On certain subclass of Dirichlet series absolutely convergent in half-plane
Denote by $\mathfrak{D}_0$ a class of absolutely convergent in half-plane $\Pi_0=\{s\colon \text{Re}\,s<0\}$ Dirichlet series $F(s)=e^{sh}-\sum_{k=1}^{\infty}f_k\exp\{s(\lambda_k+h)\},\, s=\sigma+it$, where $h> 0$, $h<\lambda_k\uparrow+\infty$ and $f_k>0$. For $0\le\alpha<h$ and $l\ge...
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Ivan Franko National University of Lviv
2022-03-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/309 |
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| author | M. M. Sheremeta |
| author_facet | M. M. Sheremeta |
| author_sort | M. M. Sheremeta |
| collection | DOAJ |
| description | Denote by $\mathfrak{D}_0$ a class of absolutely convergent in half-plane $\Pi_0=\{s\colon \text{Re}\,s<0\}$ Dirichlet series
$F(s)=e^{sh}-\sum_{k=1}^{\infty}f_k\exp\{s(\lambda_k+h)\},\, s=\sigma+it$, where $h> 0$, $h<\lambda_k\uparrow+\infty$ and $f_k>0$.
For $0\le\alpha<h$ and $l\ge 0$ we say that $F$ belongs to the class $\mathfrak{DF}_h(l,\alpha)$ if and only if
$\text{Re}\{e^{-hs}((1-l)F(s)+\frac{l}{h}F'(s))\}>\frac{\alpha}{h}$,
and belongs to the class $\mathfrak{DG}_h(l,\alpha)$ if and only if
$\text{Re}\{e^{-hs}((1-l)F'(s)+\frac{l}{h}F''(s))\}>\alpha$ for all $s\in \Pi_0$. It is proved
that $F\in \mathfrak{DF}_h(l,\alpha)$ if and only if $ \sum_{k=1}^{\infty}(h+l\lambda_k)f_k\le h-\alpha$, and
$F\in \mathfrak{DG}_h(l,\alpha)$ if and only if $\sum_{k=1}^{\infty}(h+l\lambda_k)(\lambda_k+h)f_k\le h(h-\alpha)$.
If $F_j\in \mathfrak{DF}_h(l_j,\alpha_j)$, $j=1, 2$, where $l_j\ge0$ and $0\le \alpha_j<h$, then Hadamard composition
$(F_1*F_2)\in \mathfrak{D}F_h(l,\alpha)$, where $l=\min\{l_1,l_2\}$ and
$\alpha=h-\frac{(h-\alpha_1)(h-\alpha_2)}{h+l\lambda_1}$. Similar statement is correct for the class $F_j\in \mathfrak{DG}_h(l,\alpha)$.
For $j>0$ and $\delta>0$ the neighborhood of the function $F\in \mathfrak{D}_0$ is defined as follows
$O_{j,\delta}(F)=\{G(s)=e^{s}-\sum_{k=1}^{\infty}g_k\exp\{s\lambda_k\}\in \mathfrak{D}_0\colon
\sum_{k=1}^{\infty}\lambda^j_k|g_k-f_k|\le\delta\}$. It is described the neighborhoods of functions from classes $\mathfrak{DF}_h(l,\alpha)$ and $\mathfrak{DG}_h(l,\alpha)$.
