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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 357203, 13 pages
http://dx.doi.org/10.1155/2012/357203
Research Article

The Zeros of Difference Polynomials of Meromorphic Functions

1Department of Mathematics, Wuyi University, Jiangmen, Guangdong 529020, China
2Department of Mathematics, Civil Aviation University of China, Tianjin 300300, China

Received 4 May 2012; Accepted 10 September 2012

Academic Editor: Yong Zhou

Copyright © 2012 Junfeng Xu and Xiaobin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the value distributions of difference polynomials and which related to two well-known differential polynomials, where is a meromorphic function.

1. Introduction and Main Results

In this paper, we will assume that the reader is familiar with the fundamental results and the standard notation of the Nevanlinna value distribution theory of meromorphic functions (see [1, 2]). The term “meromorphic function” will mean meromorphic in the whole complex plane . In addition, we will use notations to denote the order of growth of a meromorphic function , to denote the exponents of convergence of the zero-sequence of a meromorphic function , to denote the exponents of convergence of the sequence of distinct poles of .

Hayman [3] proved the following famous result.

Theorem A. If is a transcendental meromorphic function, is an integer, and is a constant, then assumes all finite values infinitely often.

He also conjectured in [1] that the same result holds for and 4. However, Mues [4] proved that the conjecture is not true for by providing a counterexample and proved that has infinitely many zeros. If is a transcendental entire function, holds in Theorem A.

Recently, many papers have focused on complex difference, giving many difference analogues in value distribution theory of meromorphic functions.

It is well known that , where is a constant satisfying which is regarded as the difference counterpart of , so that is regarded as the difference counterpart , where .

In 2011, Chen [5] considered the difference counterpart of Theorem A under the condition that is transcendental entire.

Theorem B. If is a transcendental entire function of finite order, and let be constants, with such that . Set and is an integer. Then assumes all finite values infinitely often, and for every one has .

Theorem C. If is a transcendental entire function of finite order with a Borel exceptional values . Let be constants, with such that . Then assumes the value infinitely often, and .

Theorem D. If is a transcendental entire function of finite order with a nonzero Borel exceptional values . Let be constants, with such that . Then for with , assumes the value infinitely often, and .

In this paper, we will extend and improve the above results from entire functions to meromorphic functions.

Theorem 1.1. If is a transcendental meromorphic function with exponent of convergence of poles , and let be constants, with such that . Set and is an integer. Then assumes all finite values infinitely often, and for every one has .

Remark 1.2. Compared with Theorem 1.2 in [6], our result not only gives a Picard type result but also gives an estimate of numbers of -points, namely, for every . Our method bases on [5], which is different from the method in [6, 7].

In the same paper, Chen gave the following example to show the Borel exceptional value may arise in Theorem D.

Example 1.3. For , , , we have . Here , which shows that the Borel exceptional value may arise.

Naturally, it is an interesting question to find the conditions which can remove the Borel exceptional value of when .

Example 1.4. For , , , we have , which assumes all finite values infinitely often. This is, has no Borel exceptional value.

Theorem 1.5. If is a transcendental meromorphic function of finite order with two Borel exceptional values . Let be constants, with such that . Then for with , assumes the value infinitely often, and .
Moreover, if satisfies , we can remove the condition .

Remark 1.6. By the simple calculation, we can see in Example 1.4, has no finite Borel exceptional value. Hence, the conclusion of Theorem 1.5 is sharp.

From the proof of Theorem 1.5, we can obtain the following.

Corollary 1.7. If is a transcendental meromorphic function of finite order with two Borel exceptional values . Let be constants, with such that . Then assumes every value infinitely often, and .

Example 1.8. For , , , it is easy to see are not Borel exceptional values, and has no zeros. Thus our condition in Corollary 1.7 is sharp.

Remark 1.9. In fact, by the definition of Borel exceptional value, we know the condition is a transcendental meromorphic function of finite order with two Borel exceptional values equivalent to .

Hayman also posed the following conjecture: if is a transcendental meromorphic function and , then takes every finite nonzero value infinitely often. This conjecture has been solved by Hayman [1] for , by Mues [4] for , by Bergweiler and Eremenko [8] for .

Recently, for an analog of Hayman conjecture for difference, Laine and Yang [9] proved the following.

