Relative growth of Dirichlet series with different abscissas of absolute convergence
UDC 517.537.72 We study the growth of a Dirichlet series $F(s)=\sum _{n=1}^{\infty}f_n\exp\{s\lambda_n\}$ with zero abscissa of absolute convergence with respect to the entire Dirichlet series $G(s)=\sum _{n=1}^{\infty}g_n\exp\{s\lambda_n\}$ by using the generalized quantities of order $\varrho^0_{\...
Gespeichert in:
| Datum: | 2020 |
|---|---|
| Hauptverfasser: | , , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6168 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | UDC 517.537.72
We study the growth of a Dirichlet series $F(s)=\sum _{n=1}^{\infty}f_n\exp\{s\lambda_n\}$ with zero abscissa of absolute convergence with respect to the entire Dirichlet series $G(s)=\sum _{n=1}^{\infty}g_n\exp\{s\lambda_n\}$ by using the generalized quantities of order $\varrho^0_{\beta,\beta}[F]_G=\varlimsup\nolimits_{\sigma\uparrow 0}\dfrac{\beta(M^{-1}_G(M_F(\sigma)))}{\beta(1/|\sigma|)}$ and lower order $\lambda^0_{\beta,\beta}[F]_G=\varliminf_{\sigma\uparrow 0} \dfrac{\beta(M^{-1}_G(M_F(\sigma)))}{\beta(1/|\sigma|)},$ where $M_F(\sigma)=\sup\{|F(\sigma+it)|\colon t\in{\Bbb R}\},$ $M^{-1}_G(x)$ is the function inverse to $M_G(\sigma),$ and $\beta$ is a positive increasing function growing to $+\infty.$
  |
|---|---|
| DOI: | 10.37863/umzh.v72i11.6168 |