Về hàm phân hình fp (f) và gp (g) chung nhau một hàm nhỏ

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

▼❆ ❚❍➚ ◆❍❯◆●

❱➋ ❍⑨▼ P❍❹◆ ❍➐◆❍ f P (f ) ❱⑨ g P (g)
❈❍❯◆● ◆❍❆❯ ▼❐❚ ❍⑨▼ ◆❍➘

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

▼❆ ❚❍➚ ◆❍❯◆●

❱➋ ❍⑨▼ P❍❹◆ ❍➐◆❍ f P (f ) ❱⑨ g P (g)
❈❍❯◆● ◆❍❆❯ ▼❐❚ ❍⑨▼ ◆❍➘

❈❤✉②➯♥ ♥❣➔♥❤✿ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷

▲❯❾◆

ữớ ữợ ồ

P ❚❘❺◆ P❍×❒◆●

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✐

▲í✐ ❝❛♠ ✤♦❛♥
❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣
t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝ ✤➣ ❝ỉ♥❣ ❜è ð ❱✐➺t ◆❛♠✳ ❚ỉ✐
❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②
✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ữủ ró
ỗ ố

t
ữớ ✈✐➳t ▲✉➟♥ ✈➠♥

▼❛ ❚❤à ◆❤✉♥❣

❳→❝ ♥❤➟♥

❳→❝ ♥❤➟♥

❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❝❤✉②➯♥ ổ

ừ ữớ ữợ ồ

P r Pữỡ

✐✐

▲í✐ ❝↔♠ ì♥
✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥✱ tỉ✐ ❧✉ỉ♥ ữủ sỹ ữợ ú
ù t t ừ P●❙✳❚❙ ❍➔ ❚r➛♥ P❤÷ì♥❣ ✭❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠
❚❤→✐ ◆❣✉②➯♥✮✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛②
✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ♥❤➜t ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ t❤➛② ✤➣ ❞➔♥❤ ❝❤♦ tỉ✐✳
❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ▲➣♥❤ ✤↕♦ ♣❤á♥❣ ✣➔♦ t↕♦✱ t
t ổ trỹ t q ỵ t s ồ qỵ t ổ
ợ ❤å❝ ❑✷✶ ✭✷✵✶✸✲ ✷✵✶✺✮ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐
◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ ❦✐➳♥ tự qỵ ụ ữ t
tổ t❤➔♥❤ ❦❤â❛ ❤å❝✳
❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ t tợ ỳ
ữớ ổ ở ✈✐➯♥✱ ❤é trđ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ tr♦♥❣ s✉èt
q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦

❚❤→✐ ♥❣✉②➯♥✱ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✺
◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥

▼❛ ❚❤à ◆❤✉♥❣


✐✐✐

▼ư❝ ❧ư❝
▲í✐ ❝❛♠ ✤♦❛♥

✐

▲í✐ ❝↔♠ ì♥

✐✐

▼ư❝ ❧ư❝

✐✐✐

▼ð ✤➛✉

✶

✶

▼ët sè t➼♥❤ ❝❤➜t ✈➲ ♣❤➙♥ ❜è ❣✐→ trà ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤

✸

✶✳✶

❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✶✳✶

❍➔♠ ✤➳♠ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

ỵ ỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ợ





rữớ ❤đ♣ ✤❛ t❤ù❝ ❝❤ù❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ✤↕♦ ❤➔♠

✼

✶✳✷✳✷

❇ê ✤➲ ❝❤➻❛ ❦❤â❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷

✷

✶✶

❱➜♥ ✤➲ ❞✉② ♥❤➜t ❦❤✐ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝❤✉♥❣ ♥❤❛✉
♠ët ❤➔♠ ♥❤ä

✷✶

✷✳✶

✷✶

❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✶

❚r÷í♥❣ ❤đ♣ P = b(x − a1 )n

l

(x − ai )ki

i=2

✳ ✳ ✳ ✳ ✳ ✳

✷✶


✐✈
✷✳✶✳✷

❚r÷í♥❣ ❤đ♣ P = b(x − a1 )

n

l

(x − ai ) ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✶

i=2

✷✳✷

❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✶

❚r÷í♥❣ ❤đ♣ P = b(x − a1 )

n

l

✸✻

✳ ✳ ✳ ✳ ✳ ✳

✸✻

(x − ai ) ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

(x − ai )ki

i=1

✷✳✷✳✷

❚r÷í♥❣ ❤đ♣ P = b(x − a1 )n

l

i=1

❑➳t ❧✉➟♥

✹✻

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✼





ởt ự ử q trồ ừ ỵ tt ❜è ❣✐→ trà ◆❡✈❛♥❧✐♥♥❛
❧➔ ♥❣❤✐➯♥ ❝ù✉ sü ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝õ❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ t❤ỉ♥❣ q✉❛
↔♥❤ ♥❣÷đ❝ ❝õ❛ ♠ët t➟♣ ❤ú✉ ❤↕♥✳ ◆➠♠ ✶✾✷✻✱ ❘✳ ◆❡✈❛♥❧✐♥♥❛ ✤÷đ❝ ❝❤ù♥❣
tä ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤
❞✉② ♥❤➜t ❜ð✐ ↔♥❤ ♥❣÷đ❝ ❦❤ỉ♥❣ t➼♥❤ ❜ë✐ ❝õ❛ ✺ ♣❤➙♥ ❜✐➺t ❝→❝ ❣✐→ trà✳ ❈æ♥❣
tr➻♥❤ ♥➔② ừ 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♦→♥ ❤å❝ tr ữợ
rss t r ọ tỗ t ởt t ủ ỳ

S ✤✐➲✉ ❦✐➺♥ E(S, f ) = E(S, f ) ❦➨♦ t❤❡♦ f ≡ g ❄✧✳ ◆➠♠ ✶✾✾✺✱ ❍✳❳✳ ❨✐
✭❬✶✹❪✮ tr↔ ❧í✐ ❝➙✉ tr↔ ❧í✐ ❝➙✉ ❤ä✐ ❝õ❛ ●r♦ss tr♦♥❣ tr÷í♥❣ ❤ñ♣ ❤➔♠ ♥❣✉②➯♥
✈➔ ♥➠♠ ✶✾✾✽✱ ●✳ ❋r❛♥❦ ✈➔ ▼✳❘❡✐♥❞❡rs ✭❬✻❪✮ ✤➣ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ tr÷í♥❣
❤đ♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ❚r♦♥❣ t❤ü❝ t➳✱ ❝➙✉ ❤ä✐ ❝õ❛ ●r♦ss ❝â t❤➸ ✤÷đ❝ ♣❤→t
❜✐➸✉ ♥❤÷ s tỗ t ổ tự P s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t
❝ù ❝➦♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f ✈➔ g t❛ ❝â f ≡ g ♥➳✉ P (f ) ✈➔ P (g)
❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ♠ët ❣✐→ trà ❈▼❄ ▼ët ❝→❝❤ tü ♥❤✐➯♥✱ t❛ ✤÷❛ r❛
❝➙✉ ọ s tỗ t ổ tự ự ❤➔♠ P s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t
❝ù ❝➦♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f ✈➔ g t❛ ❝â f ≡ g ♥➳✉ P (f ) ✈➔ P (g)
❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ❈▼❄ ✣➣ ❝â ♠ët sè ❝æ♥❣ tr➻♥❤ ❝æ♥❣ ❜è t ữợ
ự ❋❛♥❣ ❛♥❞ ❲✳ ❍♦♥❣ ✭❬✼❪✮ ✤➣
❝❤ù♥❣ ♠✐♥❤✿ ❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t✱ n ≥ 11 ❧➔ ♠ët sè
♥❣✉②➯♥ ❞÷ì♥❣✳ ◆➳✉ f n(f − 1)f ✈➔ gn(g − 1)g ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà 1 ❦➸ ❝↔
❜ë✐ t❤➻ f = g. ◆➠♠ ✷✵✵✹✱ ❲✳ ❈✳ ▲✐♥ ✈➔ ❍✳ ❳✳ ❨✐ ✭❬✶✷❪✮ ❝❤ù♥❣ ♠✐♥❤✿ ❈❤♦
f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t✱ n ≥ 13 ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳


