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The growth and value distribution of Laplace-Stieltjes transformations with infinite order in the right half-plane
Journal of Inequalities and Applications volume 2013, Article number: 273 (2013)
Abstract
By introducing the concept of -order functions, we study the growth of analytic functions defined by Laplace-Stieltjes transformations which converge on the right half-plane. Some necessary and sufficient conditions on finite -order of these functions have been obtained. We also investigate the value distribution of Laplace-Stieltjes transformations with finite -order and obtain the existence of -points and -points dealing with multiple values of two Laplace-Stieltjes transformations which converge on the right half-plane. The main results of this paper are improvement and extension of some theorems given by Shang and Gao.
MSC:44A10, 30D15.
1 Introduction and basic notes
Consider the Laplace-Stieltjes transforms
where is a bounded variation on any interval (), and σ and t are two real variables. We choose a sequence
which satisfies the following conditions:
where
Remark 1.1 The Dirichlet series is regarded as a special example of Laplace-Stieltjes transformations, and considerable attention has been paid to the growth and the value distribution of analytic functions defined by the Dirichlet series; see [1–3] for some recent results.
In 1963, Yu [4] proved the Valiron-Knopp-Bohr formula of the associated abscissas of bounded convergence, absolute convergence and uniform convergence of Laplace-Stieltjes.
Theorem A Suppose that Laplace-Stieltjes transformations (1) satisfy the first formula of (3) and . Then
where is called the abscissa of uniformly convergent .
It follows from (3), (4) and Theorem A that , i.e., is analytic in the right half-plane. Put
Remark 1.2 The concepts of , of analytic functions represented by Laplace-Stieltjes transformations convergent in the complex plane were first introduced by Yu.
Remark 1.3 From (4), for any , we have
This implies that exists.
Many problems of analytic functions defined by Laplace-Stieltjes transformations have been studied and some important results have been obtained in [5–11]. In those papers, the authors mainly used the technique of a type function to control the denominator in the definition of order. In 2012, Kong [12] investigated the growth of the Laplace-Stieltjes transforms convergent in the right half-plane by using a type function of the infinite order. In [13], the authors also investigated the growth and value distribution of infinite order analytic functions represented by Laplace-Stieltjes transformations convergent in the right half-plane. They introduced a completely new technique based on the concept of to control the growth order of the numerator or , and obtained the main theorems as follows.
Theorem B (see [13])
If the Laplace-Stieltjes transformation of infinite order has finite X-order, and sequence (2) satisfies (3) and (4), then we have
Theorem C (see [13])
If the Laplace-Stieltjes transformation of infinite order and sequence (2) satisfies (3) and (4), then we have
where .
Remark 1.4 In Theorems B and C, the definitions of X-order and the function are introduced in Section 2.
Thus, a question arises naturally: What will happen when in Theorems B and C?
In this paper, we investigate the above question by using the type functions to enlarge the growth of the denominator , where satisfies the following conditions:
-
(i)
is monotone and ;
-
(ii)
, where .
Theorem 1.1 If the Laplace-Stieltjes transformation of infinite order has infinite X-order, and sequence (2) satisfies (3) and (4), then we have
where .
Remark 1.5 If the Laplace-Stieltjes transformation of infinite order has infinite X-order and satisfies
then T is called the -order of the Laplace-Stieltjes transform .
Remark 1.6 From Lemma 2.1 and Lemma 2.2 in Section 2, we can prove the conclusion of Theorem 1.1 easily.
Theorem 1.2 If the Laplace-Stieltjes transformation has infinite X-order and sequence (2) satisfies (3) and (4), then we have
From Theorem 1.2, we further investigate the value distribution of analytic functions with infinite X-order represented by Laplace-Stieltjes transformations convergent in the right half-plane and obtain the following theorems.
Theorem 1.3 Suppose that sequence (2) satisfies (3) and (4) and the Laplace-Stieltjes transformation has infinite order. Let , where is an increasing function, and for any positive number and , satisfies (5) and
Then is the -point of with finite -order , that is, for any , the inequality
holds for any with one exception, where is the counting function of distinct zero of the function in the strip .
Theorem 1.4 Suppose that sequence (2) satisfies (3) and (4), and the Laplace-Stieltjes transformation is of infinite order. Let , where is a continuous function on , , is a positive real number, and if satisfies (5), then is the -point of with finite -order , that is, for any , the inequality
holds for any with one exception, where is the counting function of distinct zeros of the function in the strip .