Theorem E. Let be a transcendental entire function with finite order and be a nonzero complex constant. Then for , assumes every nonzero value infinitely often.

Liu et al. [10] consider the question when is a transcendental meromorphic function.

Theorem F. Let be a transcendental meromorphic function with finite order and be a nonzero complex constant. Then for , assumes every nonzero value infinitely often.

In this paper, we improved the above result by reducing the condition .

Theorem 1.10. Let be a transcendental meromorphic function with finite order with two Borel exceptional values , and be a nonzero complex constant. Then for , assumes every value infinitely often and .

From Theorem 1.10, we can obtain the following.

Corollary 1.11. If is a transcendental meromorphic function of finite order with two Borel exceptional values , and let be a nonzero complex constant. Then for , assumes every nonzero value infinitely often.

Example 1.12. For , and , it is easy to see are not Borel exceptional values, and has no zeros. Thus our condition in Corollary 1.11 is sharp.

Remark 1.13. Theorem 1.10 also improved the result in [11, Theorem 1.2], where they considered the case of entire function and .

For the analog of Hayman conjecture on , can be replaced by in Theorem 1.10. Then a simple modification of the proof of Theorem 1.10 yields the following result.

Theorem 1.14. Let be a transcendental meromorphic function with finite order with two Borel exceptional values , and be a nonzero complex constant with such that . Then for , assumes every value infinitely often and .

Corollary 1.15. If is a transcendental entire function of finite order with a Borel exceptional values , and be a nonzero complex constant with such that . Then for , assumes every value infinitely often and .

Remark 1.16. Theorem 1.14 also improved the result in [7, Theorem 1.4], where they consider the case of entire function and . the value can be a polynomial in their result. In fact, our results also can allow the value to be a polynomial, even be a meromorphic function satisfying .

Example 1.17. For , , , it is easy to see that has no Borel exceptional value, we have , which has no zeros. Hence, has a Borel exceptional value necessary in Corollary 1.15.

2. Lemmas

The following lemma, due to Gross [12], is important in the factorization and uniqueness theory of meromorphic functions, playing an important role in this paper as well. We give a slight changed form.

Lemma 2.1 (see [13]). Suppose that are meromorphic functions and are entire functions satisfying the following conditions.(i).(ii)If , , the order of is less than the order of . If , , , and the order of is less than the order of .
Then .

Lemma 2.2 (see [14, 15]). Let be a moermorphic function of finite order, and let . Then where .

Lemma 2.3 (see [14]). Let be a meormorphic function of finite order , and let . Then, for each , one has

We are concerned with functions which are polynomials in , where , with coefficients such that except possibly for a set of having finite logarithmic measure. Such functions will be called difference polynomials in . Similarly, we are concerned with functions which are polynomials in and the derivatives of , with coefficients such that (2.3) holds, except possibly for a set of having finite logarithmic measure. Such functions will be called differential-difference polynomials in . We also denote .

Halburd and Korhonen proved the following difference analogous to the Clunie Lemma [16], which has numerous applications in the study of complex differential equations, and beyond.

Lemma 2.4 (see [15]). Let be a nonconstant meromorphic solution of where and are difference polynomials in , and let and . If the degree of as a polynomial in and its shifts is at most , then for all outside of a possible exceptional set with finite logarithmic measure.

We can obtain the following differential-difference analogous to the Clunie lemma by the same method as Lemma 2.4.

Lemma 2.5. Let be a non-constant meromorphic solution of (2.4), where and are differential-difference polynomials in , and let and . If the degree of as a polynomial in and its shifts is at most , then (2.5) holds.

Remark 2.6. If the coefficients of and are satisfying the order less than , from the proof of Lemma 2.5, we can obtain

Lemma 2.7. If is a transcendental meromorphic function with exponent of convergence of poles , and let . Then, for each , one has

From the above lemma, we can obtain the following important result.

Lemma 2.8. If is a transcendental meromorphic function with exponent of convergence of poles , and let . Then, for each , one has

Proof. By the definition of exponent of convergence of poles, we can easily prove it by Lemma 2.7.

Remark 2.9. The second result is similar with .

Lemma 2.10 (see [17]). Let be a nonconstant meromorphic function, be a positive integer. where is a meromorphic function satisfying . Then

Lemma 2.11. If is a transcendental meromorphic function with exponent of convergence of poles , and let and be an integer. Set , then .