✷

◆➳✉ f n(f − 1)2f ✈➔ gn(g − 1)2g ❝❤✉♥❣ ♥❤❛✉ z ❦➸ ❝↔ ❜ë✐ t❤➻ f = g.✳✳✳
❱ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ✈➜♥ ✤➲ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët
❝→❝❤ ❞✉② ♥❤➜t ❜ð✐ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝â ❝❤ù❛ ✤↕♦ ❤➔♠ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐
✏❱➲ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä✑ ✳
▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✤÷đ❝ ❝ỉ♥❣ ❜è
✈➔♦ ♥➠♠ ✷✵✶✸ ❜ð✐ ❑✳ ❇♦✉ss❛❢✱ ❆✳ ❊s❝❛ss✉t ✈➔ ❏✳ ❖❥❡❞❛ tr
ỗ õ ữỡ ữ s
ữỡ ởt số tự ỡ tr ỵ tt ◆❡✈❛♥❧✐♥♥❛✳ ❚r♦♥❣
❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ tự ỡ tr ỵ tt
ố tr ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè
❜ê ✤➲ sû ❞ö♥❣ tr♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ tr♦♥❣ ❈❤÷ì♥❣ ✷✳

❈❤÷ì♥❣ ✷✿ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❦❤✐ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝❤✉♥❣ ♥❤❛✉ ♠ët
❤➔♠ ♥❤ä✳ ✣➙② ❧➔ ❝❤÷ì♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧↕✐ ♠ët
sè ❦➳t q✉↔ ♥❣✉②➯♥ ❝ù✉ ❝õ❛ ❑✳ ❇♦✉ss❛❢✱ ❆✳ ❊s❝❛ss✉t ✈➔ ❏✳ ❖❥❡❞❛ ✈➲ ✤✐➲✉
❦✐➺♥ ✤↕✐ sè ❝õ❛ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ✤➸ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❧➔ ❜➡♥❣ ♥❤❛✉✳


✸

❈❤÷ì♥❣ ✶

▼ët sè t➼♥❤ ❝❤➜t ✈➲ ♣❤➙♥ ❜è ❣✐→ trà
❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤
✶✳✶

❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛

✶✳✶✳✶ ❍➔♠ ✤➳♠ ✈➔ t➼♥❤ ❝❤➜t
✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦ ✈✐➺❝ t❤❡♦ ❞ã✐ ❝→❝ ✈➜♥ ✤➲ tr➻♥❤ tr
trữợ t ú tổ ởt số tr ỵ tt ố
tr ừ ◆❡✈❛♥❧✐♥♥❛✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ❝â t❤➸ t➻♠ t❤➜② tr♦♥❣ ♥❤✐➲✉ t➔✐
❧✐➺✉✱ ❝❤➥♥❣ ❤↕♥ tr♦♥❣ ❬✷❪✳ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ DR = {z ∈
C : |z| ≤ R} ✈➔ ♠ët sè t❤ü❝ r > 0✱ tr♦♥❣ ✤â 0 < R ≤ ∞ ✈➔ 0 < r < R✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳

❍➔♠

m(r, f ) =
❧➔

1

2π

2π

log+ |f (reiϕ )|dϕ
0

❤➔♠ ①➜♣ ①➾ ❝õ❛ ❤➔♠ f ✳

❚❛ ❦➼ ❤✐➺✉ n(r, f ) ❧➔ sè ❝ü❝ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✱ n(r, f ) ❧➔ sè ❝ü❝ ✤✐➸♠ ♣❤➙♥
❜✐➺t ❝õ❛ ❤➔♠ f tr♦♥❣ Dr ợ ởt số ữỡ n[∆] (r, f )
❧➔ sè ❝ü❝ ✤✐➸♠ ❜ë✐ ❝❤➦♥ ❜ð✐ ∆ ❝õ❛ ❤➔♠ f ✭tù❝ ❧➔ ❝ü❝ ✤✐➸♠ ❜ë✐ k > ∆ ❝❤➾
✤÷đ❝ t➼♥❤ ∆ ❧➛♥ tr♦♥❣ tê♥❣ n[∆] (r, f )✮ tr♦♥❣ Dr .


✹
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳

❍➔♠
r

N (r, f ) =
0

n(t, f ) − n(0, f )
dt + n(0, f ) log r,
t

❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ ❝õ❛ ❤➔♠ f
❝→❝ ❝ü❝ ✤✐➸♠✮✳ ❍➔♠

✤÷đ❝ ❣å✐ ❧➔

r

N (r, f ) =
0

✤÷đ❝ ❣å✐ ❧➔

❤➔♠ ✤➳♠ t↕✐

n(t, f ) − n(0, f )
dt + n(0, f ) log r
t

❤➔♠ ✤➳♠ ❦❤ỉ♥❣ ❦➸ ❜ë✐✳ ❍➔♠
r

N[∆] (r, f ) =
0

✤÷đ❝ ❣å✐ ❧➔

✭❝á♥ ✤÷đ❝ ❣å✐ ❧➔

n[∆] (t, f ) − n[∆] (0, f )
dt + n[∆] (0, f ) log r
t

❤➔♠ ✤➳♠ ❜ë✐ ❝❤➦♥ ❜ð✐

∆✱ tr♦♥❣ ✤â n(0, f ) = lim n(t, f )❀
t→0

n(0, f ) = lim n(t, f )❀ n[∆] (0, f ) = lim n[∆] (t, f )✳ ❙è ∆ tr♦♥❣ N[∆] (r, f )
t→0

✤÷đ❝ ❣å✐ ❧➔

t→0

❝❤➾ sè ❜ë✐ ❝❤➦♥✳ ❚❛ ❦➼ ❤✐➺✉

Z(r, f ) = N (r, 1/f );
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳

Z(r, f ) = N (r, 1/f );

Z[∆] (r, f ) = N[∆] (r, 1/f ).

