On the relative growth of Dirichlet series with zero abscissa of absolute convergence
Let $F$ and $G$ be analytic functions given by Dirichlet series with exponents increasing to $+\infty$ and zero abscissa of absolute convergence. The growth of $F$ with respect to $G$ is studied through the generalized order $$\varrho^0_{\alpha,\beta}[F]_G=\varlimsup\limits_{\sigma\uparrow 0}\dfrac{...
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| Format: | Article |
| Language: | deu |
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Ivan Franko National University of Lviv
2021-03-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/193 |
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| author | O. M. Mulyava |
| author_facet | O. M. Mulyava |
| author_sort | O. M. Mulyava |
| collection | DOAJ |
| description | Let $F$ and $G$ be analytic functions given by Dirichlet series with exponents increasing to $+\infty$ and zero abscissa of absolute convergence.
The growth of $F$ with respect to $G$ is studied through the generalized order
$$\varrho^0_{\alpha,\beta}[F]_G=\varlimsup\limits_{\sigma\uparrow 0}\dfrac{\alpha(1/|M^{-1}_G(M_F(\sigma)|)}{\beta(1/|\sigma|)}$$
and the generalized lower order $$\lambda^0_{\alpha,\beta}[F]_G=\varliminf\limits_{\sigma\uparrow 0} \dfrac{\alpha(1/|M^{-1}_G(M_F(\sigma)|)}{\beta(1/|\sigma|)},$$ where $M_F(\sigma)=\sup\{|F(\sigma+it)|:\,t\in{\mathbb R}\},$ $M^{-1}_G(x)$ is the function inverse to $M_G(\sigma)$ and $\alpha$ and $\beta$ are positive increasing to $+\infty$ functions.
Formulas are found for the finding these quantities. |
| format | Article |
| id | doaj-art-2ac0507626f04bcaa4fcc29baef9cfa6 |
| institution | DOAJ |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2021-03-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-2ac0507626f04bcaa4fcc29baef9cfa62025-08-20T02:41:33ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202021-03-01551445010.30970/ms.55.1.44-50193On the relative growth of Dirichlet series with zero abscissa of absolute convergenceO. M. Mulyava0Kyiv National University of Food Technologies, Kyiv, UkraineLet $F$ and $G$ be analytic functions given by Dirichlet series with exponents increasing to $+\infty$ and zero abscissa of absolute convergence. The growth of $F$ with respect to $G$ is studied through the generalized order $$\varrho^0_{\alpha,\beta}[F]_G=\varlimsup\limits_{\sigma\uparrow 0}\dfrac{\alpha(1/|M^{-1}_G(M_F(\sigma)|)}{\beta(1/|\sigma|)}$$ and the generalized lower order $$\lambda^0_{\alpha,\beta}[F]_G=\varliminf\limits_{\sigma\uparrow 0} \dfrac{\alpha(1/|M^{-1}_G(M_F(\sigma)|)}{\beta(1/|\sigma|)},$$ where $M_F(\sigma)=\sup\{|F(\sigma+it)|:\,t\in{\mathbb R}\},$ $M^{-1}_G(x)$ is the function inverse to $M_G(\sigma)$ and $\alpha$ and $\beta$ are positive increasing to $+\infty$ functions. Formulas are found for the finding these quantities.http://matstud.org.ua/ojs/index.php/matstud/article/view/193dirichlet series, relative growth, generalized order. |
| spellingShingle | O. M. Mulyava On the relative growth of Dirichlet series with zero abscissa of absolute convergence Математичні Студії dirichlet series, relative growth, generalized order. |
| title | On the relative growth of Dirichlet series with zero abscissa of absolute convergence |
| title_full | On the relative growth of Dirichlet series with zero abscissa of absolute convergence |
| title_fullStr | On the relative growth of Dirichlet series with zero abscissa of absolute convergence |
| title_full_unstemmed | On the relative growth of Dirichlet series with zero abscissa of absolute convergence |
| title_short | On the relative growth of Dirichlet series with zero abscissa of absolute convergence |
| title_sort | on the relative growth of dirichlet series with zero abscissa of absolute convergence |
| topic | dirichlet series, relative growth, generalized order. |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/193 |
| work_keys_str_mv | AT ommulyava ontherelativegrowthofdirichletserieswithzeroabscissaofabsoluteconvergence |