The zeros on complex differential-difference polynomials of certain types

Changjiang Song,Lei Ma,Kai Liu

doi:10.1186/s13662-018-1712-x

Abstract

In this paper, we consider the zeros distribution of f(z)P(z,f) -q(z), where P(z,f) is a linear differential-difference polynomial of a finite-order transcendental entire function f(z), and q(z) is a nonzero polynomial. To a certain extent, Theorem 1.1 generalizes the recent results (Latreuch and Belaïdi in Arab. J. Math. 7(1):27–37, 2018; Lü et al. in Kodai Math. J. 39(3):500–509, 2016) related to Hayman conjecture (Hayamn in Ann. Math. 70:9–42, 1959).

Highlights

1 Introduction We assume that the readers are familiar with the basic symbols and fundamental results of Nevanlinna theory [8, 9, 18]
In view of Lemma 2.8 and Remark 6, we assume that z0 is a simple zero of f such that z0 is not a zero or pole of bi (i = 1, 2)
It is obvious that H is a small function of f and f = b2(a2n – 2m – 2n ) + b2n f – H f

Summary

Introduction

We assume that the readers are familiar with the basic symbols and fundamental results of Nevanlinna theory [8, 9, 18]. P(z, f ) is a linear differential-difference polynomial in f (z), which is a transcendental entire function of finite order, and a(z) is a small function with respect to f (z). Lemma 2.3 Let f be a transcendental meromorphic function of finite order, and let P(z, f ) be a linear differential-difference polynomial. Lemma 2.5 ([4, Theorems 2.1, 2.2]) Let f (z) be a transcendental meromorphic function with finite order ρ(f ) = ρ, and let η be a fixed nonzero complex number. Lemma 2.8 Let f (z) be a transcendental meromorphic function of finite order, let F(z) be a linear differential-difference polynomial of f (z), and let and a(z) and b(z) be nonzero small functions with respect to f (z).

We also obtain

Note that

For the