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On the iterated exponent of convergence of zeros of
Advances in Difference Equations volume 2013, Article number: 71 (2013)
Abstract
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 ().
MSC:34A20, 30D35.
1 Introduction
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]).
Definition 1.1 The p-iterated order of a meromorphic function is defined by
Remark 1.1 If is an entire function, the p-iterated order of is defined by
It is easy to see that if . If , the hyper-order of is defined by (see [9])
Definition 1.2 The p-iterated type of an entire function with is defined by
Definition 1.3 The finiteness degree of the iterated order of an entire function is defined by
Definition 1.4 Suppose that is an entire function satisfying or , then the p-iterated order exponent of convergence of zero-sequence of is defined by
Especially, if , the p-iterated order exponent of convergence of fixed points of is defined to be
If , the p-iterated exponent of convergence of zero-sequence of is defined to be
Definition 1.5 The p-iterated exponent of convergence of distinct zero-sequence of and the p-iterated exponent of convergence of distinct fixed points of are respectively defined to be
Definition 1.6 If is an entire function satisfying or , then the finiteness degree of the iterated exponent of convergence of zero-sequence of is defined by
Remark 1.2 From Definitions 1.5 and 1.6, we can similarly give the definitions of , , and .
2 Main result
In [10], 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 [11], 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 [12], 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 [13] from (2.5) to (2.6). In the following, we list Theorems A-F which have been mentioned above.
Theorem A [10]
Let be a polynomial with degree n (≥1). Then every non-trivial solution of
has infinitely many fixed points and satisfies .
Theorem B [10]
Let be a transcendental entire function with . Then every non-trivial solution of
has infinitely many fixed points and satisfies .
Theorem C [11]
Let (≢0) () be entire functions with . a, b are complex numbers and satisfy and or (). If is an entire function of finite order, then every non-trivial solution f of
satisfies .
Theorem D [11]
Let , , be entire functions with and . Then every non-trivial solution f of
satisfies , where is a complex number.
Theorem E [12]
Let and be entire functions with finite order. If or and , then for every solution of
and for any entire function satisfying , we have
-
(i)
;
-
(ii)
(, ).
Theorem F [13]
Let () be entire functions of finite order and satisfy one of the following conditions:
-
(i)
;
-
(ii)
and .
Then for every solution of
and for any entire function satisfying , we have
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 2.1 Let and be entire functions of finite iterated order satisfying or and . Then for every solution of (2.5) and for any entire function satisfying , we have
-
(i)
;
-
(ii)
, , .
Theorem 2.2 Let , be entire functions satisfying . Then for every solution of (2.5) and for any entire functions with , we have
-
(i)
();
-
(ii)
().
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 .
Corollary 2.1 Under the hypotheses of Theorem 2.1, if , we have
-
(i)
;
-
(ii)
, , .
Corollary 2.2 Under the hypotheses of Theorem 2.2, if , we have
-
(i)
();
-
(ii)
().
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 .
Remark 2.2 Nevanlinna’s second fundamental theorem is an important tool to investigate the distribution of zeros of meromorphic functions. From Nevanlinna’s second fundamental theorem [[1], p.47, Theorem 2.5], we have that
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 .
3 Lemmas
Let be an entire function with , and denote the central index of . Then
Lemma 3.2 Let be an entire function with , then there exists a set with infinite logarithmic measure such that for all , we have
Proof By Definition 1.1 , there exists a sequence tending to ∞ satisfying and
There exists an such that for and for any , we have
Set , by (3.4) and Definition 1.1, then for any , we have
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.
Lemma 3.3 Let be an entire function with and denote the central index of . Then there exists a set with infinite logarithmic measure such that for all , we have
Let , be meromorphic functions. If f is a meromorphic solution of the equation
then we have the following two statements:
-
(i)
If , then ;
-
(ii)
If , then .
Lemma 3.5 [14]
Let be an entire function of finite iterated order with . Then there exist entire functions and such that
and
Moreover, for any , we have
where is a set of r of finite linear measure.
Lemma 3.6 Let be an entire function of finite iterated order with . Then for any given , there is a set that has finite linear measure such that for all z satisfying , we have
Proof Let be an entire function of finite iterated order with . By Definition 1.1, it is easy to obtain that holds for all sufficiently larger . By Lemma 3.5, there exist entire functions and such that
For any , we have
hold outside a set of finite linear measure. Since , by Definition 1.1, we have that holds for all sufficiently large r. From and (3.9), we have
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 [[15], 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 .
