Tính Siêu Lồi, Tính Taut Và Tính K - Đầy Của Các Tập Mở Không Bị Chặn Trong C

✣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✉ r tổ ữợ sỹ ữợ t t ❝❤✉ ✤→♦ ❝õ❛ ❚❙✳ ❚r➛♥ ❍✉➺ ▼✐♥❤✳ ❚r♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❧✉➟♥ ✈➠♥ tæ✐ ✤➣ ❦➳ t❤ø❛ t❤➔♥❤ q✉↔ ❦❤♦❛ ỗ ợ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤✳ ❍å❝ ✈✐➯♥ ❱❛♥❤♥❛s♦♥❡ ❚❍❊PP❍❆❱❖◆● ✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ t ữợ sỹ ữợ t t sỹ ❝❤➾ ❜↔♦ ♥❣❤✐➯♠ ❦❤➢❝ ❝õ❛ ❚❙✳ ❚r➛♥ ❍✉➺ ▼✐♥❤✱ tæ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ✤➳♥ ❝ỉ ❣✐→♦✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❦➼♥❤ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ❝ơ♥❣ ♥❤÷ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ❦❤â❛ ❤å❝ ✷✵✶✺✲✷✵✶✼✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ✤❡♠ ❤➳t t➙♠ ❤✉②➳t ✈➔ sü ♥❤✐➺t t➻♥❤ ✤➸ ❣✐↔♥❣ ❞↕② ✈➔ tr❛♥❣ ❜à ❝❤♦ ❝❤ó♥❣ tỉ✐ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠✳ ❱➔ ❝✉è✐ ❝ị♥❣✱ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❝↔♠ ì♥ ỗ ổ ỗ ú ✤ï tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ❝ơ♥❣ ♥❤÷ tr♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✶✼ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ❱❛♥❤♥❛s♦♥❡ ❚❍❊PP❍❆❱❖◆● ✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✐ ▲í✐ ❝↔♠ ì♥ ✐✐ ▼ư❝ ❧ư❝ ✐✐✐ ▼ð ✤➛✉ ✶ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✸ ✶✳✶ ⑩♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỏ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ỏ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺ ❍➔♠ ✤❛ ỏ ữợ t ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✻ ●✐↔ ♠➯tr✐❝ ✈✐ ♣❤➙♥ ❘♦②❞❡♥ ✕ ❑♦❜❛②❛s❤✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✼ ●✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✽ ❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♠ët ♠✐➲♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✾ ▼✐➲♥ t❛✉t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ s ỗ t tt t k✲✤➛② ❝õ❛ ❝→❝ t➟♣ ♠ð tr♦♥❣ ✶✵ Cn ✐✐✐ ✷✳✶ s ỗ ởt t ổ 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 ✤➛✉ ◆❤÷ ú t t ỵ tt ổ ự ❤②♣❡r❜♦❧✐❝ r❛ ✤í✐ ✈➔♦ ❝✉è✐ ♥❤ú♥❣ ♥➠♠ ✻✵ ❝õ❛ t❤➳ trữợ s ỳ ổ tr ự t t s ỵ t❤✉②➳t ♥➔② ✤➣ trð t❤➔♥❤ ♠ët ♥❣➔♥❤ ♥❣❤✐➯♥ ❝ù✉ q✉❛♥ trå♥❣ ❝õ❛ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ❤②♣❡r❜♦❧✐❝✳ ◆❤✐➲✉ ❦➳t q✉↔ s➙✉ s➢❝ ✈➔ ✤➭♣ ✤➩ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ♥❤ú♥❣ t ợ tr t ợ ữ s r ỵ tt ữủ ự rë♥❣ r➣✐ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ♥❤÷ ❍➺ ỹ ự ỵ tt ố tr ①➜♣ ①➾ ❉✐♦♣❤❛♥t✐♥❡✳ ❚✉② ♥❤✐➯♥ ✤❛ sè ❝→❝ ❦➳t q✉↔ ❝❤➾ ✤↕t ✤÷đ❝ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ❝â t➼♥❤ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ❝õ❛ ❝→❝ ♠✐➲♥✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❤➻♥❤ ❤å❝ ❝õ❛ ❝→❝ ♠✐➲♥ ❦❤æ♥❣ ❜à ỹ t s ỗ 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♥❣ tr Cn ỗ 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 