Some results on difference polynomials sharing values

In this article, we shall assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna theory (e.g., see [1–3]). In addition, we will use the notations λ(f) to denote the exponent of convergence of zero sequences of meromorphic function f(z); σ(f) to denote the order of f(z). We say that meromorphic functions f and g share a finite value a CM when f - a and g - a have the same zeros with the same multiplicities. For a non-zero constant c, the forward difference ${\Delta }_{c}f\left(z\right)=f\left(z+c\right)-f\left(z\right)$, ${\Delta }_c^{n+1}f\left(z\right)={\Delta }_c^{n}f\left(z+c\right)-{\Delta }_c^{n}f\left(z\right)$, n = 1, 2,.... In general, we use the notation C to denote the field of complex numbers.

Currently, there has been an increasing interest in studying difference equations in the complex plane. Halburd and Korhonen [4, 5] established a version of Nevanlinna theory based on difference operators. Ishizaki and Yanagihara [6] developed a version of Wiman-Valiron theory for difference equations of entire functions of small growth.

Recently, Liu and Yang [7] establish a counterpart result to the Brück conjecture [8] valid for transcendental entire function for which σ(f) < 1. The result is stated as follows.

Theorem A. Let f be a transcendental entire function such that σ(f) < 1. If f and ${\Delta }_c^{n}f$ share a finite value a CM, n is a positive integer, and c is a fixed constant, then

for some non-zero constant τ.

Heittokangas et al. [9], prove the following result which is a shifted analogue of Brück conjecture valid for meromorphic functions.

Theorem B. Let f be a meromorphic function of order of growth σ(f) < 2, and let c ∈ C. If f(z) and f(z + c) share the values a ∈ C and ∞ CM, then

for some constant τ.

Here, we also study the shift analogue of Brück conjecture, and obtain the results as follows.

Theorem 1.1. Let f(z) be a non-constant entire function, σ(f) < 1 or 1 < σ(f) < 2 and λ(f) < σ(f) = σ. Set L₁(f) = a _n (z) f(z + n) + a _{n -1}(z) f(z + n - 1) +... + a₁(z) f(z + 1) + a₀(z) f(z), where a _j (z)(0 ≤ j ≤ n) are entire functions with a _n (z)a₀(z) ≢ 0. Suppose that if σ(f) < 1, then max{σ(a _j )} = α < 1, and if 1 < σ(f) < 2, then max{σ(a _j )} = α < σ - 1. If f and L₁(f) share 0 CM, then

where c is a non-zero constant.

Theorem 1.2. Let f(z) be a non-constant entire function, 2 < σ(f) < ∞ and λ(f) < σ(f). Set L₂(f) = a _n (z) f(z + n) + a _{n -1}(z) f(z + n - 1) +... + a₁(z) f(z + 1) + e^zf(z), a _j (z)(1 ≤ j ≤ n) are entire functions with σ(a _j ) < 1 and a _n (z) ≢ 0. If f and L₂(f ) share 0 CM, then

where h(z) is an entire function of order no less than 1.

Theorem 1.3. Let f(z) be a non-constant entire function, σ(f) < 1 or 1 < σ(f) < 2, λ(f) < σ(f). Set L₃(f) = a _n (z) f(z + n) + a _{n- 1}(z) f(z + n - 1) + ... + a₁(z) f(z + 1) + a₀(z) f(z), a _j (z)(0 ≤ j ≤ n) are polynomials and a _n (z) ≢ 0. If f and L₃(f ) share a polynomial P(z) CM, then

where c is a non-zero constant.

Theorem 1.4. Let f(z) be a non-constant entire function, σ(f) < 1 or 1 < σ(f) < 2, λ(f) < σ(f). Set a(z) is an entire function with σ(a) < 1. If f and a(z)f(z + n) share a polynomial P(z) CM, then

where c is a non-zero constant.

The method of the article is partly from [10].

Lemma 2.1. [11] Let f(z) be a meromorphic function with σ(f) = η < ∞. Then for any given ε > 0, there is a set E₁ ⊂ (1, +∞) that has finite logarithmic measure, such that

holds for |z| = r ∉ [0, 1] ∪ E₁, r → ∞.

Applying Lemma 2.1 to $\frac{1}{f}$, it is easy to see that for any given ε > 0, there is a set E₂ ⊂ (1, ∞) of finite logarithmic measure, such that

holds for |z| = r ∉ [0, 1] ∪ E₂, r → ∞.

