❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❍⑩■ ◆●❯❨➊◆ ❑❍❖❆ ❚❖⑩◆ ❈❍❆◆❚❍❖◆❊ ❑❊❖▼❆◆■❙❆❨ ❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ❈❍❖ ❍⑨▼ P❍❹◆ ❍➐◆❍ ❈❍❯◆● ◆❍❆❯ ▼❐❚ ❍⑨▼ ◆❍➘ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❍⑩■ ◆●❯❨➊◆ ❑❍❖❆ ❚❖⑩◆ ❈❍❆◆❚❍❖◆❊ ❑❊❖▼❆◆■❙❆❨ ❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ❈❍❖ ❍⑨▼ P❍❹◆ ❍➐◆❍ ❈❍❯◆● ◆❍❆❯ ▼❐❚ ❍⑨▼ ◆❍➘ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ữớ ữợ P Pì ◆●❯❨➊◆ ✲ ✷✵✶✼ ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ sü ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ tổ ữợ sỹ ữợ P r Pữỡ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝æ♥❣ ❜è tr♦♥❣ ❝→❝ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝ ð ❱✐➺t ◆❛♠✳ ❍å❝ ✈✐➯♥ ❈❤❛♥t❤♦♥❡ ❑❡♦♠❛♥✐s❛② ❳→❝ ♥❤➟♥ ❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❚♦→♥ ❳→❝ ♥❤➟♥ ❝õ❛ ♥❣÷í✐ ữợ P r Pữỡ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔② tæ✐ ❧✉æ♥ ữủ sỹ ữợ ú ù t t ❝õ❛ P●❙✳❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚æ✐ ①✐♥ ❜➔② tä ❧á♥❣ t ỡ ổ tợ P r Pữỡ ữớ t t t tổ tứ ỳ ữợ ❝❤➟♣ ♥❤ú♥❣ ✤➛✉ t✐➯♥ tr➯♥ ❝♦♥ ✤÷í♥❣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈ỵ✐ t➜t ❝↔ ♥✐➲♠ s❛② ♠➯ ❦❤♦❛ ❤å❝ ✈➔ t➙♠ ❤✉②➳t ❝õ❛ ♥❣÷í✐ t❤➛②✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ P❤á♥❣ ✣➔♦ t↕♦ ✭❜ë q ỵ t tở rữớ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ✈➲ t➔✐ ❧✐➺✉ ✈➔ t❤õ tư❝ ❤➔♥❤ ❝❤➼♥❤ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥✱ ❱✐➺♥ ❚♦→♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕②✱ tr❛♥❣ ❜à ❝❤♦ tỉ✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð tr➯♥ ❝♦♥ ✤÷í♥❣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ❚ỉ✐ ❝ơ♥❣ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✷✸✱ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ị♥❣ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ t ỡ s s tợ ỳ ữớ t tr ✤➻♥❤ ❝õ❛ ♠➻♥❤✳ ◆❤ú♥❣ ♥❣÷í✐ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥ ❝❤✐❛ s➫ ❦❤â ❦❤➠♥ ✈➔ ❧✉æ♥ ♠♦♥❣ ♠ä✐ tæ✐ t❤➔♥❤ ❝æ♥❣✳ ❇↔♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ t ổ ỗ ◆❣✉②➯♥✱ t❤→♥❣ ✸ ♥➠♠ ✷✵✶✼ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❈❤❛♥t❤♦♥❡ ❑❡♦♠❛♥✐s❛② ▼ư❝ ❧ư❝ ▼Ð ✣❺❯ ✶ ✶ ❑✐➳♥ t❤ù❝ ❝ì s ỵ ỡ tr ỵ tt ✸ ✶✳✶✳✶✳ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷✳ ❍❛✐ ỵ ỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ❤❛② ❤➔♠ ♥❤ä ✳ ✳ ✶✸ ✶✳✷✳✶✳ ❑❤→✐ ♥✐➺♠ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤✐ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä ✷✶ ✷✳✶✳ ❚r÷í♥❣ ❤đ♣ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝➜♣ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳ ❚r÷í♥❣ ❤đ♣ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ❑➳t ❧✉➟♥ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ é ởt tr ỳ ữợ ự q trồ ỵ tt ❝ù✉ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ◆➠♠ ✶✾✷✻✱ ❘✳ ◆❡✈❛♥❧✐♥♥❛ ✤÷đ❝ ❝❤ù♥❣ tä ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ✺ ❣✐→ trà r✐➯♥❣ ❜✐➺t ❦❤æ♥❣ ❦➸ ❜ë✐ t❤➻ s➩ trị♥❣ ♥❤❛✉✳ ❈ỉ♥❣ tr➻♥❤ ♥➔② ❝õ❛ ➷♥❣ ữủ ỗ ự ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ❱➲ s❛✉✱ t