On belonging of entire Dirichlet series to a modified generalized convergence class

Author
O. M. Mulyava
Kyiv National University of Food Technologies, Kyiv, Ukraine
Abstract
For entire Dirichlet series $F(s)=\sum\nolimits_{n=0}^{+\infty}a_n e^{s\lambda_n}$ we found conditions on $a_n$, $\lambda_n$ and on positive functions $\alpha$ and $\beta$ continuous increasing to $+\infty$ on $[0,\,+\infty)$ are found, under which the condition $\int\nolimits_{\sigma_0}^{+\infty}\frac{1}{\beta(\sigma)} \alpha\left(\frac1{\sigma}{\ln M(\sigma, F)}\right)d\sigma\le+\infty$ is equivalent to the condition $$\sum\limits_{n=1}^{+\infty}(\alpha(\lambda_{n})-\alpha(\lambda_{n-1})) \beta_1\left(\frac{1}{\lambda_n}\ln\frac{1}{|a_n|}\right) \le+\infty$$, where $\beta_1(x)=\int\nolimits_{x}^{+\infty}\frac{dt}{\beta(t)}$, and $M(\sigma, F)=\sup\{|F(\sigma+it)|\colon t\in {\mathbb R}\}$.
Keywords
entire Dirichlet series; convergence class
DOI
doi:10.15330/ms.50.2.135-142
Reference
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Pages
135-142
Volume
50
Issue
2
Year
2018
Journal
Matematychni Studii
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