Lemma 2.2. [11] Let

where n is a positive integer and $b_n={\alpha }_n{e}^{i{\theta }_n},{\alpha }_n>0,{\theta }_n\in \left[0,2\pi \right)$. For any given $\epsilon \left(0<\epsilon <\frac{\pi }{4n}\right)$, we introduce 2n open sectors

Then there exists a positive number R = R(ε) such that for |z| = r > R,

if z ∈ S _j where j is even; while

if z ∈ S _j where j is odd.

Now for any given θ ∈ [0, 2π), if $\theta \ne -\frac{{\theta }_n}{n}+\left(2j-1\right)\frac{\pi }{2n}$, (j = 0, 1,..., 2n - 1), then we take ε sufficiently small, there is some S _j , j ∈ {0, 1,...,2n - 1} such that θ ∈ S _j .

Lemma 2.3. [12] Let f(z) be a meromorphic function of order σ = σ(f) < ∞, and let λ' and λ'' be, respectively, the exponent of convergence of the zeros and poles of f. Then for any given ε > 0, there exists a set E ⊂ (1, ∞) of |z| = r of finite logarithmic measure, so that

or equivalently,

holds for r ∉ E ∪ [0, 1], where n _z,η is an integer depending on both z and η, β = max{σ - 2, 2λ - 2} if λ < 1 and β = max{σ - 2, λ - 1} if λ ≥ 1 and λ = max{λ', λ''} .

Lemma 2.4. [2] Let f(z) be an entire function of order σ, then

where ν(r) be the central index of f.

Lemma 2.5. [2, 13, 14] Let f be a transcendental entire function, let $0<\delta <\frac{1}{4}$ and z be such that|z| = r and that

holds. Then there exists a set F ⊂ R₊ of finite logarithmic measure, i.e., ${\int }_F\frac{dt}{t}<\infty$, such that

holds for all m ≥ 0 and all r ∉ F.

Lemma 2.6. [10] Let f(z) be a transcendental entire function, σ(f) = σ < ∞, and G = {ω₁, ω₂,..., ω _n }, and a set E ⊂ (1, ∞) having logarithmic measure lmE < ∞. Then there is a positive number $B\left(\frac{3}{4}\le B\le 1\right)$, a point range $\left\{z_k=r_k{e}^{i{\omega }_k}\right\}$ such that |f(z _k )| ≥ BM(r _k , f ), ω _k ∈ [0, 2π), lim _{k →∞} ω _k = ω₀ ∈ [0, 2π), r _k ∉ E ∪ [0, 1], r _k → ∞, for any given ε > 0, we have

Under the hypothesis of Theorem 1.1, see [3], it is easy to get that

where Q(z) is an entire function. If σ(f) < 1, we get Q(z) is a constant. Then Theorem 1.1 holds. Next, we suppose that 1 < σ(f) < 2 and λ(f) < σ(f) = σ. We divide this into two cases (Q(z) is a constant or a polynomial with deg Q = 1) to prove.

Case (1): Q(z) is a constant. Then Theorem 1.1 holds.

Case (2): deg Q = 1. By Lemma 2.3 and λ(f) < σ(f) = σ, for any given $0<\epsilon <\text{min}\left\{\frac{\sigma -1}{2},\frac{1-\alpha }{2},\frac{\sigma -\lambda \left(f\right)}{2},\frac{\sigma -1-\alpha }{2}\right\}$, there exists a set E₁ ⊂ (1, ∞) of |z| = r of finite logarithmic measure, so that

holds for r ∉ E₁ ∪ 0[1].

By Lemma 2.5, there exists a set E₂ ⊂ (0, ∞) of finite logarithmic measure, such that

holds for |z| = r ∉ E₂ ∪ [0, 1], where z is chosen as in Lemma 2.5.

By Lemma 2.1, for any given ε > 0, there exists a set E₃ ⊂ (1, ∞) that has finite logarithmic measure such that

holds for |z| = r ∉ [0, 1] ∪ E₃, r → ∞.

Set E = E₁∪E₂∪E₃ and $G=\left\{-\frac{{\phi }_n}{n}+\left(2j-1\right)\frac{\pi }{2n}|j=0,1\right\}\cup \left\{\frac{\pi }{2},\frac{3\pi }{2}\right\}$. By Lemma 2.6, there exist a positive number $B\in \left[\frac{3}{4},1\right]$, a point range $\left\{z_k=r_k{e}^{i{\theta }_k}\right\}$ such that |f(z _k )| ≥ BM (r _k , f], θ _k ∈ [0, 2π), lim _{k →∞} θ _k = θ₀ ∈ [0, 2π) \ G, r _k ∉ E ∪ [0, 1], r _k → ∞, for any given ε > 0, as r _k → ∞, we have

By (3.1)-(3.3), we have that

Let $Q\left(z\right)=\tau e^{i{\theta }_1}z+b_0$, τ > 0, θ₁ ∈ [0, 2π). By Lemma 2.4, there are two opened angles for above ε,

For the above θ₀, there are two cases: (i) θ₀ ∈ S₀; (ii) θ₀ ∈ S₁.