❍➔♠

T (r, f ) = m(r, f ) + N (r, f )
❣å✐ ❧➔

❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ f ✳

❈→❝ ❤➔♠ N (r, f ), m(r, f ), T (r, f ) ✤÷đ❝ ❣å✐ ❝❤✉♥❣ ❧➔ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛
❈→❝ ❜ê ✤➲ s❛✉ ✤➙② ❧➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✿

✺
❇ê ✤➲ ✶✳✶✳✶✳

❈❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f1, . . . , fp ✈➔ r > 0✳ ❑❤✐ ✤â✿
p

1. m r,

p

fv (z)

≤

v=1
p

2. m r,

v=1
p

fv (z)

≤

v=1
p

3. N

r,

fv (z)

r,

≤

fv (z)

r,

≤

fv (z)

r,

≤

T (r, fv (z)) + log p,
v=1
p

fv (z)
v=1

N (r, fv (z)),

v=1
p

v=1
p

6. T

N (r, fv (z)),
v=1
p

v=1
p

5. T

m(r, fv (z)),
v=1
p

v=1
p

4. N

m(r, fv (z)) + log p,

≤

T (r, fv (z)).
v=1

❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❦❤✐ p = 2✱ f1 = f (z), f2 = a a số
tũ ỵ t s r T (r, f + a) ≤ T (r, f ) + log+ |a| + log 2✳ ❱➔ tø ✤â t❛ ❝â
t❤➸ t❤❛② t❤➳ f + a, f ❜ð✐ f, f − a ✈➔ a ❜ð✐ −a✱ s✉② r❛

|T (r, f ) − T (r, f − a)| ≤ log+ |a| + log 2.
❚❛ ❦➼ ❤✐➺✉ A(C) ❧➔ ✈➔♥❤ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ C✱ M(C) ❧➔ tr÷í♥❣ ❝→❝
❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C✳
❇ê ✤➲ ✶✳✶✳✷✳

❜➟❝ q✳ ❑❤✐ ✤â

❈❤♦ f, g ∈ M(C)✱ a ∈ C ✈➔ P (f ) ∈ C[x] ❧➔ ♠ët ✤❛ t❤ù❝
T (r, f + g) ≤ T (r, f ) + T (r, g) + O(1),
T (r, f g) ≤ T (r, f ) + T (r, g),
T (r, f − a) = T (r, f ) + O(1),
T (r, 1/f ) = T (r, f ) + O(1),
T (r, P (f )) = qT (r, f ) + O(1).


✻
❇ê ✤➲ ✶✳✶✳✸✳

❈❤♦ f, g ∈ M(C)✱ ❦❤✐ ✤â
Z(r, f − a) ≤ T (r, f ) + O(1), ∀a ∈ C,
m(r, f g) ≤ m(r, f ) + m(r, g),
N (r, f ) = N (r, f ) + N (r, f ),
Z(r, f ) ≤ Z(r, f ) + N (r, f ) + Sf (r).

❍ì♥ ♥ú❛✱ ❝❤♦ Q ∈ C[x] ❝â ❜➟❝ q✳ ❚❤➻
Z(r, f (Q(f ))) ≥ Z(r, Q(f )),
N (r, f Q(f )) = (q + 1)N (r, f ) + N (r, f ).

❑❤✐ ✤â m(r, ff ) = Sf (r) ✈➔ T (r, f )
T (r, f ) + N (r, f ) + Sf (r)✳ ❍ì♥ ♥ú❛✱ ♥➳✉ f ∈ M(C) t❤➻
❇ê ✤➲ ✶✳✶✳✹✳

❈❤♦ f

∈ M(C)✳

≤

T (r, f ) ≤ T (r, f ) + Sf (r).
❇ê ✤➲ ✶✳✶✳✺✳

❈❤♦ f

∈ M(C)✳

❑❤✐ ✤â m(r, f1 ) ≥ m(r, f1 ) + Sf (r)✳ ❍ì♥

♥ú❛ T (r, f ) − Z(r, f ) ≤ T (r, f ) − Z(r, f ) + Sf (r).

❈❤♦ f ∈ M(C) t❤ä❛ ♠➣♥ f (0) = 0, ∞✱ S ❧➔ t➟♣ ❝♦♥ ❤ú✉ ❤↕♥ ❝õ❛ C
✈➔ r ∈ [0, +∞]✳ ❚❛ ❦➼ ❤✐➺✉ Z0S (r, f ) ❧➔ ❤➔♠ ✤➳♠ t↕✐ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛
f ♠➔ ♥â ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ f − s ✈ỵ✐ ♠å✐ s ∈ S ✳ ◆❣❤➽❛ ❧➔✱
♥➳✉ (γn )n∈N ❧➔ ❞➣② ❤ú✉ ❤↕♥ ❤♦➦❝ ✈æ ❤↕♥ ❝õ❛ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f tr➯♥

♠➔ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f − s ✈ỵ✐ ♠å✐ s ∈ S ✱ ✈ỵ✐ ❜ë✐ sè qn
t÷ì♥❣ ù♥❣ t❤➻
Z0S (r, f ) =
qn (log r − log |γn |).
|γn |≤r

❈❤♦ a1, ..., an ∈ C ✈ỵ✐ n ≥ 2, n ∈ N ✈➔ f ∈ M(C)✳ ✣➦t
S = {a1 , ..., an }✳ ●✐↔ sû r➡♥❣ ❦❤æ♥❣ ❝â ❣✐→ trà ♥➔♦ tr♦♥❣ f, f ✈➔ f − aj ✈ỵ✐
1 ≤ j ≤ n ❜➡♥❣ 0 ❤♦➦❝ ∞ t↕✐ ✤✐➸♠ ❣è❝ t❤➻✱ ✈ỵ✐ ♠é✐ r > 0 t õ

ỵ

n

Z(r, f aj ) + N (r, f ) − Z0S (r, f ) + Sf (r).

(n − 1)T (r, f ) ≤
j=1



ỵ ỡ

sỷ f (z) ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr♦♥❣ ❤➻♥❤ tr➯♥ DR, 0 <
R ≤ a số ự tũ ỵ õ t õ

ỵ

m r,

1
f a

+ N r,

1
f a

= T (r, f ) − log |f (0) − a| + ε(a, r),

tr♦♥❣ ✤â ε(a, r) ≤ log+ |a| + log 2.
❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ r > 0✳ ❍➔♠

Nram (r, f ) = N (r, 1/f ) + 2N (r, f ) − N (r, f ),

❣✐→ trà ♣❤➙♥ ♥❤→♥❤ ❝õ❛ ❤➔♠ f ✳ ❉➵ t❤➜② Nram(r, f ) 0
ỵ ỵ ỡ tự ●✐↔ sû f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤
❦❤→❝ ❤➡♥❣ tr➯♥ C, a1, ..., aq ∈ C, (q > 2) ❧➔ ❝→❝ ❤➡♥❣ sè ♣❤➙♥ ❜✐➺t✳ ❑❤✐ ✤â
✈ỵ✐ ♠é✐ ε > 0✱ ❜➜t ✤➥♥❣ t❤ù❝

❣å✐ ❧➔

q

(q − 1)T (r, f ) + Nram (r, f ) ≤

N (r,
j=1

1

) + N (r, f ) + log T (r, f )
f − aj

+ (1 + ε) log+ log T (r, f ) + O(1)

✤ó♥❣ ✈ỵ✐ ♠å✐ r ≥ r0 ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳
✶✳✷

▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ✤↕♦ ❤➔♠

❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ♠ët t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥✲
❧✐♥♥❛ ❝➛♥ t❤✐➳t ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ❈→❝ t➼♥❤ ❝❤➜t
♥➔② ✤÷đ❝ ❝→❝ t→❝ ❣✐↔ ❑✳ ❇♦✉ss❛❢✱ ❆✳ ❊s❝❛ss✉t ✈➔ ❏✳ ❖❥❡❞❛ ❝❤ù♥❣ ♠✐♥❤
tr♦♥❣ ❬✷❪✳

✶✳✷✳✶ ❚r÷í♥❣ ❤đ♣ ✤❛ t❤ù❝ ❝❤ù❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ✤↕♦ ❤➔♠
❈❤♦ f (z) ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ❤➔♠ α(z) ❣å✐ ❧➔ ❤➔♠ ♥❤ä ✤è✐ ✈ỵ✐ f ♥➳✉
lim TT (r,α)
(r,f ) = 0.

r→∞


✽
❚❛ ❦➼ ❤✐➺✉ Mf (C) ❧➔ t➟♣ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ C ❧➔ ❤➔♠ ♥❤ä ✤è✐ ✈ỵ✐
f ✱ Af (C) ❧➔ ❝→❝ ❤➔♠ ♥❣✉②➯♥ tr➯♥ C ❧➔ ❤➔♠ ♥❤ä ✤è✐ ✈ỵ✐ f ✳
❇ê ✤➲ ✶✳✷✳✶✳

❈❤♦ Q ∈ C[x] ✈➔ f ∈ M(C) ❧➔ ❤➔♠ s✐➯✉ ✈✐➺t✳ ❑❤✐ ✤â
T (r, Q(f )) ≤ T (r, f Q(f )) + m(r,

1
)
f

✈➔
T (r, Q(f )) ≤ T (r, f ) + N (r, f ) + T (r, f Q(f )) + Sf (r), r ∈ [0, +∞].