Theorem 1.5 Under the assumptions of Theorem 1.4, l (≥1) is a positive integer. Then is the -point dealing with multiple values of with finite -order , that is, for any , the inequality
holds for any with at most q () exceptions, where is the counting function of distinct zeros of the function in the strip , whose multiplicities are not greater than l.
Theorem 1.6 Under the assumptions of Theorem 1.4, l (≥1) is a positive integer. Then is the -point dealing with multiple values of with finite -order , that is, for any , the inequality
holds for any with at most q () possible exceptions, where is the counting function of distinct zeros of the function in the strip , whose multiplicities are not greater than l.
The structure of this paper is as follows. In Section 2, we introduce the concepts of X-order and -order. Section 3 is devoted to proving Theorem 1.2. Section 4 is devoted to proving Theorems 1.3-1.6.
2 The definitions of X-order and -order
We first introduce the concept of X-order of such functions as follows.
Definition 2.1 [14]
If the Laplace-Stieltjes transform satisfies (sequence (2) satisfies (3) and (4)) and
then is called a Laplace-Stieltjes transform of infinite order.
By studying a lot of papers, we found that to control the growth of the molecule or in the definition of order, many mathematicians proposed the type functions to enlarge the growth of the denominator or −σ (see [4, 6, 7, 10, 11]). In this paper, we investigate the growth of the Laplace-Stieltjes transform of infinite order by using a class of functions to reduce the growth of or which is different from the previous form. Thus, we should give the definition of the new function as follows.
Let be the class of all functions satisfying the following conditions:
-
(i)
is defined on , , is positive, strictly increasing, differential and tends to +∞ as ;
-
(ii)
as .
Definition 2.2 If the Laplace-Stieltjes transformation of infinite order satisfies
where , then is called the X-order of the Laplace-Stieltjes transform .
Remark 2.1 In particular, if we take , , , where and , X-order is p-order of the Laplace-Stieltjes transformations with infinite order.
Remark 2.2 In addition, X-order is more precise than p-order to some extent. In fact, for p (≥2) being a positive integer, we can find a function and a positive real function satisfying
and
For example, let , , where t is a finite positive real constant and . We can get that , and , where denotes the p-order of f and the X-order of f.
Remark 2.3 If in Definition 2.1, then is called a Laplace-Stieltjes transform of infinite X-order.
Lemma 2.1 Let and let be the function satisfying
If satisfies (>0), then we have
Proof We consider two cases as follows.
Case 1. If is not a constant. From the assumptions of Lemma 2.1, we can get that as . Thus, for sufficiently large x, we have . From , we have . Then from the Cauchy mean value theorem, there exists satisfying
that is,
Since as and (), by (6), we can get the conclusion of Lemma 2.1 easily.
Case 2. If is a constant. By using the same argument as in Case 1, we can prove the conclusion of Lemma 2.1 easily.
Thus, the conclusion of this lemma is true. □
The following lemma is very crucial in the study of the growth of analytic functions represented by Laplace-Stieltjes transforms convergent in the right half-plane which show the relation between and of such functions.
If the abscissa of the uniformly convergent Laplace-Stieltjes transformation and the sequence (2) satisfies (3), then for any given and for σ (>0) sufficiently reaching 0, we have
where is a constant depending on ε, (3) and
3 The proof of Theorem 1.2
We prove the conclusions of Theorem 1.2 by using the properties of two functions and . This method is different from the previous method of [13] to some extent.
We first prove ‘⟸’ of Theorem 1.2. Suppose that
Then, for any positive real number , for sufficiently large n, we have
where is the inverse function of . Let and be two reciprocally inverse functions, then we have
Thus, we have
For any fixed and sufficiently small , set
that is,
If , for sufficiently large n, let , from , (8), (9) and the definition of , we have
If , from (8) and (9), we have
For sufficiently large n, from (10) and (11), we have
Since τ is arbitrary, by Theorem 1.1 and Lemma 2.1, we can get
Suppose that
Thus, there exists any real number ε (). For any positive integer n and any sufficiently small , from Lemma 2.2, we have
From (7), there exists a subsequence ; for sufficiently large p, we have
Take a sequence satisfying
From (12) and (14), we get
that is,
Thus, we have
From (14) and (15), we have
Thus, from the Cauchy mean value theorem, there exists a real number ξ between and such that
Since
then for sufficiently large p, we have
where is a constant.
From (13) and (16), we can get a contradiction. Thus, we can get
Hence, the sufficiency of Theorem 1.2 is completed.