Case (i). θ₀ ∈ S₁. Since S _j is an opened set and lim _{k →∞} θ _k = θ₀, there is a K > 0 such that θ _k ∈ S _j when k > K. By Lemma 2.2, we have

where η = η(1 - ε) sin(ε) > 0. By Lemma 2.2, if Rez _k > ζr _k (0 < ζ ≤ 1). By (3.4)-(3.7), we have

which contradicts that 0 < σ(f) - 1 - α - ε.

If Rez _k < - ζr _k (0 < ζ ≤ 1), By (3.4)-(3.7), we have

which implies that 1 < 0, r → ∞, a contradiction.

Case (ii). θ₀ ∈ S₀. Since S₀ is an opened set and lim _{k →∞} θ _k = θ₀, there is K > 0 such that θ _k ∈ S _j when k > K. By Lemma 2.2, we have

where η = τ(1 - ε) sin(ε) > 0. By (3.4)-(3.6), (3.9), we obtain

From (3.11), we get that σ(f) ≥ 2, a contradiction. Theorem 1.1 is thus proved.

Under the hypothesis of Theorem 1.2, see [3], it is easy to get that

where Q(z) is an entire function. For Q(z), we discuss the following two cases.

Case (1): Q(z) is a polynomial with deg Q = n ≥ 1. Then Theorem 1.2 is proved.

Case (2): Q(z) is a constant. Using the similar reasoning as in the proof of Theorem 1.1, we get that

where a is some non-zero constant.

If Rez _k < -ηr _k (η ∈ (0, 1]), By (3.4), (3.5), (4.2), we have

which is impossible.

If Rez _k > ηr _k (η ∈ (0, 1]), By (3.4), (3.5) and (4.2), we get

which contradicts that σ(f) > 2. This completes the proof of Theorem 1.2.

Since f and L₃(f) share P CM, we get

where Q(z) is an entire function. If σ(f) < 1, we get Q(z) is a constant. Then Theorem 1.3 holds. Next, we suppose that 1 < σ(f) < 2 and λ(f) < σ(f) = σ. Set F(z) = f(z) - P(z), then σ(F) = σ(f). Substituting F(z) = f(z) - p(z) into (5.1), we obtain

where b(z) = a _n (z)P(z + n) + ... + a₁(z)P (z + 1) + a₀(z)p(z) is a polynomial. We discuss the following two cases.

Case 1. Q(z) is a complex constant. Then Theorem 1.3 holds.

Case 2. Q(z) is a polynomial with deg Q = 1. By Lemma 2.3 and λ(f) < σ(f) = σ, for any given $0<\epsilon <\text{min}\left\{\frac{\sigma -1}{2},\frac{1-\alpha }{2},\frac{\sigma -\lambda \left(f\right)}{2},\frac{\sigma -1-\alpha }{2}\right\}$, there exists a set E₁ ⊂ (1, ∞) of |z| = r of finite logarithmic measure, so that

holds for r ∉ E₁ ∪ [0, 1].

By Lemma 2.5, there exists a set E₂ ⊂ (0, ∞) of finite logarithmic measure, such that

holds for |z| = r ∉ E₂ ∪ [0, 1], where z is chosen as in Lemma 2.5.

Set E = E₁ ∪ E₂ and $G=\left\{-\frac{{\phi }_n}{n}+\left(2j-1\right)\frac{\pi }{2n}|j=0,1\right\}\cup \left\{\frac{\pi }{2},\frac{3\pi }{2}\right\}$. By Lemma 2.6, there exist a positive number $B\in \left[\frac{3}{4},1\right]$, a point range $\left\{z_k=r_k{e}^{i{\theta }_k}\right\}$ such that | f (z _k )| ≥ BM(r _k , f), θ _k ∈ [0, 2π), lim _{k →∞}θ _k = θ₀ ∈ [0, 2π) \ G, r _k ∉ E ∪ 0[1], r _k → ∞, for any given ε > 0, as r _k → ∞, we have

Since F is a transcendental entire function and |f(z _k )| ≥ BM (r _k , f), we obtain

By (5.2)-(5.6), we have that

Using similar proof as in proof of Theorem 1.1, we can get a contradiction. Hence, Theorem 1.3 holds.

Using similar proof as in proof of Theorem 1.1, we can get Theorem 1.4 holds.

YL completed the main part of this article, YL, XQ and HX corrected the main theorems. All authors read and approved the final manuscript.