❍ì♥ ♥ú❛ ♥➳✉ deg(Q) ≥ 3✱ t❤➻
Mf (C) ⊂ Mf Q(f ) (C).
❈❤ù♥❣ ♠✐♥❤✳

❚❛ ❝â

T (r, Q(f )) = N (r, Q(f )) + m(r, f

Q(f )
)
f

≤ N (r, Q(f )) + N (r, f ) + m(r, f Q(f )) + m(r,

1
).
f

❱➻ ❝→❝ ❝ü❝ ✤✐➸♠ ❝õ❛ f ❝ô♥❣ ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ Q(f ) ✭✈ỵ✐ ❜ë✐ sè ❦❤→❝ ♥❤❛✉✮✱
t❛ s✉② r❛
N (r, Q(f )) + N (r, f ) = N (r, f Q(f )).
❉♦ ✤â

N (r, Q(f )) + N (r, f ) + m(r, f Q(f )) + m(r,

1
)
f

= N (r, f Q(f )) + m(r, f Q(f )) + m(r,
= T (r, f Q(f )) + m(r,

1
)
f

1
),
f

✤â ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ✤➛✉ t✐➯♥✳ ❚✐➳♣ t❤❡♦✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝
t❤ù ❤❛✐✳ ❚❛ ❝â
1
m(r, ) ≤ T (r, f ) + Sf (r)
f

= m(r, f ) + N (r, f ) + Sf (r).
❚❤❡♦ ❇ê ✤➲ ✶✳✶✳✹ t❛ ❝â

m(r, f ) ≤ m(r, f ) + Sf (r),

✾
❞♦ ✤â

m(r,

1
) ≤ m(r, f ) + N (r, f ) + Sf (r)
f
= m(r, f ) + N (r, f ) + N (r, f ) + Sf (r).

❚ø ✤â s✉② r❛

T (r, Q(f )) ≤ T (r, f Q(f )) + T (r, f ) + N (r, f ) + Sf (r)
≤ T (r, f Q(f )) + 2T (r, f ) + Sf (r).
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✱ ♥➳✉ deg(Q) ≥ 3 t❤➻ ❦➳t q✉↔ tr➯♥ ❧➔ ❜➜t ✤➥♥❣
t❤ù❝ ❧✉ỉ♥ ✤ó♥❣✳

❈❤♦ Q ∈ C[x] ✈➔ f, g ∈ M(C) ✭t÷ì♥❣ ù♥❣ f, g ∈ A(C)✮ ❧➔
❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t✳ ❈❤♦ P (x) = xn+1Q(x) ✈ỵ✐ n ≥ deg(Q) + 3 ✭t÷ì♥❣ ù♥❣
n ≥ deg(Q) + 2✮✳ ◆➳✉ P (f )f = P (g)g t❤➻ P (f ) = P (g).
❇ê ✤➲ ✶✳✷✳✷✳

✣➦t k = deg(Q)✳ ❱➻ P (f )f = P (g)g tỗ t c ∈ C
s❛♦ ❝❤♦ P (f ) = P (g) + c
sỷ r c = 0 t ỵ ✶✳✶✳✻ t❛ ❝â
❈❤ù♥❣ ♠✐♥❤✳

T (r, P (f )) ≤ Z(r, P (f )) + Z(r, P (f ) − c) + N (r, P (f )) + Sf (r).
❘ã r➔♥❣ t❛ t❤➜② r➡♥❣

Z(r, P (f )) = Z(r, f n+1 Q(f )) = Z(r, f Q(f )) ≤ T (r, f Q(f )).
❚❤❡♦ ❇ê ✤➲ ✶✳✶✳✷ t❛ ❝â

T (r, f Q(f )) = (k + 1)T (r, f ) + O(1),
❞♦ ✤â

Z(r, P (f )) ≤ (k + 1)T (r, f ) + O(1).
❚❛ ❝ô♥❣ ❝â

Z(r, P (f ) − c) = Z(r, P (g)) ≤ Z(r, g) + Z(r, Q(g))
≤ T (r, g) + T (r, Q(g)).
❚❤❡♦ ❇ê ✤➲ ✶✳✶✳✷✱ t❛ ❝â

Z(r, P (f ) − c) ≤ (k + 1)T (r, g) + O(1).

✭✶✳✶✮



ú ỵ r

N (r, P (f )) = N (r, f ) ≤ T (r, f ) + O(1).
❚ø ✭✶✳✶✮ t❛ ❝â

T (r, P (f )) ≤ (k + 2)T (r, f ) + (k + 1)T (r, g) + Sf (r),

✭✶✳✷✮

✭t÷ì♥❣ ù♥❣

T (r, P (f )) ≤ (k + 1)T (r, f ) + (k + 1)T (r, g) + Sf (r)).

✭✶✳✸✮

❚❤❡♦ ❇ê ✤➲ ✶✳✶✳✷ t❛ ❝â

T (r, P (f )) ≤ (n + k + 1)T (r, f ) + O(1).
❚ø ✭✷✳✷✮ t❛ s✉② r❛

nT (r, f ) ≤ T (r, f ) + (k + 1)T (r, g) + Sf (r),

✭✶✳✹✮

nT (r, f ) ≤ (k + 1)T (r, g) + Sf (r)).

✭✶✳✺✮

✭t÷ì♥❣ ù♥❣

❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝â

nT (r, g) ≤ T (r, g) + (k + 1)T (r, f ) + Sf (r),

✭✶✳✻✮

nT (r, g) ≤ (k + 1)T (r, f ) + Sf (r)).

✭✶✳✼✮

✭t÷ì♥❣ ù♥❣

❈ë♥❣ ợ t ữủ

n(T (r, f ) + T (r, g)) ≤ (k + 2)(T (r, f ) + T (r, g)) + Sf (r),
❞♦ ✤â

0 ≤ (k + 2 − n)(T (r, f ) + T (r, g)) + Sf (r),
✭t÷ì♥❣ ù♥❣

0 ≤ (k + 1 − n)(T (r, f ) + T (r, g)) + Sf (r)).
▼➙✉ t ợ n k + 3 tữỡ ù♥❣ n ≥ k + 2✮ ✳ ❱➟② c = 0
tø ✤â s✉② r❛ P (f ) = P (g)✳


✶✶
❈❤♦ ✷ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f, g ∈ M(C)✳ ✣➦t

Ψf,g =

f
2f
g
2g
−
−
+
.
f
f −1 g
g−1

❈❤♦ F, G ∈ M(C) ❦❤→❝ ❤➡♥❣✱ ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❦❤æ♥❣
❝â ❝ü❝ ✤✐➸♠ t↕✐ 0 ✈➔ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ❦➸ ❝↔ ❜ë✐✳ ●✐↔ sû Ψf,g = 0 ✈➔
❇ê ✤➲ ✶✳✷✳✸✳

lim sup
r→+∞

Z(r, f ) + Z(r, g) + N (r, f ) + N (r, g)
max(T (r, f ), T (r, g))

