On Borel's type relation for the Laplace-Stieltjes integrals

A. O. Kuryliak; O. B. Skaskiv; D. Yu. Zikrach

Let $F\colon\mathbb{R}_{+}\to \mathbb{R}_{+}$ be a function of the form $F(x)=\int_{\mathbb{R}_{+}}f(u)e^{xu}\nu(du)$, where $\nu$ is a Borel measure with unbounded support, $f$ some measurable positive function, $\mu_{*}(x,F)=\max\{f(u)e^{xu}\colon u\in\mathop{\rm supp}\nu\}$. We obtain necessary and sufficient conditions for the relation $\ln F(x)\leq (1+o(1))\ln\mu_{*}(x,F)$ to be held as $x\to +\infty$, for each function $F$ outside some set $E$ of zero lower linear density.

1. Posiko O.S., Skaskiv O.B., Sheremeta M.M. Estimates of the Laplace-Stieltjes integral// Mat. Stud. - 2004. - V.21, №2. - P. 179-186. (in Ukrainian)

2. Posiko O.S., Sheremeta M.M. Asymptotic estimates for Laplace-Stieltjes integrals// Ukr. Math. Visn. - 2005. - V.2, №4. - P. 541-549 (in Ukrainian); English transl. in Ukr. Math. Bull., V.2, №4. - P. 547-555.

3. Skaskiv O.B. On certain relations between the maximum modulus and the maximal term of an entire Dirichlet series// Mat. zametki. - 1999. - V.66, №2. - P. 282-292 (in Russian); English transl. in Math. Notes. - 1999. - V.66, №2. - P. 223-232.

4. Skaskiv O.B., Trakalo O.M. On the stability of the maximum term of the entire Dirichlet series// Ukr. Mat. Zh. - 2005. - V.57, №4. - P. 571-576 (in Ukrainian); English transl. in Ukr. Math. J. - 2005. - V.57, №4. - P. 686-693.

5. Skaskiv O.B. Behavior of the maximum term of a Dirichlet series that defines an entire function// Mat. Zametki. - 1985. - V.37, №1. - P. 41-47 (in Russian); English transl. in Math. Notes. - 1985. - V.37, №1 - P. 24-28.

6. Skaskiv O.B., Trakalo O.M. Asymptotic estimates for Laplace integrals// Mat. Stud. - 2002. - V.18, №2. - P. 125-146. (in Ukrainian)

7. Skaskiv O.B., Asymptotic property of analytic functions represented by power series end Dirichlet series: Doctoral thesis, Lviv, 1995. - 299 p.

8. Zikrach D.Yu., Skaskiv O.B. Asymptotic external estimation of the exeptional sets of Laplace-Stieltjes integrals// Nauk. Visn. Chernivets'kogo Univ. Math. - 2011. - V.1, №3. - P. 38-43.

9. Skaskiv O.B. On the central exponent of absolutely convergent Dirichlet series// Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. - 2000. - №10. - P. 27-30.

10. Skaskiv O.B. A generalized Borel relation for entire Dirichlet series// Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. - 2004. - №6. - P. 32-36.

11. Zikrach D.Yu., Skaskiv O.B. On the description of an exceptional set in Borel's relation for multiple Dirichlet series with upper restriction on the growth// Mat. Stud. - 2009. - V.32, №2. - P. 139-147. (in Ukrainian)

12. Salo T.M., Skaskiv O.B., Trakalo O.M. On the best possible description of exceptional set in Wiman' Valiron theory for entire function// Mat. Stud. - 2001. - V.16, №2. - P. 131-140.

13. Filevych P.V. Asymptotic relations between the means of Dirichlet series and their applications// Mat. Stud. - 2002. - V.19, №2. - P. 127-140.

14. Skaskiv O.B., Zikrach D.Yu. The best possible description of exceptional set in Borel's relation for multiple Dirichlet series// Mat. Stud. - 2008. - V.30, №2. - P. 189-194.

15. Skaskiv O.B., Zikrach D.Yu. On the best possible description of an exceptional set in asymptotic estimates for Laplace-Stieltjes integrals // Mat. Stud. - 2011. - V.35, №2. - P. 131-141.

