The growth and approximation for an analytic function represented by Laplace–Stieltjes transforms with generalized order converging in the half plane

By utilizing the concept of generalized order, we investigate the growth of Laplace–Stieltjes transform converging in the half plane and obtain one equivalence theorem concerning the generalized order of Laplace–Stieltjes transforms. Besides, we also study the problem on the approximation of this Laplace–Stieltjes transform and give some results about the generalized order, the error, and the coefficients of Laplace–Stieltjes transforms. Our results are extension and improvement of the previous theorems given by Luo and Kong, Singhal, and Srivastava.

(σ , t are real variables), a n are nonzero complex numbers. Obviously, if α(t) is an increasing continuous function which is not absolutely continuous, then the integral (1) defines a class of functions F(s) which cannot be expressed either in the form (2) or (4).
it is easy to get σ F u = 0, that is, F(s) is analytic in the left half plane; if lim sup n→+∞ log A * n λ n = -∞, it follows that σ F u = +∞, that is, F(s) is analytic in the whole plane. For convenience, we use L β to be a class of all the functions F(s) of the form (1) which are analytic in the half plane s < β (-∞ < β < ∞) and the sequence {λ n } satisfies (3) and (5); L 0 to be the class of all the functions F(s) of the form (1) which are analytic in the half plane s < 0 and the sequence {λ n } satisfies (3), (5), and (6); and L ∞ to be the class of all the functions F(s) of the form (1) which are analytic in the whole plane s < +∞ and the sequence {λ n } satisfies (3), (5), and (7). Thus, if -∞ < β < 0 and F(s) ∈ L β , then F(s) ∈ L 0 .
In 1963, Yu [26] first proved the Valiron-Knopp-Bohr formula of the associated abscissas of bounded convergence, absolute convergence, and uniform convergence of Laplace-Stieltjes transform. Moreover, Yu [26] also estimated the growth of the maximal molecule M u (σ , F), the maximal term μ(σ , F), by introducing the concepts of the order of F(s), and investigated the singular direction-Borel line of entire functions represented by Laplace-Stieltjes transforms converging in the whole complex plane. After his wonderful works, considerable attention has been paid to the value distribution and the growth of analytic functions represented by Laplace-Stieltjes transforms converging in the whole plane or the half plane (see [1, 3, 4, 6-8, 11-15, 18-25, 27]).
For F(s) ∈ L 0 , in view of M u (σ , F) → +∞ as σ → 0 -, the concepts of order and type can be usually used in estimating the growth of F(s) precisely.
Remark 1.1 However, if ρ = 0 and ρ = +∞, we cannot estimate the growth of such functions precisely by using the concept of type.
In this paper, the first aim is to investigate the growth of analytic functions represented by Laplace-Stieltjes transforms with generalized order converging in the half plane, and we obtain some theorems about the generalized order A * n and λ n , which are improvements of the previous results given by Luo and Kong [9,10]. To state our results, we first introduce the following notations and definitions.

Results and discussion
For generalized order of Laplace-Stieltjes transform (1), we obtain the following.
Theorem 2.1 Let F(s) ∈ L 0 , A ∈ 0i and B ∈ 0i be continuously differentiable, and the function B increase more rapidly than A such that, for any constant η ∈ (0, +∞), and and .
, and B ∈ 0i be continuously differentiable, and the function A increase more rapidly than B such that, for any constant η ∈ (0, +∞), and If F(s) satisfies (10) and If Laplace-Stieltjes transform (1) satisfies A * n = 0 for n ≥ k + 1 and A * k = 0, then F(s) will be said to be an exponential polynomial of degree k usually denoted by p k , i.e., p k (s) = λ k 0 exp(sy) dα(y). If 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 denote k to be the class of all exponential polynomials of degree almost k, that is, For F(s) ∈ L β , -∞ < β < 0, 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 In 2017, Singhal and Srivastava [17] studied the approximation of Laplace-Stieltjes transforms of finite order converging on the whole plane and obtained the following theorem. Theorem 2.3 (see [17]) If Laplace-Stieltjes transform F(s) ∈ L ∞ and is of order ρ (0 < ρ < ∞) and of type T, then for any real number -∞ < β < +∞, we have In the same year, the author and Kong [20] investigated the approximation of Laplace-Stieltjes transform F(s) ∈ L 0 with infinite order and obtained the following.  (13) where 0 < ρ * < ∞, X(·)-order can be seen in [20].
The second purpose of this paper is to study the approximation of Laplace-Stieltjes transform F(s) ∈ L 0 with generalized order, and our results are listed as follows.
Theorem 2.5 Let F(s) ∈ L 0 , A ∈ 0i , and B ∈ 0i be continuously differentiable satisfying (8) and (9), and let the function B increase more rapidly than A. If F(s) satisfies (10) and then for any real number -∞ < β < 0, we have .

Theorem 2.6
Under the assumptions of Theorem 2.2, then for any real number -∞ < β < 0, we have

Conclusions
Regarding Theorems 2.1 and 2.2, the generalized order of Laplace-Stieltjes transforms are discussed by using the more abstract functions, and some related theorems among λ n , A * n and the generalized order are obtained. Moreover, we also investigate some properties of approximation on analytic functions defined by Laplace-Stieltjes transforms of generalized order. For the topic of the growth and approximation of Laplace-Stieltjes transforms of generalized order, it seems that this topic has never been treated before. Our theorems are generalization and improvement of the previous results given by Luo and Kong [9,10], Singhal and Srivastava [17].

The proof of Theorem 2.1
Suppose that ρ := ρ AB (F) < +∞ and In view of the definition of generalized order and Lemma 4.2, for any ε > 0, there exists a constant σ 0 < 0 such that, for all 0 > σ > σ 0 , Choosing , we conclude from (9) and (16) that )) + log p , as n → +∞, which implies Since A ∈ 0i , B ∈ 0i and let ε → 0 + , we can conclude from (17) that ϑ ≤ ρ. Assume ϑ < ρ, then we can choose a constant ρ 1 such that ϑ < ρ 1 < ρ. Since B -1 ( A(x) ρ 1 ) is an increasing function, then there exists a positive integer n 0 such that, for n ≥ n 0 , where here and further K j is a constant.
If ρ AB (F) = +∞, by using the same argument as above, it is easy to prove that the conclusion is true. Therefore, this completes the proof of Theorem 2.1.
If ρ AB (F) = +∞, by using the same argument as above, it is easy to prove that the conclusion is true. Therefore, this completes the proof of Theorem 2.2.

The proof of Theorem 2.6
By combining the arguments as in the proofs of Theorems 2.2 and 2.5, we can easily prove the conclusion of Theorem 2.6.