Sharing values of q-difference-differential polynomials

This paper is devoted to the uniqueness of q-difference-differential polynomials of different types. Using the idea of common zeros and common poles (Chin. Ann. Math., Ser. A 35:675–684, 2014), we improve the conditions of the former theorems and obtain some new results on the uniqueness of q-difference-differential polynomials of meromorphic functions.


Introduction and main results
In this paper, a meromorphic function is assumed meromorphic in the whole complex plane. We assume that the reader is familiar with the basic symbols and fundamental results of Nevanlinna theory; see, for example, [2,3,10]. We say that two meromorphic functions f and g share a point a CM (IM) if f (z)a and g(z)a have the same zeros counting multiplicities (ignoring multiplicities). The logarithmic density of the set E is defined by lim sup r→∞ 1 log r [ Denote by S(r, f ) a quantity of o(T(r, f )) as r → ∞ outside a possible exceptional set E of logarithmic density 0. Yang and Hua [9] obtained an important result on the uniqueness when the differential polynomials f n f and g n g share one value CM. Recently, many studies are devoted to the uniqueness of difference and q-difference polynomials; see [4][5][6][11][12][13][14]. Zhang [12] obtained the following result.
Theorem A ( [12]) Let f (z) and g(z) be transcendental entire functions of zero order, and let n, m, d be positive integers. If n ≥ m + 5d and f (z) n (f (z) m -1) d i=1 f (q i z) and g(z) n × (g(z) m -1) d i=1 g(q i z) share 1 CM, then f (z) ≡ tg(z), t n+d = t m = 1.
Liu, Liu, and Cao [4] and Zhang and Korhonen [11] obtained the following two theorems.
Theorem B ([4, Theorem 1.5]) Let f (z) and g(z) be transcendental zero-order entire functions, and let m be a positive integer. If n ≥ m + 5 and f (z) n (f (z) ma)f (qz + c) and g(z) n (g(z) ma)g(qz + c) share a nonzero polynomial p(z) CM, then f (z) ≡ g(z).
Zhao and Zhang [13] proved the following theorem.
Wang and Ye [8] improved the conditions of Theorems B and C to n ≥ m + 4 and n ≥ 6, respectively, by using the idea of common zeros and common poles. Here we give the main idea of common zeros and common poles. Let f , g be two nonconstant meromorphic functions. Denote by n 0 (r) or n 1 (r) the numbers of common zeros or poles of fg and g, ignoring multiplicities. Let p, q be positive integers. We assume that the Laurent series of f and g at z 0 are as follows: where f 1 (z) and g 1 (z) are analytic functions at z 0 , and f 1 (z 0 ) = 0, g 1 (z 0 ) = 0; the other cases can be discussed in a similar way. So z 0 is a zero of g(z) with multiplicities q. If q > p, then z 0 is a zero of f (z)g(z) with multiplicity qp, and thus the contribution to n 0 (r) is 1 at z 0 . If q ≤ p, then z 0 is a pole of f (z)g(z) with multiplicity p-q or an analytic point of f (z)g(z), and thus the contribution to n 0 (r) is 0 at z 0 . A similar method can be discussed for n 1 (r). As usual, denote by N 0 (r) or N 1 (r) the counting functions of the common zeros or poles of fg and g, ignoring multiplicities. Thus we have N(r, 1 fg ) ≤ N(r, 1 f ) + N 0 (r) and N(r, fg) ≤ N(r, f ) + N 1 (r). In this paper, we continue to consider the uniqueness of q-difference-differential polynomials. Firstly, we improve the condition n ≥ m + 5d in Theorem A to n ≥ m + d + 3 in Theorem 1.1. Set where c i and q i = 0 (i = 1, . . . , d) are constants, and d is a positive integer. In the following theorem, we improve the condition n ≥ 2k + 6 in Theorem D to n ≥ 6.

Theorem 1.2 Let f (z) and g(z) be transcendental zero-order meromorphic functions, and let k be a positive integer. If n ≥ 6 and
We also consider the following theorems for q-difference polynomials of different types. The following theorem is also an improvement of Theorem C. Theorem 1.4 Let f (z) and g(z) be transcendental zero-order meromorphic functions, q, c ∈ C, and q = 0. If n ≥ 7, and f (z) n (f (qz + c)f (z)) and g(z) n (g(qz + c)g(z)) share 1 and ∞ CM, then

Lemma 2.1
Let f (z) be a transcendental zero-order meromorphic function, q, c ∈ C, and q = 0. Then on a set of logarithmic density 1.

