Relative Growth of Series in Systems of Functions and Laplace—Stieltjes-Type Integrals

: For a regularly converging-in- C series A ( z ) = ∑ ∞ n = 1 a n f ( λ n z ) , where f is an entire transcendental function, the asymptotic behavior of the function M − 1 f ( M A ( r )) , where M f ( r ) = max {| f ( z ) | : | z | = r } , is investigated. It is proven that, under certain conditions on the functions f , α , and the coefﬁcients a n , the equality lim r → + ∞ α ( M − 1 f ( M A ( r ))) α ( r ) = 1 is correct. A similar result is obtained for the Laplace–Stiltjes-type integral I ( r ) = (cid:82) ∞ 0 a ( x ) f ( rx ) dF ( x ) . Unresolved problems are formulated.


Introduction
Let f (z) = ∞ ∑ k=0 f k z k (1) be an entire function, M f (r) = max{| f (z)| : |z| = r}, and Φ f (r) = ln M f (r). For an entire function g with Taylor coefficients g n , the study of growth of the function Φ −1 f (ln M g (r)) in terms of the exponential type was initiated in papers [1,2] and was continued in [3]. As a result, it is proven that, if | f k−1 / f k | +∞ as k → ∞, then of the function f with respect to the function g are used in Reference [4]. Research on the relative growth of entire functions was continued by many mathematicians (an incomplete bibliography is given in [5]).
Let (λ n ) be a sequence of positive numbers increasing to +∞. Suppose that the series A(z) = ∞ ∑ n=1 a n f (λ n z) (2) in the system f (λ n z) is regularly convergent in C, i.e., ∑ ∞ n=1 |a n |M f (rλ n ) < +∞ for all r ∈ [0, +∞). Many authors have studied the representation of analytic functions by series in the system f (λ n z) and the growth of such functions. Here, we specify only the monographs of A.F. Leont'ev [6] and B.V. Vinnitskyi [3], which are references to other papers on this topic.
Since series (2) is regularly convergent in C and the function A is an entire function, a natural question arises about the asymptotic behavior of the function M −1 f (M A (r)).
We suppose that the function F is nonnegative, nondecreasing, unbounded, and continuous on the right on [0, +∞); that f is positive, increasing, and continuous on [0, +∞); and that a positive-on-[0, +∞) function a is such that the Laplace-Stietjes-type integral exists for every r ∈ [0, +∞). The asymptotic behavior of such integrals in the case f (x) = e x is studied in the monograph [7]. A question arises again about the asymptotic behavior of the function f −1 (I(r)). Here, we present some results that indicate the possibility of solving these problems.
We need also some well-known (see, for example, [10]) properties of the function ln M f (r).
We remark that, if f k ≥ 0 for all k ≥ 0, then M f (r) = f (r). Therefore, from Theorem 1, we obtain the following statement.
Corollary 1. Let f be an entire transcendental function, f k ≥ 0 for all k ≥ 0, a n ≥ 0 for all n ≥ 1, and series (2) be regularly convergent in C. Suppose that f (r)/ f (r) ≥ h > 0 for all r ≥ r 0 , ln n = O(λ n ) as n → ∞ and lim σ→+∞
Theorem 2 implies the following statement.

Remark 1.
If ρ = 1, then E ρ (r) = E 1 (r) = e r , and we have a usual Laplace-Stieltjes integral I 1 (r) = Similarly, we can prove the following statement.

Discussion Open Problems
1. The natural problem studied was the relative growth when the domain of regular convergence of series (2) is the disk D R = {z : |z| < R < +∞} and the function f is either entire or analytic in D R .