A remark to the growth of positive functions and its application
to Dirichlet series

O. M. Mulyava; M. M. Sheremeta

doi:10.15330/ms.44.2.161-170

For a function $\Phi$ continuous on $(-\infty, +\infty)$ increaing to $+\infty$ the lower and upper estimates for $\frac{\alpha(\Phi(q\sigma))}{\alpha^p(\Phi(\sigma))}$ are found, where $p>1, q>1$ and $\alpha$ is positive function continuous on $[x_0,+\infty)$, increasing to $+\infty$. The above results applied to Dirichlet series with positive exponents.

1. Goodstein R.L. Complex functions. – New York, 1965.

2. Singh S.K. On the maximum modulus and the means of an entire function// Matem. Visnyk. – 1976. – V.13(28). – P. 211–213. (in Ukrainian)

3. Sheremeta M.N. Asymptotic behaviour of Mittag-Leffler type functions and its application 1// Teoriya funcii, func. anal., ich prilozh. – 1970. – V.10. – P. 97–111. (in Russian)

4. Sheremeta M.N. Asymptotic behaviour of Mittag-Leffler type functions and its application 2// Teoriya funcii, func. anal., ich prilozh. – 1971. – V.11. – P. 41–54. (in Russian)

5. Pjanylo Ja.D., Sheremeta M.N. On the growth of entire functions represented by Dirichlet series// Izv. vuzov. Matem. – 1975. – №10. – P. 91–93. (in Russian)

6. Hal’ Yu.M., Sheremeta M.N. On the growth of analytic in a half-plane functions represented by Dirichlet series// Dokl. AN USSR. Ser. A. – 1978. – №12. – P. 1065–1067. (in Russian)

7. Filevych P.V. To the Sheremeta theorem concerning relations between the maximal term and the maximum modulus of entire Dirichlet series// Mat. Stud. – 2000. – V.13, №2. – P. 139–144.

8. Sheremeta M.N. Full equivalence of the logarithms of the maximum modulus and the maximal term of an entire Dirichlet series// Matem. Zametki. – 1990. – V.47, №6. – P. 119–123. (in Russian)

	A remark to the growth of positive functions and its application to Dirichlet series
Author	O. M. Mulyava, M. M. Sheremeta m_m_sheremeta@list.ru Ivan Franko National University of Lviv
Abstract	For a function $\Phi$ continuous on $(-\infty, +\infty)$ increaing to $+\infty$ the lower and upper estimates for $\frac{\alpha(\Phi(q\sigma))}{\alpha^p(\Phi(\sigma))}$ are found, where $p>1, q>1$ and $\alpha$ is positive function continuous on $[x_0,+\infty)$, increasing to $+\infty$. The above results applied to Dirichlet series with positive exponents.
Keywords	positive functions; Dirichlet series; maximum modulus
DOI	doi:10.15330/ms.44.2.161-170
Reference	1. Goodstein R.L. Complex functions. – New York, 1965. 2. Singh S.K. On the maximum modulus and the means of an entire function// Matem. Visnyk. – 1976. – V.13(28). – P. 211–213. (in Ukrainian) 3. Sheremeta M.N. Asymptotic behaviour of Mittag-Leffler type functions and its application 1// Teoriya funcii, func. anal., ich prilozh. – 1970. – V.10. – P. 97–111. (in Russian) 4. Sheremeta M.N. Asymptotic behaviour of Mittag-Leffler type functions and its application 2// Teoriya funcii, func. anal., ich prilozh. – 1971. – V.11. – P. 41–54. (in Russian) 5. Pjanylo Ja.D., Sheremeta M.N. On the growth of entire functions represented by Dirichlet series// Izv. vuzov. Matem. – 1975. – №10. – P. 91–93. (in Russian) 6. Hal’ Yu.M., Sheremeta M.N. On the growth of analytic in a half-plane functions represented by Dirichlet series// Dokl. AN USSR. Ser. A. – 1978. – №12. – P. 1065–1067. (in Russian) 7. Filevych P.V. To the Sheremeta theorem concerning relations between the maximal term and the maximum modulus of entire Dirichlet series// Mat. Stud. – 2000. – V.13, №2. – P. 139–144. 8. Sheremeta M.N. Full equivalence of the logarithms of the maximum modulus and the maximal term of an entire Dirichlet series// Matem. Zametki. – 1990. – V.47, №6. – P. 119–123. (in Russian)
Pages	161-170
Volume	44
Issue	2
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

A remark to the growth of positive functions and its application to Dirichlet series