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
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
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