Lemma 2.2 ([7])
Let f (z) be a zero-order meromorphic function q, c ∈ C, and q = 0. Then on a set of logarithmic density 1.

Lemma 2.3 If f is a transcendental zero-order entire function, then
on a set of logarithmic density 1.
on a set of logarithmic density 1. On the other hand, combining Lemma 2.1 with the fact that f is a transcendental zero-order function, we have on a set of logarithmic density 1.

Proofs of theorems
and G(z) share 1 and ∞ CM, we have that F-1 G-1 = B, that is, where B is a nonzero constant. If B = 1, then from the second main theorem of Nevanlinna theory, Lemma 2.1, and Lemma 2.3 we obtain Using the same method, we have Combining (2) Let , h), and then (4) can be written as Next, we prove that h(z) ≡ c 1  We also have

N r, h n+m L(z, h) ≤ N(r, h) + N 1 (r).
Here we should remark that the poles of L(z, h) may be the zeros of h and the zeros of L(z, h) may be the poles of h. Similarly, denote by N 0 (r) the counting function of the common zeros of h n+m L(z, h) and L(z, h) ignoring multiplicities, and then From the second main theorem of Nevanlinna theory and the last two inequalities we have

T r, h n+m L(z, h) ≤ N r, h n+m L(z, h) + N r, 1 h n+m L(z, h)
On the other hand, (n + m)m(r, h) = m r, h n+m ≤ m r, h n+m L(z, h) + m r,
From (7) and (8) we get From (6) and (9) we get Since n ≥ m + d + 3, the value 1 is not the Picard exceptional value of h n+m L(z, h) from (10). Proof of Theorem 1.2 Let F(z) = f (z) n f (qz + c) and G(z) = g(z) n g(qz + c). From the condition in Theorem 1.2 we know that F (k) and G (k) share 1 and ∞ CM, so where C is a nonzero constant, that is, Integrating both sides of (12), we have where p 1 (z) is a polynomial of degree at most k -1. Denote 1-C k! z k + p 1 (z) by p(z). If p(z) ≡ 0, then by the second main theorem of Nevanlinna theory, Lemma 2.1, and (13) we obtain where N 0 (r) denotes the counting function ignoring multiplicities of the common zeros of F(z) and f (qz + c), and N 1 (r) denotes the counting function ignoring multiplicities of the common poles of F(z) and f (qz + c). On the other hand, From (15), (16), and Lemma 2.1 we have Substituting (14) into (17), we obtain Using the same method, we also get Combining (18) with (19), we have (n -5) T(r, g) + T(r, f ) ≤ S(r, f ) + S(r, g), which contradicts to n ≥ 6, and thus p(z) ≡ 0. Since the degree of p 1 (z) is at most k -1, we have C = 1 and p 1 (z) ≡ 0. From (13) we get f n f (qz + c) = g n g(qz + c).
Assume that h(z) = f (z) g (z) . Then h(qz + c)h(z) n = 1, that is, h(z) n = 1 h(qz+c) , and from Lemma 2.1 we have which also contradicts to n ≥ 6, so h(z) is a nonzero constant, say c 2 . So f (z) ≡ c 2 g(z), and c n+1 2 = 1. Thus the theorem is proved.
Proof of Theorem 1.3 Since f (z) and g(z) are transcendental zero-order meromorphic functions and f (z) n L(z, f ) s and g(z) n L(z, g) s share 1 and ∞ CM, we have where E is a nonzero constant. Then (20) can be rewritten as Let F(z) = f (z) n L(z, f ) s and G(z) = g(z) n L(z, g) s . We affirm that E = 1. On the contrary, assume that E = 1. Using the second main theorem of Nevanlinna theory and Lemma 2.1 for (21), we get where N 0 (r) denotes the counting function ignoring multiplicities of the common zeros of F(z) and L(z, f ), and N 1 (r) denotes the counting function ignoring multiplicities of the common poles of F(z) and L(z, f ).
We know that T(r, H) ≤ (n + 1)T(r, h) + S(r, h) from the expression of H(z) and Lemma 2.
which also contradicts to n ≥ 7, so h(z) is a nonzero constant, say c 4 , and from (27) we get c n+1 4 = 1. Thus the theorem is proved.