Luận văn thạc sỹ tóan học: Định lý cơ bản thứ hai kiểu Cartan cho đường cong chỉnh hình ppsx

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
www.VNMATH.com
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


Chuyên ngành: GIẢI TÍCH
Mã số: 60.46.01

LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN HỌC




Người hướng dẫn khoa học:
TS. TẠ THỊ HOÀI AN



THÁI NGUYÊN – 2008
www.VNMATH.com
www.VNMATH.com
C C∪{∞}
T (r, a, f)
a a
f : C −→ P
n
(C)
H

i
i = 1, , q
www.VNMATH.com
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

www.VNMATH.com
f : C → P
n
(C)
D
j
1 ≤ j ≤ q q P
n
(C) d
j
ε > 0
M
q − (n + 1) − ε)T (r, f) ≤
q

j=1
d
−1
j
N
M
(r, D
j
, f) + o (T (r, f)) ,
r
C
P
n
(C).
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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

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
∂x
=
∂v
∂y
,
∂u
∂y
= −
∂v
∂x
, ∀ (x, y) ∈ D.
f(z)
D ⊂ C z ∈ D
f(z)
f(z) = f(z
0
) +
(z − z
0
)
1!
f

(z

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)

(z
0
) = 0.
f(z)
D ⊂ C f =
g
h
g, h D.
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D = C f(z) C f(z)
z
0
m > 0 f(z) z
0
f(z) =
1
(z − z
0
)
m
.h(z)
h(z) z
0
h(z
0
) = 0
f(z) ≡ 0
{|z| ≤ R} 0 < R < ∞
a
µ

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

+
M

µ=1
log




R(z − a
µ
)
R
2
− a
µ
z




−
N

ν=1
log





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


.
N(r, a, f),
N(r, a, f) f a
N(r, a, f) = n(0, a, f) log r +

r
0

n(t, a, f) − n(0, a, f)

dt
t
,
N(r, a, f) = n(0, a, f) log r +

r
0

n(t, a, f) − n(0, a, f)

dt
t
).
a = 0
N(r, 0, f) = (
+
0
f) log r +


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




dθ
2π
,
m(r, ∞, f) =

2π
0
log
+
| f(re
iθ
) |
dθ
2π
,
log
+
x = max{0, log x}.
m
f
(r, ∞) log |f|
|z| = r
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T (r, a, f) f
a ∈ C
T (r, a, f) = m(r, a, f) + N
f

(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),

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,
T (r, f) = q log r + O(1),
N(r, a) = q log r + O(1), a = c. k
f − c ∞
m(r, c, f) = k log r + O(1), N(r, c, f) = (q − k) log r + O(1).
T (r, f) = d log r + O(1),
d = max(p, q).
www.VNMATH.com
f(z) = e
z
.
m(r, f) =
2π

0
log
+



e

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

t
π
dt
t
+ O(log r) =
r
π
+ O(log r).
m(r, a, f) = O(log r).
sin z cos z
a
N(r, a, sin z) + O(1) = N(r, a, cos z) + O(1) =
2r
π
+ O(1).
sin z cos z e
iz
e
−iz
T (r, sin z) + O(1) = T (r, cos z) + O(1) ≤
2r
π
+ O(1).
T (r, sin z) + O(1) = T (r, cos z) + O(1) =
2r
π
+ O(1)
m(r, a, sin z) + O(1) = m(r, a, cos z) + O(1) = O(1).
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a

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

+





p

ν=1
a
ν





≤ log
+

p max
ν=1, ,p
|a
ν
|

≤
q

ν=1
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


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

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)

≤

p

ν=1
T (r, f
ν
(z)) + log p.
T

r,
p

ν=1
f
ν
(z)

≤
p

ν=1
T (r, f
ν
(z)).
p = 2 f
1
(x) = f(z), f
2
(z) = aa
T (r, f + a) ≤ T (r, f) + log
+

|a| + log 2
www.VNMATH.com
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

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

+
|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

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
1
(r) + S(r),
N
1
(r)
N
1
(r) = N

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
+log 2+log
1
|f

(0)|
S(r)
m(r, a

ν
)
www.VNMATH.com
2T (r)
S(r).
a
ν
, (1 ≤ ν ≤ q)
F (z) =
q

ν=1
1
f(z) − a
ν
.
ν |f(z) − a
ν
| <
δ
3q
µ = ν
|f(z) − a
µ
| ≥ |a
µ
− a
ν
| − |f(z) − a
ν

| ≥ 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

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

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

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

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

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
|

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

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

1
f
.
f
f

.f

F

≤

r,
1
f

+m

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

+ log




f(0)
f

(0)





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

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

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

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

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



f(0)
f

(0)




+m(r, f

F ).
q

ν=1
m(r, a
ν
) + m(r, ∞) ≤ m(r, F ) + m(r, f) + q log
+
3q
δ
+ log 2.
≤ T (r, f) − N

r,
1
f

+ N


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

δ
+ log 2
f
f

log




f(0)
f

(0)




=
1
2π
2π

0
log




f(re

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


f

− N

r,
f
f


=
1
2π
2π

0
log




f(re
iφ
)
f

(re
iφ
)


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

dφ−log |f(0)|−
1
2π
2π

0
log


f

(re
iφ
)


dφ−log |f

(0)|
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= N

r,
1
f

− N (r, f) − N

r,
1

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


+ 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

N
1
(r) = N


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

+ 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

ν=1
log




R
b
ν





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

)
k
ν
+
f

(z) f(z) =
c
−k
ν
(z − b
ν
)
k
ν
+1
+ b
v
k
v
+ 1 f

(z). b
1
, b
2
, , b
N
f


(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
|

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

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
(r, f) + S (r, f) .
S (r, f) = o(T (r, f)) r → ∞
N
1
(r, f) = N

r
1
f



+ 2N (r, f) − N (r, f

) + S (r, f)
N
0

r,
1
f


f

www.VNMATH.com
f − a
j
, j = 1, , q
www.VNMATH.com
C
P
n
(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.

f = (

f
0
,

f

1
, . . . ,

f
n
)
f
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