Nevanlinna five-value theorem for P-adic meromorphic functions and their derivatives

Nguyễn Xuân Lai Tạp chí KHOA HỌC & CÔNG NGHỆ 81(05): 91 - 95

91

NEVANLINNA FIVE-VALUE THEOREM FOR P-ADIC

MEROMORPHIC FUNCTIONS AND THEIR DERIVATIVES

Nguyen Xuan Lai

Hai Duong College

ABSTRACT

In this paper, we gave a result similar to the Nevanlinna five-value theorem.

Keywords: Unique problem, p-adic Meromorphic functions, derivative, Nevanlinna, Height of

p-adic meromorphic functions

INTRODUCTION*

In 1920, Nevanlinna proved the following

result (the Navanlinna four- value theorem):

Theorem A. Let f and g be two non-constant

meromorphic functions. If f and g share four

distinct values CM, then f is a Mobius

transformation of g.

In 1997 Yang ang Hua [17] studied the

unicity problem for meromorphic functions

and differential monomials of the form

n

f f ′ , when they share only one value, and

obtained the following theorem.

Theorem B. Let f and g be two non- constant

meromorphic (resp. entire) functions, let n ≥

11 ( resp. n≥ 6) be an integer, and a Î £ , a

≠ 0. be a non- zero finite value. If n

f f ′ and

n

g g′ share the a CM, then either f ≡ dg for

some (n+ 1)- th root of unity d, or 1

cz f c e =

and 2

cz

g c e

−

= for three non-zero constants

1

c ,

2

c and c such that ( )

1 2 2

1 2

n

c c c a

+

= − .

In this paper, by using some arguments in

[10], [16] and the Nevanlinna theory in one￾dimensional non-archimedean case,

developed in [6], [12], [13], we gave a result

similar to the Nevanlinna five-value theorem.

*

Email: Nguyenxuanlai@yahoo.com

HEIGHT OF P-ADIC MEROMORPHIC

FUNCTIONS

Let f be a nonzero holomorphic function on

p

£ . For every a p

Î £ , expanding f as

( ) i

f P z a = − ∑ with homogeneous

polynomials Pi

of degree i around a, we

define

v a i P f i ( ) min : 0 = ≡ { / } .

For a point d p

Î £ , we define function

:

d

f p v

→ by

( ) ( ) d

f f d v a v a = −

Fix real number r with 0 r < £ r . Define

1 ( , )

( , )

ln

r

f

f

n a x

N a r dx

p x ρ

= ∫

If a = 0, then set ( ) (0, ) N r N r f f = .

For l a positive integer or + ¥ , set

,

,

1 ( , )

( , )

ln

r

l f

l f

n a x

N a r dx

p x ρ

= ∫

Where

l f f a ,

( , ) min ( ), { }

z r

n a r v z l

−

≤

= ∑ .

Let k be a positive integer or+ ¥ . Define

the function ( ) k

f

v z

≤

of p

£ into ¥ by

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