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Nguyễn Thị Ngọc Ánh Tạp chí KHOA HỌC & CÔNG NGHỆ 128(14): 127 - 131

134

AN ESTIMATE FOR HOLOMORPHIC MAP RAMIFICATED OVER

HYPERPLANES IN SUBGENERAL POSITION

Pham Duc Thoan1

, Pham Hoang Ha2

, Tran Hue Minh3

1National University of Civil Engineering, 2Hanoi National University of Education, 3College of Education - TNU

SUMMARY

In this article, we study the ramification of the holomorphic map ramificate over hyperplanes in n￾subgeneral position in

( ) k

. This work is a continuation of previous work of Dethloff-Ha [1].

We thus give an improvement of the results by studying the holomorphic maps with ramification

of M. Ru [3] and Dethloff-Ha [1].

Key words: Minimal surface, Gauss map, Ramification, holomorphic map, Value distribution theory.

INTRODUCTION*

In 1993, M. Ru [3] studied the holomorphic

maps in

( ) k

with ramification. The aim of

this work is that studying the distribution of

Gauss map of Minimal surface. Using the

notations which will be introduced in

§2,

the

result of Ru can be stated as following.

Main Lemma [3] Let

0

( :...: ) : ( ) k

k R f f f   

be a

nondegenerate holomorphic map,

0

,..., H H

q

be hyperplanes in

( ) k

in

n subgeneral

position, and

( )j

be their Nochka weights.

Let

1

0

( , , ) : {0} k F f fk R



  

is

reduced representation of

f

. Assume that

q n k    2 1

and

1

( ) ( 1)

2 /

( 2)

q

j

j k

q N

k k





 







with some constant N>1. Then there exists

some positive constant

C

such that

1

4/ 1 2 /

1 0

( )

1

| | | ( ) | | |

| ( ) |

q k

S N q N

p j k

j p

q

j

j

j

F F H F

F H 





 







1

2

0

( 1) ( ) /

2

2 2

2

,

| |

k

p

k k k p q N

R

C

R z







         

* Tel: 0985 130218, Email: [email protected]

with

S

is given by

2

1

( ) ( 1) ( 2 12 / ).

q

j

 j k k k q N



     

Recently, the authors P.H. Ha and D. Dethloff

[1] gave a version on the lower dimension

spaces (

1

( ).

Theorem (Lemma 8, [1]) For every



with

1

1

2 0

q

j j

q q

m





    

and

f

which is

ramified over

j

a

with multiplicity at least

mj

for each

j (1 ),  j q

there exists a

positive constant

C0

such that

1

1

2

0 1

1 0 2 2 1

1

|| || | ( , ) | 2

.

| |

| |

q

j j

j

q q

m

q m

j j

f W f f R

C

R z

F







  

 











In this paper we will consider the

corresponding problem for the holomorphic

map on disk

{z :|z| <R } R  

(

0    R

) ramificating over hyperplanes

located n-subgeneral position in

( ). k

Our

result is stated as following:

Main Theorem. Let

f f f  : ... : :  0 k 

( ) k  R

be a nondegenerate

holomorphic map,

0

,..., H H

q

be hyperplanes

in

( ) k

in

n subgeneral position, and

( )j

be their Nochka weights. If

f

is