tr ự t ữợ t ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr ữợ rss ❋✉❥✐♠♦t♦✱ ▲✳ ❙♠✐❧❡②✱ ❍✳ ❍✳ ❑❤♦❛✐✱ ●✳ ❉❡t❤❧♦❢❢✱ ❈✳ ❈✳ ❨❛♥❣✱ ▼✳ ❘✉ ✈➔ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❦❤→❝✳ ❈❤➥♥❣ ❤↕♥✱ ◆➠♠ ✶✾✽✷✱ ❋✳●r♦ss ✈➔ ❈✳❈✳ ❨❛♥❣ ✤➣ ❝❤➾ r❛ t➟♣ ❤ñ♣ T = {z ∈ C|ez + z = 0} ❧➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❦➸ ❝↔ ❜ë✐ ❝❤♦ ❝→❝ ❤➔♠ ♥❣✉②➯♥ tr➯♥ C✱ tù❝ ❧➔ ✈ỵ✐ ❤❛✐ ❤➔♠ ♥❣✉②➯♥ f ✈➔ g ✱ ✤✐➲✉ ❦✐➺♥ Ef (T ) = Eg (T ) ❦➨♦ t❤❡♦ f ≡ g ú ỵ t T ữ tr ự ổ sè ♣❤➛♥ tû✳ ◆➠♠ ✶✾✾✺✱ ❍✳❨✐ ✤➣ ①➨t t➟♣ ❤ñ♣ SY = {z ∈ C|z n + az m + b = 0}✱ tr♦♥❣ ✤â n ≥ 15, n > m ≥ 5✱ a, b ❧➔ ❝→❝ ❤➡♥❣ sè ❦❤→❝ ❦❤æ♥❣ s❛♦ ❝❤♦ z n +az m +b = ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ❜ë✐ ✈➔ ➷♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ SY ❧➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♥❣✉②➯♥ tr➯♥ C✳ ◆➠♠ ✶✾✾✽✱ ●✳❋r❛♥❦ ✈➔ ▼✳❘❡✐♥❞❡rs ❝❤➾ r❛ ♠ët ✈➼ ❞ö ✈➲ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C✳ ♥➳✉ ♠ët ❤➔♠ ✈ỵ✐ ♠å✐ ❣✐→ trà ♥❣✉②➯♥ ❞÷ì♥❣ n t❤➻ ❧➔ ◆➠♠ ✶✾✻✼✱ ❲✳ ❑✳ ❍❛②♠❛♥ ✭❬✺❪✮ ✤➣ ✤➦t r❛ ❣✐↔ t❤✉②➳t ♥❣✉②➯♥ f t❤ä❛ ♠➣♥ f nf = ❤➔♠ ❤➡♥❣✳ ❑❤✐ ♥❣❤✐➯♥ ❝ù✉ ❣✐↔ t❤✉②➳t ♥➔②✱ ♥➠♠ ✶✾✾✼✱ ❈✳ ❈✳ ❨❛♥❣ ✈➔ ự ởt ỵ ✈➲ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤✐ ❤❛✐ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❜➟❝ ♥❤➜t ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà✳ ❚ø ✤â ✤➳♥ ♥❛② ♥❤ú♥❣ ✈➜♥ ✤➲ ự t ữợ ữủ t tr ❜ð✐ ❝→❝ ❝æ♥❣ tr➻♥❤ ❝õ❛ ♥❤✐➲✉ t→❝ ❣✐↔ tr♦♥❣ ✈➔ ữợ ữ ❈✳ ▲✐♥ ✭❬✶✷❪✮✱ ❑✳ ❇♦✉ss❛❢✱ ✷ ❆✳ ❊s❝❛ss✉t ✈➔ ❏✳ ❖❥❡❞❛ ✭❬✶❪✮✱ ❘✳ ❙✳ ❉②❛✈❛♥❛❧ ✭❬✷❪✮✱ ◆✳ ❱✳ ❚❤✐♥ ✈➔ ❍✳❚ P❤✉♦♥❣ ✭❬✾❪✮✱✳✳✳✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ✈➜♥ ✤➲ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t ❜ð✐ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝õ❛ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ✏❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä✑ ✳ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦➳t q✉↔ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ◆✳ ❱✳ ❚❤✐♥ ✈➔ ❍✳❚ P❤✉♦♥❣ ✭❬✾❪✮ ✈➔ ♠ët sè t q ỗ õ ữỡ ữ s❛✉✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝ì sð ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ỡ tr ỵ tt ố tr ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ sû ❞ư♥❣ tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ❈❤÷ì♥❣ ✷✿ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤✐ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä✳ ✣➙② ❧➔ ❝❤÷ì♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦➳t q✉↔ ♥❣✉②➯♥ ❝ù✉ ❝õ❛ ◆✳ ❱✳ ❚❤✐♥ ✈➔ ❍✳❚ P❤✉♦♥❣ ✭❬✾❪✮ ✈➔ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝ ✤➣ ❝ỉ♥❣ ❜è tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙②✳ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝ì sð ỵ ỡ tr ỵ tt t t rữợ ❤➳t t❛ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ t❤÷í♥❣ ✤÷đ❝ sỷ tr ỵ tt ố tr ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ f tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱ ✤✐➸♠ z0 ∈ C ✤÷đ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ k ∈ N∗ ❝õ❛ ❤➔♠ f (z) tỗ t ởt h(z) ổ trt t✐➯✉ tr♦♥❣ ❧➙♥ ❝➟♥ U ❝õ❛ z0 s❛♦ ❝❤♦ tr♦♥❣ õ f