LUẬN văn THẠC sĩ HAY tập duy nhất cho các hàm phân hình với giá trị khuyết

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❡✉✐ ❱❖◆●❉❆▲❆ ❚❾P ❉❯❨ ◆❍❻❚ ❈❍❖ ❈⑩❈ ❍⑨▼ P❍❹◆ ❍➐◆❍ ❱❰■ ●■⑩ ❚❘➚ ❑❍❯❨➌❚ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✺ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❡✉✐ ❱❖◆●❉❆▲❆ ❚❾P ❉❯❨ ◆❍❻❚ ❈❍❖ ❈⑩❈ ❍⑨▼ P❍❹◆ ❍➐◆❍ ❱❰■ ●■⑩ ❚❘➚ ❑❍❯❨➌❚ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ số ữớ ữợ ❞➝♥ ❦❤♦❛ ❤å❝✿ P●❙✳❚❙✳ ❍⑨ ❚❘❺◆ P❍×❒◆● ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✺ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▲í✐ ❝❛♠ ✤♦❛♥ ✐ ❇↔♥ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ sü ♥❣❤✐➯♥ ự tổ ữợ sỹ ữợ P●❙✳❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣✱ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝ ð ❱✐➺t ◆❛♠✳ ❍å❝ ✈✐➯♥ ❚❡✉✐ ❱❖◆●❉❆▲❆ ❳→❝ ♥❤➟♥ ❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❚♦→♥ ❳→❝ ♥❤➟♥ ❝õ❛ ữớ ữợ P r Pữỡ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▲í✐ ❝↔♠ ì♥ ✐✐ ▲✉➟♥ ✈➠♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ✈➔ ❤♦➔♥ t❤➔♥❤ t↕✐ trữớ ữ ữợ sỹ ữợ ❞➝♥ ❦❤♦❛ ❤å❝ ❝õ❛ P●❙✳❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣✳ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✈ỉ ❤↕♥ tỵ✐ P●❙✳❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣ ✲ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❞➻✉ ❞➢t tỉ✐ tứ ỳ ữợ ỳ t tr ữớ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈ỵ✐ t➜t ❝↔ ♥✐➲♠ s❛② ♠➯ ❦❤♦❛ ❤å❝ ✈➔ t➙♠ ❤✉②➳t ❝õ❛ ♥❣÷í✐ t❤➛②✳ ❚ỉ✐ ❝ơ♥❣ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② tr♦♥❣ ❱✐➺♥ ❚♦→♥ ❤å❝✱ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② tr❛♥❣ ❜à ❝❤♦ tỉ✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð tr➯♥ ❝♦♥ ✤÷í♥❣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ P❤á♥❣ ✣➔♦ t↕♦ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ ✈➲ t➔✐ ❧✐➺✉ ✈➔ t❤õ tư❝ ❤➔♥❤ ❝❤➼♥❤ ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚æ✐ ①✐♥ tọ ỏ t ỡ s s tợ ỳ ữớ t❤➙♥ tr♦♥❣ ❣✐❛ ✤➻♥❤ ❝õ❛ ♠➻♥❤✳ ◆❤ú♥❣ ♥❣÷í✐ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥ ❝❤✐❛ s➫ ❦❤â ❦❤➠♥ ✈➔ ❧✉æ♥ ♠♦♥❣ ♠ä✐ tæ✐ t❤➔♥❤ ❝ỉ♥❣✳ ❚ỉ✐ ❝ơ♥❣ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✷✶✱ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❇↔♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❜↕♥ ❜➧ ỗ t ❧✉➟♥ ✈➠♥ ❚❡✉✐ ❱❖◆●❉❆▲❆ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✐✐✐ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✶ ✶ ✸ ởt số tự ỡ tr ỵ tt ✶✳✶✳ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ỵ ỡ ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶✳ ❈æ♥❣ t❤ù❝ s ỵ ỡ tự t ỵ ❝ì ❜↔♥ t❤ù ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✽ ✽ ✶✵ ❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❝❤ù❛ ❣✐→ trà ❦❤✉②➳t ✶✻ ✷✳✶✳ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳ ❳→❝ ✤à♥❤ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❜ð✐ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝❤ù❛ ❣✐→ trà ❦❤✉②➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✻ ✷✵ ✷✼ ❑➳t ❧✉➟♥ ✹✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✸ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✶ ▼Ð ✣❺❯ ◆➠♠ ✶✾✷✻✱ ❘✳ ◆❡✈❛♥❧✐♥♥❛ ✤÷đ❝ ❝❤ù♥❣ tä ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t ❜ð✐ ↔♥❤ ♥❣÷đ❝ ❦❤ỉ♥❣ t➼♥❤ ❜ë✐ ❝õ❛ ✺ ♣❤➙♥ ❜✐➺t ❝→❝ ❣✐→ trà✳ ❈æ♥❣ tr➻♥❤ ữủ ỗ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t✳ ❱➲ s❛✉✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ sü ①→❝ ✤à♥❤ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❜ð✐ ↔♥❤ ♥❣÷đ❝ ❝õ❛ ♠ët t➟♣ ❤ú✉ ❤↕♥ ♣❤➛♥ tû ✤➣ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ t tr ữợ ●r♦ss ✭❬✹❪✮ ✤➲ ①✉➜t ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭❤➔♠ ♥❣✉②➯♥✮ ❜ð✐ ↔♥❤ ♥❣÷đ❝ ❝õ❛ ♠ët t➟♣ ❤ú✉ ❤↕♥✳ ❑❤✐ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ❝õ❛ ●r♦ss✱ ♥➠♠ ✶✾✾✻ ❍✳ ❨✐ ✭❬✶✶❪✮ ❝❤ù♥❣ ♠✐♥❤ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣❤↔✐ trị♥❣ ♥❤❛✉ ♥➳✉ ❝❤ó♥❣ ❝❤✉♥❣ ♥❤❛✉ t➟♣ S = {z : z n + az n−m + b = 0}✱ tr♦♥❣ ✤â m, n ❧➔ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ s❛♦ ❝❤♦ m ✈➔ n ❦❤ỉ♥❣ ❝â ÷ỵ❝ sè ❝❤✉♥❣✱ n > 2m + (m ≥ 2) ✈➔ a, b ❧➔ ❝→❝ ❤➡♥❣ sè ❦❤→❝ ❦❤æ♥❣ s❛♦ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ zn + azn−m + b = ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ❜ë✐ ◆➠♠ ✶✾✾✽✱ ❋❛♥❣ ✈➔ ❍✉❛ ✭❬✸❪✮ ✤➣ ❝❤ù♥❣ ♠✐♥❤✿ ♥➳✉ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ✈➔ g t❤ä❛ ♠➣♥ Θ(∞, f ) > 11 12 , Θ(∞, g) > 11 12 ✈➔ Ef (S) = Eg (S) ❑➳t q✉↔ tr➯♥ ❝õ❛ ❋❛♥❣ ✈➔ ❍✉❛ ❝❤♦ t❤➜② ♠ët ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ✤➸ f ≡ g, tr♦♥❣ ✤â ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝â ❝❤ù❛ ✤✐➲✉ ❦✐➺♥ ✈➲ sè ❦❤✉②➳t t↕✐ ∞✳ ❱➲ s❛✉ ❝â ♥❤✐➲✉ t t tử rở t ữợ ❝ù✉ ♥➔② ✈ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ r❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ♠ỵ✐ ❝â ❝❤ù❛ sè ❦❤✉②➳t ✤➸ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ trò♥❣ ♥❤❛✉✳ ❈❤➥♥❣ ❤↕♥✱ ▲❛❤✐r✐ ✭❬✺❪✮✱ ▲❛❤✐r✐ ✈➔ ❇❛♥❡r❥❡❡ ✭❬✻❪✮✱ ❆✳ ❇❛♥❡r❥❡❡ ✈➔ ❙✳ ▼❛❥✉♠❞❡r ✭❬✶✱ ✷❪✮ ✳✳✳✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ✈➜♥ ✤➲ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t ❜ð✐ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝â ❝❤ù❛ ❣✐→ trà ❦❤✉②➳t✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ✏❚➟♣ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ❣✐→ trà t❤➻ f ≡ g ✱ tr♦♥❣ ✤â S = {z : z − z − = 0} LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✷ ✳ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ♥➠♠ ✷✵✶✸ ❜ð✐ ❆✳ ❇❛♥❡r❥❡❡ ✈➔ ❙✳ ▼❛❥✉♠❞❡r tr♦♥❣ ❬✶❪ ✈➔ ❬✷❪✳ ỗ õ ữỡ ữ s ữỡ ởt số tự ỡ s tr ỵ tt ◆❡✈❛♥❧✐♥♥❛✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè tự ỡ tr ỵ tt ố trà ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➼ ❤✐➺✉ sû ❞ư♥❣ tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ❈❤÷ì♥❣ ✷✿ ❚➟♣ ❣✐→ trà ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ợ tr t ữỡ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❆✳ ❇❛♥❡r❥❡❡ ✈➔ ❙✳ ▼❛❥✉♠❞❡r ✈➲ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝â ❝❤ù❛ ❣✐→ trà ❦❤✉②➳t ✤➸ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❧➔ ❜➡♥❣ ♥❤❛✉✳ ❦❤✉②➳t✑ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸ ❈❤÷ì♥❣ ✶ ởt số tự ỡ tr ỵ tt t t rữợ t t ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ tr ỵ tt ố ❝❤➾♥❤ ❤➻♥❤ f tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱ ✤✐➸♠ z0 ∈ C ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ✤✐➸♠ ❜ë✐ k > f (z) tỗ t ởt ❝❤➾♥❤ ❤➻♥❤ h(z) ❦❤æ♥❣ tr✐➺t t✐➯✉ tr♦♥❣ ❧➙♥ ❝➟♥ U ❝õ❛ z0 s❛♦ ❝❤♦ tr♦♥❣ ❧➙♥ ❝➟♥ ✤â ❤➔♠ f ữủ ữợ f (z) = (z z0 )k h(z) ◆❣❤➽❛ ❧➔ f (n)(z0) = 0, ✈ỵ✐ ♠é✐ n = 1, , k − ✈➔ f (k)(z0) = 0✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✣✐➸♠ z0 ✤÷đ❝ ❣å✐ ❧➔ ❝ü❝ ✤✐➸♠ ❜ë✐ k > ❝õ❛ ❤➔♠ f (z) ♥➳✉ tr♦♥❣ ❧➙♥ ❝➟♥ U ❝õ❛ z0 ❤➔♠ f ữủ ữợ h(z) tr õ h(z) ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ f (z) = (z − z0 )k tr✐➺t t✐➯✉ tr♦♥❣ ❧➙♥ ❝➟♥ U ❝õ❛ z0 ❱ỵ✐ ♠é✐ sè t❤ü❝ x > 0✱ ❦➼ ❤✐➺✉✿ log+ x = max{log x, 0} LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✹ ❑❤✐ ✤â log x = log+ x − log+(1/x) ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C✱ r > 0✱ ✈ỵ✐ ♠é✐ ϕ ∈ [0; 2π], t❛ ❝â log |f (reiϕ )| = log+ |f (reiϕ )| − log+ ♥➯♥ 2π 2π log f (reiϕ ) dϕ = 2π 2π , f (reiϕ ) 2π log+ f (reiϕ ) dϕ− 2π ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ log+ dϕ f (reiϕ ) ❍➔♠ 2π m(r, f ) = 2π log+ f (reiϕ ) dϕ ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ①➜♣ ①➾ ❝õ❛ ❤➔♠ f ✳ ❇➙② ❣✐í t❛ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❤➔♠ ✤➳♠✳ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ r > 0✳ ❑➼ ❤✐➺✉ 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❛ f tr♦♥❣ Dr = {z ∈ C : |z| |r|}✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❍➔♠ r N (r, ∞; f ) = N (r, f ) = n(t, f ) − n(0, f ) dt + n(0, f ) log r t ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ ❝õ❛ f ✭❝á♥ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ t↕✐ ❝→❝ ❝ü❝ ✤✐➸♠✮✳ ❍➔♠ r N (r, ∞; f ) = N (r, f ) = n(t, f ) − n(0, f ) dt + n(0, f ) log r t ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❦❤ỉ♥❣ ❦➸ ❜ë✐✳ ❚r♦♥❣ ✤â n(0, f ) = lim n(t, f ), n(0, f ) = lim n(t, f ) t→0 t→0 LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✺ ✣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 ) ❧➔ ❜❛ ❤➔♠ ❝ì ❜↔♥ tr ỵ tt ố tr õ ỏ ❧➔ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✳ ❚✐➳♣ t❤❡♦ t❛ ✤➲ ❝➟♣ ✤➳♥ ♠ët sè ❤➔♠ ✤➳♠ ♠ð rë♥❣ t❤÷í♥❣ ❞ị♥❣ tr♦♥❣ ❝❤ù♥❣ ỵ t ♣❤➙♥ ❤➻♥❤✳ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ r > 0✱ ❦➼ ❤✐➺✉ nk (r, f ) ❧➔ sè ❝ü❝ ✤✐➸♠ ❜ë✐ ❝➢t ❝öt ❜ð✐ k tr♦♥❣ Dr ❝õ❛ f ✭tù❝ ❧➔ ❝→❝ ❝ü❝ ✤✐➸♠ ❜ë✐ l > k ❝❤➾ ✤÷đ❝ t➼♥❤ k ❧➛♥ tr♦♥❣ tê♥❣ nk (r, f )✮✳ ❍➔♠ r Nk (r, f ) = nk (r, f ) − nk (0, f ) dt + nk (0, f ) log r t ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❜ë✐ ❝➢t ❝öt ❜ð✐ k✱ tr♦♥❣ ✤â nk (0, f ) = limt→0 nk (r, f )✳ ❙è k tr♦♥❣ nk (r, f ) ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾ sè ❜ë✐ ❝➢t ❝öt✳ ❈❤♦ a ∈ C ∪ {∞}✱ ❦➼ ❤✐➺✉ n(r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✱ n(r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ f − a tr♦♥❣ Dr r N (r, 0; f ) = N (r, )= f −a n(t, 1 ) − n(0, ) f −a f −a dt t ) log r, f −a 1 n(t, ) − n(0, ) f −a f −a dt t + n(0, r N (r, 0; f ) = N (r, )= f −a + n(0, ) log r f −a ❈❤♦ a ∈ C ∪ {∞}✱ ❦➼ ❤✐➺✉ nk)(r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✱ nk)(r, 1/(f − a)) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ f − a tr♦♥❣ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com tứ ỵ ❝ì ❜↔♥ t❤ù ❤❛✐✱ ✈ỵ✐ ε > 0✱ t❛ ❝â nT (r, f ) ≤2 N (r, 0; f )N + N (r, ∞; f ) + N (r, 0; g) + N (r, ∞; g) + N (r, 0; f ) + N (r, ∞; f ) +T (r, f ) + T (r, g) + S(r, f ) + S(r, g) ≤ N (r, 0; f ) + N (r, ∞; f ) + N (r, 0; g) + N (r, ∞; g) + 2T (r, f ) + T (r, g) + S(r, f ) + S(r, g) ≤ (11 − 2Θ(0; f ) − 2Θ(0; g) − 2Θ(∞; f ) − 2Θ(∞; g) + ε)T (r) + S(r) ✭✷✳✷✶✮ ❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝â