The Growth of Entire Functions Defined by Laplace–Stieltjes Transforms Related to Proximate Order and Approximation

In this article, we discuss the growth of entire functions represented by Laplace–Stieltjes transform converges on the whole complex plane and obtain some equivalence conditions about proximate growth of Laplace–Stieltjes transforms with finite order and infinite order. In addition, we also investigate the approximation of Laplace–Stieltjes transform with the proximate order and obtain some results containing the proximate growth order, the error, A∗ n , and λn, which are the extension and improvement of the previous theorems given by Luo and Kong and Singhal and Srivastava.


Introduction
Our main aim of this paper is to investigate some problems about the growth and approximation of entire functions represented by Laplace-Stieltjes transforms which converge on the whole complex plane. Consider Laplace-Stieltjes transforms where α(x) is a bounded variation on any finite interval [0, Y](0 < Y < +∞). For Laplace-Stieltjes transform (1), set where the sequence λ n ∞ 0 satisfies 0 ≤ λ 1 < λ 2 < · · · < λ n < · · · , λ n ⟶ ∞ as n ⟶ ∞ if lim sup lim sup n⟶+∞ log n λ n � D < +∞, similar to the method in [1], and by using the Valiron-Knopp-Bohr formula, then it yields σ F u � +∞, that is, F(s) is an entire function in the whole plane. We denote L β to be a class of all the functions F(s) of form (1) which are analytic in the half plane Rs < β(− ∞ < β < ∞), and the sequence λ n satisfies (3) and (5), and denote L ∞ to be the class of all the functions F(s) of form (1) which are analytic in the whole plane Rs < +∞, and the sequence λ n satisfies (3), (5), and (4). us, if − ∞ < β <+∞ and F(s) ∈ L ∞ , then F(s) ∈ L β . By Widder [2], if α(t) is absolutely continuous, then F(s) is the classical Laplace integral form: (6) if α(t) is a step-function and a sequence λ n ∞ 0 satisfies (3), and α(x) � a 1 + a 2 + · · · + a n , λ n ≤ x < λ n+1 ; 0, then F(s) becomes a Dirichlet series: a n e λ n s , s � σ + it, σ, t ∈ R, (8) where a n are nonzero complex numbers; if α(t) is an increasing continuous function which is not absolutely continuous, then integral (1) defines a class of functions F(s) which cannot be expressed either in form (6) or (8) (see [2]).
In 1963, Yu [1] first proved the Valiron-Knopp-Bohr formula of the associated abscissas of bounded convergence, absolute convergence, and uniform convergence of Laplace-Stieltjes transform when s � − z. Theorem 1. If the sequence λ n satisfies (3) and (5), then Laplace-Stieltjes transform (1) satisfies where σ F u is called the abscissa of uniformly convergence of F(s), and In addition, Yu [1] introduced the concept of the order of G(z) � F(− s) to estimate the growth of the maximal molecule M u (σ, G) and the maximal term μ(σ, G) and gave the relation between the Borel line and the order of entire functions represented by Laplace-Stieltjes transform converge in the whole complex plane. After his wonderful works, considerable attention has been paid to the growth, and the value distribution of the functions represented by Laplace-Stieltjes transform or Dirichlet series converges in the half plane or the whole complex plane in the field of complex analysis (see ). Set In view of M u (σ, F) ⟶ + ∞ as σ ⟶ + ∞, in order to estimate the growth of F(s) precisely, we usual use the concepts of order and type as follows.
then it is said that F(s) is of order ρ in the whole plane, where log + x � max log x, 0 . Furthermore, if ρ ∈ (0, +∞), the type of F(s) is defined by Let F(s) be of order ρ in the whole plane and 0 < ρ < + ∞, and let ρ(σ)(σ > σ 0 ) be a nonnegative, continuous, and monotonous function, and it has a left-hand derivative and right-hand derivative in every σ( > σ 0 ) such that then ρ(σ) is called the proximate order; furthermore, if then T will be called the proximate type of F(s) with respect to proximate order ρ(σ). Set r � e σ , and let t � ψ(r) � r ρ(log r) � e σρ(σ) and r � φ(t) be two reciprocally inverse functions. en, we obtain the following theorem.