<1

✤ó♥❣ ✈ỵ✐ ♠å✐ r > 0✱ ♥➡♠ ♥❣♦➔✐ t➟♣ ❝â ✤ë ✤♦ ❤ú✉ ❤↕♥✳ ❑❤✐ ✤â✱ ❤♦➦❝ f = g
❤♦➦❝ f g = 1✳
❇ê ✤➲ ✶✳✷✳✹✳ ❈❤♦ f, g ∈ M(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t✱ t❤ä❛ ♠➣♥ (f −a)f n =
f
(g − a)g n ✈ỵ✐ a ∈ C✱ ✈➔ h = ✳ ◆➳✉ h ổ ỗ t t
g
hn 1
hn+1 h
g = n+1
, f = n+1
.
h
−1
h
−1
❇ê ✤➲ ✶✳✷✳✺✳ ❈❤♦ f, g ∈ M(C) t❤ä❛ ♠➣♥ f (0) = 0, ∞
❣✐→ tr ở f,g ổ ỗ t ✵ t❤➻

❝❤✉♥❣ ♥❤❛✉ ♠ët

max(T (r, f ), T (r, g)) ≤ N[2] (r, f ) + Z[2] (r, f ) + N[2] (r, g)
+ Z[2] (r, g) + Sf (r) + Sg (r).

✣➦❝ ❜✐➺t✱ ♥➳✉ f, g ∈ A(C) t❤➻ t❛ ❝â
max(T (r, f ), T (r, g)) ≤ Z[2] (r, f ) + Z[2] (r, g) + Sf (r) + Sg (r).

✶✳✷✳✷ ❇ê ✤➲ ❝❤➻❛ ❦❤â❛
l

❈❤♦ P (x) = (x − a1)n (x − ai)k ∈ C[x] (ai = aj , ∀i = j)
i=1
✈ỵ✐ l ≥ 2 ✈➔ n ≥ max{k2, ..., kl } ✈➔ ❝❤♦ k = li=2 ki✳ ❈❤♦ f, g ∈ M(C) ❧➔
❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ θ = P (f )f P (g)g ✳ ◆➳✉ θ ∈ Mf (C) ∩ Mg (C)✱ t❤➻ t❛
❝â ❝→❝ ✤✐➲✉ s❛✉ ✤➙②✿
♥➳✉ l = 2 t❤➻ ♥ ∈ {k, k + 1, 2k, 2k + 1, 3k + 1},
♥➳✉ l = 3 t❤➻ ♥ ∈ { k2 , k + 1, 2k + 1, 3k2 − k, 3k3 − k},
♥➳✉ l ≥ 4 t❤➻ n = k + 1.

❇ê ✤➲ ✶✳✷✳✻✳

i


✶✷

❍ì♥ ♥ú❛✱ ♥➳✉ f, g t❤✉ë❝ A(C) t❤➻ θ ❦❤ỉ♥❣ t❤✉ë❝ Af (C)✳
❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ a1 = 0✳

●✐↔ sû f, g ∈ M(C) t❤ä❛ ♠➣♥

❈❤ù♥❣ ♠✐♥❤✳

l

f

n

l
ki

(f − ai )
i=2

fg

n

(g − ai )ki

g = .



i=2

ỵ
t ổ ỹ ❝õ❛ θ✳ ✣➛✉ t✐➯♥ t❛ s➩ ❝❤➾

r❛ r➡♥❣ P (f ) ✈➔ P (g) ♥❤➟♥ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ♥❣♦➔✐ ✳ ❚❤➟t
✈➟②✱ ❣✐↔ sû r➡♥❣ ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ❝õ❛ f P (f ) t❤✉ë❝ ✳
❱➻ ✈➟②✱ ♠å✐ ❦❤æ♥❣ ✤✐➸♠ ✈➔ ♠å✐ ❝ü❝ ✤✐➸♠ ❝õ❛ f P (f ) ❦❤→❝ ♣❤↔✐ ❧➔ ♠ët
❦❤æ♥❣ ✤✐➸♠ ❤♦➦❝ ❧➔ ♠ët ❝ü❝ ✤✐➸♠ ❝õ❛ θ✳ ❉♦ ✤â

Z(r, f P (f )) + N (r, f P (f )) ≤ 2T (r, θ).
❚❤➟t ✈➟②✱ ♠é✐ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f P (f ) ❤♦➦❝ ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f ❤♦➦❝ ❧➔
❦❤æ♥❣ ✤✐➸♠ ❝õ❛ P (f ) ❜ð✐ ✈➻ ❝→❝ ❝ü❝ ✤✐➸♠ ❝õ❛ P (f ) ❧➔ ❝→❝ ❝ü❝ ✤✐➸♠ ❝õ❛
f ✳ ❙✉② r❛
Z(r, P (f )) + N (r, f ) ≤ 2T (r, θ),
✈➔ s✉② r❛
l

Z(r, f − ai ) + N (r, f ) ≤ 2T (r, θ).
i=1

❚❤❡♦ ✣à♥❤ ỵ t õ
l

(l 1)T (r, f )

Z(r, f − ai ) + N (r, f ) + Sf (r),
i=1

❞♦ ✤â (l − 1)T (r, f ) ≤ 2T (r, θ) + Sf (r), ♠➙✉ t❤✉➝♥✳ ❙✉② r❛ P (f ) ♥❤➟♥
❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❤♦➦❝ ❝ü❝ ✤✐➸♠ ❦❤æ♥❣ t❤✉ë❝ ✳ ❚÷ì♥❣ tü✱ P (g) ♥❤➟♥ ❝→❝
❦❤ỉ♥❣ ✤✐➸♠ ✈➔ ❝ü❝ ✤✐➸♠ ❦❤æ♥❣ t❤✉ë❝ ✳
❚❛ ❣✐↔ sû r➡♥❣ f, g ∈ A(C)✳ ❱➻ ❝↔ f, g ❦❤æ♥❣ ❝â ❝ü❝ ✤✐➸♠✱ ♠å✐ ❦❤æ♥❣

✶✸
✤✐➸♠ ❝õ❛ ❝↔ ❤❛✐ ❤➔♠ tr➯♥ ✤➲✉ t❤✉ë❝

✳ ❱➻ ✈➟②✱ t❛ ❝â
k

ki Z(r, f − ai )

Z(r, f P (f )g P (g)) =Z(r, f ) +
i=1

k

ki Z(r, g − ai )

+ Z(r, g ) +
i=1

✈➔

l

Z(r, f − ai ) ≥ (l − 1)T (r, f ) + Sf (r) + Sg (r),

Z(r, θ) ≥
i=1

♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t✳ ❙✉② r❛ f P (f )g P (g) ❦❤æ♥❣ t❤➸ t❤✉ë❝ Af (C)
❤♦➦❝ Ag (C)✳
❇➙② ❣✐í✱ t❛ q✉❛② ❧↕✐ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ✈➔ ❣✐↔ sû r➡♥❣ f, g ❧➔ ❝→❝ ❤➔♠

♣❤➙♥ ❤➻♥❤✳ ❈❤♦ γ ∈ C \
❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g ❝â ❜➟❝ s✳ ❘ã r➔♥❣✱ t❤❡♦
✭✶✳✽✮ γ ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ f ❝â ❜➟❝ ❣✐↔ sû ❧➔ t✳ ❉♦ γ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❦❤ỉ♥❣
✤✐➸♠ ♠➔ ❝ơ♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ θ t❛ ❝â t❤➸ s✉② r❛ ❝→❝ ♠è✐ ❧✐➯♥
❤➺ s❛✉✿

s(n + 1) = t(n + k + 1) + 2.