ởt t ổ ❝❤➦♥ tr♦♥❣ Cn ✱ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② ❝õ❛ ởt ổ tr Cn q sỹ tỗ t↕✐ ❝õ❛ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ♣❡❛❦ ✤à❛ ♣❤÷ì♥❣ t↕✐ ♠é✐ ✤✐➸♠ ❜✐➯♥ ✈➔ t↕✐ ✤✐➸♠ ∞ ❝õ❛ ♠✐➲♥ ♥➔② ỗ tớ t ố ỳ t tt ✤à❛ ♣❤÷ì♥❣ ✈➔ t➼♥❤ t❛✉t t♦➔♥ ❝ư❝ ❝õ❛ ♠ët ♠✐➲♥ tr♦♥❣ Cn ✳ P❤➛♥ ❝✉è✐ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ ❜➔② ù♥❣ ❞ö♥❣ ❝õ❛ ❝→❝ ❦➳t q✉↔ tr➯♥ ✤➸ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♠✐➲♥ ❍❛rt♦❣s ✈➔ ❝❤➾ r❛ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ởt rts tt s ỗ ❇↔♥ ❧✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❦❤✐➳♠ t rt ữủ ỳ ỵ õ ❣â♣ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❜↕♥ ✤å❝ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✷ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ sû ❞ư♥❣ ❝❤♦ ❝❤÷ì♥❣ s❛✉ ữ ỵ s ỏ ữợ ỏ ữợ ỏ ữợ t tr ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐✱ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♠ët ♠✐➲♥✱ ♠✐➲♥ t❛✉t✳ ❈→❝ ♥ë✐ ❞✉♥❣ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ✈✐➺t t❤❡♦ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✺❪✳ ✶✳✶ ⑩♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ●✐↔ sû X ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ Cn ✈➔ f : X → C ❧➔ ♠ët ❤➔♠ sè✳ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ♣❤ù❝ t↕✐ x0 ∈ X tỗ t t t : Cn → C s❛♦ ❝❤♦ |f (x0 + h) − f (x0 ) − λ (h)| = 0, |h|→0 |h| lim tr♦♥❣ ✤â h = (h1 , , hn ) ∈ C ✈➔ |h| = n n |hi | 1/2 ✳ i=1 ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ t↕✐ x0 ∈ X ♥➳✉ f ❦❤↔ ✈✐ ♣❤ù❝ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛ x0 ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ X ♥➳✉ f ❝❤➾♥❤ ❤➻♥❤ t↕✐ ♠å✐ ✤✐➸♠ t❤✉ë❝ X ✳ ✸ ▼ët →♥❤ ①↕ f : X Cm õ t t ữợ f = (f1 , , fm )✱ tr♦♥❣ ✤â fi = πi ◦ f : X → C, i = 1, , m ❧➔ ❝→❝ ❤➔♠ tå❛ ✤ë✳ ❑❤✐ ✤â f ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ X ♥➳✉ fi ❝❤➾♥❤ ❤➻♥❤ tr➯♥ X ✈ỵ✐ ♠å✐ i = 1, , m✳ ⑩♥❤ ①↕ f : X → f (X) ⊂ Cn ✤÷đ❝ ❣å✐ ❧➔ s♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ♥➳✉ f ❧➔ s♦♥❣ →♥❤✱ ❝❤➾♥❤ ❤➻♥❤ ✈➔ f −1 ❝ô♥❣ ❧➔ →♥❤ ①↕ ỵ s sû F ❧➔ ♠ët ❤å ♥➔♦ ✤â ❝→❝ →♥❤ ①↕ tø ❦❤æ♥❣ ❣✐❛♥ tæ ♣æ X ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ tæ ♣ỉ Y ✳ ❍å F ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ ỗ ts tứ x X tợ y ∈ Y ♥➳✉ ✈ỵ✐ ♠é✐ ❧➙♥ ❝➟♥ U ❝õ❛ ✤✐➸♠ y ✤➲✉ t➻♠ ✤÷đ❝ ♠ët ❧➙♥ ❝➟♥ V ❝õ❛ ✤✐➸♠ x ✈➔ ❧➙♥ ❝➟♥ W ❝õ❛ ✤✐➸♠ y s❛♦ ❝❤♦ ♥➳✉ f (x) ∈ ❲ t❤➻ f (V ) ⊂ U ✈ỵ✐ ♠å✐ f ∈ F ✳ ◆➳✉ F ❧➔ tử ỗ ợ x X ♠å✐ y ∈ Y t❤➻ F ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tử ỗ tứ X Y ỵ ỵ s ố ợ tử ỗ ✤➲✉✮ ●✐↔ sû F ❧➔ t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❝→❝ →♥❤ ①↕ ❧✐➯♥ tư❝ C(X, Y ) tø ❦❤ỉ♥❣ ❣✐❛♥ ❝❤➼♥❤ q✉✐ ❝♦♠♣❛❝t ✤à❛ ♣❤÷ì♥❣ X ✈➔♦ ❦❤ỉ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ Y ✈➔ C(X, Y ) ❝â tæ ♣æ ❝♦♠♣❛❝t ♠ð✳ ❑❤✐ ✤â F ❧➔ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ tr♦♥❣ C(X, Y ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ tọ F tử ỗ ✐✐✳✮ ❱ỵ✐ ♠é✐ x ∈ X ✱ t➟♣ ❤đ♣ Fx = {f (x)|f ∈ F } ❧➔ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ tr♦♥❣ Y ✳ ✹ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② ζ ❧➔ ♠ët ✤✐➸♠ t❤✉ë❝ K, θ ❧➔ ♠ët sè t❤ü❝ t❤✉ë❝ [0, 2π[ ✈➔ gζ,θ ❧➔ tü ✤➥♥❣ ❝➜✉ ❝õ❛ ∆ ✳ ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ inf1 |gζ,θ (λ) − |λ|= gζ,θ (0)| ❧➔ ❜à ❝❤➡♥ ✤➲✉ tr➯♥ K ✳ ❱➻ |gζ,θ (λ) − gζ,θ (0)| ≥ 21 |λ|.(1 − |ζ|2 ) t❛ ❝â t❤➸ ✤➦t rk = 41 min(1 − |ζ|2 ) ζ∈K ❑❤✐ ✤â t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✷✳✸✳✶✳ ❬✶❪ ❈❤♦ Ω ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ Cn ✱ ❣✐↔ sû r➡♥❣ Ω ❧➔ tt ữỡ t ộ tr tỗ t ỏ ữợ t t↕✐ ✤✐➸♠ ∞✳ ❑❤✐ ✤â Ω ❧➔ ♠ët ♠✐➲♥ t❛✉t✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣✳ ✲ ❚r÷í♥❣ ❤đ♣ ỗ t ởt tở ởt ❞➣② ❝♦♥ {fνk }k ❝õ❛ {fν }ν s❛♦ ❝❤♦ lim |fνk (ζ)| = ∞} k→∞ ✣➦t E = {λ ∈ ∆ : lim |fνk (ζ)| = ∞}✳ ❱➻ ✈ỵ✐ ❜➜t ❦ý ✤✐➸♠ ζ t❤✉ë❝ E t❛ k→∞ ❝â lim |fνk ◦ gζ,θ | = ∞ ✤➲✉ tr➯♥ ∆1/2 t❤❡♦ ❜ê k tỗ t ởt sè t❤ü❝ ❞÷ì♥❣ rζ s❛♦ ❝❤♦ lim |fνk (∆(ζ, rζ ))| = ∞ k→∞ ❑❤✐ ✤â E ❧➔ ♠ð tr♦♥❣ ∆✳ ❍ì♥ ♥ú❛ ♥➳✉ (ζn )n ❧➔ ♠ët ❞➣② ❝→❝ ✤✐➸♠ tr♦♥❣ E ❤ë✐ tư tỵ✐ ✤✐➸♠ ζ t❤✉ë❝ ∆✱ tø t➼♥❤ ❝♦♠♣❛❝t ❝õ❛ t➟♣ {ζn , n ≥ 0} {} tứ s r tỗ t số ữỡ r s ợ số ♥❣✉②➯♥ ❞÷ì♥❣ n✱ lim |fνk (ζ)| = ∞ ✤➲✉ tr➯♥ ∆(ζn , r) ✈➔ ❞♦ ❞â lim |fνk (ζ)| = ∞✱ k→∞ k→∞ t➟♣ E ❧➔ ✤â♥❣ tr♦♥❣ ∆✳ ❉♦ ✤â E = ∆✱ ✈➻ ✈➟② ✈ỵ✐ ♠å✐ ✤✐➸♠ ξ tở tỗ t ởt số tỹ ữỡ r s ❝❤♦ lim |fνk | = ∞ ✤➲✉ tr➯♥ ∆(ζ, rζ )✳ ✣➦❝ ❜✐➺t ✤➣② (|fνk |)k k→∞ ✷✺ ♣❤➙♥ ❦ý tỵ✐ ∞ ✤➲✉ tr➯♥ ❝→❝ t➟♣ ❝♦♥ ❝♦♠♣❛❝t ❝õ❛ ∆✱ ❞♦ ✈➟② ❞➣② {fνk }k ❧➔ ♣❤➙♥ ❦ý ❝♦♠♣❛❝t✳ ✲ rữớ ủ ợ tở ❞➣② {fν (ζ)}ν ❧➔ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ tr♦♥❣ Cn ✳ ❚❤❡♦ tr÷í♥❣ ❤đ♣ ✶✱ E ❧➔ ✤â♥❣ s✉② r❛ ❞➣② {fν }ν ❧➔ ❜à ❝❤➦♥ ✤à❛ ♣❤÷ì♥❣ tr♦♥❣ ∆✳ ❚❤❡♦ ỵ t õ ởt t tữỡ ố tỗ t ởt {fk }k {fν }ν ❤ë✐ tư tỵ✐ ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ f tứ ự tọ tỗ t↕✐ ♠ët ✤✐➸♠ ζ ∈ ∆ s❛♦ ❝❤♦ f (ζ) ∈ ∂D t❤➻ t➟♣ f (∆) ❜à ❝❤ù❛ tr♦♥❣ ∂∆✳ ❚❤➟t ✈➟②✱ ❧➜② E ❧➔ t➟♣ ✤â♥❣✳ E = {λ ∈ ∆ : f (λ) ∈ ∂∆} ❚❛ ❣✐↔ t❤✐➳t t➟♣ E ❦❤→❝ ré♥❣✳ ▲➜② λ ❧➔ ♠ët ✤✐➸♠ t❤✉ë❝ E ✳ ❱➻ Ω ❧➔ t❛✉t ✤à❛ ♣❤÷ì♥❣ t↕✐ f () tỗ t ởt V f (λ) s❛♦ ❝❤♦ Ω ∩ V ❧➔ t❛✉t✳ ❱➻ f ❧➔ ❣✐ỵ✐ ❤↕♥ ❧✐➯♥ tư❝ ✤➲✉ ❝õ❛ {fνk }k ♥➯♥ tỗ t U V ❝õ❛ f (λ) (V ⊂⊂ V ) s❛♦ ❝❤♦ t➟♣ fνk (U ) ❜à ❝❤ù❛ tr♦♥❣ Ω ∩ V ✈ỵ✐ k ✤õ ❧ỵ♥✳ ❉♦ Ω ∩ V ❧➔ t❛✉t ♥➯♥ f (U ) ⊂ ∂Ω ❞♦ ✤â E ❧➔ t➟♣ ♠ð✳ ❱➟② E = ∆✳ ❚❛ ✤➣ ❝❤➾ r❛ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✶ r➡♥❣ ❞➣② {fν }ν ❝❤ù❛ ♠ët ❞➣② ❝♦♥ ♣❤➙♥ ❦ý ❝♦♠♣❛❝t ✈➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✷ t❛ ❝❤➾ r❛ ❞➣② {fν }ν ❤♦➦❝ ♣❤➙♥ ❦ý ❝♦♠♣❛❝t ❤♦➦❝ ❤ë✐ tö ✤➲✉ tr➯♥ ❝→❝ t➟♣ ❝♦♥ ❝♦♠♣❛❝t ❝õ❛ ∆✳ ❱➟② ♠➺♥❤ ✤➲ ✷✳✸✳✶ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✻ ✷✳✹ ❚➼♥❤ k ✲ ✤➛② ❝õ❛ ♠ët t➟♣ ♠ð ❦❤æ♥❣ ❜à ❝❤➦♥ tr Cn ữỡ tỹ ố ợ rt t ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐ ❝❤♦ ♠ët t➟♣ ♠ð D tr♦♥❣ Cn ✳ ◆➳✉ z, w t❤✉ë❝ ❝ò♥❣ ♠ët t❤➔♥❤ ˜ ❝õ❛ D✱ t❤➻ t❛ ✤➦t kD (z, w) := k ˜ (z, w)✱ tr÷í♥❣ ❤đ♣ ❝á♥ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ D D ❧↕✐✱ t❛ ✤➦t kD (z, w) := ∞ ❘ã r➔♥❣ D ❧➔ k ✲ ✤➛② ✭tù❝ ❧➔ ✤➛② ✤è✐ ✈ỵ✐ ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❜➜t ❦ý t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ ♥â ❧➔ k ✲ ✤➛②✳ ❈❤♦ D ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ Cn ✳ ❚❛ ❣å✐ ✤✐➸♠ a ∈ ∂D ❧➔ ♠ët k ✲ ✤✐➸♠ ❝õ❛ D ♥➳✉ lim kD (z, b) = ∞; z→a tr♦♥❣ ✤â b ❧➔ ✤✐➸♠ tũ ỵ D a D ♠ët k ✲ ✤✐➸♠ ❝õ❛ D ♥➳✉ ❦❤æ♥❣ ❝â ❞➣② kD ✲❈❛✉❝❤② ♥➔♦ ❤ë✐ tư tỵ✐ a✳ ❘ã r➔♥❣ ✤✐➸♠ ♣❡❛❦ ❝❤➾♥❤ ❤➻♥❤ ✤à❛ ♣❤÷ì♥❣ ❧➔ ♠ët k ✲ ✤✐➸♠✱ ♠å✐ k ✲ ✤✐➸♠ ❧➔ k ✲ ✤✐➸♠✳ ❚❛ ❝â ♠➺♥❤ ✤➲ s❛✉ ✈➲ t➼♥❤ k ✲ ✤➛② ❝õ❛ ♠ët t➟♣ ♠ð tr♦♥❣ Cn ✳ ▼➺♥❤ ✤➲ ✷✳✹✳✶✳ ❬✻❪ ❈❤♦ D ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ Cn ✳ ●✐↔ sû ∞ ❧➔ ♠ët k ✕ ✤✐➸♠ ♥➳✉ D ❧➔ ❦❤æ♥❣ ❜à ❝❤➦♥✳ ❚❛ ❝â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✐✮ D ❧➔ k ✲ ✤➛②❀ ✐✐✮ ❇➜t ❦ý ✤✐➸♠ ❜✐➯♥ ❤ú✉ ❤↕♥ ❝õ❛ D ✤➲✉ ❝â ♠ët ❧➙♥ ❝➟♥ U s❛♦ ❝❤♦ D ∩ U ❧➔ k ✲ ✤➛②❀ ✐✐✐✮ ❇➜t ❦ý ✤✐➸♠ ❜✐➯♥ ❤ú✉ ❤↕♥ ❝õ❛ D ❧➔ k ✲ ✤✐➸♠ ✤à❛ ♣❤÷ì♥❣❀ ✐✈✮ ❇➜t ❦ý ✤✐➸♠ ❜✐➯♥ ❤ú✉ ❤↕♥ ❝õ❛ D ❧➔ ♠ët k ✲ ✤✐➸♠❀ ✷✼ ✈✮ ❇➜t ❦ý ✤✐➸♠ ❜✐➯♥ ❝õ❛ D ❧➔ ♠ët ❦✲ ✤✐➸♠ ❞à❛ ♣❤÷ì♥❣❀ ✈✐✮ ❇➜t ❦ý ✤✐➸♠ ❜✐➯♥ ❝õ❛ D ❧➔ ♠ët ❦✲ ✤✐➸♠✳ ❈❤ù♥❣ ♠✐♥❤✳ ❉➵ t❤➜② (i) ⇒ (ii) ⇒ (iii) ✈➔ (iv) ⇒ (v) ⇒ (iii) ụ õ (i) (vi) ỵ ứ ❣✐↔ t❤✐➳t ∞ ❧➔ k ✲✤✐➸♠ s✉② r❛ lim inf kD (z, w) > 0, z ∈ D w→∞ ❉♦ ✈➟②✱ D ❧➔ ❤②♣❡r❜♦❧✐❝ ✈➔ ✈➻ t❤➳ ❜➜t ❦ý ❞➣② kD ✲ ❝❛✉❝❤② ✤➲✉ ❝â ♥❤✐➲✉ ♥❤➜t ♠ët ✤✐➸♠ tö tr♦♥❣ D✳ ❱➻ ❜➜t ❦ý ✤✐➸♠ ❜✐➯♥ ♥➔♦ ✤➲✉ ❧➔ k ✲ ✤✐➸♠ ♥➯♥ ♠é✐ ❞➣② ♥❤÷ ✈➟② ❦❤ỉ♥❣ ❝â ❝→❝ ✤✐➸♠ tö tr➯♥ ∂D✱ ❞♦ ✈➟② (iv) ⇒ (i) ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ (iii) ⇒ (iv) ●✐↔ sû ữủ t tỗ t ởt kD {zj } ⊂ D, zj → a ∈ ∂D\{∞} ❚❛ s ự tọ tỗ t số r > s ❝❤♦ 3ε := inf{kD (zj , w) : j ∈ N, w ∈ D\B(0, r)} > (∗) ▼➦t ❦❤→❝✱ ợ t ý l tỗ t jl N wl ∈ D ♠➔ ||wl || > l ✈➔ kD (zjl , wl ) < l ❑❤✐ ✤â kD (wl , wm ) < 1 + + kD (zjl , zjm ), l, m ∈ N l m ◆➳✉ ❞➣② {jl } ❝❤➾ ❣➛♠ ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ ♣❤➛♥ tû ❝õ❛ N✱ t❤➻ ❜➡♥❣ ❝→❝❤ ①➨t ♠ët ❞➣② ❝♦♥ ♥➳✉ ❝➛♥✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ jl = jm ✱ ✈ỵ✐ ❜➜t ❦ý l ✈➔ m✱ ❞♦ ✤â {wl } ❧➔ ♠ët ❞➣② kD ✲ ❈❛✉❝❤②✳ ▼➦t ❦❤→❝✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ jl → ∞✳ ❑❤✐ ✤â {wl } ❧↕✐ ❧➔ ♠ët ❞➣② kD ✲ ❈❛✉❝❤②✱ ❞♦ ✈➟② {zjl } ❝ò♥❣ ❧➔ ❞➣② kD ổ ỵ ❧➔ ♠ët ❦✬✲✤✐➸♠✳ ✷✽ ❱➟② (∗) ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇➙② ❣✐í✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ kD (zl , zm ) < ε✱ ✈ỵ✐ ❜➜t ❦ý l, m✱ t❛ t ởt ữớ ợ C l,m : [0, 1] → D ♥è✐ zl ✈➔ zm s❛♦ ❝❤♦ kD (zl , zm ) ≤ kD (γl,m (t); γl,m (t))dt < 2kD (zl , zm ) < 2ε ˜ D ˜ := D ∩ B(0, r) ❱➻ kD (zl ; γl,m (t)) < 2ε ♥➯♥, γl,m ([0, 1]) ⊂ D, ✈➔ tanh−1 lD (γl,m (t), w) ≥ kD (γl,m (t)), w) > ε, t ∈ [0, 1], w ∈ D\B(0, r) ❇➜t ✤➥♥❣ t❤ù❝ FD˜ (z, X) inf ω∈D\B(0,r) ˜ lD (z, ω) ≤ FD (z, X), z ∈ D s✉② r❛ kD˜ (zl , zm ) < 2(coth ε)kD (zl , zm )✳ ❱➻ ✈➟② {zj } ❧➔ ♠ët ❞➣② kD˜ ✲ ❈❛✉❝❤②✳ ❈❤å♥ r > s❛♦ ❝❤♦ a ❧➔ ♠ët ˜ := D