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


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
f : C → P
n
(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
z D
D H(D)
f(z)
C
f(z) = u(x, y) + iv(x, y) D
u(x, y) v(x, y) R
2
D
u(x, y) v(x, y)
∂u

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)
(z
0
) = 0.
f(z)
D ⊂ C f =
g

log |f(z)| =
1
2π
2π

0
log


f(Re
iφ
)


R
2
− r
2
R
2
− 2Rr cos(θ − φ) + r
2
dφ
+
M

µ=1
log



z




.
f R r < R
n(r, ∞, f) n(r, ∞, f),
f
r. a ∈ C
n(r, a, f) = n

r, ∞,
1
f − a

,
n(r, a, f) = n

r, ∞,
1
f − a

.
N(r, a, f),
N(r, a, f) f a
N(r, a, f) = n(0, a, f) log r +

r
0

|,
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



dθ
2π
,
m(r, ∞, f) =


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)
r → ∞.
T (r, f) = p log r + O(1),
N(r, a, f) = p log r + O(1), m(r, a) = O(1) a = ∞.
f(z) = ∞ q
N(r, ∞, f) = q log r + O(1),
m(r, ∞, f) = (p − q) log r + O(1).
p < q
T (r, f) = q log r + O(1), N(r, a, f) = q log r + O(1),
m(r, a, f) = O(1), a = 0.
a = 0
N(r, 0, f) = p log r + O(1), m(r, a, f) = (q − p) log r + O(1).
p = q,

−
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
t
π
dt
t
+ O(log r) =
r
π
+ O(log r).
m(r, a, f) = O(log r).
sin z cos z
a

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

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

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

+ 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

r,
1
f − a

= T (r, f) + O(1),
f − a f a
T (r, f) a
m

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
q
q > 2
δ > 0 |a
µ
− a
ν
| ≥ δ 1 ≤ µ < ν ≤ q
m(r, ∞) +
q

ν=1
m(r, a
ν
) ≤ 2T (r, f) − N

f

f − a
ν

+q log
+
3q
δ
+log 2+log
1
|f

(0)|
S(r)
m(r, a
ν
)
2T (r)
S(r).
a
ν
, (1 ≤ ν ≤ q)
F (z) =
q

ν=1
1
f(z) − a
ν

≤
1
|f(z) − a
ν
|
.
|F (z)| ≥
1
|f(z) − a
ν
|
−

µ=ν
1
|f(z) − a
µ
|
≥
1
|f(z) − a
ν
|

1 −
q − 1
2q

≥
1

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

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

|
≤
3q
δ
ν
q

ν=1
log
+
1
|f(z) − a
ν
|
≤ q log
+
3q
δ
+ log 2.
log
+
|F (z)| ≥ 0 ≥
q

ν=1
log
+
1
|f(z) − a
ν

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


r,
f
f


+m (r, f

F ) .
T (r, f) = T

r,
1
f

+ log |f(0)| ,
T

r,
f
f


= T

r,
f

f


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

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

f

− N

r,
f
f


+ log




f(0)
f

(0)




+m(r, f

F ).
q

ν=1
m(r, a
ν

f

+m(r, f

F ) + log
1
|f

(0)|
+ T (r, f) − N(r, f) + q log
+
3q
δ
+ log 2
f
f

log




f(0)
f

(0)





− N

r,
f

f

.
N

r,
f

f

− N

r,
f
f


=
1
2π
2π

0
log


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

1
f


+
+ m

r,
f

f

+ m (r, f

F ) + log
1
|f

(0)|
+ q log
+
3q
δ
+ log 2.
m (r, f

F ) = m

r,
q


ν=1
f

f − a
ν

+ q log
+
3q
δ
+ log 2 + log
1
|f

(0)|
.
m(r, ∞) +
q

ν=1
m(r, a
ν
) ≤ 2T (r, f) − N
1
(r) + S(r).
N
1
(r) N (r, f) =
q

N
b
v
f(z)
f(z) =
c
k
ν
(z − b
ν
)
k
ν
+
f

(z) f(z) =
c
−k
ν
(z − b
ν
)
k
ν
+1
+ b
v
k
v

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

, , a
q
q > 2
(q − 1)T (r, f) ≤ N (r, f) +
q

j=1
N

r,
1
f − a
j

− N
1
(r, f) + S (r, f)
≤ N (r, f) +
q

j=1
N

r,
1
f − a
j

− N
0

(C)
f := (f
0
: : f
n
) : C → P
n
(C)
C f
0
, , f
n
C
f = (

f
0
:

f
1
: · · · :

f
n
)

f
i
C.