Two-member asymptotic of Laplace-Stieltjes integrals

M. M. Sheremeta; M. S. Dobushovskyy

doi:10.15330/ms.47.1.3-9

We study conditions, under which for the Laplace-Stieltjes integral $I(\sigma)=\int_{0}^{\infty}\!f(x)e^{x\sigma}dF(x)$ with the abscissa of the convergence $\sigma_c=A\in (-\infty,+\infty]$ the asymptotical equality $\ln\,I(\sigma)=\Phi_1(\sigma)+\tau(1+o(1))\Phi_2(\sigma)$ as $\sigma\uparrow A$ is true, where $\tau\in{\Bbb R}\setminus\{0\}$ and $\Phi_1,\,\Phi_2$ are some positive functions on $(-\infty,A)$.

1. Sheremeta M.M. Asymptotical behaviour of Laplace-Stiltjes integrals, Lviv: VNTL Publishers, 2010. - 211 p.

2. Sheremeta M.M., Fedynyak S.I. On the derivative of a Dirichlet series// Sibirsk. Mat. Zh. - 1998. - V.39, №1. - P. 206-223, (in Russian); Engl. Transl.: Siberian Math. J. - 1998. - V.39, №1. - P. 181-197.

3. Sheremeta M.M., Sumyk O.M. A connection detween the growth of Young conjugated function// Mat. Stud. - 1999. - V.11, №11. - P. 41-47. (in Ukrainian)

4. Sheremeta M.M. Two-member asymptotics of Young conjugated functions and problems of behaviour of positive sequences// Mat. Stud. - 2000. - V.14, №2. - P. 217-220.

5. Zabolotskyi M.V., Sheremeta M.M. Generalization of the Lindelof theorem// Ukr. Mat. J. - 1998. - V.50, №9. - P. 1177-1192, (in Ukrainian); Engl. Transl.: Ukr. Math. Zh. - 1998. - V.50, №9. - P. 1346-1364.

6. Dobushovskyy M.S., Sheremeta M.M. Estimates of Laplace-Stieltjes integrals// Ukr. Mat. J. - 2016. - V.68, №11. - P. 1467-1482, (in Ukrainian); Engl. Transl.: Ukr. Math. Zh. - 2017. - V.68, №11. - P. 1694-1714.

7. Sheremeta M.M., Dobushovskyy M.S. On the two-member asymptotics of Young conjugated functions// Mat. Stud. - 2016. - V.46, №2. - P. 178-188.

8. Sheremeta M.M. On two classes of positive functions and the belonging to them main characteristics of entire functions// Mat. Stud. - 2003. - V.19, №1. - P. 73-82.

	Two-member asymptotic of Laplace-Stieltjes integrals
Author	M. M. Sheremeta, M. S. Dobushovskyy m_m_sheremeta@list.ru Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	We study conditions, under which for the Laplace-Stieltjes integral $I(\sigma)=\int_{0}^{\infty}\!f(x)e^{x\sigma}dF(x)$ with the abscissa of the convergence $\sigma_c=A\in (-\infty,+\infty]$ the asymptotical equality $\ln\,I(\sigma)=\Phi_1(\sigma)+\tau(1+o(1))\Phi_2(\sigma)$ as $\sigma\uparrow A$ is true, where $\tau\in{\Bbb R}\setminus\{0\}$ and $\Phi_1,\,\Phi_2$ are some positive functions on $(-\infty,A)$.
Keywords	Laplace-Stieltjes integral; two-member asymptotic
DOI	doi:10.15330/ms.47.1.3-9
Reference	1. Sheremeta M.M. Asymptotical behaviour of Laplace-Stiltjes integrals, Lviv: VNTL Publishers, 2010. - 211 p. 2. Sheremeta M.M., Fedynyak S.I. On the derivative of a Dirichlet series// Sibirsk. Mat. Zh. - 1998. - V.39, №1. - P. 206-223, (in Russian); Engl. Transl.: Siberian Math. J. - 1998. - V.39, №1. - P. 181-197. 3. Sheremeta M.M., Sumyk O.M. A connection detween the growth of Young conjugated function// Mat. Stud. - 1999. - V.11, №11. - P. 41-47. (in Ukrainian) 4. Sheremeta M.M. Two-member asymptotics of Young conjugated functions and problems of behaviour of positive sequences// Mat. Stud. - 2000. - V.14, №2. - P. 217-220. 5. Zabolotskyi M.V., Sheremeta M.M. Generalization of the Lindelof theorem// Ukr. Mat. J. - 1998. - V.50, №9. - P. 1177-1192, (in Ukrainian); Engl. Transl.: Ukr. Math. Zh. - 1998. - V.50, №9. - P. 1346-1364. 6. Dobushovskyy M.S., Sheremeta M.M. Estimates of Laplace-Stieltjes integrals// Ukr. Mat. J. - 2016. - V.68, №11. - P. 1467-1482, (in Ukrainian); Engl. Transl.: Ukr. Math. Zh. - 2017. - V.68, №11. - P. 1694-1714. 7. Sheremeta M.M., Dobushovskyy M.S. On the two-member asymptotics of Young conjugated functions// Mat. Stud. - 2016. - V.46, №2. - P. 178-188. 8. Sheremeta M.M. On two classes of positive functions and the belonging to them main characteristics of entire functions// Mat. Stud. - 2003. - V.19, №1. - P. 73-82.
Pages	3-9
Volume	47
Issue	1
Year	2017
Journal	Matematychni Studii
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Table of content of issue	html