Vấn đề duy nhất của lũy thừa một hàm phân hình với đa thức đạo hàm của chúng

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ▲➊ ❱❿◆ ❈❍×❒◆● ❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ❈Õ❆ ▲Ơ❨ ❚❍Ø❆ ▼❐❚ ❍⑨▼ P❍❹◆ ❍➐◆❍ ❱❰■ ✣❆ ❚❍Ù❈ ✣❸❖ ❍⑨▼ ❈Õ❆ ❈❍Ó◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ▲➊ ❱❿◆ ❈❍×❒◆● ❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ❈Õ❆ ▲Ơ❨ ❚❍Ø❆ ▼❐❚ ❍⑨▼ P❍❹◆ ❍➐◆❍ ❱❰■ ✣❆ ❚❍Ù❈ ✣❸❖ ❍⑨▼ ❈Õ❆ ❈❍Ó◆● ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼➣ sè ✿ ✽✳✹✻✳✵✶✳✵✷ ữớ ữợ ❤å❝✿ P●❙✳ ❚❙ ❍⑨ ❚❘❺◆ P❍×❒◆● ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵ ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❝→❝ ❦➳t q✉↔ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥✱ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✈➔ ♥ë✐ ❞✉♥❣ tr➼❝❤ ❞➝♥ ✤↔♠ ❜↔♦ t➼♥❤ tr✉♥❣ t❤ü❝ ❝❤➼♥❤ ①→❝✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✷✵ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ▲➯ ❱➠♥ ❈❤÷ì♥❣ ❳→❝ trữ ữớ ữợ ❞➝♥ P●❙✳ ❚❙ ❍⑨ ❚❘❺◆ P❍×❒◆● ✐ ▲í✐ ❝↔♠ ì♥ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ P●❙✳ ❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣✱ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ✈➔ ❣✐ó♣ ✤ï tỉ✐ ❝â t❤➯♠ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝✱ ❦❤↔ ♥➠♥❣ ♥❣❤✐➯♥ ❝ù✉✱ tê♥❣ ❤ñ♣ t➔✐ ❧✐➺✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ ❤♦➔♥ ❝❤➾♥❤✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ỗ ú ✤ï tỉ✐ q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝õ❛ ♠➻♥❤✳ ❉♦ t❤í✐ ❣✐❛♥ ✈➔ tr➻♥❤ ✤ë ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❈❤ó♥❣ tỉ✐ r➜t ♠♦♥❣ ữủ sỹ õ ỵ t ổ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✷✵ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ▲➯ ❱➠♥ ❈❤÷ì♥❣ ✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ▲í✐ ❝↔♠ ì♥ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ✶✳✷ ❈→❝ ❤➔♠ ỵ ỡ ✳ ✳ ✶✳✶✳✶✳ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ỵ ỡ ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✸✳ ◗✉❛♥ ❤➺ sè ❦❤✉②➳t ✈➔ ✤✐➸♠ ❜ä ✤÷đ❝ P✐❝❛r❞ ✳ ❍➔♠ ✤➳♠ ♠ð rë♥❣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤➳♠ ♠ð rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ✐✐ ✐✐✐ ✶ ✸ ✳ ✸ ✳ ✸ ✳ ✺ ✳ ✻ ✳ ✼ ✳ ✼ ✳ ✶✵ ✷ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✶✼ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✵ ✹✷ ✷✳✶ ✷✳✷ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✐✐✐ ▼ð ✤➛✉ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱ a ∈ C ∪ {∞}✳ ❚❛ ❦➼ ❤✐➺✉ E f (a) = f −1 (a) = {z ∈ C : f (z) = a} Ef (a) = {(z, n) ∈ C × N : f (z) = a, ordf −a (z) = n} ❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C ✈➔ a ❧➔ ♠ët ❣✐→ trà ♣❤ù❝ ❤ú✉ ❤↕♥ ❤♦➦❝ ∞✳ ❚❛ ♥â✐ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ a ❦➸ ❝↔ ❜ë✐ ✭✈✐➳t ♥❣➢♥ ❣å♥ ❧➔ ❈▼✮ ♥➳✉ Ef (a) = Eg (a)✳ ❚❛ ♥â✐ f ✈➔ g ❝❤✉♥❣ ♥❤❛✉ a ❦❤æ♥❣ ❦➸ ❜ë✐ ✭✈✐➳t ♥❣➢♥ ❣å♥ ❧➔ ■▼✮ ♥➳✉ E f (a) = E g (a) ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ a(z) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♥❤ä ❝õ❛ f ♥➳✉ T (r, a) = o(T (r, f )) ❱ỵ✐ ❤➔♠ ♥❤ä a(z), t❛ ♥â✐ f, g ❝❤✉♥❣ ♥❤❛✉ ❤➔♠ a(z) ❈▼ ✭❤♦➦❝ ■▼✮ ♥➳✉ ❤➔♠ f − a ✈➔ g − a ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ❈▼ ✭■▼ t÷ì♥❣ ù♥❣✮✳ ◆➠♠ ✶✾✼✼✱ ❘✉❜❡❧ ✈➔ ❨❛♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤✿ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♥❣✉②➯♥ ❦❤→❝ ❤➡♥❣✱ ♥➳✉ f ✈➔ f ❝❤✉♥❣ ♥❤❛✉ ❤❛✐ ❣✐→ trà ❤ú✉ ❤↕♥ ♣❤➙♥ ❜✐➺t a ✈➔ b ❦➸ ❝↔ ❜ë✐ t❤➻ f = f ✳ ◆➠♠ ✶✾✼✾✱ ▼✉❡s ✈➔ ❙t❡✐♥♠❡t③ ✭❬✶✹❪✮ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ t÷ì♥❣ tü ❦❤✐ t❤❛② ✤✐➲✉ ❦✐➺♥ ❈▼ ❜ð✐ ■▼✳ ❚ø ♥❤ú♥❣ ❝æ♥❣ tr➻♥❤ ♥➔② ❝õ❛ ❝→❝ t→❝ ❣✐↔ ✤➣ ♥↔② s✐♥❤ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ✤↕♦ ❤➔♠ ❝õ❛ ❝❤ó♥❣✳ ◆➠♠ ✷✵✵✽✱ ❚✳ ❩❤❛♥❣ ✈➔ ❲✳ ▲☎ ✉ ✭❬✶✻❪✮ ✤➣ ①❡♠ ①➨t ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❧ô② t❤ø❛ ❜➟❝ n ❝õ❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä ✈ỵ✐ ✤↕♦ ❤➔♠ ❝➜♣ k ❝õ❛ ♥â ✈➔ t❤✉ ✤÷đ❝ ♠ët sè ❦➳t q✉↔ ✈➲ ✈➜♥ ✤➲ ♥➔②✳ ❈ö t❤➸✱ ❝→❝ t→❝ ❣✐↔ ✤➣ ✤÷❛ r❛ ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ✤➸ ❝→❝ ❤➔♠ f n − a ✈➔ f (k) − a ✶ ✈➔ ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ✵ ❦❤æ♥❣ ❦➸ ❜ë✐ ❤♦➦❝ ❦➸ ❝↔ ❜ë✐ t❤➻ f n = f (k) ✱ tr♦♥❣ ✤â a(z) ❧➔ ♠ët ❤➔♠ ♥❤ä✳ ❑➼ ❤✐➺✉ Mj (f ) = (f )n0i f (1) ✈➔ n1i f (k) nki t P [f ] = Mj (f ) j=1 ●➛♥ ✤➙② ❝â ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ♠ð rë♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❚✳ ❩❤❛♥❣ ✈➔ ❲✳ ▲☎ ✉ ❝❤♦ ❝→❝ tr÷í♥❣ ❤đ♣✿ t❤❛② t❤➳ ❧ơ② t❤ø❛ ❜➟❝ n ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f tr♦♥❣ ❦➳t q✉↔ ❚✳ ❩❤❛♥❣ ✈➔ ❲✳ ▲☎ ✉ ❝õ❛ ❜ð✐ ✤❛ t❤ù❝ ❜➟❝ n ❝õ❛ ❤➔♠ ✤â❀ t❤❛② t❤➳ ✤↕♦ ❤➔♠ ❝➜♣ k ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ❜ð✐ ♠ët ✤ì♥ t❤ù❝ ❝❤ù❛ ❝→❝ ✤↕♦ ❤➔♠ ❝→❝ ❝➜♣ Mj [f ] ❤♦➦❝ ✤❛ t❤ù❝ ❝❤ù❛ ❝→❝ ✤↕♦ ❤➔♠ P [f ] ❝õ❛ ❤➔♠ ✤â✳ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ❣✐ỵ✐ t❤✐➺✉ ♠ët sè ♥❣❤✐➯♥ ❝ù✉ ❣➛♥ ✤➙② ❝õ❛ ❚✳ ❩❤❛♥❣✱ ❲✳ ▲☎ ✉✱ ❆✳ ❇❛♥❡r❥❡❡✱ ❇✳ ❈❤❛❦r❛❜♦rt② ✈➔ ♠ët số t t ữợ ự õ tr ▲✉➟♥ ✈➠♥ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣✱ tr♦♥❣ ❈❤÷ì♥❣ ✶ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝➛♥ ❝❤✉➞♥ ❜à✱ ❝➛♥ t❤✐➳t ❝❤♦ ❝→❝ ♥ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❈❤÷ì♥❣ ✷ ❧➔ ❝❤÷ì♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤✐ ❧ô② t❤ú❛ ❝õ❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝â ❝❤✉♥❣ ♠ët ❣✐→ trà ❤❛② ❤➔♠ ♥❤ä ✈ỵ✐ ✤ì♥ t❤ù❝ ❤♦➦❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ♥â✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✷✵ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ▲➯ ❱➠♥ ❈❤÷ì♥❣ ✷ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ỵ ỡ r ỵ tt ố tr ❤➔♠ ①➜♣ ①➾✱ ❤➔♠ ✤➳♠✱ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✱ ✤â♥❣ ởt trỏ q trồ sốt ỵ tt r ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ♥➔② ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ✶✳✶✳✶✳ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✈➔ t➼♥❤ ❝❤➜t ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤ù❝ C ✈➔ r > ❧➔ ♠ët sè t❤ü❝ ❞÷ì♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❍➔♠ 2π m(r, f ) = 2π log+ f (reiϕ ) dϕ ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ①➜♣ ①➾ ❝õ❛ ❤➔♠ f ✳ ❇➙② ❣✐í t❛ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❤➔♠ ✤➳♠✳ ❑➼ ❤✐➺✉ 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❛ ❢ tr♦♥❣ Dr = {z ∈ C : |z| ≤ |r|}✳ ✸ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❍➔♠ r n(t, f ) − n(0, f ) dt + n(0, f ) log r t N (r, ∞; f ) = N (r, f ) = ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ ❝õ❛ ❢ ✭❝á♥ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ t↕✐ ❝→❝ ❝ü❝ ✤✐➸♠✮✳ ❍➔♠ r n(t, f ) − n(0, f ) dt + n(0, f ) log r t N (r, ∞; f ) = N (r, f ) = ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐✳ ❚r♦♥❣ ✤â n(0, f ) = lim n(t, f ) n(0, f ) = lim n(t, f ), t→0 t→0 ✣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 ) ỡ tr ỵ tt ố ❣✐→ trà✱ ♥â ❝á♥ ❣å✐ ❧➔ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✳ ▼➺♥❤ ✤➲ ✶✳✶✳✹ ✭▼ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✮✳ ❈❤♦ f1 , f2 , , fp ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C, ❦❤✐ ✤â p (1) p fν ) ≤ m(r, ν=1 p (2) ν=1 p fν ) ≤ m(r, ν=1 p (3) m(r, fν ); ν=1 p fν ) ≤ N (r, ν=1 p (4) m(r, fν ) + log p; N (r, fν ); ν=1 p fν ) ≤ N (r, ν=1 N (r, fν ); ν=1 ✹ p (5) p fν ) ≤ T (r, ν=1 p (6) T (r, fν ) + log p; ν=1 p fν ) ≤ T (r, ν=1 T (r, fν ) =1 ỵ ỡ ỵ ỵ ỡ tự t f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ r > 0✱ t❛ ❝â 1 + N r, + log |cj | ✭✶✮ T (r, f ) = m r, f f ✭✷✮ ❱ỵ✐ ♠é✐ sè ♣❤ù❝ a ∈ C, T (r, f ) − m r, 1 + N r, f −a f −a ≤ log c1 +log+ |a|+log 2, f −a tr♦♥❣ ✤â cf ❧➔ ❤➺ sè ❦❤→❝ ♥❤ä ♥❤➜t tr♦♥❣ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛ ❤➔♠ f tr♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ 0, c1 /(f − a) ❧➔ ❤➺ sè ❦❤→❝ ♥❤ä ♥❤➜t tr♦♥❣ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛ ❤➔♠ 1/(f − a) tr♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ 0✳ ◆❤➟♥ ①➨t ✶✳✶✳✻✳ ❚❛ t❤÷í♥❣ ❞ị♥❣ ỵ ỡ tự t ữợ = T (r, f ) + O(1), f −a tr♦♥❣ ✤â O(1) ❧➔ ✤↕✐ ❧÷đ♥❣ ❜à ❝❤➦♥ ❦❤✐ r → ∞ T r, ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ r > 0✳ ❑➼ ❤✐➺✉ Nram (r, f ) = N r, + 2N (r, f ) − N (r, f ) f ✈➔ ❣å✐ ❧➔ ❤➔♠ ❣✐→ trà ♣❤➙♥ ♥❤→♥❤ ❝õ❛ ❤➔♠ f ✳ ❍✐➸♥ ♥❤✐➯♥ Nram (r, f ) ỵ ỵ ❝ì ❜↔♥ t❤ù ❤❛✐✮✳ ●✐↔ sû f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ C✱ a1 , , aq ∈ C, (q > 2) ❧➔ ❝→❝ ❤➡♥❣ sè ♣❤➙♥ ❜✐➺t✱ ❦❤✐ ✤â ✈ỵ✐ ♠é✐ ε > 0✱ ❜➜t ✤➥♥❣ t❤ù❝ q (q − 1)T (r, f ) ≤ N r, j=1 + N (r, f ) − Nram (r, f ) + log T (r, f ) f − aj ✺ ≤ N (r, f ) + N r, 1 + N2+k r, + S(r, f ), f f tữỡ ữỡ ợ (, f ) + Θ(0, f ) + δ2+k (0, f ) ≤ − n ✭✷✳✷✸✮ ❑➳t ❤đ♣ ✭✷✳✷✸✮ ✈ỵ✐ ✭✷✳✷✮ t❤✉ ✤÷đ❝ − n + 2Θ(0, f ) ≥ Θ(∞, f ) + 2Θ(0, f ) + δ2+k (0, f ) > (8 − n), tù❝ ❧➔ Θ(0, f ) > + n/4✱ ♠➙✉ t❤✉➝♥✳ t ủ ợ t ữủ n + Θ(0, f ) ≥ Θ(∞, f ) + 2Θ(0, f ) + δ2+k (0, f ) > − n, tù❝ ❧➔ Θ(0, f ) > 1, ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â✱ C = ✈➔ f n = f (k) ỵ ữủ ự r ♥➔②✱ ❩❤❛♥❣ ✈➔ ▲☎ ✉ ✤÷❛ r❛ ❝➙✉ ❤ä✐✿ ✣✐➲✉ ❣➻ ①✉➜t ❤✐➺♥ ♥➳✉ f n ✈➔ [f (k) ]s ❝❤✉♥❣ ♥❤❛✉ ❤➔♠ ♥❤ä❄ ◆➠♠ ✷✵✶✵✱ ❈❤❡♥ ✈➔ ❩❤❛♥❣ ✭❬✾❪✮ ✤➣ ✤÷❛ r❛ ✤→♣ →♥ ❝❤♦ ❝➙✉ ❤ä✐ ♥➔②✱ tr♦♥❣ ✤â ❝â ♠ët sè ❤↕♥ ❝❤➳ ✈➲ ❝❤ù♥❣ ♠✐♥❤ ✈➔ ✤÷đ❝ ❇❛♥❡r❥❡❡ ✈➔ ▼❛❥✉♠❞❡r ✭❬✹❪✮ ❦❤➢❝ ♣❤ư❝✳ ◆➠♠ ✷✵✶✵ ❇❛♥❡r❥❡❡ r ự ỵ t ỵ tr ❤ä✐ ♠ð ❝õ❛ ❩❤❛♥❣ ✈➔ ▲☎ ✉ ♥❤÷ s❛✉✳ ✣à♥❤ ỵ k ( 1) n ( 1) ❧➔ ❝→❝ sè ♥❣✉②➯♥ ✈➔ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✳ ●å✐ a(z) (≡ 0, ∞) ❧➔ ❤➔♠ ♥❤ä ❝õ❛ f ✳ ●✐↔ sû f n − a ✈➔ f (k) − a ❝❤✉♥❣ ♥❤❛✉ (0, l)✳ ◆➳✉ l ≥ ✈➔ (3 + k)Θ(∞, f ) + 2Θ(0, f ) + δ2+k (0, f ) > + k − n ❤♦➦❝ ♥➳✉ l = ✈➔ + k Θ(∞, f ) + Θ(0, f ) + δ2+k (0, f ) > + k − n 2 ❤♦➦❝ ♥➳✉ l = ✈➔ (6 + 2k)Θ(∞, f ) + 4Θ(0, f ) + δ2+k (0, f ) + δ1+k (0, f ) > 12 + 2k − n, t❤➻ f n ≡ f (k) ỵ k ( 1) n (≥ 1), m (≥ 2) ❧➔ ❝→❝ sè ♥❣✉②➯♥ ✈➔ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✳ ●å✐ a(z) (≡ 0, ∞) ❧➔ ❤➔♠ ♥❤ä ❝õ❛ f ✳ ●✐↔ sû f n − a ✈➔ [f (k) ]m − a ❝❤✉♥❣ ♥❤❛✉ (0, l)✳ ◆➳✉ l = ✈➔ (3 + 2k)Θ(∞, f ) + 2Θ(0, f ) + 2δ1+k (0, f ) > + 2k − n ✭✷✳✷✹✮ ❤♦➦❝ ♥➳✉ l = ✈➔ + 2k Θ(∞, f ) + Θ(0, f ) + 2δ1+k (0, f ) > + 2k − n 2 ✭✷✳✷✺✮ ❤♦➦❝ ♥➳✉ l = ✈➔ (6 + 3k)Θ(∞, f ) + 4Θ(0, f ) + 3δ1+k (0, f ) > 13 + 3k − n, ✭✷✳✷✻✮ t❤➻ f n ≡ [f (k) ]m ✳ ❱ỵ✐ m = t ỵ t q tốt ỡ ợ ỵ r ổ ữ tợ m t ữủ ỵ s ỵ f ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✱ k (≥ 1), l (≥ 0) ❧➔ ❝→❝ sè ♥❣✉②➯♥ ✈➔ ❣å✐ a(z) (≡ 0, ∞) ❧➔ ❤➔♠ ♥❤ä ❝õ❛ f ✳ ●✐↔ sû f − a ✈➔ f (k) − a ❝❤✉♥❣ ♥❤❛✉ (0, l)✳ ◆➳✉ l ≥ ✈➔ (3 + k)Θ(∞, f ) + 2δ2 (0, f ) + δ2+k (0, f ) > k + ❤♦➦❝ ♥➳✉ l = ✈➔ + k Θ(∞, f ) + Θ(0, f ) + δ2 (0, f ) + δ2+k (0, f ) > k + 2 ❤♦➦❝ ♥➳✉ l = ✈➔ (6 + 2k)Θ(∞, f ) + 2Θ(0, f ) + δ2 (0, f ) + δ1+k (0, f ) + δ2+k (0, f ) > 2k + 10, t❤➻ f ≡ f (k) ✳ ❉ü❛ ✈➔♦ ❇ê t s t ỵ tốt ỡ ỵ tr trữớ ủ n = ❚❛ ♥❤➢❝ ❧↕✐✱ ❝❤♦ n0j , n1j , , nkj ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✳ ❇✐➸✉ t❤ù❝ Mj [f ] = (f )n0j (f (1) )n1j · · · (f (k) )nkj ✤÷đ❝ ❣å✐ ❧➔ ✤ì♥ t❤ù❝ ✈✐ ♣❤➙♥ s✐♥❤ ❜ð✐ f ✈ỵ✐ ❜➟❝ dMj = d(Mj ) = ✈➔ ✤ë ❝❛♦ ΓMj = k i=0 (i k i=0 nij + 1)nij ✳ ◆➠♠ ✷✵✶✺✱ ❆✳ r rrt ự ỵ s ❝❤♦ t❤➜② ♠ët ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ✤➸ ❧ô② t❤ø❛ ởt ỗ t ợ ởt ỡ tự s õ ỵ ✷✳✷✳✼ ✭❬✷❪✮✳ ❈❤♦ k (≥ 1)✱ n (≥ 1) ❧➔ ❝→❝ sè ♥❣✉②➯♥ ✈➔ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ ✈➔ M [f ] ❧➔ ✤ì♥ t❤ù❝ ✈✐ ♣❤➙♥ ❝â ❜➟❝ dM ✈➔ ✤ë ❝❛♦ ΓM ✈➔ k ❧➔ ❜➟❝ ❝❛♦ ♥❤➜t ❝õ❛ ❤➔♠ M [f ]✳ ●å✐ a(z) (≡ 0, ∞) ❧➔ ❤➔♠ ♥❤ä ❝õ❛ f ✳ ●✐↔ sû f n − a ✈➔ M [f ] − a ❝❤✉♥❣ ♥❤❛✉ (0, l)✳ ◆➳✉ l ≥ ✈➔ (3 + λ)Θ(∞, f ) + µ2 δµ∗2 (0, f ) + dM δ2+k (0, f ) > + ΓM + µ2 − n ✭✷✳✷✼✮ ❤♦➦❝ ♥➳✉ l = ✈➔ + λ Θ(∞, f ) + Θ(0, f ) + µ2 δµ∗2 (0, f ) + dM δ2+k (0, f ) > + ΓM + µ2 − n 2 ❤♦➦❝ ♥➳✉ l = ✈➔ (6 + 2λ)Θ(∞, f ) + 2Θ(0, f ) + µ2 δµ∗2 (0, f ) + dM δ2+k (0, f ) + dM δ1+k (0, f ) > + 2ΓM + µ2 − n, ✭✷✳✷✽✮ t❤➻ f n ≡ M [f ]✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t F = fn a ✈➔ G = M [f ] a ✳ ❑❤✐ ✤â✱ fn − a M [f ] − a F −1= ,G − = a a ❉♦ f n ✈➔ M [f ] ❝❤✉♥❣ ♥❤❛✉ (a, l)✱ s✉② r❛ F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ (1, l) ♥❣♦↕✐ trø ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ❝õ❛ a(z)✳ ❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✳ ❚r÷í♥❣ ❤đ♣ ✶✳ ●✐↔ sû H ≡ ✸✶ ❚r÷í♥❣ ❤đ♣ ✶✳✶✳ ●✐↔ sû l ≥ õ sỷ ỵ ỡ tự ❤❛✐ ✈➔ ❇ê ✤➲ ✷✳✶✳✺✱ ✷✳✶✳✸✱ t❛ ❝â T (r, F ) + T (r, G) ≤ N (r, ∞; F ) + N (r, ∞; G) + N (r, 0; F ) + N (r, 0; G) (2 + N (r, H) + N E (r, 1; F ) + N L (r, 1; F ) + N L (r, 1; G) + N (r, 1; G) − N (r, 0; F ) − N (r, 0; G ) + S(r, f ) ≤ 2N (r, ∞; F ) + N (r, ∞; G) + N2 (r, 0; F ) + N2 (r, 0; G) (2 + N E (r, 1; F ) + 2N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ✭✷✳✷✾✮ ❚r÷í♥❣ ❤đ♣ ✶✳✶✳✶✳ ❱ỵ✐ l = 1✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✾✮ ✈➔ ❞ü❛ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✹✱ ✶✳✷✳✶✺ t❛ ❝â T (r, F ) + T (r, G) ≤ 2N (r, ∞; F ) + N (r, ∞; G) + N2 (r, 0; F ) + N2 (r, 0; G) (2 + N E (r, 1; F ) + 2N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ N (r, ∞; F ) + N (r, ∞; G) + N (r, 0; F ) + µ2 Nµ∗2 (r, 0; f ) 2 (2 + N (r, 0; G) + N E (r, 1; F ) + N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ N (r, ∞; F ) + N (r, ∞; G) + N (r, 0; F ) + µ2 Nµ∗2 (r, 0; f ) 2 + N (r, 0; G) + N (r, 1; G) + S(r, f ) tù❝ ❧➔ ✈ỵ✐ ❜➜t ❦ý ε > t❛ ❝â N (r, ∞; f ) + N (r, 0; f ) + µ2 Nµ∗2 (r, 0; f ) 2 + dM N2+k (r, 0; f ) + S(r, f ) 7 1 ≤ λ+ − λ + Θ(∞, f ) + − Θ(0, f ) + µ2 2 2 − µ2 δµ∗2 (0, f ) + dM − dM δ2+k (0, f ) + ε T (r, f ) + S(r, f ) nT (r, f ) ≤ λ + ✸✷ tù❝ ❧➔ Θ(∞, f ) + Θ(0, f ) + µ2 δµ∗2 (0, f ) + dM δ2+k (0, f ) − ε T (r, f ) 2 ≤ (ΓM + µ2 + − n)T (r, f ) + S(r, f ), λ+ ♠➙✉ t❤✉➝♥✳ ❚r÷í♥❣ ❤đ♣ ✶✳✶✳✷✳ ●✐↔ sû l ≥ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✾✮ ✈➔ ❇ê ✤➲ ✶✳✷✳✶✺✱ t❛ t❤✉ ✤÷đ❝ T (r, F ) + T (r, G) ≤ 2N (r, ∞; F ) + N (r, ∞; G) + N2 (r, 0; F ) + N2 (r, 0; G) (2 + N E (r, 1; F ) + 2N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ 2N (r, ∞; F ) + N (r, ∞; G) + µ2 Nµ∗2 (r, 0; f ) + N (r, 0; G) + N (r, 1; G) + S(r, f ), tù❝ ❧➔ ✈ỵ✐ ❜➜t ❦ý ε > t❛ ❝â nT (r, f ) ≤ (λ + 3)N (r, ∞; f ) + µ2 Nµ∗2 (r, 0; f ) + dM N2+k (r, 0; f ) + S(r, f ) ≤ (λ + 3) − (λ + 3)Θ(∞, f ) + µ2 − µ2 δµ∗2 (0, f ) + dM − dM δ2+k (0, f ) + ε T (r, f ) + S(r, f ), tù❝ ❧➔ {(λ + 3)Θ(∞, f ) + µ2 δµ∗2 (0, f ) + dM δ2+k (0, f ) − ε}T (r, f ) ≤ (ΓM + + µ2 − n)T (r, f ) + S(r, f ), ♠➙✉ t❤✉➝♥✳ ❚r÷í♥❣ ❤đ♣ ✶✳✶✳✸ ●✐↔ sû l = õ sỷ ỵ ỡ ❜↔♥ t❤ù ❤❛✐ ✈➔ ❝→❝ ❇ê ✤➲ ✷✳✶✳✺✱ ✷✳✶✳✸✱ ✷✳✶✳✹✱ ✶✳✷✳✶✺ t❛ t❤✉ ✤÷đ❝ T (r, F ) + T (r, G) ≤ N (r, ∞; F ) + N (r, 0; F ) + N (r, 1; F ) + N (r, ∞; G) + N (r, 0; G) + N (r, 1; G) − N (r, 0; F ) − N (r, 0; G ) + S(r, F ) + S(r, G) ✸✸ ≤ N (r, ∞; F ) + N (r, 0; F ) + N (r, ∞; G) + N (r, 0; G) + N (r, ∞; H) (2 + N E (r, 1; F ) + N L (r, 1; F ) + N L (r, 1; G) + N (r, 1; G) − N (r, 0; F ) − N (r, 0; G ) + S(r, F ) + S(r, G) ≤ 2N (r, ∞; F ) + N (r, ∞; G) + N2 (r, 0; F ) + N2 (r, 0; G) (2 + N E (r, 1; F ) + 2N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ 2N (r, ∞; F ) + N (r, ∞; G) + µ2 Nµ∗2 (r, 0, f ) + N2 (r, 0; G) + 2(N (r, ∞; F ) + N (r, 0; F )) + N (r, ∞; G) + N (r, 0; G) (2 + N E (r, 1; F ) + N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ 4N (r, ∞; F ) + µ2 Nµ∗2 (r, 0; f ) + N2 (r, 0; G) + 2N (r, ∞; G) + N (r, 0; G) + 2N (r, 0; F ) + T (r, G) + S(r, f ), ✭✷✳✸✵✮ tù❝ ❧➔ ✈ỵ✐ ❜➜t ❦ý ε > nT (r, f ) ≤ (2λ + 6)N (r, ∞, f ) + 2N (r, 0; f ) + µ2 Nµ∗2 (r, 0, f ) + dM N1+k (r, 0; f ) + dM N2+k (r, 0; f ) + S(r, f ) ≤ {(2λ + 6) − (2λ + 6)Θ(∞, f ) + − 2Θ(0, f ) + µ2 − µ2 δµ∗2 (0, f ) + 2dM − 2dM δ1+k (0, f ) − dM δ2+k (0, f ) + ε}T (r, f ) + S(r, f ), tù❝ ❧➔ {(2λ + 6)Θ(∞, f ) + 2Θ(0, f ) + µ2 δµ∗2 (0, f ) + 2dM δ1+k (0, f ) + dM δ2+k (0, f ) − ε}T (r, f ) ≤ (2ΓM + + µ2 − n)T (r, f ) + S(r, f ), ♠➙✉ t❤✉➝♥✳ ❚r÷í♥❣ ❤đ♣ ✷✳ ●✐↔ sû H ≡ 0✳ ❉ü❛ ✈➔♦ t➼❝❤ ♣❤➙♥ t❛ t❤✉ ✤÷đ❝ A ≡ + B, G−1 F −1 tr♦♥❣ ✤â A (= 0), B ❧➔ ❝→❝ ❤➡♥❣ sè ♣❤ù❝✳ ❑❤✐ ✤â F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ (1, ∞)✳ ◆❣♦➔✐ r❛✱ t❤❡♦ ❝→❝❤ ①➙② ❞ü♥❣ F ✈➔ G t❛ t❤➜② r➡♥❣ F ✈➔ G ❝ô♥❣ ❝❤✉♥❣ ♥❤❛✉ (∞, 0)✳ ✸✹ ❉♦ ✤â✱ sû ❞ö♥❣ ❇ê ✤➲ ✶✳✷✳✶✺ ✈➔ ✤✐➲✉ ❦✐➺♥ ✭✷✳✷✼✮✱ t❛ t❤✉ ✤÷đ❝ N2 (r, 0; F ) + N2 (r, 0; G) + N (r, ∞; F ) + N (r, ∞; G) + N L (r, ∞; F ) + N L (r, ∞; G) + S(r) ≤ µ2 Nµ∗2 (r, 0; f ) + dM N2+k (r, 0; f ) + (λ + 3)N (r, ∞; f ) + S(r) ≤ {(3 + λ + dM + µ2 ) − ((λ + 3)Θ(∞, f ) + δµ∗2 (0, f ) + dM δ2+k (0, f ))}T (r, f ) + S(r) < T (r, F ) + S(r) ❉♦ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✮ ❝õ❛ ❇ê ✤➲ ✷✳✶✳✻ ❦❤æ♥❣ t❤ä❛ ♠➣♥✳ ▼ët ❧➛♥ ♥ú❛ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✷✱ t❛ t❤✉ ✤÷đ❝ F ≡ G✱ tù❝ ❧➔ f n ≡ M [f ]✳ ❱➼ ❞ö s❛✉ ❝❤➾ r❛ tr♦♥❣ ✣à♥❤ ỵ tt a(z) 0, t❤✐➳t✳ ❱➼ ❞ö ✷✳✷✳✽✳ ▲➜② f (z) = ee z ✈➔ M = f ✱ ❦❤✐ ✤â M ✈➔ f ❝❤✉♥❣ ♥❤❛✉ ✭❤♦➦❝ ∞✮ ✈➔ ✤✐➲✉ ❦✐➺♥ sè ❦❤✉②➳t t tr ỵ ữủ tọ ❝↔ 0, ∞ ❧➔ ❣✐→ trà ❜ä ✤÷đ❝ ❝õ❛ f ♥❤÷♥❣ f ≡ M ❱➼ ❞ư t✐➳♣ t❤❡♦ ❝❤➾ r❛ r số t tr ỵ ❦❤ỉ♥❣ ❝➛♥ t❤✐➳t✳ ❱➼ ❞ư ✷✳✷✳✾✳ ▲➜② f (z) = Aez + Be−z , AB = ❑❤✐ ✤â N (r, f ) = S(r, f ) ✈➔ B N (r, 0; f ) = N (r, − ; e2z ) ∼ T (r, f ) A ❉♦ ✤â Θ(∞, f ) = ✈➔ Θ(0, f ) = δp (0, f ) = 0✳ ❘ã r➔♥❣ M [f ] = f ✈➔ f ❝❤✉♥❣ ♥❤❛✉ a(z) = z số t tr ỵ ổ t❤ä❛ ♠➣♥ ♥❤÷♥❣ M ≡ f ✳ ❚r♦♥❣ ✈➼ ❞ư t t t t r tr ỵ t ❦❤æ♥❣ t❤➸ t❤❛② f n ❜➡♥❣ ✤❛ t❤ù❝ ❜➜t ❦ý P [f ] = a0 f n + a1 f n−1 + · · · + an tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝❤✉♥❣ ♥❤❛✉ IM ✭l = 0✮✳ ❱➼ ❞ö ✷✳✷✳✶✵✳ ▲➜② f (z) = ez , P [f ] = f + 2f ✈➔ M [f ] = f (3)✱ ❦❤✐ ✤â P + = (M + 1)2 ✳ ❉♦ ✤â P ✈➔ M ❝❤✉♥❣ ♥❤❛✉ (−1, 0) ◆❣♦➔✐ r❛ Θ(0, f ) = Θ(∞, f ) = δp (0, f ) = δ(0, f ) = ✸✺ ❞♦ ✈➔ ∞ ❧➔ ♥❤ú♥❣ ❣✐→ trà ❝❤➜♣ ♥❤➟♥ ữủ f õ tr ỵ ✷✳✷✳✼ ✤÷đ❝ t❤ä❛ ♠➣♥ ♥❤÷♥❣ P ≡ M ❈ơ♥❣ tr♦♥❣ ❜➔✐ ❜→♦ ✭❬✷❪✮✱ ❝→❝ t→❝ ❣✐↔ ✤➣ ✤➦t r❛ ❝➙✉ ❤ä✐✿ ❈➙✉ ❤ä✐ ✷✳✷✳✶✶✳ ❚❛ ❝â t❤➸ ♠ð rë♥❣ ✣à♥❤ ỵ trữớ ủ tự ổ ố ợ trữớ ủ tự r ự rở ỵ t ữợ ọ ✉ ❝❤♦ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✳ ❍å ✤➣ ❝❤ù♥❣ ♠✐♥❤ ỵ s ỵ f ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✱ n ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ a(z) (≡ 0, ∞) ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ t❤ä❛ ♠➣♥ T (r, a) = o(T (r, f )) ❦❤✐ r → ∞✳ ▲➜② P [f ] ❧➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❦❤→❝ ❤➡♥❣ t❤❡♦ f ✳ ●✐↔ sû f n ✈➔ P [f ] ❝❤✉♥❣ ♥❤❛✉ (a, l)✳ ◆➳✉ l ≥ ✈➔ (3 + Q)Θ(∞, f ) + 2Θ(0, f ) + d(P )δ(0, f ) > Q + + 2d(P ) − d(P ) − n ❤♦➦❝ ♥➳✉ l = ✈➔ + Q Θ(∞, f ) + Θ(0, f ) + d(P )δ(0, f ) > Q + + 2d(P ) − d(P ) − n 2 ❤♦➦❝ ♥➳✉ l = ✈➔ (6+2Q)Θ(∞, f )+4Θ(0, f )+2d(P )δ(0, f ) > 2Q+4d(P )−2d(P )+10−n, t❤➻ f n ≡ P [f ]✳ rrt ự ỵ ✭❬✼❪✮✳ ❈❤♦ k (≥ 1)✱ n (≥ 1) ❧➔ ❝→❝ sè ♥❣✉②➯♥ ✈➔ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✳ ●✐↔ sû P [f ] ❧➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ t❤✉➛♥ ♥❤➜t ❝â ❜➟❝ d(P ) ✈➔ ✤ë ❝❛♦ ΓP t❤ä❛ ♠➣♥ ΓP > (k + 1)d(P ) − 2, tr♦♥❣ ✤â k ❧➔ ❜➟❝ ✤↕♦ ❤➔♠ ❝❛♦ ♥❤➜t tr♦♥❣ P [f ]✳ ◆❣♦➔✐ r❛✱ ❣å✐ a(z) (≡ 0, ∞) ❧➔ ❤➔♠ ♥❤ä ❝õ❛ f ✳ ●✐↔ sû f n − a ✈➔ P [f ] − a ❝❤✉♥❣ ♥❤❛✉ (0, l)✳ ◆➳✉ l ≥ ✈➔ (ΓP −d(P )+3)Θ(∞, f )+µ2 δµ∗2 (0, f )+d(P )δ2+ΓP −d(P ) (0, f ) ≤ ΓP +µ2 +3−n ✭✷✳✸✶✮ ✸✻ ❤♦➦❝ ♥➳✉ l = ✈➔ Θ(∞, f ) + Θ(0, f ) + µ2 δµ∗2 (0, f ) + d(P )δ2+ΓP −d(P ) (0, f ) 2 ≤ ΓP + µ2 + − n, ✭✷✳✸✷✮ ΓP − d(P ) + ❤♦➦❝ ♥➳✉ l = ✈➔ (2(ΓP − d(P )) + 6)Θ(∞, f ) + 2Θ(0, f ) + µ2 δµ∗2 (0, f ) + d(P )δ1+ΓP −d(P ) (0, f ) + d(P )δ2+ΓP −d(P ) (0, f ) ✭✷✳✸✸✮ ≤ 2ΓP + µ2 + − n, t❤➻ f n ≡ P [f ]✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû P [f ] fn ✈➔ G = F = a(z) a(z) ❑❤✐ ✤â F − = f n −a(z) a(z) , G − = P [f ]−a(z) a(z) ✳ ❱➻ f n ✈➔ P [f ] ❝❤✉♥❣ ♥❤❛✉ (a, l), s✉② r❛ F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ (1, l) ♥❣♦↕✐ trø ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ❝õ❛ a(z)✳ ❚✐➳♣ t❤❡♦ t❛ ①➨t ❤❛✐ tr÷í♥❣ s❛✉✳ ❚r÷í♥❣ ❤đ♣ ✶✳ ●✐↔ sû H ≡ ❚r÷í♥❣ ❤đ♣ ✶✳✶✳ ◆➳✉ l ≥ t❤➻ sû ❞ư♥❣ ỵ ỡ tự ✈➔ ✷✳✶✳✸✱ t❛ t❤✉ ✤÷đ❝ T (r, F ) + T (r, G) ≤ N (r, ∞; F ) + N (r, ∞; G) + N (r, 0; F ) + N (r, 0; G) + N (r, ∞; H) (2 + N E (r, 1; F ) + N L (r, 1; F ) + N L (r, 1; G) + N (r, 1; G) − N (r, 0; F ) − N (r, 0; G ) + S(r, F ) (2 ≤ 2N (r, ∞; F ) + N (r, ∞; G) + N2 (r, 0; F ) + N2 (r, 0; G) + N E (r, 1; F ) + 2N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ✭✷✳✸✹✮ ❚r÷í♥❣ ❤đ♣ ✶✳✶✳✶ ◆➳✉ l ≥ t❤➻ sû ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✸✹✮✱ t❛ t❤✉ ✤÷đ❝ T (r, F ) + T (r, G) ✸✼ (2 ≤ 2N (r, ∞; F ) + N (r, ∞; G) + N2 (r, 0; F ) + N2 (r, 0; G) + N E (r, 1; F ) + 2N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ 2N (r, ∞; F ) + N (r, ∞; G) + µ2 Nµ∗2 (r, 0; f ) + N2 (r, 0; G) + N (r, 1; G) + S(r, f ), tù❝ ❧➔ ✈ỵ✐ ❜➜t ❦ý ε > 0✱ t❤❡♦ ❇ê ✤➲ ✶✳✷✳✷✷✱ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ trð t❤➔♥❤ nT (r, f ) ≤ (ΓP − d(P ) + 3)N (r, ∞; f ) + µ2 Nµ∗2 (r, 0; f ) + N2+ΓP −d(P ) (r, 0; f d(P ) ) + S(r, f ) ≤ {(ΓP − d(P ) + 3) − (ΓP − d(P ) + 3)Θ(∞, f ) + µ2 − µ2 δµ∗2 (0, f ) + d(P ) − d(P )δ2+ΓP −d(P ) (0, f ) + ε}T (r, f ) + S(r, f ), tù❝ ❧➔ (ΓP −d(P )+3)Θ(∞, f )+µ2 δµ∗2 (0, f )+d(P )δ2+ΓP −d(P ) (0, f ) P +à2 +3n, t ợ ỵ rữớ ủ l = t❤➻ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✸✹✮ ✈➔ ❇ê ✤➲ ✷✳✶✳✹✱ t❛ t❤✉ ✤÷đ❝ T (r, F ) + T (r, G) (2 ≤ 2N (r, ∞; F ) + N (r, ∞; G) + N2 (r, 0; F ) + N2 (r, 0; G) + N E (r, 1; F ) + 2N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ N (r, ∞; F ) + N (r, ∞; G) + N (r, 0; F ) + µ2 Nµ∗2 (r, 0; f ) + N (r, 0; G) 2 (2 + N E (r, 1; F ) + N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ N (r, ∞; F ) + N (r, ∞; G) + N (r, 0; F ) + µ2 Nµ∗2 (r, 0; f ) 2 + N2 (r, 0; G) + N (r, 1; G) + S(r, f ) tù❝ ❧➔ ✈ỵ✐ ❜➜t ❦ý ε > 0✱ t❤❡♦ ❇ê ✤➲ ✶✳✷✳✷✷ ❜➜t ✤➥♥❣ tr➯♥ trð t❤➔♥❤ nT (r, f ) ≤ (ΓP − d(P ) + )N (r, ∞; f ) + N (r, 0; f ) + µ2 Nµ∗2 (r, 0; f ) 2 