On the belonging of an entire Dirichlet series to the logarithmic convergence class (in Ukrainian)

O.M.Mulyava; M.M.Sheremeta

For entire Dirichlet series, $F(s)=\sum\limits_{n=0}^{\infty}a_n\exp\{s\lambda_n\}$ let $M(\sigma)=\sup\{|F(\sigma+it)|:t\in \mathbb{R}\}$ and $\mu(\sigma)=\max\{|a_n|\exp{(\sigma\lambda_n)}:n\ \ge 0\}$. Conditions on $\lambda_n$ for the equivalence of the relations $\int\nolimits_{0}^{\infty}{{\sigma^{-p-1}}{\ln\,M(\sigma)}d\sigma}<+\infty$ and $\int\nolimits_{0}^{\infty}{\sigma^{-p-1}{\ln\,\mu(\sigma)}d\sigma}<+\infty$ $(p>1)$ are established.

1. Kamthan P.K. A theorem on step functions (III)// Istambul Univ. Fen. Fac. Mecm., A. - 1963. - V. 28. - P.65-69.

2. Мулява О.М. Про класи збіжності рядів Діріхле// Укр. матем. журн. - 1999. - Т. 51, №11. - С.1485-1494.

3. Filevych P.V., Fedynyak S.I. On belonding of entire Dirichlet series to convergence class// Mat. Stud. - 2001. - V. 16, №1. - P.57-60.

4. Sumyk O.M., Sheremeta M.M. On connection between the growth of maximum modulus and maximal term of entire Dirichlet series in term of m-member asymptotics// Mat. Stud. -2003. - V. 19, №1. - P.83-88.

5. Мулява О.М. Про абсцису збіжності ряду Діріхле// Мат. Cтудії. - 1998. - Т. 9, №2. - C.171-176.

	On the belonging of an entire Dirichlet series to the logarithmic convergence class (in Ukrainian)
Author	O.M.Mulyava, M.M.Sheremeta Kyiv National University of Food Technologies, Ivan Franko National University of Lviv
Abstract	For entire Dirichlet series, $F(s)=\sum\limits_{n=0}^{\infty}a_n\exp\{s\lambda_n\}$ let $M(\sigma)=\sup\{\|F(\sigma+it)\|:t\in \mathbb{R}\}$ and $\mu(\sigma)=\max\{\|a_n\|\exp{(\sigma\lambda_n)}:n\ \ge 0\}$. Conditions on $\lambda_n$ for the equivalence of the relations $\int\nolimits_{0}^{\infty}{{\sigma^{-p-1}}{\ln\,M(\sigma)}d\sigma}<+\infty$ and $\int\nolimits_{0}^{\infty}{\sigma^{-p-1}{\ln\,\mu(\sigma)}d\sigma}<+\infty$ $(p>1)$ are established.
Keywords	entire Dirichlet series, exponent, logarithm of maximum modulus, maximal term
DOI	doi:10.30970/ms.33.1.17-21
Reference	1. Kamthan P.K. A theorem on step functions (III)// Istambul Univ. Fen. Fac. Mecm., A. - 1963. - V. 28. - P.65-69. 2. Мулява О.М. Про класи збіжності рядів Діріхле// Укр. матем. журн. - 1999. - Т. 51, №11. - С.1485-1494. 3. Filevych P.V., Fedynyak S.I. On belonding of entire Dirichlet series to convergence class// Mat. Stud. - 2001. - V. 16, №1. - P.57-60. 4. Sumyk O.M., Sheremeta M.M. On connection between the growth of maximum modulus and maximal term of entire Dirichlet series in term of m-member asymptotics// Mat. Stud. -2003. - V. 19, №1. - P.83-88. 5. Мулява О.М. Про абсцису збіжності ряду Діріхле// Мат. Cтудії. - 1998. - Т. 9, №2. - C.171-176.
Pages	17-21
Volume	33
Issue	1
Year	2010
Journal	Matematychni Studii
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