- Open Access
On the iterated exponent of convergence of zeros of
© Tu et al.; licensee Springer. 2013
- Received: 27 November 2012
- Accepted: 27 February 2013
- Published: 22 March 2013
In this paper, the authors investigate the iterated exponent of convergence of zeros of (), where f is a solution of some second-order linear differential equation, is an entire function satisfying or (). We obtain some results which improve and generalize some previous results in (Chen in Acta Math. Sci. Ser. A 20(3):425-432, 2000; Chen and Shon in Chin. Ann. Math. Ser. A 27(4):431-442, 2006; Tu et al. in Electron. J. Qual. Theory Differ. Equ. 23:1-17, 2011) and provide us with a method to investigate the iterated exponent of convergence of zeros of ().
- linear differential equations
- iterated order
- iterated type
- iterated exponent of convergence of zero-sequence
In this paper, we assume that readers are familiar with the fundamental results and standard notation of Nevanlinna’s theory of meromorphic functions (see [1, 2]). First, we introduce some notations. Let us define inductively, for , and , . For all sufficiently large r, we define and , ; we also denote and . Moreover, we denote the linear measure and the logarithmic measure of a set by and respectively. Let , be meromorphic functions in the complex plane satisfying except possibly for a set of r having finite logarithmic measure, then we call that is a small function of . We use p to denote a positive integer throughout this paper, not necessarily the same at each occurrence. In order to describe the infinite order of fast growing entire functions precisely, we recall some definitions of entire functions of finite iterated order (e.g., see [3–8]).
Remark 1.2 From Definitions 1.5 and 1.6, we can similarly give the definitions of , , and .
In , Chen firstly investigated the fixed points of the solutions of equations (2.1) and (2.2) with a polynomial coefficient and a transcendental entire coefficient of finite order and obtained the following Theorems A and B. Two years later in , Chen investigated the zeros of () and obtained the following Theorems C and D, where is a solution of equation (2.3) or (2.4), is an entire function satisfying . In , Tu, Xu and Zhang investigated the hyper-exponent of convergence of zeros of () and obtained the following Theorem E, where is a solution of (2.5), is an entire function satisfying . One year later, Xu, Tu and Zheng improved Theorem E to Theorem F in  from (2.5) to (2.6). In the following, we list Theorems A-F which have been mentioned above.
Theorem A 
has infinitely many fixed points and satisfies .
Theorem B 
has infinitely many fixed points and satisfies .
Theorem C 
Theorem D 
satisfies , where is a complex number.
Theorem E 
Theorem F 
The main purpose of this paper is to improve Theorem E from entire coefficients of finite order in (2.5) to entire coefficients of finite iterated order. And we obtain the following results.
, , .
Theorem 2.3 Under the hypotheses of Theorem 2.1, let , where () are entire functions which are not all equal to zero and satisfy . Then for any solution of (2.5), we have .
, , .
Remark 2.1 Theorem 2.1 is an extension and improvement of Theorem E. As for Theorem C, if (), it is easy to see that and . By Theorem E, for every solution of (2.3) and for any entire function with , we have , therefore Theorem E is also a partial extension of Theorem C. Theorem B is a special case of Corollary 2.1 for .
where is a small function of . For example, set , is a transcendental entire function with , then we have if . Our Theorem 2.1 and Theorem 2.2 also provide us with a method to investigate the iterated exponent of zero sequence of (), where and are entire functions satisfying or . If we can find equation (2.5) with entire coefficients , satisfying or and such that is a solution of (2.5), then we have (). By the above example, set , is transcendental with , then is a solution of . Since and by Theorem 2.1, we have () for any entire function satisfying or .
and . Thus, we complete the proof of this lemma. □
By Lemma 3.1 and the same proof in Lemma 3.2, we have the following lemma.
If , then ;
If , then .
Lemma 3.5 
where is a set of r of finite linear measure.
where is a set of r of finite linear measure. By Definition 1.1 and (3.10), we obtain the conclusion of Lemma 3.6. □
Remark 3.1 Lemma 3.6 gives the modulus estimation of an entire function with finite iterated order and extends the conclusion of [, p.84, Lemma 4].
Lemma 3.7 Let be an entire function of finite iterated order with (), and let , where , , are entire functions of finite iterated order which are not all equal to zero and satisfy , then .
By (3.19), we can obtain that . On the other hand, it is easy to get . Hence . □
Remark 3.2 The assumption in Lemma 3.7 is necessary. For example, if is an entire function satisfying (), set , , then we have and , i.e., .
By a similar proof to that in Lemma 3.7, we can easily get the following lemma.
Lemma 3.8 Let be an entire function with () and , where are entire functions which are not all equal to zero satisfying . Then .
Lemma 3.10 
Lemma 3.11 
Let () be entire functions with finite iterated order satisfying , then every solution of (2.6) satisfies .
Let () be entire functions with finite iterated order satisfying () and . Then every solution of (2.6) satisfies .
Remark 3.4 The conclusion of Lemma 3.12 also holds if .
where . By (3.25), we have . □
Lemma 3.14 Let , be entire functions satisfying and and let , be meromorphic functions satisfying , and () outside of a set of finite logarithmic measure, where . If is an entire solution of (3.22), then .
By (3.30), we have . □
By the above proof, we can easily obtain that Lemma 3.14 also holds if .
Now we divide the proof of Theorem 2.1 into two cases: case (i) and case (ii) and .
- (2)We prove that . Set , then and(4.2)
- (3)We prove that . Set , then and(4.6)
- (4)We prove that . Set , then and(4.11)
- (5)We prove that (). Set (), then , () and . By successive derivation on (4.14), we can also get the following equation which has a similar form to (4.16):(4.17)
where G, H are meromorphic functions which have the same form as , and satisfy and . By Lemma 3.13, we have . Since , by Lemma 3.4, we have ().
We prove that . Set , then . By the same proof as that of (2) in case (i), we have (4.5). Set , we affirm , if , then by Lemma 3.14, we have , which is a contradiction to ; therefore . Since , by Lemma 3.4 and (4.5), we have .
- (3)We prove that . Set , then and , , By the same proof as that of (3) in case (i), we can obtain (4.10). Set , where , . In the following we prove that . By Definition 1.2 and Lemma 3.10, for all sufficiently large and for any , we have(4.18)
If , by (4.20) and by a similar proof to that in Lemma 3.14, we have , which is a contradiction. Therefore , then by Lemma 3.4 and , we have .
By following the proof of (4)-(5) in case (i) and the proof of (3) in case (ii), we can obtain ().
Using a similar proof to that in case (i) of Theorem 2.1 and by Lemma 3.4, we can easily obtain Theorem 2.2. Theorem 2.3 is a direct result of Theorem 2.1 and Lemma 3.8.
The authors thank the referee for his/her valuable suggestions to improve the present article. This project is supported by the National Natural Science Foundation of China (11171119, 61202313, 11261024, 11271045), and by the Natural Science Foundation of Jiangxi Province in China (20122BAB211005, 20114BAB211003, 20122BAB201016) and the Foundation of Education Bureau of Jiang-Xi Province in China (GJJ12206). Zuxing Xuan is supported in part by NNSFC (No. 11226089, 60972145), Beijing Natural Science Foundation (No. 1132013) and the Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges Under Beijing Municipality (IDHT20130513).
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