ữủ ữợ f (z) = (z − z0 )k h(z) ◆❣❤➽❛ ❧➔ f (n) (z0 ) = 0, ✈ỵ✐ ♠é✐ n = 1, , k − ✈➔ f (k) (z0 ) = 0✳ ✣✐➸♠ z0 ✤÷đ❝ ❣å✐ ❧➔ ❝ü❝ ✤✐➸♠ ❜ë✐ k ∈ N∗ ❝õ❛ ❤➔♠ f (z) ♥➳✉ ♥â ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ k ❝õ❛ ❤➔♠ f (z) ❱ỵ✐ ♠é✐ sè t❤ü❝ x > 0✱ ❦➼ ❤✐➺✉✿ log+ x = max{log x, 0} ❑❤✐ ✤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+ , f (reiϕ ) ♥➯♥ 2π 2π log f (reiϕ ) dϕ = 2π 2π 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| ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ |r|}✳ ❍➔♠ r N (r, ∞; f ) = N (r, f ) = n(t, f ) − n(0, f ) dt + n(0, f ) log r t ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ ❝õ❛ f ✤÷đ❝ ❣å✐ ❧➔ ✭❝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 ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ t→0 ❍➔♠ 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 ố tr õ ỏ ỵ s❛✉ ✤➙② ❝❤♦ t❤➜② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ỵ f1, f2, · · · , fp✱ ❦❤✐ ✤â✿ p (1) p fν ) ≤ m(r, ν=1 p (2) ν=1 p fν ) ≤ m(r, ν=1 p (3) (5) fν ) ≤ N (r, N (r, fν ); ν=1 p fν ) ≤ N (r, N (r, fν ); ν=1 ν=1 p p fν ) ≤ T (r, ν=1 p (6) m(r, fν ); ν=1 p ν=1 p (4) m(r, fν ) + log p; T (r, fν ) + log p; ν=1 p fν ) ≤ T (r, ν=1 T (r, fν ) ν=1 ❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t ♥➔② ❧➔ ✤ì♥ ❣✐↔♥✱ t❛ ❝❤➾ ❝➛♥ ❞ü❛ t❤❡♦ t➼♥❤ ❝❤➜t ✿ ♥➳✉ a1 , , ap ❧➔ ❝→❝ sè ♣❤ù❝ ♣❤➙♥ ❜✐➺t t❤➻ p log p + ν=1 ✈➔ log+ |aν | aν ν=1 p log + p aν + log+ |aν | + log p log (p max |aν |) ν=1 ν=1, ,p ν=1 ỵ ỡ r ú tổ tr ỵ ỡ tr ỵ tt ố tr trữợ t ổ t❤ù❝ P♦✐ss♦♥ ✲ ❏❡♥s❡♥✳ ✷✾ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ N (r, 1 ) + N (r, ) f g N (r, 1 ) + N (r, ) + N (r, f ) + N (r, g) f − a2 g − a2 + S(r, f ) + S(r, g) 1 N (r, ) + N (r, ) + (N (r, f ) + N (r, g)) f − a2 g − a2 p + S(r, f ) + S(r, g) ✈➔ ) Q(f ) N (r, ) Q(g) N (r, kT (r, f ) + S(r, f ); kT (r, g) + S(r, g) ❑❤✐ ✤â tø ✭✷✳✶✺✮ t❛ t❤✉ ✤÷đ❝ T (r, F ) + T (r, G) 1 (1 + )(N (r, ) + N (r, )) s f g l 1 ) + N (r, )) f − a g − a i i i=3 1 ) + N (r, )) + (5 − k2 )(N (r, f − a2 g − a2 + (N (r, f ) + N (r, g)) + k(T (r, f ) + T (r, g)) p + (4 − ki )(N (r, ✭✷✳✷✶✮ + S(r, f ) + S(r, g) ◆➳✉ ki = ✈ỵ✐ ♠å✐ i ∈ {2, , l}✱ tø ✭✷✳✶✽✮ t❛ ❝â l T (r, F ) + T (r, G) 1 (1 + )(N (r, ) + N (r, )) + (N (r, ) s f g f − a i i=2 1 + N (r, )) + (N (r, ) + N (r, )) g − f − a2 g − a2 + (N (r, f ) + N (r, g)) + (l − 1)(T (r, f ) + T (r, g)) p + S(r, f ) + S(r, g) ✭✷✳✷✷✮ ✸✵ ◆❣♦➔✐ r❛ T (r, F ) = (n + k + 1)T (r, f ) + S(r, f ); ✭✷✳✷✸✮ T (r, G) = (n + k + 1)T (r, g) + S(r, g) ✭✷✳✷✹✮ ◆❤÷ ✈➟②✱ tø ✭✷✳✷✶✮ ✈➔ ✭✷✳✷✸✮✱ ✭✷✳✷✹✮ t❛ ❝â (n − ( + + max(0, − k2 ) + s p l max(0, − ki )))(T (r, f ) + T (r, g)) i=3 S(r, f ) + S(r, g), + +max(0, 5−k2 )+ s p ❱➔ ♥➳✉ ki = ✈ỵ✐ ♠å✐ i ∈ {2, , l}✱ t❛ t ữủ t ợ n > (n − (l + + ))(T (r, f ) + T (r, g)) s p ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ n > l + l i=3 max(0, 4−ki ) S(r, f ) + S(r, g), ✭✷✳✷✺✮ + ✳ ◆❤÷ ✈➟②✱ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✱ s p F ≡ G ❤♦➦❝ F G ≡ ◆➳✉ F G ≡ 1✱ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✹ t❛ s➩ ❝â ♠➙✉ t❤✉➝♥✳ ◆➳✉ F ≡ G ✱ t❤➻ F = G + c✱ tr♦♥❣ ✤â c ❧➔ ♠ët ❤➡♥❣ sè✳ ❚ø ✤➥♥❣ t❤ù❝ ♥➔② t❛ s✉② r❛ T (r, f ) = T (r, g) + O(1) ◆➳✉ c = 0✱ t ỵ ỡ tự t õ T (r, F ) 1 ) + N (r, ) + S(r, f ) F F −c 1 N (r, f ) + N (r, ) + N (r, ) f Q(f ) 1 + N (r, ) + N (r, ) + S(r, f ) g Q(g) N (r, F ) + N (r, (2k + 3)T (r, f ) + S(r, f ) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ (n − (k + 2))T (r, f ) S(r, f ), ✸✶ ♠➙✉ t❤✉➝♥ ✈ỵ✐ n k + ❇ð✐ ✈➟② c ≡ 0✱ ❞♦ ✤â P (f ) ≡ P (g)✳ ❚ø P (z) ❧➔ ✤❛ t❤ù❝ ❞✉② t M(C) tr ỵ t s r f g ỵ ữủ ự P ❧➔ ✤❛ t❤ù❝ ❞✉② ♥❤➜t ❝❤♦ A(C) s❛♦ ❝❤♦ P = b(x − a1 )n li=2 (x − )k ✈ỵ✐ b ∈ C∗ ✈➔ l 2✱ ki ki+1 ✈ỵ✐ ♠é✐ i l − ❦❤✐ l > ✈➔ n k + ✈ỵ✐ k = li=2 ki ✳ ●✐↔ sû P t❤ä❛ ♠➣♥ n > + max(0, − k2 ) + li=3 max(0, − ki ) ❈❤♦ f, g ∈ A(C) ❧➔ ❤❛✐ s ❤➔♠ s✐➯✉ ✈✐➺t t❤ä❛ ♠➣♥ ❝→❝ a1− ✤✐➸♠ ❝õ❛ ❝❤ó♥❣ ❝â ❜ë✐ ➼t ♥❤➜t ❧➔ s✱ tr♦♥❣ ✤â s ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ α ∈ Af (C) Ag (C) ổ ỗ t ổ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ α ❈▼✱ t❤➻ f g ỵ i ự t ự tữỡ tỹ ữ ỵ ợ ú ỵ f g ♥❣✉②➯♥✳ ❉♦ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✶✮ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ỵ t t t t t❤ù❝ T (r, F ) + T (r, G) 1 (1 + )(N (r, ) + N (r, )) + k(T (r, f ) + T (r, g)) s f g l 1 ) + N (r, )) f − a g − a i i i=3 1 + (5 − k2 )(N (r, ) + N (r, )) f − a2 g − a2 + (4 − ki )(N (r, ✭✷✳✷✻✮ + S(r, f ) + S(r, g) ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ (n − ( + max(0, − k2 ) + s l max(0, − ki )))(T (r, f ) + T (r, g)) i=3 S(r, f ) + S(r, g) ❇➜t ✤➥♥❣ t❤ù❝ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ n > + max(0, − k2 ) + s l max(0, − ki ) i=3 ✸✷ ❱➔ ♥➳✉ ki = 1✱ ∀i ∈ {2, , l}✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✷✮ ✤÷đ❝ t❤❛② t❤➳ ❜ð✐ (n − (l + ))(T (r, f ) + T (r, g)) S(r, f ) + S(r, g), s ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ n > l + ▲➟♣ ❧✉➟♥ t÷ì♥❣ tỹ ữ ỵ s t s t ự ỵ t ỵ sỹ rở t q ❝õ❛ ❑✳ ❇♦✉ss❛❢✱ ❆✳ ❊s❝❛ss✉t ✈➔ ❏✳ ❖❥❡❞❛✳ ❈ö t❤➸✱ tr ỵ t s = p = t s t ữủ ỵ ✷✳✷ ✈➔ ✷✳✸ t÷ì♥❣ ù♥❣✳ ❈❤♦ f (z) ✈➔ g(z) ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t tr♦♥❣ ✤â ❝→❝ ❦❤ỉ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ❝õ❛ ❝❤ó♥❣ ❝â ❜ë✐ ➼t ♥❤➜t s ✈➔ p t÷ì♥❣ ù♥❣✱ tr♦♥❣ ✤â s ✈➔ p ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ α(z) ∈ Mf (C) Mg (C) ổ ỗ t ổ a C∗ ✳ ◆➳✉ f f n (f −a) ✈➔ g g n (g− a) ❝❤✉♥❣ ♥❤❛✉ ❤➔♠ α ❈▼ ✈➔ ♥➳✉ n > + + t❤➻ ❤♦➦❝ f g s p tỗ t h M(C) tọ ỵ a(n + 2)(1 − hn+1 ) f= h; (n + 1)(1 − hn+2 ) a(n + 2)(1 − hn+1 ) g= (n + 1)(1 − hn+2 ) ▲➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr ỵ ợ ú ỵ r ✭✷✳✷✺✮ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ n > + + tr ỵ s p ỡ ỳ t ❝â f n+2 f n+1 g n+2 g n+1 P (f ) = −a , P (g) = −a n+2 n+1 n+2 n+1 ❚ø ✤✐➲✉ ❦✐➺♥ P (f ) = P (g)✱ t❛ ❝â n+2 n+2 f n+1 (f − a ) = g n+1 (g − a ) ✭✷✳✷✼✮ n+1 n+1 f ✣➦t h = ◆➳✉ h ≡ t❤➻ f ≡ g ◆➳✉ h ≡ 1, t❤➻ f ≡ g ❍ì♥ ♥ú❛✱ tø g ✭✷✳✷✼✮✱ t❛ s✉② r❛ ❈❤ù♥❣ ♠✐♥❤✳ a(n + 2)(1 − hn+1 ) f= h, (n + 1)(1 hn+2 ) ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ a(n + 2)(1 − hn+1 ) g= (n + 1)(1 − hn+2 ) ✸✸ ✷✳✷✳ ❚r÷í♥❣ ❤ñ♣ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ◆➠♠ ✷✵✵✽✱ ❳✳ ❨✳ ❩❤❛♥❣✱ ❏✳ ❋✳ ❈❤❡♥✱ ❲✳ ❈✳ ▲✐♥ ❬✶✷❪ ❝❤ù♥❣ ♠✐♥❤ ❈❤♦ f (z) ✈➔ g(z) ❧➔ ❤❛✐ ❤➔♠ ♥❣✉②➯♥ ❦❤→❝ ❤➡♥❣✱ ❝❤♦ n✱ k ✈➔ m ❧➔ ❜❛ số ữỡ ợ n 3m + 2k + ✈➔ P (z) = am z m + am−1 z m−1 + · · · + a1 z + a0 ❤♦➦❝ P (z) ≡ c0 ✱ tr♦♥❣ ✤â a0 = 0, a1 , , am−1 ✱ am = 0✱ c0 = ❧➔ ❝→❝ sè ♣❤ù❝✳ ◆➳✉ [f n P (f )](k) ✈➔ [g n P (g)](k) ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ❈▼✱ t❤➻✿ (i) ❑❤✐ P (z) = am z m + am−1 z m−1 + · · · + a1 z + a0 ✱ ❤♦➦❝ f (z) ≡ tg(z) ✤è✐ ✈ỵ✐ ♠ët ❤➡♥❣ sè t s❛♦ ❝❤♦ td = 1✱ tr♦♥❣ ✤â d = (n + m, , n + m − i, , n)✱ am−i = ✤è✐ ✈ỵ✐ i = 0, 1, , m✱ ❤♦➦❝ f ✈➔ g t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè R(f, g) = 0✱ tr♦♥❣ ✤â R(w1, w2) = w1n(amw1m + am−1 w1m−1 + · · · + a0 ) − w2n (am w2m + am−1 w2m−1 + · · · + a0 )❀ √ √ (ii) ❑❤✐ P (z) ≡ c0 ✱ ❤♦➦❝ f (z) = c1 / c0 ecz ✱ g(z) = c2 / c0 e−cz ✱ tr♦♥❣ ✤â c1, c2 ✈➔ c ❧➔ ❜❛ ❤➡♥❣ sè t❤ä❛ ♠➣♥ (−1)k (c1c2)n(nc)2k = 1✱ ❤♦➦❝ f ≡ tg ✈ỵ✐ ♠ët ❤➡♥❣ sè t t❤ä❛ ♠➣♥ tn = ✣à♥❤ ỵ n n tử ự t ữợ P❤✉♦♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤✿ ❈❤♦ f (z) ✈➔ g(z) ❧➔ ❤❛✐ ❤➔♠ ♥❣✉②➯♥ s✐➯✉ ✈✐➺t t❤ä❛ ♠➣♥ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ❝❤ó♥❣ ❝â ❜ë✐ ➼t ♥❤➜t s✱ tr♦♥❣ ✤â s ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣❀ ❝❤♦ n✱ k ✈➔ m ❧➔ ❜❛ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ P (z) = amz m + am−1 z m−1 + · · · + a1 z + a0 ❤♦➦❝ P (z) ≡ c0 ✱ tr♦♥❣ ✤â a0 = 0, a1 , , am−1 ✱ am = 0✱ c0 = ❧➔ ❝→❝ sè ♣❤ù❝✳ ◆➳✉ [f n P (f )](k) ✈➔ [g n P (g)](k) ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ❈▼ t❤➻✿ k+2 (i) ❑❤✐ P (z) = am z m + am−1 z m−1 + · · · + a1 z + a0 ✈➔ n > m + ✱ s t❤➻ ❤♦➦❝ f (z) ≡ tg(z) ✈ỵ✐ ♠ët ❤➡♥❣ sè t t❤ä❛ ♠➣♥ td = 1✱ tr♦♥❣ ✤â d = (n + m, , n + m − i, , n)✱ am−i = ✈ỵ✐ ♠ët sè i = 0, 1, , m✱ ❤♦➦❝ f ✈➔ g t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè R(f, g) = 0✱ tr♦♥❣ ✤â R(w1 , w2 ) = w1n (am w1m + am−1 w1m−1 + · · · + a0 ) − w2n (am w2m + am−1 w2m−1 + · · · + a0 ) ỵ 2k + ✱ t❤➻ ❤♦➦❝ f (z) = c1 / n c0 ecz ✱ g(z) = (ii) ❑❤✐ P (z) ≡ c0 ✈➔ n > s √ c2 / n c0 e−cz ✱ tr♦♥❣ ✤â c1 , c2 ✈➔ c ❧➔ ❜❛ ❤➡♥❣ sè t❤ä❛ ♠➣♥ (−1)k (c1 c2 )n (nc)2k = ❤♦➦❝ f ≡ tg ✈ỵ✐ ♠ët ❤➡♥❣ sè t s❛♦ ❝❤♦ tn = ❈❤ù♥❣ ♠✐♥❤✳ (i) ❑➼ ❤✐➺✉ P (z) = am z m + am−1 z m−1 + · · · + a1 z + a0 ❚❛ ✤➦t F = f n P (f )✱ G = g n P (g)✳ ❚❤❡♦ ❣✐↔ t❤✐➳t t❛ t❤➜② r➡♥❣F (k) ✈➔ G(k) ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ❈▼✳ ❉♦ ✤â ) G T (r, G) Nk+2 (r, δk+2 (0, G) = − lim sup r→∞ ) g n P (g) = − lim sup (n + m)T (r, g) r→∞ (k + 2)N (r, ) + mT (r, g) g − lim sup (n + m)T (r, g) r→∞ k+2 T (r, g) + mT (r, g) s − lim sup (n + m)T (r, g) r→∞ k+2 m+ s =1− n+m Nk+2 (r, ✭✷✳✷✽✮ ❚÷ì♥❣ tü t❛ ❝â k+2 s n+m m+ δk+2 (0, F ) 1− ✭✷✳✷✾✮ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✼✱ t❛ ❝â k+2 s δk+2 (0, G) + δk+2 (0, F ) − n+m k+2 2n − s = n+m m+ ✭✷✳✸✵✮ ✸✺ ❚ø n > m + k+2 ✱ t❛ t❤➜② r➡♥❣ s δk+2 (0, G) + δk+2 (0, F ) > ◆❤÷ ✈➟② F ≡ G ♦r F (k) G(k) ≡ ◆➳✉ F (k) G(k) ≡ 1✱ tù❝ ❧➔ [f n (am f m + · · · + a0 )](k) [g n (am g m + · · · + a0 )](k) ≡ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✸✱ tr÷í♥❣ ❤đ♣ ♥➔② ❦❤ỉ♥❣ t❤➸ ①↔② r❛✳ ◆➳✉ F ≡ G✱ tù❝ ❧➔ f n (am f m + · · · + a0 ) = g n (am g m + · · · + a0 ) ✣➦t h = ✤÷đ❝ ✭✷✳✸✶✮ f ✳ ◆➳✉ h ❧➔ ❤➡♥❣ sè t❤➻ t❛ t❤❛② t❤➳ f = hg ✈➔♦ ✭✷✳✸✶✮ t❛ t❤✉ g am g n+m (hn+m − 1) + am−1 g n+m−1 (hn+m−1 − 1) + · · · + a0 g n (hn − 1) = 0, ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ hd = 1✱ tr♦♥❣ ✤â d = (n + m, , n + m − i, , n)✱ am−i = ✈ỵ✐ ♠ët ❣✐→ trà i = 0, 1, m✳ ❑❤✐ ✤â f ≡ tg ✈ỵ✐ ♠ët ❤➡♥❣ sè t t❤ä❛ ♠➣♥ td = 1✳ ◆➳✉ h ❧➔ ❤➡♥❣ sè✱ tø ✭✷✳✸✶✮✱ t❛ t❤➜② r➡♥❣ f ✈➔ g t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè R(f, g) = 0✱ tr♦♥❣ ✤â R(w1 , w2 ) = w1n (am w1m + am−1 w1m−1 + · · · + a0 ) − w2n (am w2m + am−1 w2m−1 + · · · + a0 ) (ii) ▲➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ tr÷í♥❣ ❤đ♣ (i)✱ t❛ ✤➦t F = f n ✱ ✸✻ G = g n ✳ ❑❤✐ ✤â ) G δk+2 (0, G) = − lim sup T (r, G) r→∞ Nk+2 (r, n ) g = − lim sup nT (r, g) r→∞ Nk+2 (r, (k + 2)N (r, ) g − lim sup nT (r, g) r→∞ k+2 T (r, g) s − lim sup nT (r, g) r→∞ k+2 =1− sn ✭✷✳✸✷✮ ❚÷ì♥❣ tü t❛ ❝â δk+2 (0, F ) 1− k+2 sn ✭✷✳✸✸✮ 2k + ✱ t÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤đ♣ (i) ✈➔ →♣ ❞ư♥❣ ▼➺♥❤ ✤➲ ✶✳✸✱ s t❛ t❤✉ ✤÷đ❝ ❦❤➥♥❣ ✤à♥❤ ỵ ứ n > ỵ ởt trữớ ủ t ỵ ①➨t ✷✳✷✳ ❦❤✐ f ✈➔ g ❧➔ ❝→❝ ❤➔♠ ♥❣✉②➯♥ s✐➯✉ ✈✐➺t✳ ❚✉② ♥❤✐➯♥✱ ②➯✉ ❝➛✉ ✈➲ n tr♦♥❣ k+2 ỵ n > m + ọ ỡ tr ỵ s f (z) ✈➔ g(z) ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t tr♦♥❣ ✤â ❝→❝ ❦❤ỉ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ❝õ❛ ❝❤ó♥❣ ❝â ❜ë✐ ➼t ♥❤➜t s ✈➔ p t÷ì♥❣ ù♥❣✱ tr♦♥❣ ✤â s ✈➔ p ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ α(z) ∈ Mf (C) Mg (C) ổ ỗ t ổ n✱ m✱ v ✈➔ k ❧➔ ❜è♥ sè ♥❣✉②➯♥ ữỡ s ỵ n k + 1, n+m> k+4 k+2 + 2{v(k + 2) + deg Q + } p s ✸✼ ❈❤♦ P (z) = am z m + am−1 z m−1 + · · · + a1 z + a0 = (z − b1 )m1 (z − bv )mv Q(z), tr♦♥❣ ✤â mi k + ✈ỵ✐ ♠ët sè i = 1, , v✱ v + p1 ✱ m = deg Q + v i=1 mi ✱ tr♦♥❣ ✤â a0 = 0, a1 , , am−1 ✱ am = 0✱ c0 = ❧➔ ❝→❝ ❤➡♥❣ sè ♣❤ù❝✳ ◆➳✉ [f nP (f )](k) ✈➔ [gnP (g)](k) ❝❤✉♥❣ ♥❤❛✉ α(z) CM ✱ t❤➻ ❤♦➦❝ f (z) ≡ tg(z) ✈ỵ✐ t ❧➔ ❤➡♥❣ sè t❤ä❛ ♠➣♥ td = 1✱ tr♦♥❣ ✤â d = (n + m, , n + m − i, , n)✱ am−i = ✈ỵ✐ ♠ët sè i = 0, 1, , m✱ ❤♦➦❝ f ✈➔ g t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè R(f, g) = 0✱ tr♦♥❣ ✤â R(w1, w2) = w1n (am w1m + am−1 w1m−1 + · · · + a0 ) − w2n (am w2m + am−1 w2m−1 + · · · + a0 ) ✣➦t P (z) = (z − b1 )m1 (z − bv )mv Q(z)✱ tr♦♥❣ ✤â mi k + ❢♦r i = 1, , v ✱ v + ✱ m = deg Q + vi=1 mi ✣➦t p n n F = f P (f ) ✈➔ G = g P (g)✳ ❚❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â F (k) ✈➔ G(k) ❝❤✉♥❣ ❈❤ù♥❣ ♠✐♥❤✳ ♥❤❛✉ α(z) ❈▼✳ ❚❛ ❝â ) G T (r, G) Nk+2 (r, δk+2 (0, G) = − lim sup r→∞ ) g n P (g) = − lim sup (n + m)T (r, g) r→∞ (k + 2)N (r, ) + (v(k + 2) + deg Q)T (r, g) g − lim sup (n + m)T (r, g) r→∞ k+2 T (r, g) + (v(k + 2) + deg Q)T (r, g) s − lim sup (n + m)T (r, g) r→∞ Nk+2 (r, ❉♦ ✤â (v(k + 2) + deg Q) + δk+2 (0, G) 1− n+m k+2 s ✭✷✳✸✹✮ ✸✽ ❚÷ì♥❣ tü t❛ ❝â (v(k + 2) + deg Q) + δk+2 (0, F ) 1− n+m k+2 s ✭✷✳✸✺✮ N (r, G) r→∞ T (r, G) N (r, g) = − lim sup r→∞ (n + m)T (r, g) N (r, g) p − lim sup r→∞ (n + m)T (r, g) T (r, g) p − lim