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

P❍Ò◆● ❚❍➚ ❍×❒◆●

✣➚◆❍ ▲Þ ◆❊❱❆◆▲■◆◆❆✲❈❆❘❚❆◆ P✲❆❉■❈
❱⑨ ⑩P ❉Ö◆●
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


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

P❍Ò◆● ❚❍➚ ❍×❒◆●

✣➚◆❍ ▲Þ ◆❊❱❆◆▲■◆◆❆✲❈❆❘❚❆◆ P✲❆❉■❈
❱⑨ ⑩P ❉Ö◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿
❚❙✳ ❱Ô ❍❖⑨■ ❆◆

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✐


t

Pũ ữỡ


✐✐✐

▼ö❝ ❧ö❝
▼ð ✤➛✉
✶ ▲þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝
✶✳✶

✶✳✷

▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥

✶
✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

♣ ✲❛❞✐❝

✶✳✶✳✶


✶✳✷✳✶

❍➔♠ ✤➦❝ tr÷♥❣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✷✳✷

❍❛✐ ✣à♥❤ ❧þ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✷✳✸

❇ê ✤➲ q✉❛♥ ❤➺ sè ❦❤✉②➳t

✷✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝ ✈➔ →♣ ❞ö♥❣✳
♣ ✲❛❞✐❝

✷✳✶

✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥

✷✳✷


• Nf (a, r)✿ ❍➔♠ ✤➳♠ ❝õ❛ ❢ t↕✐ ❛
• mf (∞, r) ✿ ❍➔♠ ①➜♣ ①➾ ❝õ❛ ❢
• Tf (r)✿ ❍➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❢
•

• O(1)✿

✣↕✐ ❧÷ñ♥❣ ❣✐î✐ ♥ë✐

• Nf (r), Nk (f, r)✿
• W (f )
• Hj ✿

❍➔♠ ✤➳♠✱ ❤➔♠ ✤➳♠ ♠ù❝

❲r♦♥s❦✐❛♥ ❝õ❛ ❤➔♠

f

❙✐➯✉ ♣❤➥♥❣

• Fj (z) = 0✿

P❤÷ì♥❣ tr➻♥❤ ❝õ❛ s✐➯✉ ♣❤➥♥❣

k


✶


✤÷ñ❝ ❝→❝ ✣à♥❤ ❧þ ❝❤➼♥❤ ❝õ❛ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛

♣ ✲❛❞✐❝ tr♦♥❣ tr÷í♥❣ ❤ñ♣


✷

♠ët ❝❤✐➲✉✳ ◆➠♠ ✶✾✾✸✱ ❲✳❈❤❡rr② ✤➣ ①➙② ❞ü♥❣ ♠ët ❜↔♥ s❛♦

♣ ✲❛❞✐❝ ❤➛✉

❤➳t ❝→❝ ❦➳t q✉↔ ❝õ❛ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✤è✐ ✈î✐ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ①→❝
✤à♥❤ tr➯♥ ✤➽❛ t❤õ♥❣ ❝õ❛ ♠➦t ♣❤➥♥❣

♣ ✲❛❞✐❝ Cp✳ ✣➸ ❣â♣ ♣❤➛♥ ❧➔♠ ♣❤♦♥❣

♣❤ó t❤➯♠ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ✈î✐ ❝❤✐➲✉ ❝❛♦ tr♦♥❣ tr÷í♥❣ ❤ñ♣

♣ ✲❛❞✐❝✱ ✈➔♦ ♥➠♠ ✶✾✾✺ ❍❛ ❍✉② ❑❤♦❛✐ ✈➔ ▼❛✐ ❱❛♥ ❚✉ ❬✺❪ ✤➣ ♣❤→t ❜✐➸✉ ✈➔
❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝✳
❚❤❡♦ ❤÷î♥❣ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ tæ✐ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ✿

✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝ ✈➔ →♣ ❞ö♥❣✳
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② tæ✐ s➩ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❧↕✐ ✣à♥❤ ❧þ ◆❡✈❛♥❧✐♥♥❛✲

