A new result for entire functions and their shifts with two shared values

1 Introduction and main results

We assume the reader is familiar with the fundamental concepts and standard notations of the classical Nevanlinna theory [1,2], such as T ( r , f ) , N ( r , f ) , and m ( r , f ) . In addition, we denote by S ( r , f ) a quantity of o ( T ( r , f ) ) as r tends to infinity outside of a possible exceptional set of finite logarithmic measure.

For a given value a ∈ C ∪ { ∞ } and two meromorphic functions f and g , we denote by E ¯ ( a , f ) (or E ( a , f ) ) the set of all zeros of f − a ignoring multiplicity (or counting multiplicity, respectively), and let us abbreviate this with IM (or CM). If E ¯ ( a , f ) = E ¯ ( a , g ) , then we say that f and g share a IM; if E ( a , f ) = E ( a , g ) , then we say that f and g share a CM; if E ¯ ( a , f ) ⊆ E ¯ ( a , g ) , then we say that f and g partially share a IM; and if E ( a , f ) ⊆ E ( a , g ) , then we say that f and g partially share a CM.

In recent decades, by applying the difference analog of classical Nevanlinna theory [3–6] for meromorphic functions f ( z ) and their shifts f ( z + c ) or difference operators Δ c f = f ( z + c ) − f ( z ) , many uniqueness theorems between functions satisfying some certain shared conditions were obtained. Here, we recall some previous results as follows.

The main idea of the proof of Theorem 1.1 is based on the difference analog of classical Nevanlinna’s theory for meromorphic functions of finite order, which is established by Halburd and Korhonen [3,4], Chiang and Feng [5], independently, and developed by Halburd et al. [6] to hyperorder strictly less than 1. Correspondingly, the assumption “finite order” in Theorem 1.1 and Corollary 1.2 can naturally be replaced by “hyperorder ρ 2 ( f ) < 1 .”

It is a natural question whether the shared conditions can be weakened or not. In 2017, one of the authors [8] of this article improved the shared conditions of Theorem 1.1 from “3CM” to “2CM + 1IM.” In 2016, Li and Yi [9] improved Corollary 1.2 from “2CM” to “2IM.” We recall those theorems as follows.

In this article, we shall replace the shared value conditions of Theorem 1.4 by two partially shared values.

The following example shows that the word “entire function” cannot be replaced by “meromorphic function.”

2 Some lemmas

Next, we shall recall the following lemmas that are needed in the sequel.

3 Proof of Theorem 1.5

The proof is based on the idea of Lin and Ishizaki [10]. Without loss of generality, we can suppose that a 1 = 0 and a 2 = 1 . Assume, on the contrary, that f ( z ) ≢ f ( z + c ) .

Set

According to the shared value conditions, we know that any zero of f ( f − 1 ) is a regular point of φ . And since f is an entire function, one can see that φ is an entire function. In addition, using the lemma on logarithmic derivatives and Lemma 2.1, we have

which implies that

Next, we denote by N 0 r , 1 h ′ the counting function of those zeros of h ′ ( z ) , which are not the zeros of h ( z ) ( h ( z ) − 1 ) . Noting that φ ≢ 0 by the assumptions, from (3.1) and (3.2), we have

Rewrite (3.1) as

Then, we can deduce from (3.2), (3.4), and Lemma 2.2 that

Set

Immediately, we obtain

Moreover, we claim that N ( r , F ) = S ( r , f ) . Since f is an entire function, we know that all the poles of F ( z ) in (3.6) could only come from the zeros of f ′ ( z + c ) [ f ( z ) ( f ( z ) − 1 ) ] . We consider the following two cases.

Case 1. Let z 0 be a zero of f ( z ) (or f ( z ) − 1 , respectively) with multiplicity m , and of f ( z + c ) (or f ( z + c ) − 1 , respectively) with multiplicity n . Then, z 0 is a zero of f ′ ( z + c ) [ f ( z ) ( f ( z ) − 1 ) ] in (3.6) with multiplicity n − 1 + m , and a zero of f ′ ( z ) [ f ( z + c ) ( f ( z + c ) − 1 ) ] in (3.6) with multiplicity m − 1 + n too. Thus, z 0 is not a pole of F ( z ) .

Case 2. Let z 1 be a zero of f ′ ( z + c ) . We consider the following subcases.

Subcase 2.1. If z 1 is a common zero of f ( z + c ) ( f ( z + c ) − 1 ) and f ( z ) ( f ( z ) − 1 ) , then z 1 is not a pole of F ( z ) by the same argument as in Case 1.

Subcase 2.2. If z 1 is a zero of f ( z + c ) ( f ( z + c ) − 1 ) but not of f ( z ) ( f ( z ) − 1 ) , then since the multiplicity at z 1 of f ′ ( z + c ) is less than the multiplicity of f ( z + c ) ( f ( z + c ) − 1 ) , it is obvious that z 1 is also not a pole of F ( z ) .

