Results on the deficiencies of some differential-difference polynomials of meromorphic functions

Abstract In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z)mf(z+c)f′(z), f(z+c)nf′(z), f(z)mf(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying lim sup r→+∞ T(r, f) T(r,  f ′ ) <+∞, $$\mathop {\lim \,{\rm sup}}\limits_{r \to + \infty } {{T(r,\,f)} \over {T(r,\,f')}}{\rm{ < }} + \infty ,$$ and c be a non-zero complex constant, then δ(∞, f(z)m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c)nf′(z))≥ δ(∞, f′). We also investigate the value distribution of some differential-difference polynomials taking small function a(z) with respect to f(z).


Introduction and main results
The fundamental theorems and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see Hayman [1], Yang [2] and Yi-Yang [3]). In addition, for a meromorphic function f .z/, we use ı.a; f / to denote the Nevanlinna deficiency of a 2 e C D C [ f1g, where ı.a; f / D lim inf r!C1 m.r; 1 f a / T .r; f / D 1 lim sup r!C1 N.r; 1 f a / T .r; f / : We also use S.r; f / to denote any quantity satisfying S.r; f / D o.T .r; f // for all r outside a possible exceptional set E of finite logarithmic measure lim r!1 R OE1;r/\E dt t < 1. Throughout this paper, we assume m; n; k; t are positive integers. Many people were interested in the value distribution of different expressions of meromorphic functions and obtained lots of important theorems (see [1,[4][5][6]).
Recently, the topic of complex differences has attracted the interest of many mathematicians, and a number of papers have focused on the value distribution of complex differences and difference analogues of Nevanlinna theory (including [7][8][9][10][11]). By combining complex differentiates and complex differences, we proceed in this way in this paper.
Firstly, we study the Nevanlinna deficiencies related to a meromorphic function f .z/, its derivative f 0 .z/ and its differential-difference monomials and obtain the following theorems.  The following ideas derive from Hayman [5], Laine-Yang [12], Zheng-Chen [13]. In 1959, Hayman [5] studied the value distribution of meromorphic functions and their derivatives, and obtained the following famous theorems. Recently, some authors studied the zeros of f .z C c/f .z/ n a and f .z C c/ af .z/ n b, where a.¤ 0/; b are complex constants or small functions. Some related results can be found in [12][13][14][15][16][17]. Especially, Laine-Yang [12] and Zheng-Chen [13] proved the following result, which is regarded as a difference counterpart of Theorem 1.8. 12,13]). Let f .z/ be a transcendental entire function of finite order, and c be a non-zero complex constant. Then (i) for n 2,ˆ1.z/ D f .z C c/f .z/ n assumes every a 2 Cnf0g infinitely often.
In the following, we investigate the zeros of some differential-difference polynomials of a meromorphic function f .z/ taking small function a.z/ with respect to f .z/, where and in the following a.z/ is a non-zero small function of growth S.r; f /, and obtain some theorems as follows.
Theorem 1.10. Let f .z/ be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set If m n C 8 or n m C 8, then G 1 .z/ a.z/ has infinitely many zeros.
Example 1.11. An example shows that the conclusion can not hold if f .z/ is of infinite order. Let f .z/ D 2e e z , a.z/ D e z , m D 9; n D 1 and e c D 10, then then G 1 .z/ a.z/ has finite many zeros.
Theorem 1.12. Let f .z/ be a transcendental meromorphic function of finite order, and c 1 ; c 2 ; : : : ; c n be non-zero complex constants. Set z/ a.z/ has infinitely many zeros.
Let P n .z/ D a n z n C a n 1 z n 1 C C a 1 z C a 0 be a non-zero polynomial, where a 0 ; a 1 ; : : : ; a n are complex constants and t is the number of the distinct zeros of P n .z/. Then we further obtain the following results. Theorem 1.13. Let f .z/ be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set If m n C t C k.k C 3/ C 4, then G 3 .z/ a.z/ has infinitely many zeros. Theorem 1.14. Let f .z/ be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set If m n C t C k.k C 3/ C 4, then G 4 .z/ a.z/ has infinitely many zeros.

Some lemmas
To prove the above theorems, we will require some lemmas as follows.
where we assume m n without loss of generality. Thus, (2) is proved.
Using the similar method as in Lemma 2.4, we get the following lemmas.
Lemma 2.5. Let f .z/ be a transcendental meromorphic function of finite order, and Lemma 2.6. Let f .z/ be a transcendental meromorphic function of finite order, and Lemma 2.7. Let f .z/ be a transcendental meromorphic function of finite order, and 3 Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1
We firstly give the following elementary inequalities for˛;ˇ; 0 and˛Ä˛1. Since it follows that

Proofs of Theorems 1.5 and 1.6
Using the similar method as in the proof of Theorem 1.4, we can prove Theorems 1.5 and 1.6 easily.

Proof of Theorem 1.14
If f .z/ is a transcendental meromorphic function of finite order, then by Lemma 2.7, we have S.r; f / D S.r; G 4 /. Thus, by using the second fundamental theorem and Lemmas 2. Consequently, G 4 .z/ a.z/ has infinitely many zeros. This completes the proof of Theorem 1.14.

Proofs of Theorems 1.12 and 1.13
Using the similar method as in the proofs of Theorems 1.10 and 1.14 and combining Lemmas 2.5 and 2.6, we can prove Theorems 1.12 and 1.13 easily.