On the growth of Laplace-Stieltjes integrals

M. M. Sheremeta1, A. O. Kuryliak2
1) Ivan Franko National University of Lviv, Ukraine; 2) Ivan Franko National University of Lviv, Ukraine
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.
LaplaceStieltjes integral; regular growth; asymptotic estimate
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Matematychni Studii
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