d(P ) + N2+ΓP −d(P ) (r, 0; f ) + S(r, f ) ✸✽ 7 ≤ {(ΓP − d(P ) + ) − (ΓP − d(P ) + )Θ(∞, f ) 2 1 + − Θ(0, f ) + µ2 − µ2 δµ∗2 (0, f ) + d(P ) 2 − d(P )δ2+ΓP −d(P ) (0, f ) + ε}T (r, f ) + S(r, f ) tù❝ ❧➔ (ΓP − d(P ) + )Θ(∞, f ) + Θ(0, f ) + µ2 δµ∗2 (0, f ) + d(P )δ2+ΓP −d(P ) (0, f ) 2 ≤ ΓP + µ2 + − n, ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✷✮ ỵ rữớ ủ l = õ sỷ ỵ ỡ tự ❤❛✐ ✈➔ ❝→❝ ❇ê ✤➲ ✷✳✶✳✺✱ ✷✳✶✳✸✱ ✷✳✶✳✹✱ ✶✳✷✳✶✺ t❛ t❤✉ ✤÷đ❝ T (r, F ) + T (r, G) ≤ N (r, ∞; F ) + N (r, 0; F ) + N (r, 1; F ) + N (r, ∞; G) + N (r, 0; G) + N (r, 1; G) − N (r, 0; F ) − N (r, 0; G ) + S(r, F ) + S(r, G) ≤ N (r, ∞; F ) + N (r, 0; F ) + N (r, ∞; G) + N (r, 0; G) + N (r, ∞; H) (2 + N E (r, 1; F ) + N L (r, 1; F ) + N L (r, 1; G) + N (r, 1; G) − N (r, 0; F ) − N (r, 0; G ) + S(r, F ) + S(r, G) (2 ≤ 2N (r, ∞; F ) + N (r, ∞; G) + N2 (r, 0; F ) + N2 (r, 0; G) + N E (r, 1; F ) + 2N L (r, 1; F ) + 2N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ 2N (r, ∞; F ) + N (r, ∞; G) + µ2 Nµ∗2 (r, 0, f ) + N2 (r, 0; G) + 2(N (r, ∞; F ) + N (r, 0; F )) + N (r, ∞; G) + N (r, 0; G) (2 + N E (r, 1; F ) + N L (r, 1; G) + N (r, 1; G) + S(r, f ) ≤ 4N (r, ∞; F ) + µ2 Nµ∗2 (r, 0; f ) + N2 (r, 0; G) + 2N (r, ∞; G) + N (r, 0; G) + 2N (r, 0; F ) + T (r, G) + S(r, f ), ✭✷✳✸✺✮ tù❝ ❧➔ ✈ỵ✐ ❜➜t ❦ý ε > 0✱ t❤❡♦ ❇ê ✤➲ ✶✳✷✳✷✷ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ trð t❤➔♥❤ nT (r, f ) ≤ (2(ΓP − d(P )) + 6)N (r, ∞; f ) + 2N (r, 0; f ) + µ2 Nµ∗2 (r, 0; f ) ✸✾ + N2+ΓP −d(P ) (r, 0; f d(P ) ) + N1+ΓP −d(P ) (r, 0; f d(P ) ) + S(r, f ) ≤ {(2(ΓP − d(P )) + 6) − (2(ΓP − d(P )) + 6)Θ(∞, f ) + − 2Θ(0, f ) + µ2 − µ2 δµ∗2 (0, f ) + d(P ) − d(P )δ1+ΓP −d(P ) (0, f ) + d(P ) − d(P )δ2+ΓP −d(P ) (0, f ) + ε}T (r, f ) + S(r, f ), tù❝ ❧➔ (2(ΓP − d(P )) + 6)Θ(∞, f ) + 2Θ(0, f ) + µ2 δµ∗2 (0, f ) + d(P )δ1+ΓP −d(P ) (0, f ) + d(P )δ2+ΓP −d(P ) (0, f ) ≤ 2P + à2 + n, t ợ ỵ rữớ ủ ●✐↔ sû H ≡ 0✳ ❉ü❛ ✈➔♦ ✭✷✳✹✮ A ≡ + B, G−1 F −1 tr♦♥❣ ✤â A (= 0), B ❧➔ ❝→❝ ❤➡♥❣ sè ♣❤ù❝✳ ❑❤✐ ✤â F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ (1, ∞)✳ ◆❣♦➔✐ r❛✱ t❤❡♦ ❝→❝❤ ①➙② ❞ü♥❣ F ✈➔ G t❛ t❤➜② r➡♥❣ F ✈➔ G ❝ơ♥❣ ❝❤✉♥❣ ♥❤❛✉ (∞, 0)✳ ❉♦ ✤â✱ sû ❞ư♥❣ ỵ ✷✳✷✳✶✸✱ t❛ t❤✉ ✤÷đ❝ N2 (r, 0; F ) + N2 (r, 0; G) + N (r, ∞; F ) + N (r, ∞; G) + N L (r, ∞; F ) + N L (r, ∞; G) + S(r) ≤ µ2 Nµ∗2 (r, 0; f ) + N2+ΓP −d(P ) (r, 0; f d(P ) ) + (ΓP − d(P ) + 3)N (r, ∞; f ) + S(r) ≤ {(ΓP + µ2 + 3) − ((ΓP − d(P ) + 3)Θ(∞, f ) − µ2 δµ∗2 (0, f ) − d(P )δ2+ΓP −d(P ) (0, f ) + ε}T (r, f ) + S(r) < T (r, F ) + S(r), tr♦♥❣ ✤â ε > ❧➔ ✤↕✐ ữủ ọ tũ ỵ õ sỷ ✷✳✶✳✼ ✈➔ ❇ê ✤➲ ✷✳✶✳✽✱ t❛ ❦➳t ❧✉➟♥ r➡♥❣ F ≡ G✱ tù❝ ❧➔ f n ≡ P [f ] ✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✹✵ ❑➌❚ ▲❯❾◆ ❱ỵ✐ ♠ư❝ ✤➼❝❤ t tử ự ỳ ự ỵ tt ◆❡✈❛♥✲ ❧✐♥♥❛ tr♦♥❣ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❧ô② t❤ø❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ✤❛ t❤ù❝ ✤↕♦ ❤➔♠ ❝õ❛ ❝❤ó♥❣✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ✈➲ ❝→❝ ❤➔♠ t t ỵ ỡ ♠ët sè ❜ê ✤➲ ❧✐➯♥ q✉❛♥✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ✤÷đ❝ ①❡♠ ❧➔ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✱ ❝➛♥ t❤✐➳t ❝❤♦ ✈✐➺❝ ❧➔♠ rã ✈➜♥ ✤➲ ❞✉② ♥❤➜t tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ✷✳ ●✐ỵ✐ t❤✐➺✉ ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❣➛♥ t tr ữợ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❧ô② t❤ø❛ ❝õ❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ s✐♥❤ ❜ð✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✤â✳ ✣➦❝ ❜✐➺t ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ ♠✐♥❤ ❧↕✐ ♠ët tt ỵ ỵ ú t ởt số ❦✐➺♥ ✤↕✐ sè ✤➸ ❦❤✐ ❧ô② t❤ø❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ❤❛② ❤➔♠ ♥❤ä ✈ỵ✐ ✤ì♥ t❤ù❝ ✈✐ ♣❤➙♥ ❤♦➦❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ õ t ụ tứ s ỗ t ợ ỡ tự ❤❛② ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✤â✳ ❈â t❤➸ t❤➜②✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❦❤→ ✤➦❝ ❜✐➺t ✭❝→❝ ✤ì♥ t❤ù❝ ✈✐ ♣❤➙♥ t❤❛♠ ❣✐❛ ✤➲✉ ❝ị♥❣ ❜➟❝ k ✮✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐ ❝❤ó♥❣ tỉ✐ s➩ ♥❣❤✐➯♥ ❝ù✉ s➙✉ ❤ì♥ ✈➲ ✈➜♥ ✤➲ ♥➔② ✤è✐ ✈ỵ✐ ♠ët tự tũ ỵ t ❦❤↔♦ ❬✶❪ ❇❛♥❡r❥❡❡ ❆✳ ✭✷✵✵✼✮✱ ✏❯♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s t❤❛t s❤❛r❡ t✇♦ s❡ts✑✱ ❙♦✉t❤❡❛st ❆s✐❛♥ ❇✉❧❧✳ ▼❛t❤✳✱ ✸✶✭✶✮✱ ♣♣✳ ✼✲✶✼✳ ❬✷❪ ❇❛♥❡r❥❡❡ ❆✳✱ ❈❤❛❦r❛❜♦rt② ❇✳ ✭✷✵✶✺✮✱ ✏❋✉rt❤❡r ✐♥✈❡st✐❣❛t✐♦♥s ♦♥ ❛ q✉❡st✐♦♥ ♦❢ ❩❤❛♥❣ ❛♥❞ ▲☎ ✉✑✱ ❆♥♥✳ ❯♥✐✈✳ P❛❡❞❛❣♦❣✳ ❈r❛❝✳ ❙t✉❞✳ ▼❛t❤✳✱ ✶✹✱ ♣♣✳ ✶✵✺✲✶✶✾✳ ❬✸❪ ❇❛♥❡r❥❡❡ ❆✳✱ ❈❤❛❦r❛❜♦rt② ❇✳ ✭✷✵✶✻✮✱ ✏❖♥ t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ❇r☎ ✉❝❦ ❝♦♥❥❡❝t✉r❡✑✱ ❈♦♠♠✉♥✳ ❑♦r❡❛♥ ▼❛t❤✳ ❙♦❝✳✱ ✸✷✭✷✮✱ ♣♣✳ ✸✶✶✲✸✷✼✳ ❬✹❪ ❇❛♥❡r❥❡❡ ❆✳✱ ▼❛✐❥✉♠❞❡r ❙✳ ✭✷✵✶✵✮✱ ✏❖♥❡ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ ♣♦✇❡r ♦❢ ❛ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ s❤❛r✐♥❣ ❛ s♠❛❧❧ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤❡ ♣♦✇❡r ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡✑✱ ❈♦♠♠❡♥t✳ ▼❛t❤✳ ❯♥✐✈✳ ❈❛r♦❧✐♥✳✱ ✺✶ ✭✹✮✱ ♣♣✳ ✺✻✺✲✺✼✻✳ ❬✺❪ ❇r✉❝❦ ❘✳ ✭✶✾✾✻✮✱ ✏❖♥ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ s❤❛r❡ ♦♥❡ ✈❛❧✉❡ ❈▼ ✇✐t❤ t❤❡✐r ❢✐st ❞❡r✐✈❛t✐✈❡✑✱ ❘❡s✉❧ts ▼❛t❤✳✱ ✸✵ ✭✶✮✱ ♣♣✳ ✷✶✲✷✹✳ ❬✻❪ ❈❤❛❦r❛❜♦rt② ❇✳ ✭✷✵✶✼✮✱ ✏❆ s✐♠♣❧❡ ♣r♦♦❢ ♦❢ t❤❡ ❈❤✉❛♥❣✬s ✐♥❡q✉❛❧✐t②✑✱ ❆♥✳ ❯♥✐✈✳ ❱❡s ❚✐♠✐s✳ ❙❡r✳ ▼❛t✳ ■♥❢♦r♠✳✱ ✺✺ ✭✷✮✱ ♣♣✳ ✽✺✲✽✾✳ ❬✼❪ ❈❤❛❦r❛❜♦rt② ❇✳ ✭✷✵✶✾✮✱ ✏❯♥✐q✉❡♥❡ss ♦❢ ♣♦✇❡r ♦❢ ❛ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝✲ t✐♦♥s ✇✐t❤ ✐ts ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧✑✱ ❚❛♠❦❛♥❣ ❏♦✉r✳ ♦❢ ▼❛t❤✳✱ ✺✵ ✭✷✮✱ ♣♣✳ ✶✸✸✲✶✹✼✳ ❬✽❪ ❈❤❛r❛❦ ❑✳❙✳✱ ▲❛❧ ❇✳ ✭✷✵✶✺✮✱ ✏❯♥✐q✉❡♥❡ss ♦❢ f n ❛♥❞ P [f ]✑✱ ❛r❳✐✈✿ ✶✺✵✶✳✵✺✵✾✷✈✶✳ ✹✷ ❬✾❪ ❈❤❡♥✳ ❆✳✱ ❩❤❛♥❣✳ ●✳ ✭✷✵✶✵✮✱ ✏❯♥✐❝✐t② ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡✑✱ ❑②✉♥❣♣♦♦❦ ▼❛t❤✳ ❏✳✱ ✺✵ ✭✶✮✱ ♣♣✳ ✼✶✲✽✵✳ ❬✶✵❪ ▲❛❤✐r✐ ■✳✱ ❙❛r❦❛r ❆✳ ✭✷✵✵✹✮✱ ✏❯♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡✑✱ ❏✳■♥❡q✉❛❧✳ P✉r❡ ❆♣♣❧✳ ▼❛t❤✳✱ ✹✭✶✮✱ ❆rt✐❝❧❡ ■❞✳✷✵✳ ❬✶✶❪ ▲✐ ❏✳❉✳✱ ❍✉❛♥❣✳ ●✳❳✳ ✭✷✵✶✺✮✱ ✏❖♥❡ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s t❤❛t s❤❛r❡ ♦♥❡ s♠❛❧❧ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤❡✐r ❞❡r✐✈❛t✐✈❡✑✱ P❛❧❡st✐♥❡ ❏✳ ▼❛t❤✳✱ ✹ ✭✶✮✱ ♣♣✳ ✾✶✲✾✻✳ ❬✶✷❪ ▲✐ ◆✳✱ ❨❛♥❣ ❩✳ ✭✷✵✶✵✮✱ ✏▼❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ t❤❛t s❤❛r❡s ♦♥❡ s♠❛❧❧ ❢✉♥❝t✐♦♥ ✇✐t❤ ✐ts ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧✑✱ ❑②✉♥❣♣♦♦❦ ▼❛t❤✳ ❏✳✱ ✺✵ ✭✶✮✱ ♣♣✳ ✹✹✼✲✹✺✹✳ ❬✶✸❪ ▼♦❦❤♦♥✬❦♦ ❆✳❩✳ ✭✶✾✼✶✮✱ ✏❖♥ t❤❡ ◆❡✈❛♥❧✐♥♥❛ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ s♦♠❡ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥✱ ✐♥ ❚❤❡♦r② ♦❢ ❋✉♥❝t✐♦♥s✱ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✑✱ ■③❞ ✲ ✈♦ ❑❤❛✬❦♦✈s❦✱ ❯♥✲t❛✱ ✶✹✱ ♣♣✳ ✽✸✲✽✼✳ ❬✶✹❪ ▼✉❡s ❊✳✱ ❙t❡✐♥♠❡t③ ◆✳ ✭✶✾✼✾✮✱ ✏▼❡r♦♠♦r♣❤❡ ❋✉♥❦t✐♦♥❡♥✱ ❞✐❡ ♠✐t ✐❤r❡r ❆❜❧❡✐t✉♥❣ ❲❡rt❡t❡✐❧❡♥✑✱ ▼❛♥✉s❝r✐♣t❛ ▼❛t❤✳✱ ✷✾✱ ♥♦✳ ✷✲✹✱ ♣♣✳ ✶✾✺✲✷✵✻✳ ❬✶✺❪ ❘✉❜❡❧ ▲✳❆✳✱ ❨❛♥❣ ❈✳❈✳ ✭✶✾✼✻✮✱ ✏❱❛❧✉❡s s❤❛r❡❞ ❜② ❛♥ ❡♥t✐r❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡✑✱ ❈♦♠♣❧❡① ❛♥❛❧②s✐s ✭Pr♦❝✳ ❈♦♥❢✳✱ ❯♥✐✈✳ ❑❡♥t✉❝❦②✱ ▲❡①✐♥❣t♦♥✱ ❑②✳✮✱ ✺✾✾✱ ♣♣✳✶✵✶✲✶✵✸✳ ❬✶✻❪ ❩❤❛♥❣ ❚✳❉✳✱ ▲☎ ✉ ❲✳❘✳ ✭✷✵✵✽✮✱ ✏◆♦t❡s ♦♥❡ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s s❤❛r✐♥❣ ♦♥❡ s♠❛❧❧ ❢✉♥❝t✐♦♥ ✇✐t❤ ✐ts ❞❡r✐✈❛t✐✈❡✑✱ ❈♦♠♣❧❡① ❱❛r✳❊❧✐♣✳ ❊q♥✳✱ ✺✸ ✭✾✮✱ ♣♣✳ ✽✺✼✲✽✻✼✳ ✹✸