sup r→∞ (n + m)T (r, g) =1− p(n + m) Θ(∞, G) = − lim sup ✭✷✳✸✻✮ ❚÷ì♥❣ tü t❛ ❝â Θ(∞, F ) 1− p(n + m) ✭✷✳✸✼✮ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✻✱ t❛ t❤✉ ✤÷đ❝ 2Θ(∞, G) + δk+2 (0, G) + (k + 2)Θ(∞, F ) + δk+2 (0, F ) = (k + 4)(1 − ) + 2(1 − p(n + m) (v(k + 2) + deg Q) + n+m k+2 s ) ✭✷✳✸✽✮ ❚ø m + n > k+4 k+2 + 2{v(k + 2) + deg Q + }✱ t❛ t❤✉ ✤÷đ❝ p s 2Θ(∞, G) + δk+2 (0, G) + (k + 2)Θ(∞, F ) + δk+2 (0, F ) > k + ❇ð✐ ✈➟②✱ F ≡ G ♦r F (k) G(k) ≡ (α(z))2 ◆➳✉ F ≡ G✱ tù❝ ❧➔ f n (am f m + · · · + a0 ) = g n (am g m + · · · + a0 ) ✭✷✳✸✾✮ ✸✾ ✣➦t h = ✤÷đ❝ f ✳ ◆➳✉ h ❧➔ ❤➡♥❣ sè t❤➻ t❛ t❤❛② t❤➳ f = hg ✈➔♦ ✭✷✳✸✾✮✱ t❛ t❤✉ g am g n+m (hn+m − 1) + am−1 g n+m−1 (hn+m−1 − 1) + · · · + a0 g n (hn − 1) = 0, ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ hd = 1✱ tr♦♥❣ ✤â d = (n + m, , n + m − i, , n)✱ am−i = ✈ỵ✐ ♠ët sè i = 0, 1, m✳ ❑❤✐ ✤â f ≡ tg ✈ỵ✐ ♠ët ❤➡♥❣ sè t t❤ä❛ ♠➣♥ td = 1✳ ◆➳✉ h ❧➔ ♠ët ❤➡♥❣ sè✱ tø ✭✷✳✸✾✮✱ t❛ ❞➵ ❞➔♥❣ t❤✉ ✤÷đ❝ f ✈➔ g t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè R(f, g) = 0✱ tr♦♥❣ ✤â R(w1 , w2 ) = w1n (am w1m + am−1 w1m−1 + · · · + a0 ) − w2n (am w2m + am−1 w2m−1 + · · · + a0 ) ✣➦t ✭✷✳✹✵✮ F (k) G(k) ≡ (α(z))2 ❚❛ ❦➼ ❤✐➺✉ ❧➔ t➟♣ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ❝õ❛ (α(z))2 ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ F (k) ✈➔ G(k) s➩ ❝â ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ♥➡♠ ♥❣♦➔✐ ❚❤➟t ✈➟②✱ ❣✐↔ sû t➜t ❝↔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ❝õ❛ F (k) ✤➲✉ t❤✉ë❝ ✳ ❑❤✐ ✤â N (r, F (k) ) + N (r, ) F (k) 2T (r, α(z))2 ) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ v N (r, f ) + N (r, i=1 ) f − bi S(r, f ) ❚❤❡♦ ✤à♥❤ ỵ ỡ tự t õ v (v 1)T (r, f ) N (r, f ) + N (r, i=1 ❦❤✐ ✤â (v − 1)T (r, f ) ) + S(r, f ), f − bi S(r, f ) ✣➙② ❝❤➼♥❤ ❧➔ ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ◆❤÷ ✈➟②✱ F (k) ✈➔ G(k) ❝â ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ♥➡♠ ♥❣♦➔✐ ❚❛ ❣✐↔ t❤✐➳t r➡♥❣ z0 ❧➔ ♠ët ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f ❜➟❝ s1 ❦❤æ♥❣ t❤✉ë❝ ✱ ❦❤✐ ✤â z0 ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ [f n P (f )](k) ✳ ◆❤÷ ✈➟② z0 ❧➔ ♠ët ❝ü❝ ✤✐➸♠ ❝õ❛ [g n P (g)](k) ✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ z0 ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ g ❜➟❝ p1 p✳ ❚ø ✭✷✳✹✵✮✱ t❛ ❝â ✹✵ (n + m)p + 2k ❚÷ì♥❣ tü✱ t❛ ❣✐↔ n sû r➡♥❣ zi ❧➔ bi − ✤✐➸♠ ❝õ❛ f ✈ỵ✐ ❜➟❝ si ✱ ❦❤✐ ✤â zi ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g (n + m)p + 2k ✈ỵ✐ ❜➟❝ pi ✱ i = 1, , v ✳ ❑❤✐ ✤â si , i = 1, , v mi ỵ ❝ì ❜↔♥ t❤ù ❤❛✐ t❛ ❝â ns1 − k = (n + m)p1 + k ✱ ❦❤✐ ✤â s1 (1 + )T (r, f ) p vT (r, f ) N (r, f ) + N (r, ) + f n + ( + p (n + m)p + 2k v i=1 v N (r, i=1 ) + S(r, f ) f − bi mi )T (r, f ) + S(r, f ) (n + m)p + 2k ❇ð✐ ✈➟② n + vi=1 mi (1 − )T (r, f ) (n + m)p + 2k S(r, f ) ✣✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ◆❤÷ ✈➟② ✭✷✳✹✵✮ ❦❤ỉ♥❣ t❤➸ ①↔② r❛✳ ✹✶ ❑➳t ❧✉➟♥ ❱ỵ✐ ♠ư❝ ✤➼❝❤ t tử ự ỳ ự ỵ tt ♣❤➙♥ ❜è ❣✐→ trà tr♦♥❣ ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥ tr♦♥❣ ỵ tt ố tr ❤➻♥❤ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❧↕✐ ♠ët sè t➼♥❤ ❝❤➜t ✈➲ ♣❤➙♥ ❜è ❣✐→ trà ❝õ❛ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ✤÷đ❝ ①❡♠ ❧➔ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✱ ❝➛♥ t❤✐➳t ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ✷✳ P❤→t ❜✐➸✉ ự ởt số ỵ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤✐ ✤❛ t❤ù❝ ❝❤ù❛ ❝→❝ ✤↕♦ ❤➔♠ ❝➜♣ ✶ ❤❛② ❝➜♣ ❝❛♦ ❝õ❛ ❝→❝ ❤➔♠ ♥➔② ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä✳ ❈→❝ ỵ tứ t q tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❜➟❝ ✶✱ ỵ tứ ỵ t tr trữớ ủ tự ự ✤↕♦ ❤➔♠ ❜➟❝ ❝❛♦✳ ❈→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ♠ỵ✐ ❝❤➾ ❞ø♥❣ ❧↕✐ ð ✈✐➺❝ ❦❤↔♦ s→t ♠ët ✈➔✐ ❞↕♥❣ ✤➦❝ ❜✐➺t ❝õ❛ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝➜♣ ✶ ❤❛② ❝➜♣ ❝❛♦✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐ ❝❤ó♥❣ tỉ✐ s➩ t✐➳♣ tư❝ ♣❤→t tr✐➸♥ ✈➜♥ ✤➲ ♥➔② ✤è✐ ✈ỵ✐ ♠ët sè ❞↕♥❣ ✤❛ t❤ù❝ ❦❤→❝✳ ✹✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❇♦✉ss❛❢✱ ❑✳✱ ❊s❝❛ss✉t✱ ❆✳ ❛♥❞ ❖❥❡❞❛✱ ❏✳ ✭✷✵✶✸✮✱ ✧❈♦♠♣❧❡① ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s f P (f )✱ g P (g) s❤❛r✐♥❣ ❛ s♠❛❧❧ ❢✉♥❝t✐♦♥✧✱ ■♥❞❛❣❛t✐♦♥❡s ▼❛t❤❡♠❛t✐❝❛❡✱ ✷✹✱ ✶✺✲✹✶✳ ❬✷❪ ❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❏✳ ✸✼✹✱ ❬✸❪ ✭✷✵✶✶✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ❞✐❢❢❡r✲ ❉②❛✈❛♥❛❧✱ ❘✳ ❙✳ ✸✸✺✲✸✹✺✳ ❋r❛♥❦✱ ●✳ ❛♥❞ ❘❡✐♥❞❡rs✱ ▼✳ ✭✶✾✾✽✮✱ ✧❆ ✉♥✐q✉❡ r❛♥❣❡ s❡t ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✇✐t❤ ✶✶ ❡❧❡♠❡♥ts✧✱ ❆♣♣❧✱ ✸✼✱ ✶✽✺✲✶✾✸✳ ❬✹❪ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✱ ●r♦ss✱ ❋✳ ❈♦♠♣❧❡①✳ ❱❛r✳ ❚❤❡♦r②✳ ✭✶✾✼✼✮✱ ✧❋❛❝t♦r✐③❛t✐♦♥ ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ s♦♠❡ ♦♣❡♥ ♣r♦❜❧❡♠s✧✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✺✾✾✱ ✺✶✲✻✾✱ ❙♣r✐♥❣❡r ✭❇❡r❧✐♥✮✳ ❬✺❪ ❍❛②♠❛♥✱ ❲✳ ❑✳ ✭✶✾✻✼✮✱ ❘❡s❡❛r❝❤ Pr♦❜❧❡♠s ✐♥ ❋✉♥❝t✐♦♥ ❚❤❡♦r②✱ ❆t❤❧♦r❡ Pr❡ss ✭❯♥✐✈✳ ♦❢ ▲♦♥❞♦♥✮✳ ❬✻❪ ▲✐✱ P✳ ❛♥❞ ❨❛♥❣✱ ❈✳ ❈✳ ✭✶✾✾✺✮✱ ✧❙♦♠❡ ❢✉rt❤❡r r❡s✉❧ts ♦♥ t❤❡ ✉♥✐q✉❡ r❛♥❣❡ s❡ts ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❑♦❞❛✐ ▼❛t❤✳ ❏✱ ✶✽✱ ✹✼✸✲✹✺✵✳ ❬✼❪ ▲✐♥✱ ❲✳ ❛♥❞ ❨✐✱ ❍✳ ✭✷✵✵✹✮✱ ✧❯♥✐q✉❡♥❡ss t❤❡♦r❡♠s ❢♦r ♠❡r♦♠♦r✲ ♣❤✐❝ ❢✉♥❝t✐♦♥s ❝♦♥❝❡r♥✐♥❣ ❢✐①❡❞ ✲ ♣♦✐♥ts✧✱ ❆♣♣❧✳✱ ✹✾✱ ✼✾✸✲✽✵✻✳ ❬✽❪ ▲✐♥✱ ❲✳ ❛♥❞ ❨✐✱ ❍✳ ♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❈♦♠♣❧❡①✳ ❱❛r✳ ❚❤❡♦r② ✭✷✵✵✹✮✱ ✧❯♥✐q✉❡♥❡ss t❤❡♦r❡♠s ❢♦r ♠❡r♦♠♦r✲ ■♥❞✐❛♥✳ ❏✳ P✉r❡ ❆♣♣❧✳ ▼❛t❤✳✱ ✸✺✱ ✶✷✶✲✶✸✷✳ ✹✸ ❬✾❪ ❚❤✐♥✱ ◆✳ ❱✳ ❛♥❞ P❤✉♦♥❣✱ ❍✳ ❚✳ ✭✷✵✶✻✮✱ ✧❯♥✐q✉❡♥❡ss ♦❢ ♠❡r♦✲ ♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s s❤❛r✐♥❣ ❛ ✈❛❧✉❡ ♦r s♠❛❧❧ ❢✉♥❝t✐♦♥✧✱ ▼❛t❤✳ ❙❧♦✈❛❝❛✳ ✻✻ ✭✹✮✱ ❬✶✵❪ ✽✷✾✕✽✹✹✳ ❨❛♥❣✱ ❈✳ ❈✳ ❛♥❞ ❍✉❛✱ ❳✳ ❍✳ s❤❛r✐♥❣ ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥✧✱ ✷✵✷ ✭✷✮✱ ❬✶✶❪ ❆♥♥✳ ❆❝❛❞✳ ❙❝✐✳ ❋❡♥♥✳ ▼❛t❤✱ ✸✾✺✲✹✵✻✳ ❨✐✱ ❍✳ ❨✳ ✭✶✾✾✺✮✱ ✧❆ q✉❡st✐♦♥ ♦❢ ●r♦ss ❛♥❞ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❬✶✷❪ ✭✶✾✾✼✮ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳✱ ✶✸✽✱ ✶✻✾✲✶✼✼✳ ❩❤❛♥❣✱ ❳✳ ❨✳✱ ❈❤❡♥✱ ❏✳ ❋✳ ❛♥❞ ▲✐♥✱ ❲✳ ❈✳ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s s❤❛r✐♥❣ ♦♥❡ ✈❛❧✉❡✧✱ ♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ✺✻✱ ✶✽✼✻✲✶✽✽✸✳ ✭✷✵✵✽✮✱ ✧❊♥t✐r❡ ♦r ❈♦♠♣✉t❡rs ❛♥❞ ▼❛t❤❡✲