (sj ) → b.
◆➳✉ 0 < s := z→∞
lim inf lD (z, w) t❤➻ fj (0) → b✱ ❞♦ ✈➟② {fj } ❧➔ ❜à ❝❤➦♥ ✤à❛
w→b
♣❤÷ì♥❣ tr➯♥ r∆✱ t❤❡♦ ✤à♥❤ ❧þ ▼♦♥t❡❧ t❛ ❝â fj (sj ) → b✳ ✣✐➲✉ ♥➔② ❧➔ ♠➙✉
t❤✉➝♥✳ ❱➟② ♠➺♥❤ ✤➲ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❑➳t q✉↔ t✐➳♣ t❤❡♦ s➩ tr➻♥❤ ❜➔② ✈➲ t➼♥❤ t❛✉t ❝õ❛ ♠ët t➟♣ ♠ð tr♦♥❣ Cn ✱
t❛ ❝â ♠➺♥❤ ✤➲ s❛✉✿
▼➺♥❤ ✤➲ ✷✳✷✳✷✳ ❬✻❪ ❈❤♦ D ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ Cn ✳ ●✐↔ sû r➡♥❣ ∞ ❧➔
♠ët t✲✤✐➸♠ ♥➳✉ D ❦❤æ♥❣ ❜à ❝❤➦♥✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✐✮ D ❧➔ t❛✉t❀
✐✐✮ ❇➜t ❦ý ✤✐➸♠ ❜✐➯♥ ❤ú✉ ❤↕♥ ❝õ❛ D ✤➲✉ ❧➔ ✤✐➸♠ t❛✉t ✤à❛ ♣❤÷ì♥❣❀
✐✐✐✮ ❇➜t ❦ý ✤✐➸♠ ❜✐➯♥ ❤ú✉ ❤↕♥ ❝õ❛ D ✤➲✉ ❧➔ t ✲ ✤✐➸♠ ✤à❛ ♣❤÷ì♥❣❀
✶✼
✐✈✮ ❇➜t ❦ý ✤✐➸♠ ❜✐➯♥ ❤ú✉ ❤↕♥ ❝õ❛ D ❧➔ t ✲ ✤✐➸♠✳
❈❤ù♥❣ ♠✐♥❤✳
✰✭✐✈✮ ⇒ ✭✐✮✿
●✐↔ sû D ❧➔ t❛✉t ✈➔ {fj }∞
j=1 ∈ Hol(∆, D) ❧➔ ♠ët ❞➣② ❝→❝ ❤➔♠ ❝❤➾♥❤
❤➻♥❤✳ ▲➜② a ∈ ∂D\{∞}✱ ❣✐↔ sû lim fj (0) = a✱ ❞♦ D ❧➔ t❛✉t ♥➯♥ s✉② r❛
j→∞
❞➣②
{fj }∞
j=1
❧➔ ♣❤➙♥ ❦ý ❝♦♠♣❛❝t✳ ❱➟② t❤❡♦ ♥❤➟♥ ①➨t ✷✳✷✳✶✱ t❛ ❝â a ❧➔ ♠ët
t ✲ ✤✐➸♠✳
◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû ♠å✐ ✤✐➸♠ a ∈ ∂D\ {∞} ✤➲✉ ❧➔ t ✲ ✤✐➸♠✳ ▲➜② ♠ët ❤å
❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ {fj }∞
j=1 ∈ Hol(∆, D)✱ ♥➳✉ tç♥ t↕✐ λ ∈ ∆ s❛♦ ❝❤♦
lim fj (λ) = a ∈ ∂D,
j→∞
t❤➻ tç♥ t↕✐ θj ∈ Aut(∆) s❛♦ ❝❤♦ fj ◦ θj ∈ Hol(∆, D),
◆➳✉ ✤➦t gj = fj ◦ θj t❤➻ lim gj (0) = a✳ ❱➻ a ❧➔ t ✲ ✤✐➸♠ ♥➯♥ t❤❡♦ ✤à♥❤
j→∞
♥❣❤➽❛ ❤å ❤➔♠
{gj }∞
j=1
❧➔ ♣❤➙♥ ❦➻ ❝♦♠♣❛❝t✳ ❉♦ θj ❧➔ ✤➥♥❣ ❝➜✉ ❝❤➾♥❤ ❤➻♥❤
tr➯♥ ∆ ♥➯♥ t❛ s✉② r❛ {fj }∞
j=1 ❧➔ ♣❤➙♥ ❦➻ ❝♦♠♣❛❝t✳ ❱➟② D ❧➔ t❛✉t✳
✰✭✐✮ ⇒ ✭✐✐✮✿ ❍✐➸♥ ♥❤✐➯♥✱ ✈➻ ❝❤➾ ❝➛♥ ❧➜② ❧➙♥ ❝➟♥ U ❝õ❛ a ❧➔ t❛✉t✱ s✉② r❛ D ∩U
❧➔ t❛✉t ✈î✐ ♠å✐ a ∈ ∂D\ {∞}✳
✰✭✐✐✮ ⇒ ✭✐✐✐✮✿ ❍✐➸♥ ♥❤✐➯♥✳
❇➙② ❣✐í t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✭✐✐✐✮ ⇒ ✭✐✮✿
●✐↔ sû {fj }∞
j=1 ∈ Hol(∆, D) ❧➔ ♠ët ❞➣② ❜➜t ❦➻ ❦❤æ♥❣ ♣❤➙♥ ❦ý ❝♦♠♣❛❝t✳
❱➻ ∞ ❧➔ ♠ët t ✲ ✤✐➸♠ ❝õ❛ D ♥➯♥
1 = z→∞
lim lD (z, w) = z→∞
lim {inf{|λ| , ∃ϕ ∈ Hol(∆, D) : ϕ(0) = z,
w→b
w→b
ϕ(λ) = w}}.
◆➳✉ lim fj (λ) = ∞ ✈î✐ λ ∈ ∆ ♥➔♦ ✤â✱ t❛ ❧➜② θ ∈ Aut(∆) s❛♦ ❝❤♦
j→∞
θ(0) = λ t❤➻ fj ◦θ(0) = fj (λ) → ∞✳ ❉ ♦ ✤â tç♥ t↕✐ gj = fj ◦θ ∈ Hol(∆, D)
✶✽
rt lD (z, ) ợ z , b D
z
lim lD (z, w) < 1, ợ ồ b D õ ổ ởt t
wb
tr tt
{fj }
j=1 ữỡ tr ỵ t s r
{fj }
j=1 ở tử f Hol(, D) ự r
a = f (0) D t f () D
U ởt ừ a s a ởt t ừ D U t
t r tỗ t r (0, 1] s fj (r) U ợ j ừ ợ
t t f (r) D r0 (0, 1] ởt số ợ
t tọ tự
r0 := sup{r (0, 1] : f (r) D}.
sỷ r0 < 1, t t f (r ) D ợ t ý s |s| = r0 , f (s)
ởt t ữỡ tứ tt t t t s r tỗ t
r1 (r0 , 1] s f (r, ) D, t t õ
ự
s r ởt ừ ởt t
t ý ừ ởt t r D
tr Cn ởt t
ự a D ởt t a sỷ r
a ổ t õ tỗ t {fj }
j=1 Hol(, D)
{tj } fj (0) a, tj t0 fj (tj ) b D
D r ợ ộ V ừ b tỗ t ởt U
ừ t0 s fj (U ) V ợ j ừ ợ õ t tt r
V D, U r ợ r (0, 1) fj (U ) V