♣ ✲❛❞✐❝ ❬✺❪✳ ❙❛✉ ✤â ❝❤➾ r❛ ♠ët sè ù♥❣ ❞ö♥❣ q✉❛♥ trå♥❣ ❝õ❛ ✣à♥❤ ❧þ
◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝ ✈➔♦ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ sü s✉② ❜✐➳♥ ❝õ❛ ✤÷í♥❣
❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ♣ ✲❛❞✐❝✳ ▼➦t ❦❤→❝✱ ✣à♥❤ ❧þ ▼❛s♦♥ ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥
❈❛rt❛♥


♣ ✲❛❞✐❝ ❬✶❪ ✤➣ ✤÷ñ❝ ✤÷❛ ✈➔♦ ❣✐↔♥❣ ❞↕②✳ ◆❣♦➔✐ r❛✱ ❝ô♥❣ ❝â ♠ët

sè t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❜➡♥❣ t✐➳♥❣ ❆♥❤ ❬✷❪✱ ❬✸✲✹❪ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❦✐➳♥
t❤ù❝ ❝ì ❜↔♥ ♥➔②✳ ❚ø ✤â ❝→❝ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝✱ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤ ✈➔ ♥❤ú♥❣
♥❣÷í✐ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✱ ❝â t❤➸ t❤❛♠ ❦❤↔♦ ❜ê s✉♥❣ ✈➔ ♠ð rë♥❣ t❤➯♠
❦✐➳♥ t❤ù❝ ✈➲ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛

♣ ✲❛❞✐❝✳ ❚❤æ♥❣ q✉❛ ❝→❝ t➔✐ ❧✐➺✉ ♥➔②✱

tr➯♥ ❝ì sð ❝→❝ ❦✐➳♥ t❤ù❝ ✤➣ ❜✐➳t✱ tr♦♥❣ ❈❤÷ì♥❣ ✶ tæ✐ ①✐♥ tr➻♥❤ ❜➔② ♠ët
sè ❦✐➳♥ t❤ù❝ ✈➲ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤

♣ ✲❛❞✐❝ ✤➸ ❞ò♥❣

❝❤♦ ❈❤÷ì♥❣ ✷✳

✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥
✶✳✶✳✶ ❚r÷í♥❣ ❝→❝ sè ♣ ✲❛❞✐❝
❱î✐

p

❧➔ ♠ët sè ♥❣✉②➯♥ tè ❝è ✤à♥❤✱ ❖str♦✇s❦✐ ✤➣ ❦❤➥♥❣ ✤à♥❤✿

❈❤➾ ❝â ❤❛✐ ❝→❝❤ tr❛♥❣ ❜à ❝❤✉➞♥ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝❤♦ tr÷í♥❣ ❤ú✉ t➾
▼ð rë♥❣ t❤❡♦ ❝❤✉➞♥ t❤æ♥❣ t❤÷í♥❣ t❛ ❝â tr÷í♥❣ sè t❤ü❝
❝❤✉➞♥

♣ ✲❛❞✐❝ t❛ ❝â tr÷í♥❣ sè Qp✳


Cp

tr Qp



Dr = {z Cp : |z| r} , D<r> = {z Cp : |z| = r} .
sỷ

f (z)

Dr

tr

ữủ

f (z) =

an z n
n0


lim |an | |z n | = 0

n

tỗ t

n N



n=0

õ t

f (z)

tr ừ tờ ộ

an z n

ợ ộ

z Cp

n=0




|an z n | 0



n

õ ộ ở tử ở tử

ừ ộ ữủ t ổ tự

số tr t

(r, f ) = max {n : |an |rn = à(r, f )} .
n0







t ộ

an z n

ở tử t

z Cp : |z| = r <

t

n=0

lim |an |rn = 0

n

t

{|an |rn }

0

(t, f ) (0, f )
dt+(0, f ) log r, (0 < r < ),
t

tr õ log rt tỹ ỡ số

ỵ [1]

