Some unity results on entire functions and their difference operators related to 4 CM theorem


<jats:p>This paper is to consider the unity results on entire functions sharing two values with their difference operators and to prove some results related to 4 CM theorem. The main result reads as follows: Let <jats:inline-formula><jats:alternatives><jats:tex-math>$f(z)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>z</mml:mi>
<mml:mo>)</mml:mo>
</mml:math></jats:alternatives></jats:inline-formula> be a nonconstant entire function of finite order, and let <jats:inline-formula><jats:alternatives><jats:tex-math>$a_{1}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$a_{2}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula> be two distinct finite complex constants. If <jats:inline-formula><jats:alternatives><jats:tex-math>$f(z)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>z</mml:mi>
<mml:mo>)</mml:mo>
</mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\Delta _{\eta }^{n}f(z)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msubsup>
<mml:mi>Δ</mml:mi>
<mml:mi>η</mml:mi>
<mml:mi>n</mml:mi>
</mml:msubsup>
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>z</mml:mi>
<mml:mo>)</mml:mo>
</mml:math></jats:alternatives></jats:inline-formula> share <jats:inline-formula><jats:alternatives><jats:tex-math>$a_{1}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$a_{2}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula> “CM”, then <jats:inline-formula><jats:alternatives><jats:tex-math>$f(z)\equiv \Delta _{\eta }^{n} f(z)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>z</mml:mi>
<mml:mo>)</mml:mo>
<mml:mo>≡</mml:mo>
<mml:msubsup>
<mml:mi>Δ</mml:mi>
<mml:mi>η</mml:mi>
<mml:mi>n</mml:mi>
</mml:msubsup>
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>z</mml:mi>
<mml:mo>)</mml:mo>
</mml:math></jats:alternatives></jats:inline-formula>, and hence <jats:inline-formula><jats:alternatives><jats:tex-math>$f(z)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>z</mml:mi>
<mml:mo>)</mml:mo>
</mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\Delta _{\eta }^{n}f(z)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msubsup>
<mml:mi>Δ</mml:mi>
<mml:mi>η</mml:mi>
<mml:mi>n</mml:mi>
</mml:msubsup>
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>z</mml:mi>
<mml:mo>)</mml:mo>
</mml:math></jats:alternatives></jats:inline-formula> share <jats:inline-formula><jats:alternatives><jats:tex-math>$a_{1}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$a_{2}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula> CM.</jats:p>


Introduction and main results
It is well known that a monic polynomial is uniquely determined by its zeros and a rational function by its zeros and poles ignoring a constant factor. But it becomes much more complicated to deal with the transcendental meromorphic function case. In 1929, Nevanlinna proved his famous 5 IM theorem and 4 CM theorem (see e.g. [20,23]): if meromorphic functions f (z) and g(z) share five (respectively, four) distinct values in the extended complex plane IM (respectively, CM), then f (z) ≡ g(z) ((respectively, f (z) = T(g(z)), where T is a Möbius transformation). Here and in what follows, we say that f (z) and g(z) share the finite value a CM(IM) if f (z)a and g(z)a have the same zeros with the same multiplicities (ignoring multiplicities), and we say that f (z) and g(z) share the ∞ CM(IM) if f (z) and g(z) have the same poles with the same multiplicities (ignoring multiplicities).
To relax those shared conditions in Nevanlinna's 4 CM theorem, Gundersen provided an example to show that 4 CM shared values cannot be replaced with 4 IM shared values, but with 3 CM shared values and 1 IM shared value in [5]. That is, "4 IM = 4 CM" and "3 CM + 1 IM = 4 CM". In addition, he showed that "2 CM + 1 IM = 4 CM" in [6] (see correction in [8]), as well as by Mues in [17]. The problem that "1 CM + 3 IM = 4 CM" is still open. We recall the following result by Mues in [17], which mainly inspired us to write this paper.
In Theorem A, f and g share the value a "CM" means that where N E (r, a) is defined to be the reduced counting function of common zeros of f (z)a and g(z)a with the same multiplicities. Similarly, N 1) E (r, a) used later is defined to be the reduced counting function of common simple zeros of f (z)a and g(z)a.
Applying Theorem A, one can get (see Theorem 4.8 in [23]) the following.
Then we see that the condition N(r, 1 f -a j ) = S(r, f ) (j = 1, 2) in Theorem B means that δ(a j , f ) = 1 (j = 1, 2). And we say that a is a Nevanlinna exceptional value of f (z), provided that δ(a, f ) > 0.
To reduce the number of shared values, Rubel and Yang appear to be the first to consider the unity of the entire function sharing two values with its first derivative in [21]. They proved that, for a nonconstant entire function f , if f and f share values a, b CM, then f ≡ f . Mues and Steinmetz [18] improved Rubel and Yang's result by replacing "2 CM" with "2 IM" in 1979, and then by replacing "entire function" with "meromorphic function" in [19] (see also Gundersen [7]). In 2013, Li [15] improved these results by adding some condition on the poles of the meromorphic function f . This paper is to consider replacing the "derivative" with "difference operator", which is defined as follows: where η is always a nonzero complex constant. This idea is partly due to the work by Heittokangas et al. in [12]. They were the first to consider a nonconstant meromorphic function f (z) sharing values with its shift f (z + η) and to prove the following.
Theorem C ( [12]) Let f (z) be a meromorphic function of finite order, and let η ∈ C. If f (z) and f (z + η) share three distinct periodic functions a 1 , a 2 , a 3 ∈ S(f ) with period η CM, then f (z) = f (z + η) for all z ∈ C.
In Theorem C, where S(f ) is the set containing all meromorphic functions a(z) satisfying where E is an exceptional set of finite logarithmic measure. Theorem C can be read as a "3 CM" theorem and it has been improved to "2 CM + 1 IM" by Heittokangas et al. [13].
The key theory used in their research consists of the difference counterparts of Nevanlinna theory of meromorphic functions (see e.g. [3,10,11]). In 2013, Chen and Yi [2] proved the following Theorem D, which was then extended to Theorem E by Cui and Chen in [4], and to Theorem F by Zhang and Liao in [24].
Theorem D ( [2]) Let f (z) be a transcendental meromorphic function such that its order ρ(f ) is not integer or infinite, and let η be a constant such that f (z + η) ≡ f (z). If f (z) and

