On zeros and deficiencies of differences of meromorphic functions
© Long et al.; licensee Springer. 2014
Received: 22 January 2014
Accepted: 31 March 2014
Published: 6 May 2014
For a transcendental entire function in the complex plane, we study its divided differences . We partially prove a conjecture posed by Bergweiler and Langley under the additional condition that the lower order of is smaller than . Furthermore, we prove that if zero is a deficient value of , then , where .
1 Introduction and main results
In this paper, we assume that the reader is familiar with the standard notations of Nevanlinna theory of meromorphic functions (see [1, 2] or ). In particular, for a meromorphic function in the complex plane ℂ, we use and to denote its order and lower order respectively, and to denote the exponent of the convergence of the zero-sequences.
Recently, a number of papers including [4–10] have focused on the complex difference equations and differences. In  Bergweiler and Langley investigated the existence of zeros of and . Their result may be viewed as discrete analogs of the following theorem on the zeros of .
Then has infinitely many zeros.
Theorem A is sharp, as shown by , tanz and examples of arbitrary order greater than 1 constructed in . For as in the hypotheses of Theorem A, it follows from Hurwitz’s theorem that, if is a zero of then has a zero near for all sufficiently small . Thus it is natural to ask, for such functions , whether must always have infinitely many zeros or not. In , Bergweiler and Langley answered this problem and obtained Theorem B and Theorem C.
Theorem B ()
Let be a transcendental entire function of order . If defined by (1.2) is transcendental, then has infinitely many zeros. In particular, if , then is transcendental and has infinitely many zeros.
Theorem C ()
There exists with the following property. Let be a transcendental entire function with order . Then defined by (1.2) has infinitely many zeros.
In , Bergweiler and Langley conjecture that the conclusion of Theorem C holds for . In this paper, we will prove this conjecture under the additional condition that .
Theorem 1 Let and let f be a transcendental entire function of order . If and , then defined by (1.2) is transcendental and has infinitely many zeros.
Using Theorem 1, we easily obtain the following corollary.
Corollary 1 Let and let f be a transcendental entire function of order . If , then is transcendental and has infinitely many zeros.
In , Bergweiler and Langley also proved that, for a transcendental meromorphic of order , if has finitely many poles , such that , then has infinitely many zeros (see , Theorem 1.4). Furthermore, for a transcendental entire function f of order , Chen and Shon proved that , and if has finitely many zeros , such that , then has infinitely many zeros and (see ). This result implies that zero is not the Borel exceptional value of .
In , Langley investigated the deficiency of divided difference defined by (1.2). He obtained that if is a transcendental entire function of order and , then . In particular, if , then (see , Theorem 1.4). The proof of his result depends on theorem which is invalid for .
We get the following results on the deficiency .
If , then there exists a set of positive upper logarithmic density such that , as , , where .
If zero is a deficient value of , then .
It is clear that, for a given transcendental entire function , all but countably many such that has at most finitely many zeros , such that . Furthermore, we know that, for an entire function , if has a finite deficient value then . Hence, Theorem 2 implies that, for some particular functions of order , we obtain a similar conclusion.
Then and (see [, p.252]). If we let , then it follows from Theorem 2 that .
The paper is organized as follows. In Section 2, we shall collect some notations and give some lemmas which will be used later. In Section 3, we shall prove Theorem 1. In Section 4, we shall prove Theorem 2.
2 Preliminaries and lemmas
where , is the maximum term and is the central index. It is well known that is a nondecreasing and right continuous function. Furthermore, if is transcendental entire, then as .
For a set , we define its Lebesgue measure by and its logarithmic measure by .
Following Hayman [, pp.75-76], we say that a set E is an ε-set if E is a countable union of open discs not containing the origin and subtending angles at the origin whose sum is finite. If E is an ε-set, then the set of for which the circle meets has finite logarithmic measure and hence zero upper logarithmic density. Moreover, for almost all real θ, the intersection of E with the ray is a bounded set.
The following lemma contains a basic property of meromorphic functions of finite order.
Lemma 2.1 ()
Let be a meromorphic function with . Then, for given real constants and H (), there exists a set such that , where and .
The following lemma is a version of the celebrated theorem of .
Lemma 2.2 ()
Let be a transcendental entire function with lower order . Then, for each , there exists a set such that , where , , and .
We collect some important properties of the differences of meromorphic functions in the following lemmas.
Lemma 2.3 ()
Let be a transcendental meromorphic function in ℂ which satisfies (1.3). Then, with the notation (1.1) and (1.2), and are both transcendental.
Lemma 2.4 ()
Lemma 2.5 ()
3 Proof of Theorem 1
In order to prove Theorem 1, we need one more lemma. This lemma can be proved in a similar way to the proof of Lemma 4 in ; we shall omit the proof.
This contradicts the assumption that . Thus must be transcendental.
Since as , (3.9) gives a contradiction. Therefore, must have infinitely many zeros and the proof of Theorem 1 is completed. □
4 Proof of Theorem 2
To prove Theorem 2, we first prove the following lemma.
hold for all .
hold for all . Let . By (4.5) and (4.11), we see that (4.1) holds for all . The proof of Lemma 4.1 is completed. □
To finish the proof of (i), we need to consider the following two cases.
This gives (i).
Obviously, (4.29) gives a contradiction and the proof of Theorem 2 is completed. □
The authors would like to thank the referee for his/her valuable suggestions, which greatly improved the present article. The authors wish to thank professor Wu Shengjian in Peking University for valuable suggestions and numerous important helps. The work was supported by the United Technology Foundation of Science and Technology Department of Guizhou Province and Guizhou Normal University (Grant No. LKS12), and National Natural Science Foundation of China (Grant No. 11171080).
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