ợ r > 0 à(r, .) : Ar (Cp) R+ tọ t t s
à(r, f ) = 0 f 0
à(r, f + g) max {à(r, f ), à(r, g)}
à(r, f g) = à(r, f )à(r, g)


✻

✶✳✶✳✸ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝
❚r÷í♥❣ ❝→❝ ❤➔♠ ♣❤➙♥ t❤ù❝ ❝õ❛ ❝→❝ ❤➔♠ tr♦♥❣

H(D)

M(D)✳
❉✳ ◆➳✉ f

❦➼ ❤✐➺✉ ❧➔

f ∈ M(D) ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥
❦❤æ♥❣ ❝â ❝ü❝ ✤✐➸♠ tr➯♥ D t❤➻ t❛ ❣å✐ f ❧➔ ❝❤➾♥❤ ❤➻♥❤✳

✶✳✷✳✶ ❍➔♠ ✤➦❝ tr÷♥❣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t
●✐↔ sû

❢=

f ❧➔ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ Cp ✳ ❱î✐ ♠é✐ a ∈ Cp ✱ ✈✐➳t
Pi (z − a) ✈î✐ Pi ❝→❝ ✤❛ t❤ù❝ ❜➟❝ i✳

✣à♥❤ ♥❣❤➽❛

vf (a) = min {i : Pi = 0} .
d ∈ Cp ✱ ✤à♥❤ ♥❣❤➽❛
vfd (a) = vf −d (a)✳ ❈è ✤à♥❤
❈❤♦

vfd : a ∈ Cp −→ N
ρ0 ✈î✐ 0 < ρ0 ≤ r✳

♠ët ❤➔♠
sè t❤ü❝

①→❝ ✤à♥❤ ❜ð✐


✼

✣à♥❤ ♥❣❤➽❛

r


dx,
x

1
Nl,f (a, r) =
log p
ρ0
ð ✤â

min {vf −a (z), l} .

nl,f (a, x) =
|z|≤r
❈❤♦

k

❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠

✤à♥❤ ❜ð✐✿


0
≤k
vf (z) =
v (z)
f

vf≤k


=
log p

r
ρ0

n≤k
f (a, x)
x

dx.

t❤➻ ✤➦t

Nf≤k (r) = Nf≤k (0, r).
❚❛ ✤➦t

Nf≤k (a, r)

1
=
log p

r
ρ0

n≤k
f (a, x)
x


●✐↔ sû f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ Cp ✱ ❦❤✐ ✤â tç♥ t↕✐ ❤❛✐ ❤➔♠ f2 , f1
f1
s❛♦ ❝❤♦ f1 , f2 ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ✈➔ f =
✳ ❱î✐ a ∈ Cp ∪ {∞}✱
f2
t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➳♠ sè ❦❤æ♥❣ ✤✐➸♠ nf (a, r) ❝õ❛ ❢ t↕✐ ❛ ❤❛② ❝á♥ ❣å✐
❤➔♠ ✤➳♠ sè

❛ ✲ ✤✐➸♠ ❝õ❛ ❢ ❜ð✐✿


n (∞, r) = n (0, r)
f
f2
nf (a, r) =
n
(0, r).
f1 −af2

Nf (a, r) ❝õ❛ ❢ t↕✐ ❛ ❜ð✐✿

N (∞, r) = N (0, r)
f
f2
Nf (a, r) =
N
f −af (0, r).

✣à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➳♠



am1 = 0✱ bm2 = 0✳

❚❛ ❝â

Nf (0, r) = Nf1 (0, r) = log |f1 |r − log |am1 | ,
Nf (∞, r) = Nf2 (0, r) = log |f2 |r − log |bm2 | .
❑➨♦ t❤❡♦

