Abstract

We study the value distribution of a special class difference polynomial about finite order meromorphic function. Our methods of the proof are also different from ones in the previous results by Chen (2011), Liu and Laine (2010), and Liu and Yang (2009).

1. Introduction and Results

A function is called meromorphic function, if it is analytic in the complex plane except at isolated poles. It is assumed that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory such as the characteristic function , proximity function , counting function , and the first and second main theorem (see [1–3]). The notation denotes any quantity that satisfies the condition: as possibly outside an exceptional set of of finite linear measure. We use the notation to denote the exponent of convergence of zeros of , and use the notation to denote the order of growth of the meromorphic function . Also, we give an estimate of numbers of -points, namely, for every .

Next, we will introduce the notation of Borel exceptional value (see [1]).

Definition 1.1. Let be a transcendental meromorphic function in with the order . A complex number is said to be a Borel exceptional value if Here can be replaced by .

In 1959, Hayman [4] proved the following Theorem.

Theorem A. Let be a meromorphic function in , if , where is a positive integer and , are two finite complex numbers such that and , then is a constant.

On the other hand, Mues [5] showed that for the conclusion is not valid.

Recently, as the significant results on Nevanlinna theory with respect to difference operators, see the papers [6, 7] by Halburd and Korhonen and [8] by Chiang and Feng. Many papers (see [2–4, 9–17]) have focused on complex differences and given many difference analogues in value distribution theory of entire functions.

In 2010, replacing by in Theorem A, Liu and Laine [17] obtained the following result.

Theorem B (see [17]). Let be a transcendental entire function of finite order, not of period , where is a nonzero constant, and let be a nonzero small function of . Then the difference polynomial has infinitely many zeros in the complex plane, provided that .

In 2011, Chen [18] considered the difference counterpart of Theorem A and proved an almost direct difference analogue of Hayman's Theorem.

Theorem C (see [18, Theorem  1.1]). Let be a transcendental entire function of finite order, not of period , and let , , be three complex numbers. Then assumes all finite values infinitely often, provided that and for every .

In 1994, Ye [19] considered a similar problem and obtained that if is a transcendental meromorphic function and is a nonzero finite complex number, then assumes every finite complex value infinitely often for . Ye [19] also asked whether the conclusion remains valid for .

In 2008, Fang and Zalcman [20] solved this problem and obtained the following result.

Theorem D. Let be a transcendental meromorphic function and be a nonzero complex number. Then assumes every complex value infinitely often for each positive integer .

Just like Theorem B, it is natural to ask whether Theorem D can be improved by the ideas of difference operator. The purpose of this paper is to study value distribution of meromorphic function with respect to difference. Our methods of proof are also different from ones in previous Theorems (see [17, 18, 21]). We obtain the following results.

Theorem 1.2. Let be a transcendental meromorphic function of finite order, not of period , where is a nonzero constant, and let be a small function of , let be a nonzero constant. Then the difference polynomial has infinitely many zeros in the complex plane, provided that .

Corollary 1.3. Let be a transcendental entire function of finite order, not of period , where is a nonzero constant, and let be a small function of , let be a nonzero constant. Then the difference polynomial has infinitely many zeros in the complex plane, provided that .

Recently, Qi and Liu [22] obtained the following result.

Theorem E (see [22, Theorem  2]). Let be a transcendental entire function of finite order, be a nonzero constant, and be integers satisfying , and let , be two complex numbers such that . If , then either assumes every nonzero finite value infinitely often or , where , and is periodic function with period .

Thus, it is natural to ask, what happens if in Theorem E?

By the same method of [18, 23], we investigate this problem and obtain the following results.

Theorem 1.4. Let be a transcendental entire function with finite order with a Borel exceptional value , be a nonzero complex constant, and let , be two complex numbers such that and , then assumes every nonzero value infinitely often and .

Theorem 1.5. Let be a transcendental entire function of finite order, be a complex constant, and let , be two complex numbers such that and . If has infinitely many multiple zeros, then takes every value infinitely often.

Example 1.6. satisfies . However, cannot assume any nonzero value .

Remark 1.7. From the Example 1.6, the condition is necessary in Theorems 1.4 and 1.5.

Remark 1.8. Some ideas in this paper are based on [18, 23–25].

2. Some Lemmas

In order to prove our theorems, we need the following Lemmas.

The Lemma 2.1 is a difference analogue of the logarithmic derivative lemma, given by Halburd and Korhonen [7] and Chiang and Feng [8], independently.

Lemma 2.1 (see [7, Theorem  2.1]). Let be a meromorphic function of finite order, and let and . Then

Lemma 2.2 (see [1, Theorem  1.12]). Let be a nonconstant meromorphic function, and let , where , are small function of . Then

By using the formulation (12) in [13], it is easy to get the following lemma.

Lemma 2.3. Let be a meromorphic function of finite order, . Then

Lemma 2.4. Let be a transcendental entire function of finite order with a Borel exceptional value , be complex constant, and let , be two complex numbers such that and , then , where .

Proof. Rewrite as the form
For each , by Lemma 2.1 and (2.4), we get that
Because is a transcendental entire function of finite order with a Borel exceptional value . Then we obtain Thus, (2.5) and (2.7) give that .

