Uniqueness of meromorphic function and its order K concerning the difference polynomials

UNIQUENESS OF MEROMORPHIC FUNCTION AND ITS

ORDER K CONCERNING THE DIFFERENCE POLYNOMIALS

PHAM TUYET MAI

Abstract. In this paper, we study the uniqueness problem on difference poly￾nomials and its differential of meromorphic function sharing a common value.

1. Introduction

A meromorphic function means meromorphic in the whole complex plane. We

assume that the reader is used to doing the standard notations and fundamental

results of Nevanlinna theory. Let be two meromorphic function f, g and a ∈

C ∪ {∞}. We say that f and g share a − CM if f − a and g − a have the same

zero with multiplicities . We denote by

Em)

(a; f) = {z ∈ C : f(z) = a}

the set of all a-points of f with multiplicities not exceeding m, where a-point is

counted according to it’s multiplicity.

In 2011, K. Liu, X. Ling and T. B. Cao proved the following:

Theorem A. Let f and g be transcendental meromorphic functions with finite

order, c ∈ C be a nonzero constant and n ∈ N. If n > 14, f

n

(z)f(z + c) and

g

n

(z)g(z + c) share 1 − CM, then f = tg, or fg = t, where t

n+1 = 1.

The results of this paper was suggested thinking of ideal differential order k. We

will consider the functions (f

n

(z)f(z + c))(k) and (g

n

(z)g(z + c))(k)

. Our result

is stated as follows:

Theorem 1. Let f and g be transcendental meromorphic functions with finite

order, c ∈ C is a nonzero constant and n ∈ N, k be a positive integer. If one of

the following conditions is hold

1. n > 10k + 24 and E1)(1,(f

n

(z)f(z + c))(k)

) = E1)(1,(g

n

(z)g(z + c))(k)

);

2. n > 4k + 15 and (f

n

(z)f(z + c))(k)

, (g

n

(z)g(z + c))(k)

share 1 − CM;

then f = tg or (f

n

(z)f(z + c))(k)

.(g

n

(z)g(z + c))(k) = 1, where t

n+1 = 1.

2000 Mathematics Subject Classification. Primary 32H30.

Key words: Uniqueness theorem, difference polynomials.

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