Nf (0, r) − Nf (∞, r) = log |f |r − log

|am1 |
= log |f |r − log |f ∗ ( 0)|,
|bm2 |


✾

❚r♦♥❣ ✤â

am1
✳
bm2

f ∗ (0) =

❚❛ ❝â

f ∗ (0) = lim z m2 −m1 f (z) ∈ Cp∗ .
z−→0


❚❛ ❝â

mf (0, r) = log+ µf (0, r) = max {0, − log |f |r } .
❙❛✉ ✤➙② t❛ ❝â ♠ët sè t➼♥❤ ❝❤➜t ✤ì♥ ❣✐↔♥ ❝õ❛ ❤➔♠ ✤➳♠ ✈➔ ❤➔♠ ①➜♣ ①➾✳

▼➺♥❤ ✤➲ ✶✳✹✳ [1]

●✐↔ sû ❢i ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ✵ tr➯♥ Cp✱ i = 1, 2, ..., k✳
❑❤✐ ✤â ✈î✐ ♠é✐ r > 0✱ t❛ ❝â
k

k

(∞, r) ≤

Nk
fi

Nfi (∞, r) + O(1); N k
i=1

i=1

(∞, r) ≤
fi

Nfi (∞, r) + O(1);
i=1



fi1 , fi2 ∈ A (Cp )✳

✤â✱ ✈✐➳t

k

i=1

F
fi =
;
f12 ...fk2

k

fi =
i=1

mfi (∞, r)+O(1).

G
,
f12 ...fk2

❑❤✐


✶✵


nfi (∞, r);

(∞, r) ≤

nk
fi

i=1

i=1

❙✉② r❛

k

(∞, r) ≤

nk

fi ✳

nfi (∞, r).
i=1

i=1

✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦

k


fi |r ≤ log max |fi |r = max log |fi |r ,
i∈{1,...,k}

i=1

i∈{1,...,k}

♥➯♥

mk
fi

(∞, r) ≤ max mfi (∞, r).
i∈{1,...,k}

i=1

❱➔

k

k

fi |r =

log|
i=1
❉♦ ✤â

log|fi |r .

lim


✶✶

▼➺♥❤ ✤➲ ✶✳✺✳ [1]

●✐↔ sû ❢i ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ✵ tr➯♥ Cp✱ i = 1, 2, ..., k✳
❑❤✐ ✤â ✈î✐ ♠é✐ ρ0 < r✱ t❛ ❝â
k
k
Tf (r) + O(1)✳
(r) ≤
Tf (r) + O(1)❀ T
(r) ≤
T
f
f
k

k

i

i=1

i

i=1


+
Γf (T ) = {t ∈ R : (n−
f (t) − nf (t)) = 0, t ≥ T },
ð ✤â

T = − log r

❱î✐ ❝→❝ ❦➼ ❤✐➺✉ ✤➣ ✤÷ñ❝ ①→❝ ✤à♥❤ ♥➔② ✈➔ ❝❤ó þ r➡♥❣ sè ❝→❝ ♣❤➛♥ tû ❝õ❛

Γf (T )

❧➔ ❤ú✉ ❤↕♥✱ ❝❤ó♥❣ t❛ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ s❛✉ ✤➙②

✣à♥❤ ❧þ ✶✳✻

✳

✭❈æ♥❣ t❤ù❝ P♦✐ss♦♥✲❏❡♥s❡♥✮

●✐↔ sû ❢ ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ♣✲❛❞✐❝ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ❦❤æ♥❣ tr➯♥ Dr ✳ ❑❤✐
✤â
+
−
(n−
f (t) − nf (t))(t − T ) + nf (c)(c − T ).

Tf (r) − Tf (ρ) = Nf (r) =
c>t≥T

❈❤ù♥❣ ♠✐♥❤✳


nf (0, x) −
dx + log ρ,
x

1
M1 =
log p
0
r

1
M2 =
log p

nf (0, x) −
r
dx + log ,
x
ρ

ρ
r

1
M3 =
log p

nf (0, x) −
dx,

✭✶✳✷✮

✣➸ ❝❤ù♥❣ ♠✐♥❤ ✭✶✳✷✮✱ tr÷î❝ t✐➯♥ t❛ ❝❤ù♥❣ ♠✐♥❤

Tf (r) − a = M = M5 .

❚r÷í♥❣ ❤ñ♣ ✶✳

= 0.