Lemma 2.5 (see [1]). Let      be meromorphic functions,    be entire functions, and satisfy(i), (ii)when , is not a constant,(iii)when , , ,where is of finite linear measure or finite logarithmic measure.
Then .

Lemma 2.6 (see [1]). Let be a transcendental meromorphic function of order and be the convergence exponent of its zeros. Then .

Lemma 2.7 (see [1], Hadamard's factorization theorem). Let be a transcendental entire function of finite order with zeros and a k-fold zero at origin. Then where is the canonical product of formed with the nonnull zeros of , and is a polynomial of degree.

Lemma 2.8 (see [1]). Let be the order of the canonical product . We use to denote the exponent of convergence of zeros of . Then .

3. Proofs of Theorems

Proof of Theorem 1.2. Set . Obviously, . If it is false, then . Thus we have that where . Using Lemmas 2.1 and 2.3, we deduce that Equations (3.1) and (3.2) imply , a contradiction, therefore .
Furthermore, we claim that Otherwise, . By integration, we obtain , where is a constant, hence .
If , we can deduce . This contradicts the hypothesis.
If , by the same arguments of the proof of Case , we get the same contradiction.
By a simple calculation we get that
From Lemmas 2.1 and 2.2 and some results of Nevanlinna Theory, we obtain that
Next, we will estimate and .
The poles of come from the zeros of , the poles of , the poles of , and the poles of . By the hypothesis, we ignore the poles of . If is a zero of or a pole of but not a pole of , then is a simple pole of . If is a common pole of and , and the multiplicity is and , respectively, then is a pole of with the multiplicity of no more than . If is a pole of but not a pole of , we obtain that is at most a simple pole of because of (3.4). Using the Lemma 2.3, we can get that
We deal with the poles of as above. The zeros of , the poles of , the poles of , and the zeros of compose the poles of . If is a zero of , zero of , or pole of , then is a simple pole of . If is a pole of but not a pole of , using the Laurent series, we can get that is analytic at . Therefore, we conclude that
Combining (3.2), (3.4), (3.5), and (3.6), we have that
From (3.2) and Lemma 2.2, we deduce that . Therefore, we get that
By (3.2), (3.7), (3.9), and the First Fundamental Theorem, we can simplify (3.8) to be
Because , we deduce that
If has finite zeros, then , a contradiction.
We complete the proof of the Theorem 1.2.

Proof of Corollary 1.3. The proof of Corollary 1.3 is the same as the proof of Theorem 1.2; note that is entire, some different places are stated below.
The poles of come from the zeros of . By the hypothesis, we ignore the poles of . If be a zero of , then is a simple pole of . Using the Lemma 2.3, we can get that
The zeros of and the zeros of compose the poles of . If is a zero of or zero of , then is a simple pole of . Therefore, we conclude that
Combining (3.2), (3.4), (3.5), and (3.12), we have that
By (3.2), (3.13), (3.9), and the First Fundamental Theorem, we can simplify (3.14), to be
Because , we deduce that
If has finite zeros, then , a contradiction.
The proof of Corollary 1.3 is complete.

Proof of Theorem 1.4. By Lemma 2.7, we write as follows where is the canonical product of formed with the nonnull zeros of , and is a polynomial of degree.
Since is the Borel exceptional value of , by Definition 1.1 and Lemmas 2.6 and 2.8, we can rewrite as the form where is an entire function with , is a constant, is a positive integer. Thus where Now we suppose that . By Lemma 2.1, we see that , so that and can be rewritten as the form where is a constant, is an entire function of By (3.18)–(3.20), we get Since and , by comparing growths of both sides of (3.23), we see that . Thus, by (3.23), we have By Lemma 2.5 and (3.24), we get that . This contradicts our assumption that . Hence .
The proof of Theorem 1.4 is complete.

Proof of Theorem 1.5. We suppose that has infinitely many multiple zeros. If , then has obviously infinitely many zeros. Now we suppose that . If has only finitely many zeros, then can be rewritten as the form where , are polynomials, and , .
Differentiating (3.25), we obtain
From (3.25), we get . Substituting into (3.26), we have
Since has infinitely many multiple zeros, there is a multiple zero such that is sufficiently large and , . Thus, the right side of (3.27) has a multiple pole at , but the left side of (3.27) has only a simple pole at . This is a contradiction.
Hence takes any value infinitely often.
The proof of Theorem 1.5 is complete.

At last, for further study, we pose a question.

Question 3. If in Theorem 1.2, what will happen?

Acknowledgments

The second author would like to thank the Department of Mathematics, Kiel University of Germany, for its hospitality during the study period there and would like to express his hearty thanks to Professor Walter Bergweiler for his valuable advice. This work was supported by the NNSF of China (no. 11171184, 10771220) and supported by project 10XKJ01, 12C401 and 12C104 from the Leading Academic Discipline Project of Shanghai Dian Ji University. This work was supported partially by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the authors worked as visiting scholars. The authors would like to express their hearty thanks to Chern Institute of Mathematics provided very comfortable research environments to them.