Value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations

Abstract In this paper, we investigate the value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations, and obtain the results on the relations between the order of the solutions and the convergence exponents of the zeros, poles, a-points and small function value points of the solutions, which show the relations in the case of non-homogeneous equations are sharper than the ones in the case of homogeneous equations.


Introduction and main results
In this paper, we use the standard notations and basic results of Nevanlinna's value distribution theory (see [1][2][3]). In addition, we use the notation .f / to denote the order of a meromorphic function f .z/ in the whole complex plane, and the notations .f / and . 1 f / to denote the convergence exponent of the zeros and the poles of f .z/ respectively. Many scholars applied Nevanlinna theory and its difference analogues to study the properties of meromorphic solutions of complex differential equations and complex difference equations, and obtained fruitful achievement (see [4][5][6][7][8]). Especially, it is an essential respect to study the oscillation property of meromorphic solutions of complex differential equations and complex difference equations.
For complex linear differential equations, Chen, et al. (see [9]) considered the non-homogeneous equation where A j .z/.j D 0; 1; ; k 1/ and F .z/.6 Á 0/ are polynomials and obtained that every solution f .z/ of (1) satisfies .f / D .f /: ( Meanwhile, they pointed out that every solution of the homogeneous linear differential equation  (2). The above facts illustrate that the result on the relation between .f / and .f / in the case of the non-homogeneous equation is better than the one in the case of the homogeneous equation. We note that for homogeneous and non-homogeneous complex linear difference equations, as complex linear differential equations, there is also such important character. Especially, Chen (see [10]) studied the relations between the order and the convergence exponent of entire solutions of homogeneous and non-homogeneous complex linear difference equations, and obtained the following results.
Theorem A ( [10]). Let h j .z/.j D 1; 2; ; n/ be polynomials such that h n .z/ 6 Á 0; c i .i D 1; 2; ; n/ be constants which are unequal to each other. Suppose that f .z/ is a finite-order transcendental entire solution of the homogeneous linear difference equation Then: ; n/ and F .z/ be polynomials such that F .z/h n .z/ 6 Á 0, c i .i D 1; 2; ; n/ be constants which are unequal to each other. Suppose that f .z/ is a finite-order transcendental entire solution of the non-homogeneous linear difference equation Then: ng; then f .z/ assumes every non-zero finite value a infinitely often, and .f a/ D .f /.
We note that in Theorems A(i) and B(i), the result on the relation between .f / and .f / in the case of the nonhomogeneous equation is better than the one in the case of the homogeneous equation. Meantime, we note that the value distribution properties of the zeros and a-points of the solutions are distinct in Theorem A(i) and (ii).
In this paper, we continue to consider the oscillation property of meromorphic solutions of complex linear differential equations and complex linear difference equations. Firstly, we consider more general equations than (1) and (3)-(5), namely homogeneous and non-homogeneous complex linear differential-difference equations. Secondly, since the above conclusions are just on entire solutions, we consider meromorphic solutions of the involved equations. Meanwhile, we also consider the value distribution of the poles of meromorphic solutions, and generalize the relative results into the case in which meromorphic solutions assume a small function.
When the coefficients of the complex linear differential-difference equation are zero-order meromorphic functions (including the polynomials and rational functions), we obtain the following Theorem 1.1, Corollary 1.2 and Remark 1.3.
From Theorem 1.1 and Corollary 1.2, we see that the conclusion "maxf . 1 f /; .f /g D .f /" may not hold when F .z/ Á 0, that is, (6) reduces into the corresponding homogeneous equation. Therefore, we consider the corresponding homogeneous equation further and obtain the following Theorem 1.4 and Remark 1.5.

Lemmas for proofs of main results
It is shown in [11] that for any arbitrary complex number c ¤ 0, the following inequalities