Uniqueness theorem on meromorphic functions and their difference operators

Dan Liu,Mingliang Fang,Bingmao Deng

doi:10.1186/s13662-018-1878-2

Abstract

In this paper, we study the uniqueness problems of meromorphic functions and their difference operators. Our main result is a difference analogue of a result of Jank–Mues–Volkmann, which is concerned with the uniqueness of an entire function sharing one finite value with its derivatives. Some recent papers studied the case of entire functions of finite order sharing a periodic small function to f. We consider the case of meromorphic functions of finite order sharing a polynomial, which is a more popular case. Examples are provided for our results.

Highlights

Introduction and main results LetC denote the complex plane and f a meromorphic function in the whole complex plane C
T(r, f ) is called the characteristic function of f, and it plays a cardinal role in the whole theory of meromorphic functions
Problem 1 In this paper, we study the uniqueness of meromorphic functions of finite order sharing a polynomial with its differences

Summary

Liu et al Advances in Difference Equations

If ρ(f ) < ∞, we say that f is a meromorphic function of finite order. For a meromorphic function a, if T(r, a) = S(r, f ), where S(r, f ) = o(T(r, f )), as r → ∞, possibly outside of an exceptional set of finite logarithmic measure, we say that a is a small function of f. Theorem 6 ([8]) Let f (z) be a nonconstant entire function of finite order such that n c f (z). Theorem 7 ([6]) Let f be a nonconstant meromorphic function of finite order, and let p(z). Theorem 8 Let f be a nonconstant meromorphic function of finite order, let n ∈ N+ be a positive integer, and let P(z) (≡ 0) be a polynomial. Problem 1 In this paper, we study the uniqueness of meromorphic functions of finite order sharing a polynomial with its differences. Eπ iz P, nc f – P = –P, and nc +1f – P = –P have the same zeros with the same multiplicities, and f = (1 + eπiz)P(z), where eπiz is a periodic function of period c = 2 This example shows that Case (ii) in Theorem 8 exists.

If fn and n n

Then n

Define that n

Hence n