✭✶✳✾✮

❚✐➳♣ t❤❡♦✱ t❛ ❣✐↔ sû r➡♥❣ ✈ỵ✐ i ∈ {2, ..., l}, g − ai ❝â ❦❤æ♥❣ ✤✐➸♠ γ ∈ C \
❝â ❜➟❝ si ✳ ◆â ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ f ❝â ❜➟❝ ti ✳ ❱➻ ✈➟②✱ t❤❡♦ ✭✶✳✾✮ t❛ ❝â

si (ki + 1) = ti (n + k + 1) + 2.

✭✶✳✶✵✮

❚ø ✭✶✳✾✮ ✈➔ ✭✶✳✶✵✮ t❛ s✉② r❛ s > t ✈➔ si > ti ✳
❳➨t ❝ü❝ ✤✐➸♠ γ ∈ C \
❝õ❛ f ✳ ❚❛ t❤➜② γ ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g ✱ ❤♦➦❝
❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g − ai ✈ỵ✐ ♠é✐ i ∈ {2, ..., l}✱ ❤♦➦❝ ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g
♠➔ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g ♠➔ ❝ơ♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❦❤ỉ♥❣ ✤✐➸♠
❝õ❛ g − ai (∀i ∈ {2, ..., l})✳ ❈❤♦ Z0 (r, g ) ❧➔ ❤➔♠ ✤➳♠ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛
g ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ g ✈➔ ❝õ❛ g − ai ✈ỵ✐ ♠å✐ i ∈ {2, ..., l} ✭ ✤➳♠
❝↔ ❜ë✐ ✮ ✈➔ Z 0 (r, g ) ❧➔ ❤➔♠ ✤➳♠ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g ♠➔ ❦❤æ♥❣ ♣❤↔✐
❧➔ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ g ✈➔ ❝õ❛ g − ai ✈ỵ✐ ♠å✐ i ∈ {2, ..., l} ❦❤æ♥❣ ❦➸ ❜ë✐✳ ❱➻
T (r, θ) = Sf (r) = Sg (r)✱ t❛ ❝â
l

Z(r, g − ai ) + Z 0 (r, g ) + Sf (r) + Sg (r). ✭✶✳✶✶✮

N (r, f ) ≤ N (r, g) +
i=2



ỵ t õ
l

(l 1)T (r, f ) ≤Z(r, f ) +

Z(r, f − ai ) + N (r, f )
i=2

− Z0 (r, f ) + Sf (r).
❉♦ ✤â tø ✭✶✳✶✶✮ t❛ ❝â
l

Z(r, f − ai ) + Z(r, g)

(l − 1)T (r, f ) ≤ Z(r, f ) +
i=2
l

Z(r, g − ai ) + Z 0 (r, g ) − Z0 (r, f ) + Sf (r) + Sg (r).

+
i=2

✭✶✳✶✷✮

❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝â
l

Z(r, g − ai ) + Z(r, f ) + Z 0 (r, f )

(l − 1)T (r, g) ≤Z(r, g) +
i=2
l

Z(r, f − ai ) − Z0 (r, g ) + Sf (r) + Sg (r).

+

✭✶✳✶✸✮

i=2

❑➳t ❤đ♣ ✭✶✳✶✷✮ ✈➔ ✭✶✳✶✸✮ t❛ ✤÷đ❝

(l − 1)(T (r, f ) + T (r, g))
l

l

Z(r, g − ai ) + Z(r, f ) +

≤ 2 Z(r, g) +
i=2

+ Sf (r) + Sg (r).

Z(r, f − ai )
i=2

✭✶✳✶✹✮

❚r÷í♥❣ ❤đ♣ l = 2✳
❑❤ỉ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t a2 = 1✳ ●✐↔ sû r➡♥❣ t➜t ❝↔
❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f, f − 1, g, g − 1 ❝â ❜➟❝ ❧ỵ♥ ❤ì♥ ✺✱ trø r❛ ♥❤ú♥❣ ✤✐➸♠


✶✺
♥➡♠ tr♦♥❣

✱ ❦❤✐ ✤â

1
Z(r, f ) ≤ T (r, f ) + Sf (r) + Sg (r),
5
1
Z(r, f − 1) ≤ T (r, f ) + Sf (r) + Sg (r),
5
1
Z(r, g) ≤ T (r, g) + Sf (r) + Sg (r),
5
1
Z(r, g − 1) ≤ T (r, g) + Sf (r) + Sg (r),
5
♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✶✳✶✹✮✳ ❚❛ t❤➜② ❝➦♣ (n, k) ❞➝♥ ✤➳♥ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f ❤♦➦❝
❝õ❛ g ♥❣♦➔✐

❝â ❜➟❝ ≤ 4✳
❉♦ ✤â✱ t❛ ①❡♠ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ ❞➝♥ ✤➳♥ ❝→❝ ❦❤ỉ♥❣ ✤✐➸♠ ❝â ❜➟❝ ≤ 4
❝õ❛ f, f − 1, f, g − 1✳ ❚❤➟t ✈➟②✱ ✈➻ f ✈➔ g ✤â♥❣ ✈❛✐ trá ♥❤÷ ♥❤❛✉ ❧✐➯♥ q✉❛♥
✤➳♥ n ✈➔ k ✱ ♥â ❧➔ ✤✐➲✉ ❦✐➺♥ ừ trữớ ủ ú ử ợ g ❤♦➦❝
g − 1 ❝â ❦❤æ♥❣ ✤✐➸♠ ❜➟❝ s ≤ 4✳ ❚r♦♥❣ ♠é✐ tr÷í♥❣ ❤đ♣ t❛ ❦➼ ❤✐➺✉ t ❧➔ ❜➟❝
❝õ❛ ❝ü❝ ✤✐➸♠ ❝õ❛ f ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g ❤♦➦❝ g − 1✳ ◆❤➢❝ ❧↕✐ ❦❤✐ f ❝â ❝ü❝
✤✐➸♠ ❝➜♣ 4, g ❤♦➦❝ g − 1✱ ♥➳✉ ♥â ❝â ❦❤æ♥❣ ✤✐➸♠✱ t❤➻ ❜➟❝ ❝õ❛ ❦❤æ♥❣ ✤✐➸♠
♣❤↔✐ ≥ 5✳ ❉♦ ✤â✱ t❛ ❝❤➾ ①➨t ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g ✈➔ g − 1 ❧➔ ❝ü❝ ✤✐➸♠
❝õ❛ f ❝â ❜➟❝ 1, 2, 3.
✣➛✉ t✐➯♥ t❛ ❣✐↔ sû r➡♥❣ g ❝â ❦❤æ♥❣ ✤✐➸♠ γ =

❝â ❜➟❝ s = 2✳ ❚❤➻

2(n + 1) = t(k + n + 1) + 2.

✭✶✳✶✺✮

❚❤❡♦ ✭✶✳✶✺✮ ♥➳✉ t = 1 t❛ ❝â

n = k + 1.

✭✶✳✶✻✮

❚✐➳♣ t❤❡♦✱ ♥➳✉ t ≥ 2✱ tø ✤✐➲✉ ❦✐➺♥ 2n + 2 < t(k + n + 1) + 2✱ s✉② r❛ ✭✶✳✶✻✮
❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳
●✐↔ sû g ❝â ♠ët ❦❤æ♥❣ ✤✐➸♠ γ =

❝â ❜➟❝ s = 3✳ ❑❤✐ ✤â

3(n + 1) = t(k + n + 1) + 2.