We can prove the necessity of Theorem 1.2 by using a similar argument as in the proof of sufficiency of Theorem 1.2.
Thus, the proof of Theorem 1.2 is completed.
4 Proofs of Theorems 1.3-1.6
In this section, we give the definition of -order of Laplace-Stieltjes transformations in the level half-strip as follows.
Definition 4.1 Let be an analytic function with infinite X-order represented by Laplace-Stieltjes transformations convergent in the right half-plane. Set , where is a real number and l is a positive number. Let and
where . Then is called the -order of in the level half-strip .
To prove Theorems 1.3-1.6, we need some lemmas as follows.
Lemma 4.1 If the Laplace-Stieltjes transformation is of infinite X-order, sequence (2) satisfies (3) and (4), and , where is an increasing function, and for any positive number and , satisfies
then for any , we have
Proof We will prove this lemma by using a similar argument to that in [11]. From the assumptions of Lemma 4.1, for any ,
Then
Since is an analytic function with infinite X-order, from the above inequality and the definition of -order, we can get the conclusion of Lemma 4.1. □
Lemma 4.2 [[11], Lemma 2.4]
Let
Then
-
(i)
this mapping maps the horizontal half-strip B to the unit disc , and its inverse mapping is
-
(ii)
();
-
(iii)
();
-
(iv)
().
Lemma 4.3 (see [14])
Let f be an admissible function in the unit disc , let q be a positive integer, and let be pairwise distinct complex numbers. Then, for , ,
where is a possibly occurring exceptional set with , and the term is replaced by when some . We use to denote
as possibly outside the set E such that . If the order of f is finite, the remainder is an without any exceptional set.
Remark 4.1 Under the assumptions of Lemma 4.3, for a positive integer l, we can get the following inequality easily:
where is the counting function of poles of the function with multiplicities ≤l in , each point counted only once.
Lemma 4.4 [[15], p.282, (1.8)]
Let h be an analytic in the disc , then
where is the maximum modulus of h in the disc .
4.1 The proof of Theorem 1.3
Since sequence (2) satisfies (3) and (4), the Laplace-Stieltjes transformation of infinite X-order, and (), from Theorem 1.2, we have
and from Lemma 4.1 and (17), for any , we have
Thus, it follows
where and .
Set , where is stated as in Lemma 4.2. Then from Lemma 4.2, we get that is analytic in the unit disc and satisfies
where . Therefore, from (18), (19) and Lemma 2.1, we have
From (20), Lemma 4.4 and Lemma 2.1, we can get that g is an admissible function in and
Then from Lemma 4.3 and (21), we can get that at most there exists one exception a satisfying
Since
and
we have
Thus, from (22)-(23) and Lemma 2.1, we have
Hence, for any and (24), the inequality
holds for any with one exception, where is the counting function of zeros of the function in the strip .
Thus, we complete the proof of Theorem 1.3.
4.2 The proof of Theorem 1.4
Since is a continuous function on , then we can get that is a function of bounded variation on (). Set
From the assumptions of Theorem 1.6, for any real number x (), we have
that is,
From the assumption of Theorem 1.4, by (25) and Theorem 1.1, we can get
From (26), for any , we have
where .
Set , where is stated as in Lemma 4.2. Then from Lemma 4.2, we can get that is analytic in the unit disc and satisfies
where . Therefore, from the above inequality and Lemma 4.2, we have
Then, similar to the proof of Theorem 1.3, we can prove that for any , the inequality
holds for any with one exception, where is stated as in Theorem 1.6.
Thus, we complete the proof of Theorem 1.4.
4.3 Proofs of Theorems 1.5 and 1.6
From Remark 4.1, by using the same argument as in Theorems 1.3 and 1.4, we can prove the conclusions of Theorems 1.5 and 1.6 easily.
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Acknowledgements
This project is supported by the NSFC (No. 61202313, 11201395) and the Natural Science foundation of Jiangxi Province in China (No. 2010GQS0119, No. 20122BAB201016 and No. 20132BAB211001). The second author is supported in part by NNSFC (Nos. 11226089, 61271370), 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 (CIT&TCD20130513).
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HYX and ZXX completed the main part of this article, ZXX corrected the main theorems. All authors read and approved the final manuscript.
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Xu, HY., Xuan, ZX. The growth and value distribution of Laplace-Stieltjes transformations with infinite order in the right half-plane. J Inequal Appl 2013, 273 (2013). https://doi.org/10.1186/1029-242X-2013-273
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DOI: https://doi.org/10.1186/1029-242X-2013-273