ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC SƯ PHẠM NGUYỄN TRƯỜNG GIANG

VỀ DẠNG ĐỊNH LÝ CƠ BẢN THỨ HAI KIỂU
CARTAN CHO CÁC ĐƯỜNG CONG CHỈNH HÌNH LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN HỌC
THÁI NGUYÊN – 2008
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ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC SƯ PHẠM
NGUYỄN TRƯỜNG GIANG

i
i = 1, , q
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ε > 0
q

j=1
m(r, H
j
, f) ≤ (n + 1 + ε)T(r, f),
r > 0
f : C → P
n
(C)
D
j
, j = 1, , q,
d
j
(q − (n + 1) − ε)T (r, f) ≤
q

j=1
d
−1
j
N (r, D
j
, f) + o(T(r, f)),
r

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D C
f(z) = u(x, y) + iv(x, y) C z
0
∈ C
lim
h→0
f(z
0
+ h) − f(z
0
)
h
f(z) z
0
f(z) C D C
z
0
∈ D.
f(z) z
0
∈ C
C z
0
f(z) D
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z D
D H(D)
f(z)
C

0
) +
(z − z
0
)
2
2!
f

(z
0
) + . . .
f(z)
|z − z
0
| ≤ ρ D.
f(z)
z
0
.
z
0
∈ C m > 0
m > 0 f(z) f
(n)
(z
0
) = 0,
n = 1, , m − 1 f
(m)

µ = 1, , M, b
ν
, ν = 1, 2, , N,
f
z = re
iθ
(0 < r < R), f(z) = 0, f(z) = ∞
log |f(z)| =
1
2π
2π

0
log


f(Re
iφ
)


R
2
− r
2
R
2
− 2Rr cos(θ − φ) + r
2
dφ

R(z − b
ν
)
R
2
− b
ν
z




.
f R r < R
n(r, ∞, f) n(r, ∞, f),
f
r. a ∈ C
n(r, a, f) = n

r, ∞,
1
f − a

,
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n(r, a, f) = n

r, ∞,
1
f − a

z∈D(r)
z=0
(
+
z
f) log |
r
z
|,
D(r) r
+
z
f = max{0,
z
f}
m(r, a, f) f
a ∈ C
m(r, a, f) =

2π
0
log
+



1
f(re
iθ
) − a

(r, a, f),
T (r, f) = m(r, ∞, f) + N(r, ∞, f).
T (r, a, f) ≥ N(r, a, f) + O(1),
O(1) r → ∞
T (r, f) = T (r, a, f) + log |f(0)|.
f(z) = c
z
p
+ + a
p
z
q
+ + b
p
,
c = 0.
p > q f(z) → ∞ z → ∞
m(r, a, f) = 0(1) z → ∞ a f(z) = a
p
N(r, a, f) =
r

a
n(t, a)
dt
t
= p log r + O(1)
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r → ∞.
T (r, f) = p log r + O(1),

re
iθ



dθ
2π
=
π
2

−
π
2
r cos θ
dθ
2π
=
r
π
.
f N(r, ∞, f) = 0 T (r, f) = r/π.
a = 0, ∞, f(z) = a 2πi
2t
2π
t,
N(r, a, f) =
r

o

1
, a
2
, , a
p
log
+




p

ν=1
a
ν




≤
p

ν=1
log
+
|a
ν
|
log

+
|a
ν
| + log p
p f
1
(z), f
2
(z), , f
p
(z)
m

r,
p

ν=1
f
ν
(z)

≤
p

ν=1
m (r, f
ν
(z)) + log p
m


ν
(z))
N

r,
p

ν=1
f
ν
(z)

≤
p

ν=1
N (r, f
ν
(z)).
T

r,
p

ν=1
f
ν
(z)