❑❤✐ ✤â

r

nf (0, x)
dx,
x

1
M=
log p

M5 = M6 .

0
◆➳✉

Γ=∅

t❤➻


tỷ

i = 1, 2, ..., n,

s = nf (0, r),

s1 = nf (0, b2 ),

c1 = |as |,

c2 = log |f |b1 ,

c3 = log |f |b2 ,

c4 = |as1 |,

+
(1)
(n
f (t) nf (t))(t t ),

M7 =
tt(1)

+
(2)
(n
f (t) nf (t))(t t ).


n

ợ n = 1

b1 = r õ nf (0, x) = 0,
ừ Tf (r) t ữủ
t b1 < r õ
t

0 < x < r

ứ t tử

M = s(log r log b1 ) = log c1 rs log c1 bs1 ,
Tf (b1 ) = log c1 bs1 .


b1 < r



n=1



(1)
s = n
f (t ),
tỷ


t) = Tf (r) a.
f (t ) nf (t ))(t

tử


✶✹

❱➟②

M = M6 = Tf (r) − a.
❉♦ ✤â ✭✶✳✷✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✳

•

❱î✐ n ≥ 2.

●✐↔ sû ❤➺ t❤ù❝ ✭✶✳✷✮ ✤ó♥❣ ✈î✐ ♠å✐
❤➺ t❤ù❝ ✭✶✳✷✮ ✤ó♥❣ ✈î✐ ♠å✐

(1 ≤ v ≤ n − 1).

v

❚❛ ❝❤ù♥❣ ♠✐♥❤

n✳

b1 < r ✳
❑❤✐ ✤â 0 < bn < bn−1 < .... < b1 < r ✈➔


= M7 +

1
log p

nf (0, x)
dx
x
b1
r

1
= M6 + s(T − t(1) ) +
log p

nf (0, x)
dx.
x

✭✶✳✹✮

b1
▼➦t ❦❤→❝

r

1
log p


M = Tf (r) − a = M6 + s(T − t(1) ) + s(t(1) − T ) = M6 .
❳➨t

b1 = r ✳
0 < bn < ..... < b2 < b1 = r ✈➔ ❞♦ ✤â
< t(2) < .... < t(n) . ⑩♣ ❞ö♥❣ ❣✐↔ t❤✐➳t q✉②

❑❤✐ ✤â t❛ ❝â

T = t(1)

1
log p

b2

♥↕♣ t❛ ❝â

nf (0, x)
dx = c3 − a = M8 = M7 + s1 (t(1) − t(2) ).
x

0
❱➟②

1
M = c3 −a+
log p

b1


✈➔

nf (0, x)
dx = s1 (log b1 −log b2 ) = s1 (t(2) −t(1) ) = log c4 bs11 −log c4 bs21 ,
x

b2

c3 = log c4 bs21 .
❱➻

t∈
/Γ

❦❤✐

t(1) < t < t(2) ✱

✈➔

Tf (r)

❧✐➯♥ tö❝ ♥➯♥

Tf (r) = log c4 bs11 .
❚ø ✭✶✳✼✮✱ ✭✶✳✽✮ ✈➔ ✭✶✳✾✮✱ t❛ ♥❤➟♥ ✤÷ñ❝

M = Tf (r) − a = M7 + s1 (t(1) − t(2) ) + s1 (t(2) − t(1) ) = M7 .
❱➻


❑➨♦ t❤❡♦

Tf (r) = Tf1 (r) + Tf2 (r) = log r + Tf2 (r).
❍➔♠

f2

✤â♥❣ ✈❛✐ trá ♥❤÷ ❤➔♠

f

tr♦♥❣ tr÷í♥❣ ❤ñ♣ ✶✱ ❞♦ ✤â

r

nf2 (0, x)
dx = Tf2 (r) − a = M6 .
x

1
log p
0
❱➟②

M = Tf2 (r) − a + log r = Tf (r) − a = M5 .
❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ♥❤➟♥ ✤÷ñ❝

