Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type

Abstract In this paper, we introduce the concept of the perfect ϕ \phi -type to describe the growth of the maximal molecule of Laplace-Stieltjes transform by using the more general function than the usual. Based on this concept, we investigate the approximation and growth of analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms convergent in the half plane and obtain some results about the necessary and sufficient conditions on analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms with perfect ϕ \phi -type, which are some generalizations and improvements of the previous results given by Kong [On generalized orders and types of Laplace-Stieltjes transforms analytic in the right half-plane, Acta Math. Sin. 59A (2016), 91–98], Singhal and Srivastava [On the approximation of an analytic function represented by Laplace-Stieltjes transformations, Anal. Theory and Appl. 31 (2015), 407–420].


Introduction
Thus, if ( ) ∈ F s L 0 and −∞ < < β 0, then ( ) ∈ F s L β . In order to estimate the growth of ( ) F s , Yu [1] introduced the concepts of the maximal term ( ) μ σ F , , the maximal molecule ( ) M σ F , u and the order of ( ) F s and also studied that the value distribution of entire functions defined by Laplace-Stieltjes transforms converge in the complex plane. After his wonderful works, many scholars studied the value distribution and the growth of analytic functions represented by Laplace-Stieltjes transforms converge in the whole plane or the half plane, and obtained a large number of important and interesting results (see [2][3][4][5][6][7][8][9][10][11][12][13][14]). Define , the concepts of order and type can be usually used in estimating the growth of ( ) F s precisely.
, the type and the lower type of ( ) . We denote Π k to be the class of all exponential polynomials of degree almost k, that is, the error in approximating the function ( ) F s by exponential polynomials of degree n in uniform norm as is always supposed to be not null. For ( ) ∈ F s L 0 , Singhal and Srivastava [16] in 2015 studied the approximation of ( ) F s with finite order, and obtained as follows.  ,0 such that the derivative ′ ϕ is positive, continuous and increasing to +∞ on (−∞ ) ,0 . If ∈ ϕ Ξ 0 and Laplace-Stieltjes transform ( ) ∈ F s L 0 satisfies then T is called the ϕ-type of ( ) F s . Similarly, the lower ϕ-type of ( ) F s is defined by ϕ , then we say that ( ) F s is of perfect ϕ-type, that is, Let φ be the inverse function of ′ ϕ , then φ is continuous on ( +∞) 0, and increases to 0, and let ∈ ϕ Ξ 0 Now, we list our main results below to show the relations among the perfect ϕ-type, the error ( ) E F β , n , λ n and A n ⁎ for Laplace-Stieltjes transforms ( ) F s with the perfect ϕ-type.
then for any real number −∞ < < β 0 and < < +∞ T 0 ϕ , we have There exists a non-decreasing positive integer sequence { } n v satisfying (1.9) and 1 , in view of Theorems 1.2 and 1.3, we can obtain the following corollaries.
Remark 1.4. In view of Corollaries 1.1 and 1.2, this shows that our results are some generalizations and improvements of Theorem 1.1.

Some lemmas
To prove our results, we also need to give the following lemmas.
for all ≥ n 0. Thus, for any < σ 0 and < x 0, we have On the other hand, assume that , and by considering with φ being the inverse function of ′ ϕ , thus for all ≥ n 0, it leads to and > q 0, we have Proof. First, denote ( ) = ( , for ∈ ( +∞) x a, . Thus, it is easy to get that ( ) → G x 0 as → + x a . Here, we only prove that ( ) G x is a decreasing function on ( +∞) a, . In view of the definitions G 1 and G 2 , it follows and The perfect ϕ-type of Laplace-Stieltjes transforms  1819

(2.2)
Since φ is an increasing function, then for > x a, it follows In view of (2.1)-(2.3), it yields that . Therefore, this completes the proof of Lemma 2.3. □

(2.7)
Since φ is continuous on ( +∞) 0, and increases to 0, thus in view of (2.6), it yields that → η 0 k as → +∞ k  Hence, for which implies    In view of (1.7) and Lemma 2.1, it follows that  that is, , e x p , where K is a constant and only depends on h and β.
Thus, from (3.2) and (3.5), it follows that (3.14) To prove (1.8), in view of (3.14), we only need to prove In view of (3.2), we can conclude that there exists a positive integer subsequence { } n v such that