	On Borel's type relation for the Laplace-Stieltjes integrals
Author	A. O. Kuryliak, O. B. Skaskiv, D. Yu. Zikrach andriykuryliak@gmail.com, matstud@franko.lviv.ua; zikrach.dm@gmail.com Ivan Franko National University of Lviv; Ukrainian Academy of Printing
Abstract	Let $F\colon\mathbb{R}_{+}\to \mathbb{R}_{+}$ be a function of the form $F(x)=\int_{\mathbb{R}_{+}}f(u)e^{xu}\nu(du)$, where $\nu$ is a Borel measure with unbounded support, $f$ some measurable positive function, $\mu_{}(x,F)=\max\{f(u)e^{xu}\colon u\in\mathop{\rm supp}\nu\}$. We obtain necessary and sufficient conditions for the relation $\ln F(x)\leq (1+o(1))\ln\mu_{}(x,F)$ to be held as $x\to +\infty$, for each function $F$ outside some set $E$ of zero lower linear density.
Keywords	Laplace-Stieltjes integral; exceptional set; asymptotic estimate
DOI	doi:10.30970/ms.42.2.134-142
Reference	1. Posiko O.S., Skaskiv O.B., Sheremeta M.M. Estimates of the Laplace-Stieltjes integral// Mat. Stud. - 2004. - V.21, №2. - P. 179-186. (in Ukrainian) 2. Posiko O.S., Sheremeta M.M. Asymptotic estimates for Laplace-Stieltjes integrals// Ukr. Math. Visn. - 2005. - V.2, №4. - P. 541-549 (in Ukrainian); English transl. in Ukr. Math. Bull., V.2, №4. - P. 547-555. 3. Skaskiv O.B. On certain relations between the maximum modulus and the maximal term of an entire Dirichlet series// Mat. zametki. - 1999. - V.66, №2. - P. 282-292 (in Russian); English transl. in Math. Notes. - 1999. - V.66, №2. - P. 223-232. 4. Skaskiv O.B., Trakalo O.M. On the stability of the maximum term of the entire Dirichlet series// Ukr. Mat. Zh. - 2005. - V.57, №4. - P. 571-576 (in Ukrainian); English transl. in Ukr. Math. J. - 2005. - V.57, №4. - P. 686-693. 5. Skaskiv O.B. Behavior of the maximum term of a Dirichlet series that defines an entire function// Mat. Zametki. - 1985. - V.37, №1. - P. 41-47 (in Russian); English transl. in Math. Notes. - 1985. - V.37, №1 - P. 24-28. 6. Skaskiv O.B., Trakalo O.M. Asymptotic estimates for Laplace integrals// Mat. Stud. - 2002. - V.18, №2. - P. 125-146. (in Ukrainian) 7. Skaskiv O.B., Asymptotic property of analytic functions represented by power series end Dirichlet series: Doctoral thesis, Lviv, 1995. - 299 p. 8. Zikrach D.Yu., Skaskiv O.B. Asymptotic external estimation of the exeptional sets of Laplace-Stieltjes integrals// Nauk. Visn. Chernivets'kogo Univ. Math. - 2011. - V.1, №3. - P. 38-43. 9. Skaskiv O.B. On the central exponent of absolutely convergent Dirichlet series// Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. - 2000. - №10. - P. 27-30. 10. Skaskiv O.B. A generalized Borel relation for entire Dirichlet series// Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. - 2004. - №6. - P. 32-36. 11. Zikrach D.Yu., Skaskiv O.B. On the description of an exceptional set in Borel's relation for multiple Dirichlet series with upper restriction on the growth// Mat. Stud. - 2009. - V.32, №2. - P. 139-147. (in Ukrainian) 12. Salo T.M., Skaskiv O.B., Trakalo O.M. On the best possible description of exceptional set in Wiman' Valiron theory for entire function// Mat. Stud. - 2001. - V.16, №2. - P. 131-140. 13. Filevych P.V. Asymptotic relations between the means of Dirichlet series and their applications// Mat. Stud. - 2002. - V.19, №2. - P. 127-140. 14. Skaskiv O.B., Zikrach D.Yu. The best possible description of exceptional set in Borel's relation for multiple Dirichlet series// Mat. Stud. - 2008. - V.30, №2. - P. 189-194. 15. Skaskiv O.B., Zikrach D.Yu. On the best possible description of an exceptional set in asymptotic estimates for Laplace-Stieltjes integrals // Mat. Stud. - 2011. - V.35, №2. - P. 131-141.
Pages	134-142
Volume	42
Issue	2
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html