≤

|a| + log 2
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f + a, f f, f − a a −a
|T (r, f) − T (r, f − a)| ≤ log
+
|a| + log 2.
f a
m

r,
1
f − a

+ N

r,
1
f − a

= T (r, f) − log |f(0) − a| + ε(a, r),
|ε(a, r)| ≤ log
+
|a| + log 2.
m

r,
1
f − a

+ N

+
|a| + log 2.
m

r,
1
f − a

+ N

r,
1
f − a

= T (r, f) + log |f(0) − a| + ε(a, r),
|ε(a, r)| ≤ log
+
|a| + log 2.
m(r, a) m

r,
1
f − a

m(r, ∞) m(r, f).
|z| ≤ r a
1
, a
2
, , a

)
S(r) = m

r,
f

f

+m

r,
q

ν=1
f

f − a
ν

+q log
+
3q
δ
+log 2+log
1
|f

(0)|
S(r)
m(r, a

| ≥ δ −
δ
3q
≥
2
3
δ,
µ = ν
1
|f(z) − a
µ
|
≤
3
2δ
≤
1
|f(z) − a
ν
|
.
|F (z)| ≥
1
|f(z) − a
ν
|
−

µ=ν
1

|
− q log
+
2
δ
− log 2
≥
q

µ=1
log
+
1
|f(z) − a
µ
|
− q log
+
3q
δ
− log 2.
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µ = ν log
+
1
|f(z) − a
µ
|
≤ log
+

1
|f(z) − a
ν
|
+ (q − 1) log
+
2
δ
.

µ=ν
log
+
1
|f(z) − a
µ
|
≤ (q − 1) log
+
2
δ
.
log
+
|F (z)| ≥
q

µ=1
log
+

ν
|
− q log
+
3q
δ
− log 2.
|f(z) − a
ν
| ≥
δ
3q
ν
1
|f(z) − a
ν
|
≤
3q
δ
ν
q

ν=1
log
+
1
|f(z) − a
ν
|

+
1
|f(z) − a
ν
|
− q log
+
3q
δ
− log 2.
z = re
iθ
2π

0
log
+


F (re
iθ
)


dθ
≥
2π

0


1
f
.
f
f

.f

F

≤

r,
1
f

+m

r,
f
f


+m (r, f

F ) .
T (r, f) = T

r,
1

.
m

r,
f
f


+ N

r,
f
f


= m

r,
f

f

+ N

r,
f

f

+ log

f

f

−
− N

r,
f
f


+ log




f(0)
f

(0)




.
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T (r, f) = m

r,

f

+ log
1
|f(0)|
+ m

r,
f

f

+
+ N

r,
f

f

− N

r,
f
f


+ log




r,
f

f

− N

r,
f
f


+ m

r,
f

f

+m(r, f

F ) + log
1
|f

(0)|
+ T(r, f) − N(r, f) + q log
+
3q

iφ
)
f

(re
iφ
)




dφ + N

r,
f
f


− N

r,
f

f

.
N

r,
f




dφ − log




f(0)
f

(0)




=
1
2π
2π

0
log


f(re
iφ
)



f


+ N (r, f

) .
q

ν=1
m(r, a
ν
) + m(r, ∞)
≤ 2T (r, f) −

2N (r, f) − N (r, f

) + N

r,
1
f


+
+ m

r,
f

f

r,
1
f


+ 2N (r, f) − N (r, f

) ,
S(r) = m

r,
f

f

+ m

r,
q

ν=1
f

f − a
ν

+ q log
+
3q
δ




b
v
k
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k
b
1
, b
2
, , b
N
f(z)
k
1
, k
2
, , k
N
b
v
f(z)
f(z) =
c
k
ν
(z − b
ν

(z) k
1
+ 1, k
2
+ 1, , k
N
+ 1
N(r, f) =
N

ν=1
k
ν
log |
r
b
ν
| N(r, f

) =
N

ν=1
(k
ν
+ 1) log |
r
b
ν
|

+ 1)) log |
R
b
ν
| =
N

ν=1
(2k
ν
− 1)) log |
r
b
ν
| ≥ 0.
f
C a
1
, a
2
, , a
q
q > 2
(q − 1)T (r, f) ≤ N (r, f) +
q

j=1
N

r,


+ 2N (r, f) − N (r, f

) + S (r, f)
N
0

r,
1
f


f

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f − a
j
, j = 1, , q
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C
P
n
(C)
f := (f
0
: : f
n
) : C → P
n
(C)

1
, . . . ,

f
n
)
f
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