Conditions on real parameters $\gamma_0,\,\gamma_1,\,\gamma_2,\,a_1$ and $a_2$ of the differential equation
$w''+(\gamma_0e^{2hs}+\gamma_1e^{hs}+\gamma_2) w=a_1e^{hs}+a_2e^{2hs}$ are found, under which this equation has a solution
either in $\mathfrak{DF}_h(l,\alpha)$ or in $\mathfrak{DG}_h(l,\alpha)$. |
| format | Article |
| id | doaj-art-e3f4e74e4d4b4568804f04f2b87d954d |
| institution | Matheson Library |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2022-03-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-e3f4e74e4d4b4568804f04f2b87d954d2025-07-08T09:08:39ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202022-03-01571324410.30970/ms.57.1.32-44309On certain subclass of Dirichlet series absolutely convergent in half-planeM. M. Sheremeta0Ivan Franko National University of Lviv, LvivDenote by $\mathfrak{D}_0$ a class of absolutely convergent in half-plane $\Pi_0=\{s\colon \text{Re}\,s<0\}$ Dirichlet series $F(s)=e^{sh}-\sum_{k=1}^{\infty}f_k\exp\{s(\lambda_k+h)\},\, s=\sigma+it$, where $h> 0$, $h<\lambda_k\uparrow+\infty$ and $f_k>0$. For $0\le\alpha<h$ and $l\ge 0$ we say that $F$ belongs to the class $\mathfrak{DF}_h(l,\alpha)$ if and only if $\text{Re}\{e^{-hs}((1-l)F(s)+\frac{l}{h}F'(s))\}>\frac{\alpha}{h}$, and belongs to the class $\mathfrak{DG}_h(l,\alpha)$ if and only if $\text{Re}\{e^{-hs}((1-l)F'(s)+\frac{l}{h}F''(s))\}>\alpha$ for all $s\in \Pi_0$. It is proved that $F\in \mathfrak{DF}_h(l,\alpha)$ if and only if $ \sum_{k=1}^{\infty}(h+l\lambda_k)f_k\le h-\alpha$, and $F\in \mathfrak{DG}_h(l,\alpha)$ if and only if $\sum_{k=1}^{\infty}(h+l\lambda_k)(\lambda_k+h)f_k\le h(h-\alpha)$. If $F_j\in \mathfrak{DF}_h(l_j,\alpha_j)$, $j=1, 2$, where $l_j\ge0$ and $0\le \alpha_j<h$, then Hadamard composition $(F_1*F_2)\in \mathfrak{D}F_h(l,\alpha)$, where $l=\min\{l_1,l_2\}$ and $\alpha=h-\frac{(h-\alpha_1)(h-\alpha_2)}{h+l\lambda_1}$. Similar statement is correct for the class $F_j\in \mathfrak{DG}_h(l,\alpha)$. For $j>0$ and $\delta>0$ the neighborhood of the function $F\in \mathfrak{D}_0$ is defined as follows $O_{j,\delta}(F)=\{G(s)=e^{s}-\sum_{k=1}^{\infty}g_k\exp\{s\lambda_k\}\in \mathfrak{D}_0\colon \sum_{k=1}^{\infty}\lambda^j_k|g_k-f_k|\le\delta\}$. It is described the neighborhoods of functions from classes $\mathfrak{DF}_h(l,\alpha)$ and $\mathfrak{DG}_h(l,\alpha)$. Conditions on real parameters $\gamma_0,\,\gamma_1,\,\gamma_2,\,a_1$ and $a_2$ of the differential equation $w''+(\gamma_0e^{2hs}+\gamma_1e^{hs}+\gamma_2) w=a_1e^{hs}+a_2e^{2hs}$ are found, under which this equation has a solution either in $\mathfrak{DF}_h(l,\alpha)$ or in $\mathfrak{DG}_h(l,\alpha)$.http://matstud.org.ua/ojs/index.php/matstud/article/view/309dirichlet series; hadamard composition; neighborhood of the function; differential equation. |
| spellingShingle | M. M. Sheremeta On certain subclass of Dirichlet series absolutely convergent in half-plane Математичні Студії dirichlet series; hadamard composition; neighborhood of the function; differential equation. |
| title | On certain subclass of Dirichlet series absolutely convergent in half-plane |
| title_full | On certain subclass of Dirichlet series absolutely convergent in half-plane |
| title_fullStr | On certain subclass of Dirichlet series absolutely convergent in half-plane |
| title_full_unstemmed | On certain subclass of Dirichlet series absolutely convergent in half-plane |
| title_short | On certain subclass of Dirichlet series absolutely convergent in half-plane |
| title_sort | on certain subclass of dirichlet series absolutely convergent in half plane |
| topic | dirichlet series; hadamard composition; neighborhood of the function; differential equation. |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/309 |
| work_keys_str_mv | AT mmsheremeta oncertainsubclassofdirichletseriesabsolutelyconvergentinhalfplane |