ˆ ∩ B(a, r )✳ k ✲✤✐➸♠ ❝õ❛ D ˆ ❧➔ ❜à ❝❤➦♥ ♥➯♥ t❛ ❝â✿ ❱➻ D inf{FD˜ (z, w) : j ˜ B(a, r )} > 1, w ∈ D\ ữ trữợ t t❤➜② ❞➣② kDˆ ✲ ❈❛✉❝❤② {zj } ❝ô♥❣ ❧➔ ♠ët kD ổ ỵ (iii) ⇒ (iv) ◆➳✉ ∞ ❧➔ ♠ët ✤✐➸♠ ♣❡❛❦ ✤à❛ ♣❤÷ì♥❣ t❛ ❝â ♠➺♥❤ ✤➲ s❛✉ ▼➺♥❤ ✤➲ ✷✳✹✳✷✳ ❬✻❪ ởt ỏ ữợ ✈➔ ❧➔ ♠ët k ✕ ✤✐➸♠ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♠ët t➟♣ ♠ð ❦❤æ♥❣ ❜à ❝❤➦♥ D tr♦♥❣ Cn ✱ t❤➻ ∞ ❧➔ ♠ët k ✕ ✤✐➸♠✳ ✷✾ ˜ := D\B(0, r)✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② r > s❛♦ ❝❤♦ ∞ ❧➔ ♠ët k ✲ ✤✐➸♠ ❝õ❛ D ❑❤✐ ✤â t❛ ❝â FD˜ (z, X) inf lD (z, w) ˜ FD (z, X), z ∈ D w∈D∩B(0,r) ▼➦t ❦❤→❝✱ ♥➳✉ r2 > r1 > r t❤➻ inf{tanh−1 lD (z, w) :z ∈ D\B(0, r1 ), w ∈ D ∩ B(0, r)⑥ inf{kD (z, w) :z ∈ D ∩ ∂ B(0, r1 ), w ∈ D ∩ ∂ B(0, r)⑥ min{kD∪B(0,r2 ) (z, w) :z ∈ ∂ B(0, r1 ), w ∈ ∂ B(0, r)⑥ ❂ ✿ε ❱➻ ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ♣❡❛❦ ỏ ữợ ữỡ t❛ ❝â ∞ ❧➔ ♠ët ✤✐➸♠ ♣❡❛❦ ✤❛ ✤✐➲✉ ❤á❛ ữợ D B(0, r2 ) ✷✳✶✳✶✱ ❜➜t ❦ý t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ t➟♣ ♥➔② ❧➔ ❤②♣❡r❜♦❧✐❝ ✈➔ ❞♦ ✤â ε > ❱➻ ✈➟② FD˜ (z, X) (coth ε)FD (z, X), z ∈ D\B(0, r1 ) sỷ r tỗ t ởt kD ✲ ❈❛✉❝❤② ❤ë✐ tö ✤➳♥ ∞✳ ▲➦♣ ❧↕✐ ❧➟♣ ❧✉➟♥ ❝õ❛ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ♠➺♥❤ ✤➲ ✷✳✸✳✶✱ t❛ ❝â ♠å✐ ❞➣② ❝♦♥ ❝õ❛ ❞➣② ♥➔② ˜ ✤➲✉ ❧➔ ❞➣② k ổ ỵ t ❝â ✤✐➲✉ ♣❤↔✐ t❤✉ë❝ D D ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✺ ❈→❝ ♠✐➲♥ ❍❛rt♦❣s✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ s➩ tr➻♥❤ ❜➔② ❝→❝ ù♥❣ ❞ö♥❣ ❝õ❛ ❝→❝ ❦➳t q✉↔ tr➯♥ ❝❤♦ ❝→❝ ♠✐➲♥ rsts rữợ t t õ s ♥❣❤➽❛ ✷✳✺✳✶✳ ❬✻❪ ❈❤♦ G ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ Cn ởt D G ì Cm ữủ ❧➔ ♠ët ♠✐➲♥ ❍❛rt♦❣s tr➯♥ G ✈ỵ✐ t❤ỵ ❝➙♥ ❜➡♥❣ m ❝❤✐➲✉ ✸✵ ♥➳✉ D = {(z, w) ∈ G × Cm : h(z, w) < 1}, tr♦♥❣ ✤â h : G × Cm → [0, ∞) ❧➔ ♠ët ❤➔♠ ♥û❛ ❧✐➯♥ tư❝ tr➯♥ ✈ỵ✐ h(z, λ) = |λ|.h(z, w), z ∈ G, λ ∈ ∆, w ∈ Cm ❘ã r➔♥❣ D ❧➔ ❜à ❝❤➦♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ G ❧➔ ❜à ❝❤➦♥ ✈➔ inf h > 0✱ tr♦♥❣ G×S ✤â S ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à tr♦♥❣ C ỗ G m ỗ log h ỏ ữợ 2j log |1 22j1 z1 | , ◆➳✉ θ(z1 ) : = j=1 ϕ(z1 ) : = max{|z1 |, + θ(z1 )}, t❤➻ D := {(z1 , z2 ) ∈ ∆ × C : |z2 |eϕ(z1 ) < 1} ❧➔ ♠ët ♠✐➲♥ ❍❛rt♦❣s ỗ tr ợ tợ ởt ●å✐ E0 ❧➔ t➟♣ ❝→❝ ✤✐➸♠ ❜✐➯♥ (a1 , a2 ) ❝õ❛ D ✈ỵ✐ a1 = ✣➦t E1 := {(0, a2 ) ∈ C2 | |a2 | = 1}, E2 := {(0, a2 ) ∈ C2 e−1 < |a2 | < 1}, E3 := {(0, a2 ) ∈ C2 : |a2 | = e−1 } ◆❤➟♥ ①➨t r➡♥❣ ∂D = Ej ✳ j=0 ❚❛ ❝â ✈➼ ❞ö s❛✉✿ ❱➼ ❞ö ✷✳✺✳✶✳ ✐✮ ❇➜t ❦ý ✤✐➸♠ ♥➔♦ ❝õ❛ E0 ∪ E1 ✤➲✉ ❧➔ ✤✐➸♠ ❝❤➢♥ ✈➔ ❞♦ ✤â✱ t❤❡♦ ♠➺♥❤ ✤➲ ✷✳✷✳✸✱ ♥â ❧➔ ♠ët t✲ ✤✐➸♠✳ ✐✐✮ ❇➜t ❦ý ✤✐➸♠ ♥➔♦ ❝õ❛ E2 ✤➲✉ ❧➔ ✤✐➸♠ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ t✲ ✤✐➸♠ ✤à❛ ♣❤÷ì♥❣✱ ❞♦ ✤â t❤❡♦ ♠➺♥❤ ✤➲ ✷✳✷✳✸✱ ♠ët t✲✤✐➸♠ ✤à❛ ♣❤÷ì♥❣ ❦❤ỉ♥❣ ❧➔ ♠ët ✤✐➸♠ ❝❤➢♥ t♦➔♥ ❝ö❝✳ ✣➦❝ ❜✐➺t✱ t❤❡♦ ♠➺♥❤ ✤➲ ✷✳✷✳✷✱ ❦❤æ♥❣ ❧➔ ♠ët ♠✐➲♥ t❛✉t✳ ✸✶ ✐✐✐✮ ❇➜t ❦ý ✤✐➸♠ ♥➔♦ ❝õ❛ E3 ✤➲✉ ❦❤æ♥❣ ❧➔ t✲ ✤✐➸♠ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❞♦ ✤â ❦❤ỉ♥❣ ❧➔ ✤✐➸♠ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❧➔ t➟♣ ❝→❝ t✲ ✤✐➸♠ ữỡ t ợ ởt ❧➙♥ ❝➟♥ ❝õ❛ ❜➜t ❦ý ❝→❝ ✤✐➸♠ ♥➔② ∂D\E3 ✱ ❝ơ♥❣ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ✤✐➸♠ t❛✉t ✤à❛ ♣❤÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ✐✳✮ ❚❛ ❝â ϕ ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ Cn ✱ ❞♦ ✈➟② ❝→❝ ✤✐➸♠ ❝õ❛ E0 ❧➔ ❝→❝ ✤✐➸♠ ❝❤➢♥✳ ❍ì♥ ♥ú❛✱ > log |z2 | → ❦❤✐ D (z1 , z2 ) → E1 ✳ ❙✉② r❛ ❝→❝ ✤✐➸♠ ❝õ❛ E1 ❝ô♥❣ ❧➔ ✤✐➸♠ ❝❤➢♥✳ ✐✐✳✮ ✣➦t a := (0, a2 ) ∈ E2 ✳ ❚❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ a2 > 0✳ ❈❤å♥ c ∈ (a2 , 1) ✈➔ ♥❤➟♥ ①➨t r➡♥❣ ❝→❝ ✤➽❛ fj (t) := (21−2j , ct) t❤✉ë❝ ✈➔♦ Hol(∆, D) ♥➳✉ 21−2j ≤ − log c✳ ❱➻ fj (t) → f (t) := (0, ct) ✤➲✉ ✤à❛ ♣❤÷ì♥❣✱ f (0) ∈ D ✈➔ f ( ac2 ) = a ♥➯♥ a ❧➔ ♠ët t ✲ ✤✐➸♠✳ ✣➸ t❤➜② r➡♥❣ a ❧➔ ♠ët ✤✐➸♠ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣✱ ♥❤➟♥ ①➨t ✤➛✉ t✐➯♥ ❧➔ ♥➳✉ F ❧➔ ❤ñ♣ ❝õ❛ ❝→❝ ✤➽❛ rí✐ ♥❤❛✉ ∆(21−2j , 2−2j ), j = 1, 2, t❤➻ lim θ(z1 ) = F z1 →0 ❚❤➟t ✈➟②✱ log |1−22j−1 z1 | ≥ log ✈ỵ✐ ❜➜t ❦ý z1 ∈ / F ✈➔ ❜➜t ❦ý j ≥ 1✱ ❞♦ ✤â ∞ k θ(z1 ) ≥ −j 2j−1 log |1 − j=1 ✣➦t F 2−j , k ≥ z1 | − log j=k+1 z1 → 0, k → ∞✱ ❞♦ θ ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ♥➯♥ t❛ ❝â ✤➥♥❣ ¯ t❤ù❝ ♠♦♥❣ ố ú ỵ r C t ✤➽❛ ✤â♥❣ r∆ ✈ỵ✐ e−1 < r < ✈➔ z ∈ D ∩ (∆ × C), t❤➻ θ(z1 ) < log(re) < tr tỗ t ε > s❛♦ ❝❤♦ ♥➳✉ z1 t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ✈➔ |z1 | < ε✱ t❤➻ z1 ∈ F ✳ ❉♦ ✈➟② D ∩ (ε∆ × C) ⊂ F × C ❚❛ ✤à♥❤ ♥❣❤➽❛ p(z) := −j −1 tr➯♥ ∆(21−2j , 2−2j ) × C, ✤➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ p ❧➔ ♠ët ❤➔♠ ❝❤➢♥ tr➯♥ D ( ì C) t t ý a ợ a1 = ✈➔ r < |a2 | < 1✳ ❱➻ r (e1 , 1) ữủ tũ ỵ t ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ✐✐✐✳✮ ❚ø (i), (ii) ✈➔ ♠➺♥❤ ✤➲ ✷✳✷✳✷✱ t➟♣ ❝→❝ ✤✐➸♠ ❜✐➯♥ ♠➔ ❦❤æ♥❣ ❧➔ t✲ ✤✐➸♠ ❧➔ ❦❤→❝ ré♥❣ ✈➔ ♥â ❜à ❝❤ù❛ tr♦♥❣ E3 ✳ ❱➻ E3 ❧➔ ♠ët q✉ÿ ✤↕♦ ù♥❣ ✈ỵ✐ ♣❤➨♣ q✉❛② ♥➯♥ t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❚r♦♥❣ ❬✸❪ t❛ ❝â ❦➳t q✉↔ s❛✉✿ ◆➳✉ D ❧➔ ❤②♣❡r❜♦❧✐❝✱ t❤✐ G ❧➔ ❤②♣❡r❜♦❧✐❝ ✈➔ inf h > ✈ỵ✐ ❜➜t ❦ý K×S K ⊂⊂ G (∗) ❚❤➟t ✈➟②✱ ✈➻ G × {0} ⊂ D ♥➯♥ G ❧➔ ❤②♣❡r❜♦❧✐❝✳ ●✐↔ sỷ r inf h = KìS ợ K G t tỗ t ởt {aj } D ✈➔ ♠ët ✤✐➸♠ a := (a , a ) ∈ G × Cn , a = 0✱ s❛♦ ❝❤♦ aj → a ✈➔ h(aj ) → ❦❤✐ j → ∞ ❉♦ ❝→❝ ✤➽❛ fj (t) := (aj , taj (h(aj )))−1 t❤✉ë❝ ✈➔♦ Hol(∆, D), fj (0) → a ˜ := (a , 0) ∈ D ✈➔ fj (t) → ∞ ✈ỵ✐ ❜➜t ❦ý t ∈ ∆\④ 0}✳ ❱➻ ✈➟② z→∞ lim inf lD (z, w) = ♥➯♥ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥✳ w→˜ a ✸✸ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝→❝ ♠✐➲♥ ❍❛rt♦❣s tr➯♥ ∆ ✈ỵ✐ t❤ỵ ❝➙♥ ❜➡♥❣ ♠ët ❝❤✐➲✉✱ ❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❝õ❛ (∗) ✤÷đ❝ ❝❤➾ r❛ tr♦♥❣ ❬✽❪✳ ❇➡♥❣ ❝→❝ ù♥❣ ❞ö♥❣ ❝õ❛ ❝→❝ ❦➳t q trữợ t ự ữủ s❛✉✿ ▼➺♥❤ ✤➲ ✷✳✺✳✶✳ ❬✻❪ ◆➳✉ (∗) ✤↕t ✤÷đ❝✱ t❤➻ D ❧➔ ❤②♣❡r❜♦❧✐❝✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐✱ t❤➻ tø ♠➺♥❤ ✤➲ ✷✳✷✳✶✱ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ❞➣② {tj } ⊂ ∆ ✈➔ {fj } ⊂ Hol(∆, D)✱ ✈ỵ✐ tj → 0, fj (tj ) → ∞ ✈➔ fj (0) → a ∈ D ▲➜② a = (a , a ) ✈ỵ✐ a ∈ G ✈➔ a ∈ Cn ✳ ❉♦ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ G t❛ ❝â fj (tj ) → a ✳ ❱➻ fj (tj ) ⊂ K ✈ỵ✐ K ⊂⊂ D ♥➯♥ fj (tj ) → ∞ ✈➔ t❛ ❝â > h(f (tj )) = ||f (tj )||h(f (tj ), f (tj ) ) ||f (tj )|| ||f (tj )|| inf h → ∞, K×S ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ư s❛✉ ❝❤➾ r❛ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ ♠➺♥❤ ✤➲ ✷✳✷✳✶ ❝â t❤➸ ❞÷ì♥❣ ♥❤÷♥❣ ♥❤ä ❤ì♥ ỗ t ởt ❤②♣❡r❜♦❧✐❝ tr♦♥❣ C3 ✈ỵ✐ ∞ ❦❤ỉ♥❣ ❧➔ ♠ët t ✲ ✤✐➸♠✳ ❚❤➟t ✈➟②✱ t❤❡♦ ✈➼ ❞ö tr➯♥✱ ❤➔♠    max{|z1 |, − |ez2 |}, |z2 | < e−1 Φ(z1 , z2 ) :=    z1 |, |z2 | e−1 ❧➔ ❞÷ì♥❣ ✈➔ ❧✐➯♥ tư❝ tr➯♥ ♠✐➲♥ D✳ ❉♦ ✈➟②✱ t❤❡♦ ♠➺♥❤ ✤➲ ✷✳✺✳✶ ˜ := {(z, w) ∈ D× : C|w|Φ(z) < 1} D ❧➔ ♠ët ♠✐➲♥ ❍❛rt♦❣s ❤②♣❡r❜♦❧✐❝✳ ✸✹ ❇➙② ❣✐í ❝è ✤à♥❤ c ∈ (e−1 , 1)✳ ●✐↔ sû r➡♥❣ ✤➽❛ fj (t) := (21−2j , ct) t❤✉ë❝ ✈➔♦ Hol(∆, D) ✈ỵ✐ ❜➜t ❦ý j ✤õ ❧ỵ♥✳ ◆➳✉ gj (t) := (cet)j ✱ t❤➻ |gj (t)| < ≤ min{22j−1 , (1 − ce|t|)−1 } ❦❤✐ ce|t| < 1, ✈➔ |gj (t)| < 22j−1 ✈ỵ✐ t ∈ ∆ ✈➔ j ≥ 2✱ tù❝ ❧➔ |gj (t)|Φ(fj (t)) < tr➯♥ ∆ ˜ ✈ỵ✐ hj (0) → ∈ D ˜ ✳ ❍ì♥ ♥ú❛✱ ✈ỵ✐ ❉♦ ✈➟②✱ hj := (fj , gj ) ∈ Hol(∆, D) ❜➜t ❦ý t ♠➔ ce|t| > 1✱ t❛ ❝â hj (t) → ∞✳ ❙✉② r❛ −1 lim inf l (u, v) ≤ (ce) ˜ D u→∞ v→0 ❈❤♦ c → t❛ ❝â lim inf lD˜ (u, v) ≤ (e)−1 u→∞ v→0 ❙✉② r❛ ∞ ❦❤æ♥❣ t tr D ú ỵ ợ ❝✉è✐ ❝ò♥❣ ❜➡♥❣ e−1 ✳ ˜ ⊃ ((z j , wj ))j → ∞✱ t❤➻ ❞➵ t❤➜② r➡♥❣ z j → ✈➔ ❚❤➟t ✈➟②✱ ♥➳✉ D lim inf |z2j | ≥ e−1 ✳ ❱➻ D ⊂ ∆2 ♥➯♥ ♥➳✉ j→∞ ((ˆ z j , wˆ j ))j → 0, t❤➻ lim inf lD˜ ((z j , wj ), (ˆ z j , wˆ j ))j ≥ lim inf lD (z j , zˆj ) j→∞ j→∞ ≥ lim inf l∆2 (z j , zˆj ) j→∞ = lim inf |z2j | ≥ e−1 j→∞ ▼➺♥❤ ✤➲ t✐➳♣ t❤❡♦ ❝❤➾ r❛ ♠ët ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ởt rts tt s ỗ õ ✤➲ s❛✉✿ ✸✺ ▼➺♥❤ ✤➲ ✷✳✺✳✷✳ ❬✻❪ ▼ët ♠✐➲♥ ❍❛rt♦❣s D tt s ỗ G tt s ỗ log h ởt ỏ ữợ tử log h1 (−∞) = G × {0} ❈❤ù♥❣ ♠✐♥❤✳ ⇐) ●✐↔ sû h ❝â ❝→❝ t➼♥❤ ❝❤➜t ♥❤÷ t✐➯♥✳ ●✐↔ sû G s ỗ tự õ ởt t ỏ ữợ ổ (z, ˜ w) := ϕ(z) ❧➔ ♠ð rë♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ϕ tỵ✐ D❀ ❞➵ t❤➜② r➡♥❣ max{ϕ, ˜ log h} ởt rở ỏ ữợ D s r D s ỗ sỷ G tt tỗ t ởt {fi } ∈ Hol(∆, D) s❛♦ ❝❤♦ ❤â ❦❤æ♥❣ ♣❤➙♥ ❦ý ❝♦♠♣❛❝t✳ ❑❤✐ ✤â t❛ ❣✐↔ t❤✐➳t r➡♥❣ fj → f ∈ Hol(∆, D)✳ ✣➦❝ ❜✐➺t✱ ✈ỵ✐ ❜➜t ❦ý r ∈ (0, 1), ∞ f j (r∆) := Kr ⊂⊂ D j=1 ❱➻ inf h > ♥➯♥ {fj }j ✱ ❧➔ tr r Kr ìS ỵ ▼♦♥t❡❧ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ fj → f ∈ Hol(∆, Cn )✳ ✣➦t f := (f , f )✳ ❉♦ f (∆) ⊂ G ✈➔ ❞♦ h ❧✐➯♥ tư❝✱ t❛ ❝â h(f (t)) ≤ ✈ỵ✐ ❜➜t ❦ý t ỵ ỹ t õ ❤♦➦❝ h ◦ f < ❤♦➦❝ h ◦ f ≡ 1✱ tù❝ ❧➔ ❤♦➦❝ f (∆) ⊂ D ❤♦➦❝ f (∆) ∈ ∂D✳ ⇒) ✣➸ ❝❤ù♥❣ ♠✐♥❤ t❛ ❝❤ó ỵ r D ỗ r t log h ởt ỏ ữợ ✈➔ h > tr➯♥ G × S ✳ ❍ì♥ ♥ú❛✱ t❛ t❤➜② tø t➼♥❤ t❛✉t ✭t➼♥❤ ❤②♣❡r❜♦❧✐❝✮ ❝õ❛ D ❦➨♦ t❤❡♦ t➼♥❤ t❛✉t ✭t➼♥❤ ❤②♣❡r❜♦❧✐❝✮ ❝õ❛ G✳ ❇➙② ❣✐í t❛ ❣✐↔ sû r➡♥❣ h ❧➔ ❦❤ỉ♥❣ ❧✐➯♥ tư❝ t↕✐ ✤✐➸♠ a ∈ D✳ ❑❤✐ ✤â t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ♠ët ❞➣② {aj } ⊂ D ✈➔ r > s❛♦ ❝❤♦ aj → a ❦❤✐ ✸✻ j → ∞ ✈➔ (h(a))−1 < r ≤ (h(aj ))−1 ✈ỵ✐ ♠å✐ j ú ỵ r fj (t) := (aj , raj t) t❤✉ë❝ Hol(∆, D)✳ ▼➦t ❦❤→❝✱ ✈ỵ✐ ✤➽❛ ❣✐ỵ✐ ❤↕♥ fj (t) := (aj , raj t) t❛ ❝â f (0) ∈ D ✈➔ f ((h(a)r)−1 ) ∈ ∂D✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ t❛✉t ✭t➼♥❤ s ỗ D t õ ự s ỗ t t❛✉t ✈➔ t➼♥❤ ❦ ✲ ✤➛② ❝õ❛ ❝→❝ t➟♣ ♠ð ❦❤ỉ♥❣ ❜à ❝❤➦♥ tr♦♥❣ Cn ✧ ✤➣ tr➻♥❤ ❜➔② ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥ ♥❤÷✿ →♥❤ ①↕ ❝❤➾♥❤ ỵ s ỏ ữợ ỏ ữợ ỏ ữợ ❛♥t✐♣❡❛❦✱ ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❦♦❜❛②❛s❤✐✱ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♠ët ♠✐➲♥✱✳ ✳ ✳ ✷✳ ❚r➻♥❤ ❜➔② ✈➲ ❦❤→✐ ♥✐➺♠ ✈➔ ❝❤ù♥❣ ỵ t s ỗ t ✤➛②✱ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ✈➔ t➼♥❤ t❛✉t ❝õ❛ ♠ët t➟♣ ♠ð ❦❤æ♥❣ ❜à ❝❤➦♥ tr♦♥❣ Cn ✳ ✸✳ ◆❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② ❝õ❛ ♠ët ♠✐➲♥ tr♦♥❣ Cn q✉❛ sü tỗ t ữỡ t↕✐ ♠é✐ ✤✐➸♠ ❜✐➯♥ ✈➔ t↕✐ ✤✐➸♠ ∞ ❝õ❛ ♠✐➲♥ ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ t➼♥❤ t❛✉t ✤à❛ ♣❤÷ì♥❣ ✈➔ t➼♥❤ t❛✉t t♦➔♥ ❝ö❝ ❝õ❛ ♠✐➲♥ tr♦♥❣ Cn ✳ ✹✳ ❚r➻♥❤ ❜➔② ✈➲ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♠✐➲♥ ❍❛rst♦❣s ✈➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ♠✐➲♥ ❍❛rt♦❣s ❧➔ t❛✉t s ỗ t ssr ✭✶✾✾✾✮✱ ✧❚❛✉t♥❡ss ❛♥❞ ❝♦♠♣❧❡t❡ ❤②♣❡r❜♦❧✐❝✐t② ♦❢ ❞♦♠❛✐♥s ✐♥ Cn ✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✱ ✶✷✼✱ ♣♣✳ ✶✵✺✲✶✶✻✳ ❬✷❪ ❏❛r♥✐❝❦✐ ▼✳ ❛♥❞ ❋❧✉❣ P✳ P✳ ✭✶✾✾✸✮✱ ✧■♥✈❛r✐❛♥t ❉✐st❛♥❝❡s ❛♥❞ ▼❡tr✐❝s ✐♥ ❈♦♠♣❧❡① ❆♥❛❧②s✐s✧✱ ❉❡ ●r✉②t❡r ❊①♣✳ ▼❛t❤✳✱ ✈♦❧✳ ✾✱ ❲❛❧t❡r ❞❡ ●r✉②t❡r✱ ❇❡r❧✐♥✳ ❬✸❪ ❏❛r♥✐❝❦✐ ▼✳ ❛♥❞ ❋❧✉❣ P✳ P✳ ✭✷✵✵✺✮✱ ✧■♥✈❛r✐❛♥t ❉✐st❛♥❝❡s ❛♥❞ ♠❡tr✐❝s ✐♥ ❈♦♠♣❧❡① ❆♥❛❧②s✐s✲ r❡✈✐s✐t❡❞✧✱ ❉✐ss❡rt❛t✐♦♥❡s ▼❛t❤✳ ✹✸✵✱ ✶✲✶✾✷✳ ❬✹❪ ❑❡r③♠❛♥ ◆✳ ❛♥❞ ❘♦s❛② ❏✳ P✳ ✭✶✾✽✶✮✱ ✧❋♦♥❝t✐♦♥s ♣❧✉r✐s♦✉s❤❛r♠♦♥✐q✉❡s ❞✬❡①❤❛✉st✐♦♥ ❜♦r♥❡✬❡s ❡t ❞♦♠❛✐♥❡s t❛✉t✧✱ ▼❛t❤✳ ❆♥♥ ✷✺✼✱ ✶✼✶✲✶✽✹✳ ❬✺❪ ❑♦❜❛②❛s❤✐ ❙✳ ✭✶✾✾✽✮✱ ✧❍②♣❡r❜♦❧✐❝ ❝♦♠♣❧❡① s♣❛❝❡s✧✱ ●r✉♥❞❧❡❤❡r❡♥ ▼❛t❤✳ ❲✐ss✳✱ ✈♦❧✳ ✸✶✽✱ ❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣✱ ❇❡r❧✐♥✳ ❬✻❪ ◆✐❦♦❧♦✈ ◆✳ ❛♥❞ ❋❧✉❣ P✳ P✳ ✭✷✵✵✺✮✱ ✧▲♦❝❛❧ ✈s✳ ❣❧♦❜❛❧ ❤②♣❡r❝♦♥✈❡①✐t②✱ t❛✉t♥❡ss ♦r ❦ ✲ ❝♦♠♣❧❡t❡♥❡ss ❢♦r ✉♥❜♦✉♥❞❡❞ ♦♣❡♥ s❡ts ✐♥ Cn ✧✱ ❆♥♥✱ ❙❝✉♦❧❛ ◆♦r♠✳ ❙✉♣✳ P✐s❛ ❈❧✳ ❙❝✐✳ ✺ ❱♦❧ ■❱✱ ✻✵✶✲✻✶✽✳ ❬✼❪ ❘♦②❞❡♥ ❍✳ ▲✳ ✭✶✾✼✶✮✱ ✧❘❡♠❛r❦s ♦♥ t❤❡ ❑♦❜❛②❛s❤✐ ♠❡tr✐❝✧✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ▼❛t❤✳ ✶✽✺✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ♣♣✳✶✷✺ ✲ ✶✸✼✳ ❬✽❪ ❚❤❛✐ ❉✳ ❉✳ ❛♥❞ ❚r❛♥❣ P✳ ◆✳ ❚✳ ✭✷✵✵✹✮✱ ✧❚❛✉t♥❡ss ♦❢ ❧♦❝❛❧❧② t❛✉t ❞♦✲ ♠❛✐♥s ✐♥ ❝♦♠♣❧❡① s♣❛❝❡s✧✱ ❆♥♥✱ P♦❧♦♥✳ ▼❛t❤✱ ✽✸✱ ✶✹✶✲✶✹✽✳ ✳ ✸✾ ... ởt ❝♦♥ {f? ?k }k ❝õ❛ {fν }ν s❛♦ ❝❤♦ lim |f? ?k (ζ)| = ∞} k? ??∞ ✣➦t E = {λ ∈ ∆ : lim |f? ?k (ζ)| = ∞}✳ ❱➻ ✈ỵ✐ ❜➜t ❦ý ✤✐➸♠ ζ t❤✉ë❝ E t❛ k? ??∞ ❝â lim |f? ?k ◦ gζ,θ | = ∞ ✤➲✉ tr➯♥ ∆1/2 t❤❡♦ ❜ê ✤➲ k tỗ t... ▼ët ♠✐➲♥ D ⊂ Cn ✤÷đ❝ ❣å✐ ❧➔ k ✲ ❤②♣❡r❜♦❧✐❝ ♥➳✉ kD ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tr➯♥ D✳ ✲ ▼ët ♠✐➲♥ k ✲ ❤②♣❡r❜♦❧✐❝ D ✤÷đ❝ ❣å✐ ❧➔ k ✲ ❤②♣❡r❜♦❧✐❝ ✤➛② ✭❤❛② k ✲ ✤➛②✮ ♥➳✉ ♥â ✤➛② ✤è✐ ✈ỵ✐ ❦❤♦↔♥❣ ❝→❝❤ kD ✳ ▼✳▲✳❘♦②❞❡♥... ❜➜t ❦ý t t tữỡ ố K tr tỗ t ởt ❤➡♥❣ sè t❤ü❝ ❞÷ì♥❣ rk s❛♦ ❝❤♦ ♠å✐ tü ✤➥♥❣ ❝➜✉ g ❝õ❛ ∆ t❤ä❛ ♠➣♥ g(0) ∈ K ⇒ ∆(g(0), rk ) ⊂ g(∆1/2 ), ð ✤â ∆(g(0), rk ) = {λ ∈ ∆ : |λ − g(0)| < rk } ✷✹ ❈❤ù♥❣ ♠✐♥❤✳