The zeros of complex differential-difference polynomials
© Liu et al.; licensee Springer. 2014
Received: 1 April 2014
Accepted: 16 May 2014
Published: 28 May 2014
This paper is devoted to considering the zeros of complex differential-difference polynomials of different types. Our results can be seen as the differential-difference analogues of Hayman conjecture (Ann. Math. 70:9-42, 1959).
1 Introduction and main results
Let be a meromorphic function in the complex domain. Assume that the reader is familiar with standard symbols and fundamental results of Nevanlinna theory [1, 2]. Recall that is a small function with respect to , if , where is used to denote any quantity satisfying as outside of a possible exceptional set of finite logarithmic measure. Denote by and the order and the hyper-order of f. In this paper, c is a non-zero complex constant, n, k are positive integers, unless otherwise specified.
Hayman  conjectured that if f is a transcendental meromorphic function, then takes every finite non-zero value infinitely often. In fact, Hayman  proved that if f is a transcendental meromorphic function and , then takes every finite non-zero value infinitely often. Later, the case was settled by Mues . Bergweiler and Eremenko , Chen and Fang [, Theorem 1] proved the case of , respectively. In the past years, the topic on the zeros of differential polynomials has always been an important research problem in value distribution of meromorphic functions. With the development of the difference analogues of Nevanlinna theory, some authors paid their attention to the zeros of difference polynomials. Laine and Yang [, Theorem 2] firstly considered the zeros distribution of , where a is a non-zero constant, which can be seen as a difference analogue of Hayman conjecture. Recently, many authors were interested in the zeros distribution of difference polynomials of different types, such as [8–13].
A polynomial can be called a differential-difference polynomial in f whenever is a polynomial in , its shifts and their derivatives, with small functions of as the coefficients. It is interesting to consider the zeros of differential-difference polynomials. The aim of the paper is to explore the differences or analogues among the zeros of differential polynomials, difference polynomials, differential-difference polynomials. Liu et al. [, Theorems 1.1 and 1.3] considered this problem and obtained the following result, where .
Theorem A Let f be a transcendental entire function of finite order and be a non-zero small function with respect to . If , then has infinitely many zeros. If f is not a periodic function with period c and , then has infinitely many zeros.
If in Theorem A, some results can be found in . In this paper, we will consider the zeros of differential-difference polynomials of and .
Theorem 1.1 Let f be a transcendental entire function of hyper-order . If , then has infinitely many zeros, where is a non-zero small function with respect to .
The condition cannot be deleted, which can be seen by of , thus has finitely many zeros, where and is a non-zero polynomial. In fact, for any integer k, we can choose appropriate to make , is a polynomial in .
If f is a finite order transcendental entire function, we prove the following result.
Theorem 1.2 Let f be a finite order transcendental entire function. If , then has infinitely many zeros, where is an entire function with .
where is the exponent of convergence of zeros of .
Theorem 1.3 Let f be a finite order transcendental entire function with a Borel exceptional polynomial . If , then has infinitely many zeros, where b is a non-zero constant.
From the above three theorems, we can reduce the value of n with additional conditions. However, we hope that the condition can be reduced to in Theorem 1.1. Unfortunately, we have not succeeded in doing that.
If is a transcendental meromorphic function, we obtain the next result.
Theorem 1.4 Let f be a transcendental meromorphic function of hyper-order . If , then has infinitely many zeros, where is a non-zero small function with respect to .
Using the similar method of proofs of Theorems 1.1 and 1.4 below, we can get the following result.
Theorem 1.5 Let f be a transcendental meromorphic (entire) function of hyper-order . If (), then has infinitely many zeros, where is a non-zero small function with respect to .
Finally, we recall the classical results due to Hayman [, Theorems 8 and 9], which can be combined as follows.
Theorem B Let f be a transcendental meromorphic function and , b be a finite complex constant. Then has infinitely many zeros for . If f is transcendental entire, this holds for , resp. , if .
We then proceed to consider the zeros of , which can be seen as the differential-difference analogues of Theorem B.
Theorem 1.6 Let f be a transcendental entire function with finite order, let , be small functions with respect to f. Then has infinitely many zeros for , resp. , if .
Remark 3 The condition cannot be improved if , which can be seen by the function and , thus has no zeros. The condition cannot be improved if , which can be seen by the function and , thus has no zeros.
2 Some lemmas
The difference analogue of logarithmic derivative lemma, given by Chiang and Feng [, Corollary 2.5], Halburd and Korhonen [, Theorem 2.1], plays an important part in considering the difference analogues of Nevanlinna theory. Afterwards, Halburd, Korhonen and Tohge improved the condition of growth from to as follows.
Lemma 2.1 [, Theorem 5.1]
for all r outside of a set of finite logarithmic measure.
Lemma 2.2 [, Lemma 8.3]
for all r runs to infinity outside of a set of finite logarithmic measure.
From Lemma 2.2, then we get the following lemma.
for all r outside of a set of finite logarithmic measure.
Lemma 2.5 [, Theorems 1.22 and 1.24]
Thus, (2.10) follows from (2.12) and (2.13). □
Using the similar method as the proof of Lemma 2.6, we get the following result, which is important in the proof of Theorem 1.5.
Inequality (2.15) cannot be improved. If , , thus , which implies that . If , , thus , which implies that .
The following two results are due to Yang and Yi, see .
Lemma 2.8 [, Theorem 1.56]
where and , then either or .
Lemma 2.9 [, Theorem 1.52]
the order of is less than that of for , , then ().
3 Proofs of Theorem 1.1 and Theorem 1.4
which is a contradiction with .
4 Proof of Theorem 1.2
obviously, is an entire function. Thus, from (4.4) and (4.5), we get , which is a contradiction.
5 Proof of Theorem 1.3
Since is a nonzero polynomial, then we get . Let , , . Thus, we get . Since , , which implies that , we get , , are not constants, which is a contradiction with Lemma 2.8. The proof of Theorem 1.3 is completed.
6 Proof of Theorem 1.6
Since , then (6.5) implies that has infinitely many zeros. In what follows, we will prove that if f is a transcendental entire function with finite order and , then n can be reduced to .
Combining the above two estimates, we obtain , a contradiction.
where is an entire function with , β is a constant. If , which implies that , which is impossible. If , then , which also is impossible. Thus, and , from Lemma 2.9, we get , and , which are impossible. Thus, we have completed the proof of Theorem 1.6.
Remark 4 Inequality (6.1) is not valid for is a transcendental meromorphic function, which can be seen by , thus . Thus, .
The authors would like to thank the referees for valuable suggestions for improving our paper. This work was partially supported by the NSFC (No. 11301260, 11101201), the NSF of Jiangxi (No. 20132BAB211003) and the YFED of Jiangxi (No. GJJ13078) of China.
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