Proof. We can rewrite as the form For each , by Lemma 2.2 and (2.10), we get that From Lemma 2.10, we have By (2.11) and (2.12), we have This is, . Next we prove .
By Lemma 2.2 and (2.10), we have Note that , we have Thus, from (2.14) and (2.15), we have Hence, we prove . Therefore, .

Remark 2.12. If , we can prove by the inequality , without the condition .

3. Proof of Theorem 1.1

We only prove the case . For the case , we can use the same method in the proof. Suppose that and . First, we claim that is transcendental meromorphic. Suppose that , where is a rational function. Then By Lemma 2.4, for each , we have By Lemma 2.10, we have This is a contradiction. Hence, the claim holds. Thus, can be written as where is a polynomial, is an entire function with , and is the canonical product formed with the poles of . Hence, . Obviously, .

Differentiating (3.4) and eliminating , we obtain where We claim that . Suppose that Integrating (3.7), we have where is a constant. Therefore, by (3.4) and (3.8), and the definition of , we obtain therefore, We can prove that by the similar to the proof in [5]. We omit it here.

Differentiating (3.9), and then dividing by , we have Therefore, by Lemma 2.2, we obtain that Note that , we have From (3.8), we know that the poles of come from the poles of , we have From (3.13) and (3.14), we have We can obtain . It is a contradiction. Hence, the claim holds.

Since and the total of as a differential-difference polynomial in , its shift and its derivatives, , by (3.5), Lemma 2.5 and Remark 2.6, we obtain that for , for all outside of an exceptional set of finite logarithmic measure. Form , we have for all outside of an exceptional set of finite logarithmic measure. By (3.14) and (3.15), we can get a contradiction with . Hence, has infinitely many zeros and . The proof of Theorem 1.1 is complete.

4. Proof of Theorem 1.5

Since has a Borel exceptional value , we can write as where is a constant, is an integer satisfying , and are entire functions such that , , , and is the canonical product formed with the poles of satisfying . Set , it is easy to see that .

First, we prove that is transcendental. If , where is a rational function, then Thus, by Lemma 2.10, we have By Lemma 2.7, we have From (4.4) and (4.5), we can get By (4.2), (4.3), and (4.6), we have It is contradiction with .

Secondly, we prove that . By the expression of , we have . Set . Suppose that , then by (4.1), we have this is, Since , we see the order of the left-hand side of (4.9) is . Obviously, it contradicts the order of the right side of (4.9) is less than for . Hence, .

Thirdly, we prove . If , then can be written as where is a constant, is an entire function satisfying , and by Lemma 2.8. Hence, .

By (4.1), (4.10), and , we have In fact, (4.11) can be rewritten into where . Obviously, . We just consider three cases.

Case 1 (). By Lemma 2.1, we have , this is, , a contradiction.

Case 2 (, ). By Lemma 2.1, we still have , this is, , a contradiction.

Case 3 (, ). By Lemma 2.1, we have and . In order to complete our proof, we need to get a contradiction.

Subcase 1. If , then . Since , we know . We get a contradiction.

Subcase 2. If , then . By the assumption . We can get , a contradiction.

If the condition is replaced by , from , we know . We get a contradiction.

This completes the proof of the Theorem.

5. Proof of Theorem 1.10

Since has a Borel exceptional value , we can write as where is a constant, is an integer satisfying , and are entire functions such that , , , and is the canonical product formed with the poles of satisfying . Set , it is to see that .

Now we suppose that . By Lemma 2.11, we have , so that and can be rewritten into the form where is a constant, is an entire function satisfying , and by Lemma 2.8. Hence, .

By (5.1) and (5.2), we get where Note that and , by comparing growths of both sides of (5.3), we see that . Thus, by (5.3), we have By Lemma 2.1, we get . This is a contradiction with our assumption . Hence, .

6. Proof of Theorem 1.14

Similar to the proof of Theorem 1.10, we can obtain (5.1) and (5.2).

By (5.1) and (5.2), we get where Note that and , by comparing growths of both sides of (5.3), we see that . Thus, by (5.3), we have By Lemma 2.1, we get . This is a contradiction with . Hence, .

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11126327, 11171184), the Science Research Foundation of CAUC, China (no. 2011QD10X), NSF of Guangdong Province (no. S2011010000735), and STP of Jiangmen, China (no. 133).

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