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)^{m}f(z+c)f′(z), f(z+c)^{n}f′(z), f(z)^{m}f(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying
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 δ (a, f) to denote the Nevanlinna deficiency of a ∊ ℂ̃ = ℂ⋃{∞} where
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, 4–6]).
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 [7–11]). 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′(z) and its differential-difference monomials
Let f(z) be a meromorphic function of finite order satisfying
Let f(z) be a meromorphic function of finite order satisfying (1), and c be a non-zero complex constant. Then
It is easy to find meromorphic functions to make the inequalities in Theorems1.1and1.2hold. For example, let f_{1}(z) = e^{z}, then
Let
It follows that
Thus, we have
Thus, Theorems 1.1 and 1.2 are sharp.
In addition, from the above examples, we can find that
We also get the following relations between δ(∞, f) and δ(∞, F_{i}), i = 1, 2.
Let f(z) be a meromorphic function of finite order, and c be a non-zero complex constant. If
Letf(z) be a meromorphic function of finite order, and c be a non-zero complex constant. If
Let f(z) be a meromorphic function of finite order, and c be a non-zero complex constant. Set
If
From the conclusions of Theorems 1.1, 1.4, 1.5 and 1.6, we see that there may exist some meromorphic function f(z) satisfying δ(∞, F_{i}) = 0, i = 1, 2, 3 as δ(∞, f) > 0. An interesting problem arises naturally: How can we find some meromorphic function f(z) to satisfy δ(∞, F_{i}) = 0, i = 1, 2, 3 as δ(∞, f) > 0?
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.
([5]). Let f(z) be a transcendental entire function. Then
for n = 3 and a ≠ 0, Ψ(z) = f′(z) − af(z)^{n}assumes all finite values infinitely often.
for n = 2, Φ(z) = f′(z) f(z)^{n}assumes all finite values except possibly zero infinitely often.
Recently, some authors studied the zeros of f(z+c) f(z)^{n} − a and f(z+c) − af(z)^{n} − b, where a(≠0), b are complex constants or small functions. Some related results can be found in [12–17]. Especially, Laine-Yang [12] and Zheng-Chen [13] proved the following result, which is regarded as a difference counterpart of Theorem1. 8.
([12, 13]). Letf(z) be a transcendental entire function of finite order, and c be a non-zero complex constant. Then
for n ≥ 2, Φ_{1}(z) = f(z+c) f(z)^{n}assumes every a ∈ ℂ\{0} infinitely often.
for n ≥ 3, a ≠0, Ψ_{1}(z) = f(z+c)− af(z)^{n}assumes every b ∈ ℂ 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.
Letf(z) be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set
If m ≥ n + 8 or n ≥ m + 8, then G_{1}(z) − a(z) has infinitely many zeros.
An example shows that the conclusion can not hold if f(z) is of infinite order. Let f(z) = 2e^{ez}, a(z) = e^{z}, m = 9, n = 1 and e^{c} = −10, then
Let f(z) be a transcendental meromorphic function of finite order, andc_{1}, c_{2}, … c_{n} be non-zero complex constants. Set
If m ≥ σ_{1} + 2n + k(k + 3) + 4, where σ_{1} = s_{1} + s_{2} + … + s_{n}, then G_{2}(z) − a(z) has infinitely many zeros.
Let P_{n}(z) = a_{n}z^{n} + a_{n−1}z^{n−1} + … + a_{1}z + 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.
Let f(z) be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set
If m ≥ n + t + k(k + 3) + 4, then G_{3}(z) − a(z) has infinitely many zeros.
Let f(z) be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set
If m ≥ n + t + k(k + 3) + 4, then G_{4}(z) − a(z) has infinitely many zeros.
To prove the above theorems, we will require some lemmas as follows.
([7, 10]).Let f(z) be a meromorphic function of finite order ρ and c be a fixed non-zero complex number, then we have
By [18], [19, p.66] and [20], we immediately deduce the following lemma.
Let f(z) be a meromorphic function of finite order, and c be a non-zero complex constant. Then
([3, p.37]).Let f(z) be a nonconstant meromorphic function in the complex plane and l be a positive integer. Then
Let f(z) be a transcendental meromorphic function of finite order, and G_{1}(z) = f(z)^{m}f(z + c)^{n} .f^{′}(z). Then we have
From Lemmas 2.2 and 2.3, we have
On the other hand, from Lemma 2.2 again, we have
Using the similar method as in Lemma 2.4, we get the following lemmas.
Let f(z) be a transcendental meromorphic function of finite order, and
Let f(z)be a transcendental meromorphic function of finite order, and
Let f(z) be a transcendental meromorphic function of finite order, and
We firstly give the following elementary inequalities
Since
Then by Lemma 2.1, we have
Since N(r, f′) = N(r, f) = N̅ (r, f), it follows by Lemma 2.2 that
From (1), we have lim sup
It follows that δ(∞, f′) ≤ δ(∞, F′).
Thus, we complete the proof of Theorem 1.1.
Since
Let F_{4}(z) = f(z)^{m + 2} then we have
It follows that δ(∞, F_{4}) = δ(∞, f′) = δ. Since F_{4}(z) = f(z)^{m+2}, we have
From (7), (8) and Lemmas 2.1 and 2.2, we have
From (7) and (8) again, we have
Thus, from (9), (10) and
This completes the proof of Theorem 1.4.
Using the similar method as in the proof of Theorem 1.4, we can prove Theorems 1.5 and 1.6 easily.
Suppose that f(z) is a transcendental meromorphic function of finite order. Since m ≥ n + 8 or n ≥ m + 8, then by Lemma 2.4, we have S(r, f) = S(r, G_{1}). Thus, by using the second fundamental theorem and Lemmas 2.2 and 2.4, we have
Thus, we have from m ≥ n + 8 or n ≥ m + 8 that
Consequently, G_{1}(z)− a(z) has infinitely many zeros.
This completes the proof of Theorem 1.10.
If f(z) is a transcendental meromorphic function of finite order, then by Lemma 2.7, we have S(r, f) = S(r. G_{4}). Thus, by using the second fundamental theorem and Lemmas 2.2 and 2.7 again,
where γ_{1}, γ_{2},.....,γ_{t} are distinct zeros of P_{m}(z). Since m ≥ n + t + k(k + 3) + 4, we have
Consequently, G_{4}(z) − a(z) has infinitely many zeros.
This completes the proof of Theorem 1.14.
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.
The authors are grateful to the referees and editors for their valuable comments which lead to the improvement of this paper.
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