# On the growth of Laplace-Stieltjes integrals

Author
1) Ivan Franko National University of Lviv, Ukraine; 2) Ivan Franko National University of Lviv, Ukraine
Abstract
In the paper it is investigated the growth of characteristics of Laplace-Stieltjes integrals $I(\sigma)=\int_0^{+\infty}f(x)dF(x)$, where $F$ is a nonnegative nondecreasing unbounded function continuous on the right on $[0,+\infty)$ and $f$ is a nonnegative on $[0,+\infty)$ function such that there exist $a\ge0$, $b\ge0$ and $h>0$: $\int\nolimits_{x-a}^{x+b}f(t)dF(t)\ge hf(x)$ for all $x\ge a$. Assume that $\alpha, \beta$ are positive continuously differentiable functions increasing to $+\infty$ on $[0,+\infty)$ such that: a) $\alpha(cx)=(1+o(1))\alpha(x)$ $(x\to+\infty)$ for any $c>0$; b) $\beta(x(1+o(1)))=(1+o(1))\beta(x)$ $(x\to+\infty)$; c) $\frac{d\beta^{-1}(\alpha(x)/\varrho)}{d\ln x}=O(1)$ $(x\to+\infty)$ for every $\varrho\in(0,+\infty)$. 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 $\ln F(x)=o\Big(x\beta^{-1}\big(\frac{\alpha(x)}{\varrho}\big)\Big)$ $(x\to+\infty)$, then $\varrho_{\alpha\beta}(I)=k_{\alpha\beta}(f)$ (Theorem 1). If in additional the function $v(x)=-(\ln\,f(x))'$ is continuous and increasing on $[x_0,+\infty)$ and $\varrho_{\alpha\beta}(I)\le +\infty$, then $\lambda_{\alpha\beta}(I)=\varkappa_{\alpha\beta}(f)$ (Theorem 2), where $$\mathop{\overline{\underline{\lim}}}\limits_{\sigma\to +\infty}\frac{\alpha(\ln I(\sigma))}{\beta(\sigma)}:=\begin{cases}\varrho_{\alpha\beta}(I),\\ \lambda_{\alpha\beta}(I),\end{cases} \mathop{\overline{\underline{\lim}}}\limits_{x\to +\infty} \frac{\alpha(x)}{\beta\left(\frac{1}{x}\ln\frac{1}{f(x)}\right)}:=\begin{cases}k_{\alpha\beta}(f),\\ \varkappa_{\alpha\beta}(f).\end{cases}$$ Similar results are proved also for so called the modified generalized order and lower order.
Keywords
Laplace–Stieltjes integral; regular growth; asymptotic estimate
DOI
doi:10.15330/ms.50.1.22-35
Reference
1. E.G. Calys, A note on the order and type of integral functions, Riv. Mat. Univer Parma, 5 (1964), 133–137.

2. R.S.L. Srivastava, On the order and type of integral functions, Riv. Mat. Univer Parma, 1 (1959), 249– 255.

3. R.S.L. Srivastava, On the order and type of integral functions, Riv. Mat. Univer Parma, 2 (1961), 265– 270.

4. L.V. Kulavec’, O.M. Mulyava, On the growth of a class of entire Dirichlet series, Carpatian Math. Publ., 5 (2014) (2), 300–309. doi: 10.15330/cmp.6.2.300-309 (in Ukrainian).

5. O.B. Skaskiv, On certain relations between the maximum modulus and the maximal term of an entire Dirichlet series, Math. Notes 66 (1999), 223–232. doi.org/10.1007/BF02674881

6. O.B. Skaskiv, O.M. Trakalo, Asymptotic estimates for Laplace integrals, Mat. Stud., 18 (2002) (2), 125–146. (in Ukrainian)

7. M.M. Sheremeta, Asymptotical behaviour of Laplace–Stieltjes integrals, VNTL Publishers, Lviv, 2010.

8. A.J. Macintyre, Laplace’s transformation and integral functions, Proc. London Math. Soc., 45 (1939), 1–20. doi.org/10.1112/plms/s2-45.1.1

9. M.M. Sheremeta, On two classes of positive functions and belonging to them of main characteristics of entire functions, Mat. Stud., 19 (2003) (1), 73–82.

10. E. Seneta, Regularly varying functions, Lect. Notes Math., 508, Springer-Verlag, Berlin, 1976.

Pages
22-35
Volume
50
Issue
1
Year
2018
Journal
Matematychni Studii
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