✭✶✳✶✼✮

❚❤❡♦ ✭✶✳✶✼✮ ♥➳✉ t = 1 t❛ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ✈➻ k ≤ n✳
◆➳✉ t = 2 t❛ ❝â

n = 2k + 1.

✭✶✳✶✽✮


✶✻
◆➳✉ t ≥ 3 t❛ ❝â 3(n + 3) < 3(k + n + 1) + 2✱ s✉② r❛ ✭✶✳✶✽✮ ❧➔ ♥❣❤✐➺♠ ❞✉②
♥❤➜t✳
❚✐➳♣ t❤❡♦ t❛ ❣✐↔ sû g ❝â ♠ët ❦❤æ♥❣ ✤✐➸♠ ❝â ❜➟❝ s = 4✳ ❑❤✐ ✤â
✭✶✳✶✾✮

4(n + 1) = t(k + n + 1) + 2.
◆➳✉ t = 1✱ ✈➻ k ≤ n✱ ♥➯♥ t❛ ❝â 4(n + 1) > t(k + n + 1) + 2.
◆➳✉ t = 2✱ t❤❡♦ ✭✶✳✶✾✮ t❛ ❝â ♥❣❤✐➺♠

n = k.

✭✶✳✷✵✮

n = 3k + 1.

✭✶✳✷✶✮

◆➳✉ t = 3✱ t❛ ❝â ♥❣❤✐➺♠ ❦❤→❝

❉♦ ✤â✱ tø ✭✶✳✶✻✮✱ ✭✶✳✶✽✮✱ ✭✶✳✶✾✮✱ ✭✶✳✷✶✮✱ t➜t ❝↔ ❝→❝ tr÷í♥❣ ❤đ♣ ❝õ❛ g ❝â
❦❤ỉ♥❣ ✤✐➸♠ ❝â ❜➟❝ s ≤ 4 ♥❤÷ s❛✉

n = k + 1, s = 2,

n = 2k + 1, s = 3,

n = k, s = 4,

n = 3k + 1, s = 4.

❚✐➳♣ t❤❡♦✱ t❛ s➩ ①➨t ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g − 1, γ =
❜➟❝ s ❝õ❛ g − 1 t❤ä❛ ♠➣♥

s(k + 1) = t(k + n + 1) + 2.
✣➛✉ t✐➯♥ t❛ ❣✐↔ sû r➡♥❣ g − 1 ❝â ♠ët ❦❤æ♥❣ ✤✐➸♠ γ =
❚❤❡♦ ✭✶✳✷✸✮ t❛ ❝â

2(k + 1) = t(k + n + 1) + 2.

✭✶✳✷✷✮

❝â ❜➟❝ ≤ 4✳ ❱➟②
✭✶✳✷✸✮
❝â ❜➟❝ s = 2✳
✭✶✳✷✹✮

❱➻ k ≤ n t❛ ❦❤æ♥❣ t➻♠ ữủ ợ t = 1 k = n + 1
❤♦➦❝ t ≥ 2✱ ✈➻ 2(k + 1) < t(k + n + 1) + 2✳

❚✐➳♣ t❤❡♦ t❛ ❣✐↔ sû s = 3✳
◆➳✉ t = 1✱ t❛ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠

n = 2k.

✭✶✳✷✺✮

◆➳✉ t ≥ 2✱ t❛ ổ t ữủ ợ k n 3(k + 1) <
t(k + n + 1) + 2✳


✶✼
❈✉è✐ ❝ò♥❣ t❛ ❣✐↔ sû s = 4✳
◆➳✉ t = 1✱ t❛ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠

n = 3k + 1.

✭✶✳✷✻✮

n = k.

✭✶✳✷✼✮

◆➳✉ t = 2✱ t❛ ❝â ♥❣❤✐➺♠ ❦❤→❝

◆➳✉ t ≥ 3 t ổ t ữủ ợ k n ❜ð✐ ✈➻ 4(k + 1) <
t(k + n + 1) + 2.
❉♦ ✤â✱ tø ✭✶✳✷✺✮✱ ✭✶✳✷✻✮✱ ✭✶✳✷✼✮ t➜t ❝↔ ❝→❝ tr÷í♥❣ ❤đ♣ ❝õ❛ g − 1 ❝â ❝ü❝
✤✐➸♠ γ =
❝â ❜➟❝ s ≤ 4 ♥❤÷ s❛✉

n = 2k, s = 3,

n = 3k + 1, s = 4,

n = k, s = 4.

✭✶✳✷✽✮

❱➟②✱ t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦❤✐ n = k, k + 1, 2k, 2k + 1, 3k + 1 ❦❤æ♥❣ ❝â
❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f, f − 1, g, g − 1 ♥❣♦➔✐
❝â ❜➟❝ ≤ 4 ❞♦ ✤â t
tr tr ữủ ự ợ tr÷í♥❣ ❤đ♣ l = 2✳
❚r÷í♥❣ ❤đ♣ l = 3.
●✐↔ sû r➡♥❣ t➜t ❝↔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f, g, f − ai , g − ai , ∀i = 2, 3 ❝â ❜➟❝
♥❤ä ❤ì♥ 4 trø ♥❤ú♥❣ ✤✐➸♠ ♥➡♠ tr♦♥❣ ❀ t❤➻

1
Z(r, f ) ≤ T (r, f ) + Sf (r) + Sg (r) ✈➔
4
1
∀i = 2, 3, Z(r, f − ai ) ≤ T (r, f ) + Sf (r) + Sg (r).
4
1
Z(r, g) ≤ T (r, g) + Sf (r) + Sg (r) ✈➔
4
1
∀i = 2, 3, Z(r, g − ai ) ≤ T (r, g) + Sf (r) + Sg (r).
4
❚ø ✭✶✳✶✹✮ t❛ ❝â l ≤ 2✱ ♠➙✉ t❤✉➝♥✳

❉♦ ✤â✱ t❛ s➩ ❦✐➸♠ tr❛ ✈ỵ✐ ♠å✐ n ✈➔ ki (i ∈ {2, 3}) ❞➝♥ ✤➳♥ ❦❤æ♥❣ ✤✐➸♠
♥❣♦➔✐
❝â ❜➟❝ ≤ 3 ❝❤♦ f, g, f − ai , g − ai ✈ỵ✐ i = 2, 3✳ ❚❤➟t ✈➟②✱ ✈➻ f ✈➔
g ✤â♥❣ ✈❛✐ trá ♥❤÷ ♥❤❛✉✱ ✤➙② ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ tr÷í♥❣ ❤đ♣ ♥➔② ✤ó♥❣✱ ✈➼
❞ư ✈ỵ✐ g ❤♦➦❝ g − ai ♥➔♦ ✤â ❝â ♠ët ❦❤ỉ♥❣ ✤✐➸♠ ❝â ❜➟❝ ♥❤ä ❤ì♥ 3✳ ❚r♦♥❣
♠é✐ tr÷í♥❣ ❤đ♣ t❛ ❦➼ ❤✐➺✉ t ❧➔ ❜➟❝ ❝õ❛ ❝ü❝ ✤✐➸♠ ❝õ❛ f ❧➔ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛
g ❤♦➦❝ g − ai ✈ỵ✐ i ♥➔♦ ✤â✳ ◆❤➢❝ ❧↕✐ r➡♥❣ ❦❤✐ f ❝â ❝ü❝ ✤✐➸♠ ❝â ❜➟❝ 3, g


✶✽
❤♦➦❝ g − ai ✱ ♥➳✉ ♥â ❝â ❦❤æ♥❣ ✤✐➸♠✱ ♥â ♣❤↔✐ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝â ❜➟❝ ≥ 4✳
❙✉② r❛✱ t❛ ❝❤➾ ❝➛♥ ❦✐➸♠ tr❛ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g ❤♦➦❝ g − ai (∀i ∈ {2, 3})
❧➔ ❝→❝ ❝ü❝ ✤✐➸♠ ❝õ❛ f ❝â ❜➟❝ 1, 2✳
✣➛✉ t✐➯♥ t❛ ❣✐↔ sû r➡♥❣ g ❝â ♠ët ❦❤æ♥❣ ✤✐➸♠ γ =
❚❤❡♦ ✭✶✳✾✮ t❛ ❝â

❝â ❜➟❝ s = 2✳
✭✶✳✷✾✮

2(n + 1) = t(k + n + 1) + 2.
❚❤❡♦ ✭✶✳✷✾✮ ♥➳✉ t = 1 t❛ ❝â

✭✶✳✸✵✮

n = k + 1.