Proof can be written as
By the Wiman-Valiron lemma (see [2, 16]), for all z satisfying and , we have
where is a set of finite logarithmic measure. From (1.4.5) in [[16], p.26], for any given , we have
holds outside a set with finite logarithmic measure, where is the maximum term of f. By the Cauchy inequality, we have . Substituting it into (3.13), we have
By Lemma 3.1, there exists a set having infinite logarithmic measure such that for all , we have
By (3.15) and Lemma 3.6, for all and for any ε (), we have
Substituting (3.16) into (3.14), we have
By (3.11), we have
Substituting (3.12), (3.16), (3.17) into (3.18), for all z satisfying and , we have
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 .
Remark 3.3 By a similar proof to that of Lemma 2.2 in [3] or Lemma 6 in [8], we can easily get the following lemma which is a better result than that in [3] or [8] by allowing .
Lemma 3.9 Let be an entire function satisfying , , then for any given , there exists a set that has infinite logarithmic measure such that for all , we have
Lemma 3.10 [17]
Let be a transcendental meromorphic function and be a given constant, for any given , there exists a set that has finite logarithmic measure and a constant that depends only on α and () with such that for all z satisfying , we have
Lemma 3.11 [6]
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 .
Lemma 3.13 Let , be entire functions of finite iterated order satisfying . Let , be meromorphic functions with , , if is an entire solution of equation
then .
Proof From (3.22), we have
By the lemma of logarithmic derivative and (3.23), we have
where is a set having finite linear measure. By Lemma 3.2, there exists a set having infinite logarithmic measure such that for all , we have
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 .
Proof Without loss of generality, we suppose that . From (3.22), we have
By Lemma 3.9, for any given β (), there exists a set having infinite logarithmic measure such that for all , we have
By Lemma 3.10, there exists a set having finite logarithmic measure such that for all , we have
where is a constant. By the hypotheses, for all , we have
By (3.26)-(3.29), for all z satisfying and , we have
By (3.30), we have . □
By the above proof, we can easily obtain that Lemma 3.14 also holds if .
4 Proof of Theorem 2.1
Now we divide the proof of Theorem 2.1 into two cases: case (i) and case (ii) and .
Case (i): (1) We prove that . Assume that is a solution of (2.5), then by Lemma 3.11. Set , since , then , . Substituting , , into (2.5), we have
If , by Lemma 3.11, we have , which is a contradiction. Since and , by Lemma 3.4 and (4.1), we have , therefore .
-
(2)
We prove that . Set , then and
(4.2)
By (2.5), we get
The derivation of (2.5) is
Substituting (4.2), (4.3) into (4.4), we obtain
Let . We affirm that . If , by Lemma 3.13, we have , which is a contradiction; therefore . Since , by Lemma 3.4 and (4.5), we get .
-
(3)
We prove that . Set , then and
(4.6)
Substituting (4.3) into (4.4), we have
The derivation of (4.7) is
Set , , it is easy to see that
then by (4.8), we get
Substituting (4.6) into (4.9), we have
If , by Lemma 3.13, we have , which is a contradiction; therefore . Since , by Lemma 3.4 and (4.10), we have .
-
(4)
We prove that . Set , then and
(4.11)
The derivation of (4.9) is
By (4.9), we have
Substituting (4.13) into (4.12), we have
Let , , it is easy to obtain that
by (4.14), we have
Substituting (4.11) into (4.15), we have
Let . By Lemma 3.13, we have . Since , by Lemma 3.4 and (4.16), we have .
-
(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 ().
Case (ii): (1) We prove that . Assume that is a solution of (2.5), by Lemma 3.12, we know that . Set , is an entire function with , then we have , . Substituting , , into (2.6), we have (4.1). We affirm that . If , by Lemma 3.12, we have , which is a contradiction. Since and , by Lemma 3.4 and (4.1), we have ; therefore .
-
(2)
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)
By Lemma 3.9 and Lemma 3.10, for all sufficiently large and for any ε (), we have
By (4.18)-(4.19) and Lemma 3.10, it is easy to obtain
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 ().
5 Proof of Theorems 2.2-2.3
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.
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Acknowledgements
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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TJ completed the main part of this article, TJ, XZX and XHY corrected the main theorems. All authors read and approved the final manuscript.
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Tu, J., Xuan, ZX. & Xu, HY. On the iterated exponent of convergence of zeros of . Adv Differ Equ 2013, 71 (2013). https://doi.org/10.1186/1687-1847-2013-71
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DOI: https://doi.org/10.1186/1687-1847-2013-71
Keywords
- linear differential equations
- iterated order
- iterated type
- iterated exponent of convergence of zero-sequence