M1 = Tf (ρ) − a = M9 ,




✶✼

✶✳✷✳✷ ❍❛✐ ✣à♥❤ ❧þ ❝❤➼♥❤
❚r♦♥❣ ♠ö❝ ♥➔② tæ✐ s➩ ❞✐➵♥ ✤↕t ❤❛✐ ✣à♥❤ ❧þ ❝❤➼♥❤ tr♦♥❣ ❧þ t❤✉②➳t ♣❤➙♥
❤➻♥❤ ◆❡✈❛♥❧✐♥♥❛

♣ ✲❛❞✐❝✳ ❚❛ ✈➝♥ ❞ò♥❣ ❦➼ ❤✐➺✉ |.| t❤❛② ❝❤♦ |.|p tr➯♥ Cp✳ ❚❛

❝è ✤à♥❤ ❤❛✐ sè t❤ü❝

ρ

✈➔

ρ0

s❛♦ ❝❤♦

0 < ρ0 < ρ < ∞✳

✣➛✉ t✐➯♥ t❛ s➩ ✤✐

❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ❧þ ❝❤➼♥❤ t❤ù ♥❤➜t✳

✣à♥❤ ❧þ ✶✳✽✳ ✭✣à♥❤ ❧þ ❝❤➼♥❤ t❤ù ♥❤➜t✮ [1]

◆➳✉ f ❧➔ ♠ët ❤➔♠ ❦❤→❝ ❤➡♥❣ tr➯♥ Cp(0, ρ) t❤➻ ✈î✐ ♠å✐ a ∈ Cp t❛ ❝â
mf (a, r) + Nf (a, r) = Tf (r) + O(1).


❈❤ù♥❣ ♠✐♥❤✳ f ∈ A(ρ (Cp)

t❛ ❝â

f
f
❉♦ ✤â

f (k)
f

k

=
r

i=1

= |f |r ≤
r

f (i)
f (i−1)

k

=
r


❑❤✐ ✤â

h
hg − gh h
g
.
−
=
2
h
g r
g
h
1
g
h
≤ max
,
≤ .
g r h r
r

=
r

r

❚÷ì♥❣ tü t❛ ❝ô♥❣ t❤✉ ✤÷ñ❝

f (k)

i=j

❑❤✐ ✤â ✈î✐ 0 < r < ρ✱
q

(q − 1)Tf (r) ≤ Nf (r) +

Nf (aj , r) − NRamf (∞, r) − log r + Sf
j=1
q

≤ N f (r) +

N f (aj , r) − log r + Sf ,
j=1

tr♦♥❣ ✤â
q

log |f − aj |ρ0 − log |f |ρ0 + (q − 1) log

Sf =
j=1

A
.
δ


✶✾

❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ✈➔ ✤➦t

F0 = f0 , Fi = f1 − ai f0

(i = 1, 2, ..., q).

❑❤✐ ✤â

|fk (z)| ≤ A max {|F0 (z)|, |Fi (z)|} ,
i

(k = 0, 1).

❚❛ ❧✉æ♥ sû ❞ö♥❣

W = W (f0 , f1 ) =

f0 f1
f0 f1

❧➔ ❦➼ ❤✐➺✉ ❲r♦♥s❦✐❛♥ ❝õ❛

f0

✈➔

f1 ✳

✣➦t



(l = 1, 2, ..., q − 1)

β1 , ...., βq−1

✈î✐

s❛♦ ❝❤♦

0 < max {δ|f0 (z)|, |Fj (z)|} ≤ |Fβ1 (z)| ≤ .... ≤ |Fβq−1 (z)| < ∞
❑❤✐ ✤â t❛ ❝â

|fk (z)| ≤
✈î✐ ♠é✐

k = 0, 1;

A
A
max {δ|f0 (z)|, |Fj (z)|} ≤ |Fβl (z)|,
δ
δ

l = 1, 2, ....q − 1✳

|f (z)| = max |fk (z)| ≤
k

◆❤÷ ✈➟② t❛ t❤✉ ✤÷ñ❝