Remark 1.2
We will improve Theorems D-F by the following Theorem 1.1, whose proof is given with a different method from those in [2,4,24].
holds, then f (z) ≡ n η f (z), and hence f (z) and n η f (z) share a 1 and a 2 CM.
As a continuation of Theorem B and Theorem 1.2, we prove the following.

Theorem 1.3
Let f (z) be a nonconstant entire function of finite order, and let a 1 , a 2 be two distinct finite complex constants. If f (z) and n η f (z) share a 1 and a 2 IM, and there exists , and hence f (z) and n η f (z) share a 1 and a 2 CM.
Other basic concepts and fundamental results of the Nevanlinna theory of meromorphic functions (see e.g. [14,23]) may be used directly in what follows.

Lemmas
Now we recall two lemmas which are important in the proofs of our theorems. The first lemma has been used frequently in dealing with value sharing problems related to difference operators.

Lemma 2.1 ([11])
Let η ∈ C, n ∈ N, and let f (z) be a meromorphic function of finite order. Then, for any small periodic function a(z) with period η, with respect to f (z), where the exceptional set associated with S(r, f ) is of at most finite logarithmic measure.
Use the notation N k) (r, 1 f -a ) (N (k (r, 1 f -a )) to denote the counting function of the zeros of f (z)a in the disk |z| ≤ r, whose multiplicities ≤ k (≥ k) and are counted once. Then we have the following. Proof Suppose that f (z) ≡ n η f (z). Since f (z) and n η f (z) share two values a 1 and a 2 "CM", according to the second fundamental theorem and Lemma 2.1, we can easily derive that

Lemma 2.2 ([23]) Let f (z) be a nonconstant meromorphic function, a be an arbitrary complex number, and k be a positive integer. Then
Hence we prove the first conclusion.
Suppose that f (z) ≡ n η f (z) and (1.1) holds, and we prove conclusions (i)-(iv) step by step.
Step 1. Notice that f (z) and n η f (z) share the value a 1 "CM" and (1.1) imply that Then, applying the second fundamental theorem again, we have From this and the second equality in the first conclusion, we can see that Step 2. For all b ∈ C\{a 1 , a 2 }, from the second fundamental theorem, the second equality in the first conclusion, and conclusion (i), we can derive that Similarly, we can prove that the following equality holds: Step 3. Set Then we get from (2.1) and the lemma of logarithmic derivatives that

It is obvious that h(z) ≡ 0 since n η f (z) is not a constant. Hence from (2.1)-(2.3) we can deduce that
Similarly, we can prove that N(r, 1 f ) = S(r, f ).
Step 4. Consider the following function: .
The condition that f (z) and n η f (z) share two values a 1 and a 2 "CM" ensures that g(z) is a meromorphic function such that all poles of g(z) consist of zeros of f (z)a 1 and f (z)a 2 . We obtain from Lemma 2.1 and the lemma of logarithmic derivatives that (2.5) Let z ij (j = 1, 2, . . .) be the multiple common zeros of fa i and n η fa i (i = 1, 2), and let m ij and n ij be the multiplicities of the zero z ij of fa i and n η fa i , respectively. Note that m ij , n ij ≥ 2. It follows from expression (2.4) of g(z) that z ij (j = 1, 2, . . .) are zeros of g(z) with multiplicity at least min{m ij , n ij } -1 ≥ 1. This and (2.5) show that Proof It is easy to find that f (z) and n η f (z) share a 1 "CM", since f (z) and n η f (z) share the value a 1 IM and (1.1) holds.
From the second fundamental theorem and (1.1), we have (2.6) Let k = 1. Then (ii) in Lemma 2.2 can be rewritten as (2.6) and (2.7) give And due to the above inequality implies (2.8) and thus Similarly, the following equality holds: Then, by (i) in Lemma 2.3, we can derive that (2.12) Since f (z) and n η f (z) share a 2 IM, from (2.12) we obtain that Thus, f (z) and n η f (z) share the value a 2 "CM".

Remark 2.2
We can find that (2.8), (2.12), and (2.13) used in the proof of Theorem 1.1 still hold when 2 IM is replaced with 2 "CM".
Therefore, Lemma 2.3 is valid now. Let us consider the following two functions: (3.5) Notice that f (z) and n η f (z) share two values a 1 and a 2 "CM", and we see that F(z) and G(z) are meromorphic functions sharing 0 and ∞ "CM". By (3.5), (ii) in Lemma 2.3, and the Valiron-Mokhon'ko theorem (see e.g. [16,22]), we have and we get, by applying the lemma of logarithmic derivatives, = S r, F + S r, G = S(r, f ). (3.8) Clearly, (3.7) shows that the poles of ϕ(z) are simple, and they can only come from the zeros of F (z) and G (z) as well as the poles of F(z) and G(z).
In the following, suppose that z 2 is a pole of F(z) and G(z) with the same multiplicity k, which comes from the zero z 2 of fa 2 and n η fa 2 with the same multiplicity k. And suppose that the following two expansions hold in the neighborhood of zz 2 : a simple computation shows that which implies that z 2 is not the pole of ϕ(z).