nT (r, g) ≤(11 − 2Θ(0; f ) − 2Θ(0; g) − 2Θ(∞; f ) − 2Θ(∞; g) + ε)T (r) + S(r) ✭✷✳✷✷✮ ❑➳t ❤ñ♣ ✭✷✳✷✶✮ ✈➔ ✭✷✳✷✷✮ t❛ ❝â (n − 11 + 2Θ(0; f ) + 2Θ(∞; f ) + 2Θ(0; g) + 2Θ(∞; g) − ε)T (r) ≤ S(r) ✭✷✳✷✸✮ ❱➻ ε > 0✱ ✭✷✳✷✸✮ ❦❤ỉ♥❣ t❤➸ ①↔② r❛✳ ❚r÷í♥❣ ❤đ♣ ❝♦♥ ✶✳✷✳ m = 0✳ ❙û ❞ö♥❣ ❇ê ✤➲ ✷✳✻ t❛ ❝â (2) N (r, 0; g ) + N E (r, 1; F ) + 2N L (r, 1; G) + 2N L (r, 1; F ) ≤ N (r, 0; g ) + N (r, 1; G) + N L (r, 1; G) + N L (r, 1; G) + 2N L (r, 1; F ) ≤ N (r, 0; g ) + N (r, 1; G| ≥ 2) + N L (r, 1; G) + 2N L (r, 1; F ) ≤ N (r, 0; g |g = 0) + N (r, 1; G| ≥ 2) + 2N (r, 1; F | ≥ 2) ≥ N (r, 0; g) + N (r, ∞; g) + N (r, 0; f ) + N (r, ∞; f ) + S(r, f ) + S(r, g) ✭✷✳✷✹✮ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✷ ❙û ❞ö♥❣ ✷✳✷✹✱ ❇ê tứ ỵ ỡ ❜↔♥ t❤ù ❤❛✐ t❛ ❝â ✈ỵ✐ ε > 0, 1) nT (r, f ) ≤ N (r, 0; f ) + N (r, ∞; f ) + N E (r, 1; F ) + N L (r, 1; F ) 1) + N L (r, 1; G) + N E (r, 1; F ) − N (r, 0; f ) + S(r, f ) ≤ N (r, 0; f ) + N (r, ∞; f ) +N (r, 0; g) + N (r, ∞; g) (2 + T (r, f ) + T (r, g) + N E (r, 1; F ) + 2N L (r, 1; G) + 2N L (r, 1; F ) + N (r, 0; g ) + S(r, f ) + S(r, g) ≤ N (r, 0; f ) + N (r, ∞; f ) +3 N (r, 0; g) + N (r, ∞; g) + T (r, f ) + T (r, g) + S(r, f ) + S(r, g) ≤ (16 − 3Θ(0; f ) − 3Θ(∞; f ) − 3Θ(0; g) − 3Θ(∞; g) + ε)T (r) + S(r) ✭✷✳✷✺✮ ▼ët ❝→❝❤ t÷ì♥❣ tü✱ t❛ ❝â✿ nT (r, g) ≤(16 − 3Θ(0; f ) − 3Θ(∞; f ) − 3Θ(0; g) − 3Θ(∞; g) + ε)T (r) + S(r) ✭✷✳✷✻✮ ❑➳t ❤ñ♣ ✭✷✳✷✺✮ ✈➔ ✭✷✳✷✻✮ t❛ ❝â (n − 16 + 3Θ(0; f ) + 3Θ(∞; f ) + 3Θ(0; g) + 3Θ(∞; g) − ε)T (r) ≤ S(r) ✭✷✳✷✼✮ ❱➻ ε > ♥➯♥ ✭✷✳✷✼✮ ❦❤æ♥❣ t❤➸ ①↔② r❛✳ ❚r÷í♥❣ ❤đ♣ ✷✳ H ≡ ▲➜② t➼❝❤ ♣❤➙♥ ❝æ♥❣ t❤ù❝ ✭✷✳✷✮ t❛ ❝â A ≡ + B, F −1 G−1 ✭✷✳✷✽✮ tr♦♥❣ ✤â A, B ❧➔ ❝→❝ ❤➡♥❣ sè ✈➔ A = ❚ø ✭✷✳✷✽✮ t❛ t❤✉ ✤÷đ❝✿ F ≡ (B + 1)G + A − B − BG + A − B ✭✷✳✷✾✮ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✸ ❘ã r➔♥❣ ❧➔ ✭✷✳✷✾✮ ❝ị♥❣ ✈ỵ✐ ❇ê ✤➲ ✷✳✶ t❛ ❝â ✭✷✳✸✵✮ T (r, f ) = T (r, g) + O(1) ●✐↔ sû r➡♥❣ B = 0, −1 ◆➳✉ A − B − = 0, tø ✭✷✳✷✾✮ ❝❤ó♥❣ t❛ ✤➦t ✤÷đ❝✿ ❚r÷í♥❣ ❤đ♣ ✷✳✶ N (r, B+1−A ; G) = N (r, 0; F ) B+1 ❚ø tr➯♥✱ ❇ê ỵ ỡ tự t ❝â✿ nT (r, g) < N (r, ∞; G) + N (r, 0; G) + N (r, B+1−A ; G) + S(r, g) B+1 ≤ N (r, ∞; g) + N (r, 0; g) + N (r, 0; g + a) + N (r, 0; f ) + N (r, 0; f + a) + S(r, g) ≤ 2T (r, f ) + 3T (r, g) + S(r, g), ✤✐➲✉ ♥➔② ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✸✵✮ t❛ ❝â ♠➙✉ t❤✉➝♥ ❦❤✐ n ≥ ❱➻ ✈➟② A − B − = ✈➔ ❞♦ ✤â ✭✷✳✷✾✮ ✤÷❛ ✈➲ F ≡ ❚ø ✤â t❛ ❝â✿ N r, (B + 1)G BG + −1 ; G = N (r, ∞; f ) B ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✶ ♠ët ❧➛♥ ỳ sỷ ỵ ỡ tự t❛ ❝â✿ nT (r, g) < N (r, ∞; G) + N (r, 0; G) + N (r, −1 ; G) + S(r, g) B ≤ N (r, ∞; g) + N (r, 0; g) + N (r, 0; g + a) + N (r, ∞; f ) + S(r, gδ) ≤ T (r, f ) + 3T (r, g) + S(r, g), ✤✐➲✉ ♥➔② ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✸✵✮ t❛ ❝â ♠➙✉ t❤✉➝♥ ❦❤✐ n ≥ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✹ ❚r÷í♥❣ ❤đ♣ ❝♦♥ ✷✳✷✳ ●✐↔ sû r➡♥❣ B = −1 ❚ø ✭✷✳✷✾✮ t❛ ❝â F ≡ A −G + A + ✭✷✳✸✶✮ ◆➳✉ A + = 0, tø ✭✷✳✸✶✮ t❛ ❝â✿ N (r, A + 1; G) = N (r, ∞; f ) ❙û ❞ư♥❣ ❧➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ tr÷í♥❣ ❤đ♣ ❝♦♥ ✷✳✶ tr➯♥ t❛ s➩ ❝â ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â A + = ✈➔ tø ✭✷✳✸✶✮ t❛ ❝â F G ≡ 1, tù❝ ❧➔ f n−1(f + a)gn−1(g + a) ≡ b2 , ❧➔ ❦❤æ♥❣ t❤➸ ①↔② r❛ t❤❡♦ ❇ê ✤➲ ✷✳✹✳ ❚r÷í♥❣ ❤đ♣ ❝♦♥ ✷✳✸✳ ●✐↔ sû r➡♥❣ B = ❚ø ✭✷✳✷✾✮ t❛ ❝â✿ F ≡ ◆➳✉ A − = 0, tø ✭✷✳✸✷✮ t❛ ❝â✿ G+A−1 A ✭✷✳✸✷✮ N (r, − A; G) = N (r, 0; F ) ứ õ ỵ ❣✐è♥❣ ♥❤÷ tr➯♥ t❛ ❝â ♠➙✉ t❤✉➝♥✳ ❱➟②✱ A = ✈➔ ❞♦ ✤â F ≡ G, tù❝ ❧➔ f n−1 (f + a) ≡ g n−1 (g + a) t ỵ õ ữủ tứ sỷ ❞ö♥❣ ❇ê ✤➲ ✷✳✺✳ ❧➔ tèt tr♦♥❣ ❈→❝ ✈➼ ❞ư s❛✉ ✤÷đ❝ ❝❤♦ t❤➜② ✤✐➲✉ ❦✐➺♥ Θf + Θg > n1 ỵ n m ≥ 2✳ ❱➼ ❞ö ✷✳✷ ✭❬✻❪✱ ✈➼ ❞ö ✷✮✳ ❈❤♦ − hn−1 − hn−1 f = −a ; g = −ah , − hn − hn tr♦♥❣ ✤â h= α2 (ez − 1) 2πi ; α = exp ez − α n ✈➔ n (≥ 3) ❧➔ sè ♥❣✉②➯♥✳ ❑❤✐ ✤â T (r, f ) = (n − 1)T (r, h) + O(1) ✈➔ T (r, g) = (n − 1)T (r, h) + O(1) ✈➔ T (r, h) = T (r, ez ) + O(1)✳ ❍ì♥ ♥ú❛ h = α, α2 ✈➔ ✤è✐ ✈ỵ✐ ❝→❝ sè ♣❤ù❝ γ = α, α2, N (r, γh) T (r, h) ụ ú ỵ r ởt ♥❣❤✐➺♠ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✺ ❝õ❛ h = ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❝ü❝ ✤✐➸♠ ✈➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f ✈➔ g✳ ❉♦ ✤â Θ(∞; f ) = Θ(∞; g) = ✳ ▼➦t ❦❤→❝ n−1 n−2 Θ(0, f ) = − lim sup r→∞ k=1 N (r, β k ; h) + N (r, ∞; h) (n − 1)T (r, h) + O(1) =0 ✈➔ n−2 Θ(0, g) = − lim sup r→∞ k=1 N (r, β k ; h) + N (r, 0; h) (n − 1)T (r, h) + O(1) = 0, tr♦♥❣ ✤â β = exp 2πi ❘ã r➔♥❣ ❧➔ Ef (S, ∞) = Eg (S, ∞) ✈➻ f n−1 (f + n−1 a) ≡ g n−1 (g + a) ♥❤÷♥❣ f ≡ g ✳ ❱➼ ❞ư ✷✳✸ ✭❬✶❪✮✳ ❈❤♦ f ✈➔ g ❧➔ ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ❱➼ ❞ö ✷✳✷✱ tr♦♥❣ ✤â h= 2πi α(αez − 1) ; α = exp ez − n ✈➔ n (≥ 3) ❧➔ sè ♥❣✉②➯♥✳ ❚r♦♥❣ ❬✷❪✱ ❆✳ ❇❛♥❡r❥❡❡ ✈➔ ❙✳ r ự ỵ S = z : z n + az n−1 + b = ✱ tr♦♥❣ ✤â n (≥ 6) ❧➔ sè ♥❣✉②➯♥ ✈➔ a, b ❧➔ ❤❛✐ ❤➡♥❣ sè ❦❤→❝ ❦❤æ♥❣ s❛♦ ❝❤♦ z n +az n−1 +b = ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ❜ë✐✳ ●✐↔ sû r➡♥❣ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ t❤ä❛ ♠➣♥ Ef (S, m) = Eg (S, m)✳ ◆➳✉ ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ m ≥ ✈➔ Θf + Θg > max{(9 − n)/2, (n + 1)/(n − 1)}❀ ❤♦➦❝ m = ✈➔ Θf + Θg > max{(10 − n)/2, (n + 1)/(n − 1)}❀ ❤♦➦❝ m = ✈➔ Θf + Θg > max{(15 − n)/2, (n + 1)/(n − 1)}✱ ❦❤✐ ✤â f ≡ g ✱ tr♦♥❣ ✤â Θf = Θ(0, f ) + Θ(∞, f ) + 21 δ2 (−a, f ) ✈➔ Θg ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ t÷ì♥❣ tü✳ ❚ø ❣✐↔ t❤✐➳t s✉② r❛ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ zn + azn1 + b ổ ỡ ỵ ú ❜ð✐ wj , j = 1, 2, , n✳ ❈❤♦ F, G ❧➔ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✮ ✈➔ ✭✷✳✷✮✳ ❚ø Ef (S, m) = Eg (S, m) t❛ s✉② r❛ F ✈➔ G ❝❤✉♥❣ (1, m)✳ ❚❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉✿ ❈❤ù♥❣ ♠✐♥❤✳ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✻ ❚r÷í♥❣ ❤đ♣ ✶✳ H ≡ 0✳ ❚r÷í♥❣ ❤đ♣ ❝♦♥ ✶✳✶ t❛ ❝â✿ ❚❛ ①➨t m ≥ 1✳ ❑❤✐ m ≥ 2✱ sû ❞ö♥❣ ❇ê ✤➲ ✷✳✻ N (r, 0; G ) + N (r, 1; G| ≥ 2) + N ∗ (r, 1; F, G) ≤ N (r, 0; G ) + N (r, 1; G| ≥ 2) + N (r, 1; G| ≥ 3) ≤ N (r, 0; G |G = 0) + S(r, g) ✭✷✳✸✸✮ ❉♦ ✤â sû ❞ö♥❣ ✭✷✳✸✸✮ ✈➔ ❇ê ✤➲ ✷✳✶✱ ❇ê ✤➲ ✷✳✷ ✈➔ ❇ê ✤➲ t ỵ ỡ tự ợ ộ ε > 0✱ t❛ ❝â ≤ N (r, 0; g) + N (r, −a; g) + N (r, ∞; g) + S(r, g) (n + 1)T (r, f ) ≤ N (r, 0; f ) + N (r, −a; f ) + N (r, ∞; f ) + N (r, 1; F | = 1) + N (r, 1; F | ≥ 2) − N0 (r, 0; F ) + S(r, f ) ≤ N (r, 0; f ) + N (r, ∞; f ) +N (r, 0; g) + N (r, ∞; g) + N2 (r, −a; f ) + N (r, −a; g| ≥ 2) + N (r, 1; G| ≥ 2) + N ∗ (r, 1; F, G) + N (r, 0; G ) + S(r, f ) + S(r, g) ≤ N (r, 0; f ) + N (r, ∞; f ) + N (r, 0; g) + N (r, ∞; g) + N2 (r, −a; f ) + N2 (r, −a; g) + S(r, f ) + S(r, g) ≤ 10 − 2Θ(0, f ) − 2Θ(∞, f ) − 2Θ(0, g) − 2Θ(∞, g) − δ2 (−a, f ) − δ2 (−a, g) + ε T (r) + S(r) = 10 − 2Θf − 2Θg + ε T (r) + S(r), ✭✷✳✸✹✮ tr♦♥❣ ✤â Θf ✈➔ Θg ✤÷đ❝ ①→❝ ✤à♥❤ tr♦♥❣ ỵ ữỡ tỹ t õ (n + 1)T (r, g) ≤ 10 − 2Θf − 2Θg + ε T (r) + S(r) ✭✷✳✸✺✮ ❑➳t ❤ñ♣ ✭✷✳✸✹✮ ✈➔ ✭✷✳✸✺✮ t❛ ❝â LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✼ n − + 2Θf + Θg − ε T (r) ≤ S(r) ✭✷✳✸✻✮ ❱➻ ε > 0✱ ✭✷✳✸✻✮ ❞➝♥ ✤➳♥ sü ♠➙✉ t❤✉➝♥✳ ❑❤✐ m = 1✱ sû ❞ư♥❣ ❇ê ✤➲ ✷✳✻✱ ❝ỉ♥❣ t❤ù❝ ✭✷✳✸✸✮ ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉✿ N (r, 0; G ) + N (r, 1; G| ≥ 2) + N ∗ (r, 1; F, G) ≤ N (r, 0; G ) + N (r, 1; G| ≥ 2) + N L (r, 1; G) + N (r, 1; F | ≥ 3) ≤ N (r, 0; G |G = 0) + n N (r, wj ; f ) − N (r, wj ; f ) j=1 ≤ N (r, 0; g) + N (r, −a, g) + N (r, ∞, g) + N (r, 0; f ) + N (r, ∞; f ) +S(r, f ) + S(r, g) ✭✷✳✸✼✮ ❙û ❞ö♥❣ ✭✷✳✸✼✮ ✈➔ ❇ê ✤➲ ✷✳✷ ✈➔ ❇ê ✤➲ ✷✳✼✱ t❤ü❝ ❤✐➺♥ ❣✐è♥❣ ❝æ♥❣ t❤ù❝ ✭✷✳✸✹✮✱ tø ỵ ỡ tự t õ ợ ộ ε ≥ 0✱ (n + 1)T (r, f ) ≤ 2N (r, 0; f ) + N2 (r, −a; f ) + 2N (r, ∞; f ) + 2N (r, 0; g) + N2 (r, −a; g) + 2N (r, ∞; g) + N (r, 0; f ) + N (r, ∞; f ) + S(r, f ) + S(r, g) ≤ 2N (r, 0; f ) + 2N (r, ∞; f ) + N2 (r, −a; f ) + 2N (r, 0; g) + 2N (r, ∞; g) + N2 (r, −a; g) + T (r, f ) + S(r, f ) + S(r, g) ≤ (11 − 2Θf − 2Θg + ε)T (r) + S(r) ✭✷✳✸✽✮ ❚÷ì♥❣ tü t❛ ❝â (n + 1)T (r, g) ≤ 11 − 2Θf − 2Θg + ε T (r) + S(r) ✭✷✳✸✾✮ ❑➳t ❤ñ♣ ✭✷✳✸✽✮ ✈➔ ✭✷✳✸✾✮ t❛ ❝â n − 10 + 2Θf + 2Θg − ε T (r) ≤ S(r) ✭✷✳✹✵✮ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✽ ❱➻ ε > 0✱ ✭✷✳✹✵✮ ❞➝♥ ✤➳♥ sü ♠➙✉ t❤✉➝♥✳ ❚r÷í♥❣ ❤đ♣ ❝♦♥ ✷✳✷✳ ❳➨t m = 0✳ ❙û ❞ö♥❣ ❇ê ✤➲ ✷✳✻ t❛ ❝â✿ (2 N (r, 0; G ) + N E (r, 1; F ) + 2N L (r, 1; G) + 2N L (r, 1; F ) (2 ≤ N (r, 0; G ) + N E (r, 1; F ) + N L (r, 1; G) + N L (r, 1; G) + 2N L (r, 1; F ) ≤ N (r, 0; G ) + N L (r, 1; G| ≥ 2) + N L (r, 1; G) + 2N L (r, 1; F ) ≤ N (r, 0; G |g = 0) + N (r, 1; G| ≥ 2) + 2N (r, 1; F | ≥ 2) ≤ N (r, 0; g) + N (r, ∞; g) + N (r, −a; g) + N (r, 0; g) + N (r, ∞; g) + 2N (r, 0; f ) + 2N (r, ∞; f ) + S(r, f ) + S(r, g) ≤ N (r, 0; g) + N (r, ∞; g) + N (r, −a; g) + 2T (r, g) ✭✷✳✹✶✮ + 4T (r, f ) + S(r, f ) + S(r, g) ❉♦ ✤â sû ❞ö♥❣ ✭✷✳✹✶✮ ✈➔ ❇ê ✤➲ ✷✳✷ ✈➔ ❇ê ✤➲ ✷✳✼✱ tø ✤à♥❤ ỵ ỡ tự ợ ộ > t❛ ❝â 1) (n + 1)T (r, f ) ≤ N (r, 0; f ) + N (r, −a; f ) + N (r, ∞; f ) + NE (r, 1; F ) (2 + N L (r, 1; F ) + N L (r, 1; G) + N E (r, 1; F ) − N0 (r, 0; F ) + S(r, f ) ≤ N (r, 0; f ) + N (r, ∞; f ) +N (r, 0; g) + N (r, ∞; g) (2 + N2 (r, −a; f ) + N (r, −a; g| ≥ 2) + N E (r, 1; F ) + 2N L (r, 1; G) + 2N L (r, 1; F ) + N (r, : G ) + S(r, f ) + S(r, g) LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✾ ❙✉② r❛ (n + 1)T (r, f ) ≤ N (r, 0; f ) + N (r, ∞; f ) +N2 (r, −a; f ) + N (r, 0; g) + N (r, ∞; g) +N2 (r, −a; g) + 4T (r, f ) + 2T (r, g) + S(r, f ) + S(r, g) ≤ 16 − 2Θf − 2Θg + ε T (r) + S(r) ✭✷✳✹✷✮ ❚÷ì♥❣ tü t❛ ❝â (n + 1)T (r, g) ≤ 16 − 2Θf − 2Θg + ε T (r) + S(r) ✭✷✳✹✸✮ ❑➳t ❤ñ♣ ✭✷✳✹✷✮ ✈➔ ✭✷✳✹✸✮ n − 15 + 2Θf + 2Θg − ε T (r) ≤ S(r) ✭✷✳✹✹✮ ❱➻ ε > 0✱ ✭✷✳✹✹✮ ❞➝♥ ✤➳♥ sü ♠➙✉ t❤✉➝♥✳ ❚r÷í♥❣ ❤đ♣ ✷✳ H ≡ 0✳ ▲➜② t➼❝❤ ♣❤➙♥ ❝æ♥❣ t❤ù❝ ✭✷✳✷✮✱ t❛ ❝â A ≡ + B F −1 G−1 ✭✷✳✹✺✮ ❚r♦♥❣ ✤â A ✈➔ B ❧➔ ❝→❝ ❤➡♥❣ sè ✈➔ A = 0✳ ❚ø ✭✷✳✹✺✮ t❛ ❝â✿ F ≡ (B + 1)G + A − B − BG + A − B ✭✷✳✹✻✮ ❚ø ✭✷✳✹✻✮ ❝ị♥❣ ✈ỵ✐ ❇ê ✤➲ ✷✳✶ t❛ ❝â T (r, f ) = T (r, g) + O(1) ❚r÷í♥❣ ❤đ♣ ❝♦♥ ✷✳✶✳ ✭✷✳✹✼✮ B = 0, −1✳ ◆➳✉ A − B − = 0✱ tø ✭✷✳✹✻✮ t❛ ❝â✿ N r, B+1−A ; G = N (r, 0; F ) B+1 ✭✷✳✹✽✮ ❚➼♥❤ t♦→♥ ♥❤÷ tr➯♥✱ t ủ ợ ỵ ỡ ❜↔♥ t❤ù ❤❛✐✱ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✹✵ t❛ ❝â✿ nT (r, g) < N (r, ∞; G) + N (r, 0; G) + N r, B+1−A ; G +S(r, g) B+1 ≤ N (r, ∞; g) + N (r, 0; g) + N (r, 0; g + a) + N (r, 0; f ) + N (r, 0; f + a) + S(r, g) ✭✷✳✹✾✮ ❑❤✐ n ≥ 6✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✹✾✮ ❦❤æ♥❣ t❤➸ ①↔② r❛✳ ❱➻ ✈➟② A−B −1 = 0✱ ✈➔ ❞♦ ✤â ✭✷✳✹✻✮ ❝❤♦ t❛ (B + 1)G ✭✷✳✺✵✮ F ≡ BG + ❚ø ✤â t❛ ❝â✿ −1 N r, ; G = N (r, ∞; f ) ✭✷✳✺✶✮ B ≤ 2T (r, f ) + 3T (r, g) + S(r, g) ❙û ❞ö♥❣ ❇ê ✤➲ ✷✳✶ ✈➔ ✤à♥❤ ỵ ỡ tự ởt ỳ t õ nT (r, g) < N (r, ∞; G) + N (r, 0; G) + N r, −1 ; G +S(r, g) B ≤ N (r, ∞; g) + N (r, 0; g) + N (r, 0; g + a) + N (r, ∞; f ) + S(r, g) ≤ T (r, f ) + 3T (r, g) + S(r, g) ✭✷✳✺✷✮ ❑❤✐ n ≥ 6✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✺✷✮ ❦❤æ♥❣ t❤➸ ①↔② r❛✳ ❚r÷í♥❣ ❤đ♣ ❝♦♥ ✷✳✷✳ B = −1✳ ❚ø ✭✷✳✹✻✮ t❛ ❝â✿ F ≡ ◆➳✉ A + = 0✱ tø ✭✷✳✺✸✮ t❛ ❝â✿ A −G + A + ✭✷✳✺✸✮ ✭✷✳✺✹✮ ▲➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ s✉② r❛ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â A + = ✈➔ tø ✭✷✳✺✸✮ t❛ ❝â F G ≡ 1✱ tù❝ ❧➔ f n−1(f + a)gn−1(g + a) ≡ b2✱ ✤✐➲✉ ♥➔② ❦❤æ♥❣ t❤➸ ①↔② r❛ t❤❡♦ ❇ê ✤➲ ✷✳✹✳ N (r, A + 1; G) = N (r, ∞; f ) LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✹✶ ❚r÷í♥❣ ❤đ♣ ❝♦♥ ✷✳✸✳ B = 0✳ ❚ø ✭✷✳✹✻✮✱ G+A−1 F ≡ A t❛ ❝â✿ ✭✷✳✺✺✮ ◆➳✉ A − = 0✱ tø ✭✷✳✺✺✮ t❛ ❝â✿ ✭✷✳✺✻✮ ▲➟♣ ❧✉➟♥ ✈➔ t➼♥❤ t♦→♥ ♥❤÷ tr➯♥ t❛ ❝â sü ♠➙✉ t❤✉➝♥✳ ❱➟② A = 1✱ ❞♦ ✤â F ≡ G✱ ❦➨♦ t❤❡♦ f n−1 (f + a) ≡ g n1 (g + a) ỵ ữủ s r tø ❇ê ✤➲ ✷✳✽✳ N (r, − A; G) = N (r, 0; F ) ❇➙② ❣✐í t❛ ①❡♠ ①➨t ♠ët sè ✈➼ ❞ö ❝❤♦ t❤➜② ✤✐➲✉ ❦✐➺♥ Θf + g (n + 1)/(n 1) tr ỵ ✷✳✺ ❧➔ ❝➛♥ t❤✐➳t ❦❤✐ n ≥ ✈➔ m ≥ 2✳ ❱➼ ❞ö ✷✳✹✳ ❳➨t f = −a((1 − hn−1 )/(1 − hn )); > g = −ah((1 − hn−1 )/(1 − hn )), tr♦♥❣ ✤â h = ((α2(ez − 1))/(ez − α)), α = exp(2πi/n) ✈➔ n (≥ 3) ❧➔ sè ♥❣✉②➯♥✳ ❑❤✐ ✤â T (r, f ) = (n − 1)T (r, h) + O(1), T (r, g) = (n − 1)T (r, h) + O(1), tø ✤â T (r, h) = T (r, ez ) + O(1)✳ ◆❣♦➔✐ r❛✱ t❛ t❤➜② r➡♥❣ h = α, α2 ✈➔ ✈ỵ✐ ❝→❝ sè ♣❤ù❝ ❜➜t ❦ý γ = α, α2✱ N (r, γ; h) ∼ T (r, h)✳ ❚❛ ụ ú ỵ r h = ổ ❧➔ ❝ü❝ ✤✐➸♠ ✈➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f ✈➔ g✳ ❉♦ ✤â Θ(∞, f ) = Θ(∞, g) = ▼➦t ❦❤→❝ Θ(0, f ) = − lim sup r→∞ ✈➔ Θ(0, g) = − lim sup r→∞ n−1 n−2 k k=1 N (r, β ; h) + N (r, ∞; h) =0 (n − 1)T (r, h) + O(1) n−2 k k=1 N (r, β ; h) + N (r, 0; h) =0 (n − 1)T (r, h) + O(1) tr♦♥❣ ✤â β = exp(2πi/(n − 1))✳ ❍ì♥ ♥ú❛ δ2 (−a, g) = − lim sup r→∞ 2N (r, 0; h) (n − 3) = (n − 1)T (r, h) + O(1) (n − 1) LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✹✷ ✈➔ δ2 (−a, f ) = − lim sup r→∞ 2N (r, ∞; h) (n − 3) = (n − 1)T (r, h) + O(1) (n − 1) +1 ❚ø ✤â Θf + Θg = nn − ✳ ❘ã r➔♥❣ Ef (S, ∞) = Eg (S, ∞) ✈➻ f n−1(f + a) ≡ gn−1(g + a)✱ ♥❤÷♥❣ f ≡ g✳ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✹✸ ❑➳t ❧✉➟♥ ❱ỵ✐ ♠ư❝ ✤➼❝❤ t✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ ♥❤ú♥❣ ự ỵ tt ố tr tr ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➛♥ ❤➻♥❤✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s❛✉ ✤➙②✿ ✶✳ ❚ê♥❣ ❤ñ♣ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ ỵ ỡ tr ỵ tt ố trà ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ✤â ❧➔ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ sû ❞ư♥❣ tr♦♥❣ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❝❤÷ì♥❣ ✷✳ ✷✳ ●✐ỵ✐ t❤✐➺✉ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ✤➳♠✱ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ✸✳ P❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❧↕✐ ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝â ❝❤ù❛ ❣✐→ trà ❦❤✉②➳t ✤➸ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❧➔ ❜➡♥❣ ỵ t số ự tr t t ỵ ✷✳✺ ❝❤♦ t❤➜② ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝❤ù❛ ❣✐→ trà ❦❤✉②➳t t↕✐ 0✱ ∞ ✈➔ δ2(−a, f )✳ ❈â t❤➸ t❤➜②✱ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ❝❤➾ r❛ ỳ ỗ tû ✈ỵ✐ ♠ët sè ✤✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳ ✈➲ sè ❦❤✉②➳t✳ ◆❣♦➔✐ ♥❤ú♥❣ ❦➳t q✉↔ ✤➣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥✱ ❝á♥ ❝â ♠ët sè ❦➳t q✉↔ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐ ❝❤ó♥❣ tỉ✐ s➩ t✐➳♣ tử t tr ự ợ ỵ tữ ữ r tốt ỡ ợ số ❦❤✉②➳t✳ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✹✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆✳ ❇❛♥❡r❥❡❡ ❛♥❞ ❙✳ ▼❛❥✉♠❞❡r✱ ❖♥ ✉♥✐q✉❡ r❛♥❣ s❡t ♦❢ ♠❡r♦♠♦r✲ ♣❤✐❝ ❢✉♥❝t✐♦♥s ✇✐t❤ ❞❡❢✐❝✐❡♥t ♣♦❧❡s✱ ❋❛❝t❛ ❯♥✐✈✳ ❙❡r✳ ▼❛t❤✳ ■♥❢♦r♠✱ ✷✽✭✶✮✭✷✵✶✸✮✱ ✶✲✶✺✳ ❬✷❪ ❆✳ ❇❛♥❡r❥❡❡ ❛♥❞ ❙✳ ▼❛❥✉♠❞❡r✱ ❖♥ ✉♥✐q✉❡ r❛♥❣ s❡t ♦❢ ♠❡r♦✲ ♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✇✐t❤ ❞❡❢✐❝✐❡♥t ✈❛❧✉❡s✱ ❏✳ ♦❢ ❈♦♠♣❧❡① ❆♥❛❧②s✐s✱ ✷✵✶✸ ✭✷✵✶✸✮✱ ✶✲✾✳ ❬✸❪ ▼✳ ❋❛♥❣ ❛♥❞ ❳✳ ❍✉❛✱ ▼❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s t❤❛t s❤❛r❡ ♦♥❡ ❢✐✲ ♥✐t❡ s❡t ❈▼✱ ❏✳ ◆❛♥❥✐♥❣❯♥✐✈✳ ▼❛t❤✳ ❇✐q✉❛rt❡r❧②✱ ✶✺✭✶✮✭✶✾✾✽✮✱ ✶✺✲✷✷✳ ❬✹❪ ❋✳ ●r♦ss✱ ❋❛❝t♦r✐③❛t✐♦♥ ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✈➔ s♦♠❡ ♦♣❡♥ ♣r♦❜❧❡♠s✱ Pr♦❝✳ ❈♦♥❢✳ ❯♥✐✈✳ ❑❡♥t✉❝❦②✱ ▲❡✐①♥❣t♦♥✱ ❑❡♥t✉❝❦② ✭✶✾✼✻✮❀ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✺✾✾✭✶✾✼✼✮✱ ✺✶✲✻✾✱ ❙♣r✐♥❣❡r ✭❇❡r❧✐♥✮✳ ❬✺❪ ■✳ ▲❛❤✐r✐✱ ❆ q✉❡st✐♦♥ ♦❢ ●r♦ss ✈➔ ✇❡✐❣❤t❡❞ s❤❛r✐♥❣ ♦❢ ❛ ❢✐♥✐t❡ s❡t ❜② ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✱ ❆♣♣❧✐❡❞ ▼❛t❤✳ ❊✲◆♦t❡s✱ ✷✭✷✵✵✷✮✱ ✶✻✲✷✶✳ ❬✻❪ ■✳ ▲❛❤✐r✐ ❛♥❞ ❆✳ ❇❛♥❡r❥❡❡✱ ❯♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝✲ t✐♦♥s ✇✐t❤ ❞❡❢✐❝✐❡♥t ♣♦❧❡s✱ ❑②✉♥❣♣♦♦❦ ▼❛t❤✳ ❏✳✱ ✹✹✭✷✵✵✹✮✱ ✺✼✺✲✺✽✹✳ ❬✼❪ ■✳ ▲❛❤✐r✐ ✈➔ ❙✳ ❉❡✇❛♥✱ ❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ ❛ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ✈➔ ✐ts ❞❡r✐✈❛t✐✈❡✱ ❑♦❞❛✐ ▼❛t❤✳ ❏✳✱ ✷✻✭✷✵✵✸✮✱ ✾✺✲✶✵✵✳ ❬✽❪ ❆✳❩✳ ▼♦❤♦♥✬❦♦✱ ❖♥ t❤❡ ◆❡✈❛♥❧✐♥♥❛ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ s♦♠❡ ♠❡r♦✲ ♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✱ ❚❤❡♦r② ♦❢ ❋✉♥❝t✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✶✹✭✶✾✼✶✮✱ ✽✸✲✽✼✳ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✹✺ ❬✾❪ P✳ ▲✐ ❛♥❞ ❈✳❈✳ ❨❛♥❣✱ ❙♦♠❡ s❛✉ ✤â r❡s✉❧ts ♦♥ t❤❡ ✉♥✐q✉❡ r❛♥❣❡ s❡ts ♦❢ ♠❡r♦♠♦r♣❤✐❝❢✉♥❝t✐♦♥s✱ ❑♦❞❛✐ ▼❛t❤✳ ❏✳✱ ✶✸✭✶✾✾✺✮✱ ✹✸✼✲✹✺✵✳ ❬✶✵❪ ❍✳❳✳ ❨✐✱ ❆ q✉❡st✐♦♥ ♦❢ ●r♦ss ✈➔ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✱ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳✱ ✶✸✽✭✶✾✾✺✮✱ ✶✻✾✲✶✼✼✳ ❬✶✶❪ ❍✳❳✳ ❨✐✱ ❯♥✐❝✐t② t❤❡♦r❡♠ ❢♦r ♠❡r♦♠♦r♣❤✐❝ ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s ■■■✳✱ ❇✉❧❧✳ ❆✉str❛❧✳ ▼❛t❤✳ ❙♦❝✳✱ ✺✸ ✭✶✾✾✻✮✱ ✼✶✲✽✷✳ ❬✶✷❪ ❍✳❳✳ ❨✐✱ ▼❡r♦♠♦r♣❤✐❝✳ ❢✉♥❝t✐♦♥s t❤❛t s❤❛r❡ ♦♥❡ ♦r t✇♦ ✈❛❧✉❡s ■■✱ ❑♦❞❛✐ ▼❛t❤✳ ❏✳✱ ✷✷✭✶✾✾✾✮✱ ✷✻✹✲✷✼✷✳ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ... ❤ú✉ ❤↕♥✳ ❑❤✐ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ❝õ❛ ●r♦ss✱ ♥➠♠ ✶✾✾✻ ❍✳ ❨✐ ✭❬✶✶❪✮ ❝❤ù♥❣ ♠✐♥❤ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣❤↔✐ trị? ??❣ ♥❤❛✉ ♥➳✉ ❝❤ó♥❣ ❝❤✉♥❣ ♥❤❛✉ t➟♣ S = {z : z n + az n−m + b = 0}✱ tr♦♥❣ ✤â m, n ❧➔ ❤❛✐ sè ữỡ