The approximation of Laplace-Stieltjes transforms with finite order
- Hong Yan Xu1, 2Email author and
- San Yan Liu1
https://doi.org/10.1186/s13660-017-1441-9
© The Author(s) 2017
Received: 21 March 2017
Accepted: 27 June 2017
Published: 12 July 2017
Abstract
In this paper, we study the irregular growth of an entire function defined by the Laplace-Stieltjes transform of finite order convergent in the whole complex plane and obtain some results about λ-lower type. In addition, we also investigate the problem on the error in approximating entire functions defined by the Laplace-Stieltjes transforms. Some results about the irregular growth, the error, and the coefficients of Laplace-Stieltjes transforms are obtained; they are generalization and improvement of the previous conclusions given by Luo and Kong, Singhal and Srivastava.
Keywords
MSC
1 Introduction
As we know, Dirichlet series is regarded as a special example of the Laplace-Stieltjes transform. The Laplace-Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For real-valued functions, it is the Laplace transform of a Stieltjes measure, however it is often defined for functions with values in a Banach space. It can be used in many fields of mathematics, such as functional analysis, and certain areas of theoretical and applied probability.
In 1963, Yu [10] proved the Valiron-Knopp-Bohr formula of the associated abscissas of bounded convergence, absolute convergence, and uniform convergence of Laplace-Stieltjes.
Theorem A
Moreover, Yu [10] first introduced the maximal molecule \(M_{u}(\sigma ,G)\), the maximal term \(\mu (\sigma ,G)\) and the Borel line, and the order of analytic functions represented by Laplace-Stieltjes transforms convergent in the complex plane. After his works, considerable attention has been paid to the growth and value distribution of the functions represented by the Laplace-Stieltjes transform convergent in the half-plane or the whole complex plane in the field of complex analysis (see [11–15]).
Since \(M(\sigma ,F)\) and \(M_{u}(\sigma ,F)\) tend to +∞ as \(\sigma \rightarrow +\infty \), in order to estimate the growth of \(F(s)\) more precisely, we will adapt some concepts of order, lower order, type, lower type as follows.
Definition 1.1
Remark 1.1
We say that \(F(s)\) is of the regular growth, when \(\rho =\lambda \), and \(F(s)\) is of the irregular growth, when \(\rho \neq \lambda \).
Definition 1.2
Remark 1.2
The purpose of the definition of type is to compare the growth of class functions which all have the same order. For example, let \(f(s)=e^{e ^{s}}\), \(g(s)=e^{e^{2s}}\), by a simple computation, we have \(\rho (f)=1= \rho (g)\), but \(T(f)=1\) and \(T(g)=\infty \). Thus, we can see that the growth of \(g(s)\) is faster than \(f(s)\) as \(\vert s \vert \rightarrow +\infty \).
2 Results and discussion
Recently, many people studied some problems on analytic functions defined by the Laplace-Stieltjes transforms and obtained a number of interesting results. Kong, Sun, Huo and Xu investigated the growth of analytic functions with kinds of order defined by the Laplace-Stieltjes transforms (see [16–22]), and Shang, Gao, and Sun investigated the value distribution of such functions (see [23–26]). From these references, we get the following results.
Theorem 2.1
From Definition 1.2, a natural question to ask is: What happened if \(e^{\sigma \rho }\) is replaced by \(e^{\lambda \sigma }\) in the definition of lower type when \(\rho \neq \lambda \)? We are going to consider this question.
Definition 2.1
Remark 2.1
Obviously, \(\tau_{\lambda }\geq \tau \) and \(\tau_{\lambda }=\tau \) as \(\rho =\lambda \). But we cannot confirm whether \(\tau_{\lambda } \geq T\) or \(\tau_{\lambda }\leq T\).
The following results are the main theorems of this paper.