❚✐➳♣ t❤❡♦✱ ♥➳✉ t = 2 tø ✤✐➲✉ ❦✐➺♥ 2n + 2 < 2(k + n + 1) + 2 t❤➻ ✭✶✳✸✵✮ ❧➔
♥❣❤✐➺♠ ❞✉② ♥❤➜t✳
●✐↔ sû r➡♥❣ g ❝â ♠ët ❦❤æ♥❣ ✤✐➸♠ γ =

❝â ❜➟❝ s = 3✳ ❚❤➻

3(n + 1) = t(k + n + 1) + 2.

✭✶✳✸✶✮

❚❤❡♦ ✭✶✳✸✶✮ ♥➳✉ t = 1 t❛ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠
k
n= .
2
◆➳✉ t = 2 t❛ ❝â

✭✶✳✸✷✮

n = 2k + 1.

✭✶✳✸✸✮

❉♦ ✤â✱ t❤❡♦ ✭✶✳✸✵✮✱ ✭✶✳✸✷✮✱ ✭✶✳✸✸✮ t➜t ❝↔ ❝→❝ tr÷í♥❣ ❤đ♣ ❝õ❛ g ❝â ❦❤ỉ♥❣
✤✐➸♠ ❝â ❜➟❝ s ≤ 3 ♥❤÷ s❛✉

n = k + 1, s = 2, t = 1,
k
n = , s = 3, t = 1,
2
n = 2k + 1, s = 3, t = 2.
❚✐➳♣ t❤❡♦ ①➨t ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g − ai , γ ∈
/
si ❝õ❛ g − ai t❤ä❛ ♠➣♥

❝â ❜➟❝ si ≤ 3✳ ❱➟②✱ ❜➟❝

si (ki + 1) = t(k + n + 1) + 2.
✣➛✉ t✐➯♥ ❣✐↔ sû r➡♥❣ g − ai ❝â ❦❤æ♥❣ ✤✐➸♠ γ ∈
/
✭✶✳✸✹✮ t❛ ❝â

2(ki + 1) = t(k + n + 1) + 2.

✭✶✳✸✹✮
❝â ❜➟❝ si = 2✳ ❚❤❡♦
✭✶✳✸✺✮


✶✾
❱➻ ki ≤ n ✈➔ ki ≤ k t❛ ❝â 2(ki + 1) < t(k + n + 1) + 2✳ ❉♦ ✤â t❛ ❦❤ỉ♥❣
t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✸✺✮✳
❚✐➳♣ t❤❡♦ t❛ ❣✐↔ sû s = 3. ◆➳✉ t = 1 t❛ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠
✭✶✳✸✻✮

3ki = n + k.
◆➳✉ t = 2 t❛ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ✈➻ 3ki < 2(n + k)✳

❉♦ ✤â✱ ❝❤➾ ❝â ✶ tr÷í♥❣ ❤đ♣ ❞✉② ♥❤➜t ❝õ❛ g − ai ❝â ❦❤æ♥❣ ✤✐➸♠ γ ∈
/
❝â ❜➟❝ si ≤ 3 ❧➔

n + k = 3ki ,

s = 3,

t = 1.

t ự ữủ r ợ n = k + 1, k/2, 2k + 1, 3ki − k
❦❤æ♥❣ ❝â ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f, g, f − ai , g − ai (i = 2, 3) r❛ ♥❣♦➔✐
❝â
❜➟❝ ≤ 3 ✈➔ ❞♦ ✤â✱ ❝→❝ ♣❤→t ❜✐➸✉ ❝õ❛ ❜ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ tr÷í♥❣
❤đ♣ l = 3✳
❚r÷í♥❣ ❤đ♣ l ≥ 4✳
❚❛ ❣✐↔ sû r➡♥❣ t➜t ❝↔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f, g, f − ai , g − ai , ∀i ∈ {2, ..., l}
❝â ❜➟❝ ♥❤ä ❤ì♥ 3 trø ✤✐ ❝→❝ ✤✐➸♠ ♥➡♠ tr♦♥❣
t❤➻

1
Z(r, f ) ≤ T (r, f ) + Sf (r) + Sg (r) ✈➔
3
1
∀i = {2, ..., l}, Z(r, f − ai ) ≤ T (r, f ) + Sf (r) + Sg (r).
3
1
Z(r, g) ≤ T (r, g) + Sf (r) + Sg (r) ✈➔
3
1
∀i = {2, ..., l}, Z(r, g − ai ) ≤ T (r, g) + Sf (r) + Sg (r).
3
❚ø ✭✶✳✶✹✮ t❛ ❝â l ≤ 3✱ ♠➙✉ t❤✉➝♥✳
❉♦ ✤â✱ t❛ s➩ ❦✐➸♠ tr❛ ✈ỵ✐ ♠å✐ n ✈➔ ki (i ∈ {2, 3, ..., l}) ❞➝♥ ✤➳♥ ❦❤æ♥❣
✤✐➸♠ ♥❣♦➔✐
❝â ❜➟❝ ≤ 2 ❝❤♦ f, g, f − ai , g − ai ✈ỵ✐ i ∈ {2, 3, ..., l}✳ ❚❤➟t
✈➟②✱ ✈➻ f ✈➔ g ✤â♥❣ ✈❛✐ trá ♥❤÷ ♥❤❛✉✱ ✤➙② ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ tr÷í♥❣ ❤đ♣

♥➔② ✤ó♥❣✱ ✈➼ ❞ư ✈ỵ✐ g ❤♦➦❝ g − ai ♥➔♦ ✤â ❝â ♠ët ❦❤ỉ♥❣ ✤✐➸♠ ❝â ❜➟❝ ♥❤ä
❤ì♥ 2✳ ❚r♦♥❣ ♠é✐ tr÷í♥❣ ❤đ♣ t❛ ❦➼ ❤✐➺✉ t ❧➔ ❜➟❝ ❝õ❛ ❝ü❝ ✤✐➸♠ ❝õ❛ f ❧➔
❦❤æ♥❣ ✤✐➸♠ ❝õ❛ g ❤♦➦❝ g − ai ✈ỵ✐ i ♥➔♦ ✤â✳ ◆❤➢❝ ❧↕✐ r➡♥❣ ❦❤✐ f ❝â ❝ü❝
✤✐➸♠ ❝â ❜➟❝ 2, g ❤♦➦❝ g − ai ✱ ♥➳✉ ♥â ❝â ❦❤æ♥❣ ✤✐➸♠✱ ♥â ♣❤↔✐ ❝â ❦❤æ♥❣