Tương tự định lý năm điểm Nevanlinna và giả thuyết Hayman đối với đạo hàm của các hàm phân hình P- Adic

Nguyễn Xuân Lai và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 113(13): 25 - 31

25

VERSION OF NEVANLINNA FIVE-VALUE THEOREM AND HAYMAN

CONJECTURE FOR DERIVATIVES OF P-ADIC MEROMORPHIC FUNCTIONS

Nguyen Xuan Lai1,*, Tran Quang Vinh2

1Hai Duong College, 2Dai Tu High Schools, Thai Nguyen

SUMMARY

In this paper, we gave a version of Nevanlinna five-value theorem and Hayman Conjecture for

derivatives of p-adic meromorphic functions.

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

Conjecture, non- Archimedean meromorphic functions, Value distribution, compensation function,

characteristic function, counting function.

INTRODUCTION*

In 1926, Nevanlinna proved the following

result (the Nevanlinna five-value theorem).

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

meromorphic functions such that for five

distinct values a1 , a2 , a3 , a4 ,

a5 we have f(x)

= ai ⇔ g(x) = ai

, i = 1, 2, 3, 4, 5. Then f ≡ g.

In 1967, Hayman also proposed the following

conjecture.

Hayman Conjecture. If an entire function f

satisfies ( ) ( ) '

1

n

f z f z ≠ for a positive

integer n and all z∈ℂ , then f is a constant.

It has been verified for transcendental entire

functions by Hayman himself for n > 1, and

by Clunie for n ≥ 1. These results and some

related problems have become to be known as

Hayman's Alternative, and caused

increasingly attensions.

In recent years the similar problems are

investiged for functions in a non￾Archimedean fields. In 2008, J. Ojeda[16]

proved that for a transcendental meromorphic

function f in an algebraically closed fields of

characteristic zero, complete for a non￾Archimedean absolute value K, the function

'

1

n

f f − has innitely many zeros, if n ≥ 2.

Ha Huy Khoai and Vu Hoai An[12]

established a similar results for a differential

monomial of the form ( ) ( ), n k f f where f is a

meromorphic function in ℂP .

*

Email: [email protected]

In this paper, by using some arguments in

[12] we gave a result similar to the

Nevanlinna five-value theorem for derivatives

of p-adic meromorphic functions.

The main tool to be used is the non￾Archimedean Nevanlinna theory, so we first

recall some basic facts of the theory. More

details can be found in [3],[4],[8],[10], [11],

[12],[14].

VALUE DISTRIBUTION FOR NON￾ARCHIMEDEAN MEROMORPHIC

FUNCTIONS

Throughout this paper, K will denote an

algebraically closed field of characteristic

zero, complete for a non-trivial non￾Archimedean absolute value denoted by . ,

and log be a real logarithm function of base

ρ > 1, and ln be a real logarithm function of

base e.

Counting functions of non-Archimedean

entire function (see [8, pp.21-

23],[3],[4],[10], [11], [12])

Let f be a non-constant entire function on K

and b ∈ K. Then we can write f in the form

( ) ( )n

n

n q

f z b z b

∞

=

= − ∑

with bq ≠ 0 and we put 0

( ) , f ω b q = q∈ ℕ.

For a point a ∈ K we define the function

: K a ω f → ℕ by 0

( ) ( ) a

f f a ω ω b b = −