On the growth of Laplace-Stieltjes integrals
Journal Title: Математичні Студії - Year 2018, Vol 50, Issue 1
Abstract
In the paper it is investigated the growth of characteristics of Laplace-Stieltjes integrals I(σ)=∫+∞0f(x)dF(x), where F is a nonnegative nondecreasing unbounded function continuous on the right on [0,+∞) and f is a nonnegative on [0,+∞) function such that there exist a≥0, b≥0 and h>0: ∫x+bx−af(t)dF(t)≥hf(x) for all x≥a. Assume that α,β are positive continuously differentiable functions increasing to +∞ on [0,+∞) such that: a) α(cx)=(1+o(1))α(x) (x→+∞) for any c>0; b) β(x(1+o(1)))=(1+o(1))β(x) (x→+∞); c) dβ−1(α(x)/ϱ)dlnx=O(1) (x→+∞) for every ϱ∈(0,+∞). The main results of the paper are contained in Theorems 5 and 7 and are derived from the following two statements of independent interest. If F satisfies condition lnF(x)=o(xβ−1(α(x)ϱ)) (x→+∞), then ϱαβ(I)=kαβ(f) (Theorem 1). If in additional the function v(x)=−(lnf(x))′ is continuous and increasing on [x0,+∞) and ϱαβ(I)≤+∞, then λαβ(I)=ϰαβ(f) (Theorem 2), where lim−−−¯¯¯¯¯¯¯σ→+∞α(lnI(σ))β(σ):={ϱαβ(I),λαβ(I),lim−−−¯¯¯¯¯¯¯x→+∞α(x)β(1xln1f(x)):={kαβ(f),ϰαβ(f). Similar results are proved also for so called the modified generalized order and lower order.
Authors and Affiliations
M. Sheremeta, A. Kuryliak
Keywords
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- EP ID EP436191
- DOI 10.15330/ms.50.1.22-35
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