Subcase 2.3. If z 1 is not a zero of f ( z + c ) ( f ( z + c ) − 1 ) , then it follows from (3.3) and the aforementioned two subcases that N ( r , F ) = S ( r , f ) .

Therefore, we have

In what follows, we rewrite (3.6) as

Denote by N ¯ D r , 1 f ( z + c ) ( f ( z + c ) − 1 ) the reduced counting function of those zeros of f ( z + c ) ( f ( z + c ) − 1 ) but not of f ( z ) ( f ( z ) − 1 ) . It follows from (3.7), (3.8), and F ≢ 0 that

Now, we denote by N ¯ ( m , n ) ( r , a ; f ( z ) , f ( z + c ) ) = N ¯ ( m , n ) ( r , a ) the reduced counting function of those common zeros of f ( z ) − a with multiplicity m , and of f ( z + c ) − a with multiplicity n , and claim that

holds for any fixed pair ( m , n ) and a ∈ { 0 , 1 } . Otherwise, without loss of generality, we can assume there exists some pair ( m , n ) such that

Next, we discuss the following two cases. In the following, let z 0 be a common zero of f ( z ) and f ( z + c ) with multiplicity m and multiplicity n , respectively, and then, F ( z 0 ) ≠ 0 , ∞ by (3.8), where we ignore those points in N ¯ D r , 1 f ( z + c ) ( f ( z + c ) − 1 ) .

Because the coefficients of the term ( z − z 0 ) − 1 of the Laurent expansions of the both sides of (3.8) are identical, we have

Case 1. Suppose that m = n . Then, we have F ( z 0 ) = 1 by (3.11). If F ( z ) ≢ 1 , then it follows from (3.7) that

which is a contradiction to (3.10). Hence, F ( z ) ≡ 1 , and it follows from (3.8) that

which implies f ( z ) and f ( z + c ) share 0, 1 CM by the assumptions of Theorem 1.5. In fact, from the aforementioned identical equation (for convenience, notate this equation as ( * ) ), we can derive E ¯ ( 0 , f ( z ) ) ⊇ E ¯ ( 0 , f ( z + c ) ) . Otherwise, if there exists a point ξ such that f ( ξ + c ) = 0 but f ( ξ ) ≠ 0 , then we also have f ( ξ ) − 1 ≠ 0 . Moreover, ξ is a pole of the left side of (⁎) but is not a pole of the right of (⁎), which is a contradiction. Therefore, we have E ¯ ( 0 , f ( z ) ) = E ¯ ( 0 , f ( z + c ) ) . Similarly, we can obtain E ¯ ( 1 , f ( z ) ) = E ¯ ( 1 , f ( z + c ) ) . Next, by observing the Laurent expansions on both sides of (⁎), we can further deduce that f ( z ) and f ( z + c ) share 0, 1 CM.

Now, by applying Theorem 1.4 to this case, we know that f ( z ) ≡ f ( z + c ) , but this is a contradiction.

Case 2. Suppose that m ≠ n . Then, we have F ( z 0 ) = m n by (3.11). If F ( z ) ≢ m n , then it follows from (3.7) that

which is a contradiction to (3.10). Hence, F ( z ) ≡ m n , and it follows from (3.8) that

which implies

for some nonzero constant A 0 . This together with the Valiron-Mohon’ko theorem yields

It follows from Lemma 2.2 immediately that m = n , which is a contradiction in this case.

Therefore, the claim of (3.9) is true.

Finally, we will derive a contradiction in the following, and complete our proof.

Since

what is more, we can deduce that

and from (3.9) that

Hence, we can obtain from the aforementioned formulas that

Similarly, we can also obtain

By (3.12) and (3.13), we have

which is a contradiction.

Therefore, this completes the proof of Theorem 1.5.

4 Discussion

In 1989, Brosch [11] proved the following sufficient conditions for periodicity of meromorphic functions in his doctoral dissertation.

As we know, one can apply the uniqueness theorems on meromorphic functions and their shifts to study the periodicity of meromorphic functions. For example, the third author of this article improved Theorem 4.1 as follows.

An immediate consequence of Theorem 4.2 is the following result.

Now, using Theorem 1.4, we can obtain the following result.

Therefore, regarding Corollary 4.3, Theorem 4.4, and the main result Theorem 1.5 of this article, we give the following question.

Question If the condition “ f ( z ) and g ( z ) share 0, 1 IM” of Theorem 4.4 is replaced with “ E ¯ ( 0 , f ( z ) ) ⊆ E ¯ ( 0 , g ( z ) ) and E ¯ ( 1 , f ( z ) ) ⊆ E ¯ ( 1 , g ( z ) ) ”, does the conclusion of Theorem 4.4 still hold?