Theorem 2. If Laplace-Stieltjes transform F(s) ∈ L ∞ and is of proximate order ρ(σ) and order ρ(0
where τ � lim sup Furthermore, if the sequence λ n satisfies then In order to estimate the growth of Laplace-Stieltjes transform of order ρ � +∞ precisely, we will use some concepts of the p-order as follows. then we call F(s) is of p-order ρ p in the whole plane, where log p+1 x � log p log x for p ∈ N + . Let F(s) be of p-order ρ p in the whole plane and 0 < ρ p < + ∞, similar to the proximate type of F(s) with respect to proximate order ρ(σ); we can give the proximate type of F(s) with respect to proximate order of p-order ρ p as follows. Let ρ p (σ)(σ > σ 0 ) be a nonnegative, continuous, and monotonous function, and it has a lefthand derivative and right-hand derivative in every σ( > σ 0 ) such that 2 Journal of Function Spaces then ρ p (σ) is called the p-proximate order; furthermore, if then T p will be called the proximate type of F(s) with respect to p-proximate order ρ p (σ). Set r � e σ , and let t � ψ p (r) � r ρ p (log r) � e σρ p (σ) and r � φ p (t) be two reciprocally inverse functions. en, we obtain the following theorem. where Furthermore, if the sequence λ n satisfies (18), then e other purpose of this article is to study the approximation of the entire function represented by Laplace-Stieltjes transform converges in the whole plane. When Laplace-Stieltjes transform (1) satisfies A * k ≠ 0 and A * n � 0 for n ≥ k + 1, then F(s) will be called an exponential polynomial of degree k usually denoted by p k , i.e., p k (s) � λ k 0 exp(sy)dα(y). When we choose a suitable function α(y), the function p k (s) may be reduced to a polynomial in terms of exp(sλ i ), that is, k i�1 b i exp(sλ i ). We also use Π k to denote the class of all exponential polynomial of degree almost k, that is, For F(s) ∈ L ∞ , − ∞ < β < + ∞, we denote by E n (F, β) the error in approximating the function F(s) by exponential polynomials of degree n in uniform norm as where Recently, Singhal and Srivastava [15] and Xu and Liu [25] studied the approximation on Laplace-Stieltjes transforms of finite order and obtained the following theorems.
can obtain the minimum (1)) . (46) en, it means arψ ′ (r) − b � 0. In view of the first equality in (40), we deduce b � aρ(1 + o(1))ψ(r), that is, We can see that, as the value of r increases to the given value above, the value of h(r) changes from a negative value to a positive value. us, h(r) can obtain minimum (1)) .
where C and K are constants.

e Proof of eorem 3.
We firstly prove τ p ≤ T p . Assume that T p < + ∞. From then for any positive number ε and sufficiently large σ, it yields where r � e σ . By Lemmas 2 and 4, for sufficiently large σ, it follows where exp 0 (x) � x, exp p (x) � exp(exp p− 1 (x)), (p ≥ 1). In view of Lemma 2, it is easy to see that exp p− 1 ψ p (r)(T p + ε) − λ n log r + log 2 can attain the minimum when r satisfies the following equation: By Lemma 1, we have rψ p ′ (r) � ψ p (r)ρ p (1 + o(1)) as σ ⟶ + ∞. us, it follows from (79) that Hence, the function exp p− 1 ψ p (r)(T p + ε) − λ n log r + log 2 can obtain the minimum as r satisfies equation (80). From p ≥ 2 and Lemma 1, it yields 6 Journal of Function Spaces So, we can deduce from (78), (81), and (82) that that is, Since ε is arbitrary, we conclude from the above inequality that and τ p < + ∞ as T p < + ∞. Next, we prove T p ≤ e Dρ p τ p . Assume that 0 ≤ τ p < + ∞; then, for any ε > 0, there exists a constant K 2 such that us, from Lemma 3, Lemma 4, and (86), it yields By using the same argument as in Lemma 3, we can conclude that the function log e σ+D+ε τ p + ε can attain the maximum as λ n satisfies log exp(σ + D + ε) us, by Lemma 1, we can deduce from (87) and (89) Since ε is arbitrary, let ε ⟶ 0 in (89); it follows that T p ≤ e Dρ p τ p and T p < + ∞ as τ p < + ∞. Hence, we prove (23).
Furthermore, if the sequence λ n satisfies (18), then we see D � 0 from the proof of eorem 2. us, it is easy to prove that T p � τ p from (23). erefore, this completes the proof of eorem 3.

e Proof of eorem 7.
By using the same argument as in the proof of eorem 3 and combining (99), (102), and (106), we can prove the conclusions of eorem 7 easily.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
H. Y. Xu was responsible for conceptualization and writing and preparing the original draft; H. Y. Xu and W. J. Tang contributed to writing the review and editing; and W. J. Tang, H. Y. Xu, and J. Chen were responsible for funding acquisition.