Theorem 2.2
Theorem 2.3
Theorem 2.4
We denote by \(\overline{L}_{\beta }\) the class of all the functions \(F(s)\) of the form (5) which are analytic in the half-plane \(\Re s<\beta\) (\(-\infty <\beta <\infty\)) and the sequence \(\{\lambda _{n}\}\) satisfies (2) and (4); and we denote by \(L_{\infty }\) the class of all the functions \(F(s)\) of the form (5) which are analytic in the half-plane \(\Re s<+\infty \) and the sequence \(\{\lambda_{n}\}\) satisfies (2), (4), and (6). Thus, if \(-\infty <\beta <+\infty \) and \(F(s) \in \overline{L}_{\beta }\), then \(F(s)\in L_{\infty }\). If Laplace-Stieltjes transform (5) \(A^{*}_{n}=0\) for \(n\geq k+1\) and \(A^{*}_{n}\neq 0\), then \(F(s)\) will be called an exponential polynomial of degree k usually denoted by \(p_{k}\), \(i.e\)., \(p_{k}(s)= \int^{\lambda_{k}}_{0}\exp (sy)\,d\alpha (y)\). When we choose a suitable function \(\alpha (y)\), the function \(p_{k}(s)\) may be reduced to a polynomial in terms of \(\exp (s\lambda_{i})\), that is, \(\sum_{i=1} ^{k}b_{i}\exp (s\lambda_{i})\).
In this paper, we will further investigate the relation between \(E_{n}(F,\beta )\) and the growth of an entire function defined by the L-S transform with irregular growth. It seems that this problem has never been treated before. Our main result is as follows.
Theorem 2.5
3 Conclusions
From Theorems 2.2-2.5, we can see that the growth of Laplace-Stieltjes transforms is investigated under the assumption \(\rho \neq \lambda \), and that some theorems about the λ-lower type \(\tau_{\lambda }\), \(\lambda_{n}\), \(A_{n}^{*}\), and λ are obtained. In addition, we also study the problem on the error in approximating entire functions defined by the Laplace-Stieltjes transforms. This project is a new issue of Laplace-Stieltjes transforms in the field of complex analysis. Our results are generalization and improvement of the previous conclusions given by Luo and Kong [16, 27], Singhal and Srivastava [28].
4 Methods
4.1 Proofs of Theorems 2.2 and 2.3
To prove the above theorems, we require the following lemmas.
Lemma 4.1
see [27], Lemma 2.1
Lemma 4.2
see [16], Lemma 2.2
4.1.1 The proof of Theorem 2.2
Hence, this completes the proof of Theorem 2.2.
4.1.2 The proof of Theorem 2.3
Thus, this completes the proof of Theorem 2.3.
4.2 The proof of Theorem 2.4
Hence, this proves that (12) holds.
4.3 The proof of Theorem 2.5
To prove this theorem, we require the following lemma.
Lemma 4.3
Proof
Now, we are going to prove Theorem 2.5.
4.4 The proof of Theorem 2.5
Therefore, we complete the proof of Theorem 2.5.
4.5 Remarks
From the proof of Theorem 2.5, and combining those results of the Laplace-Stieltjes transforms in Ref. [14, 16, 27], we can obtain the following results on the approximation of Laplace-Stieltjes transforms, which can be found partly in [28].
Theorem 4.1
Theorem 4.2
- (i)
\(h(x)\) is defined on \([a,+\infty )\) and is positive, strictly increasing, differentiable and tends to +∞ as \(x\rightarrow + \infty \);
- (ii)
\(\lim_{x\rightarrow +\infty } \frac{d(h(x))}{d(\log^{[p]}x)}=k\in (0,+\infty )\), \(p\geq 1, p\in \mathbb{N}^{+}\), where \(\log^{[0]}x=x, \log^{[1]}x=\log x\) and \(\log^{[p]}x=\log (\log^{[p-1]}x)\).
Declarations
Acknowledgements
We thank the referee(s) for reading the manuscript very carefully and making a number of valuable and kind comments which improved the presentation.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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