On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients
UDC 517.537 In his study of the geometric properties of functions analytic in a disk ${\mathbb D} = \{z\colon |z|<1\},$ G. S. Sălăgean introduced a class $S_j(\alpha)$ of functions $f(z) = z + \sum _{k = 2}^{\infty}f_kz^k$ such that $\operatorname{Re} \frac{D^{j + 1}f(z)}{D^{j}f(z)} &...
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| Datum: | 2025 |
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| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
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Institute of Mathematics, NAS of Ukraine
2025
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/8555 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513160502444032 |
|---|---|
| author | Sheremeta, M. Mulyava, O. Medvedev, M. Шеремета, Мирослав Мулява, Оксана Медвєдєв, Микола |
| author_facet | Sheremeta, M. Mulyava, O. Medvedev, M. Шеремета, Мирослав Мулява, Оксана Медвєдєв, Микола |
| author_sort | Sheremeta, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-04-16T12:01:20Z |
| description | UDC 517.537
In his study of the geometric properties of functions analytic in a disk ${\mathbb D} = \{z\colon |z|<1\},$ G. S. Sălăgean introduced a class $S_j(\alpha)$ of functions $f(z) = z + \sum _{k = 2}^{\infty}f_kz^k$ such that $\operatorname{Re} \frac{D^{j + 1}f(z)}{D^{j}f(z)} > \alpha\in [0, 1)$ for each $ z\in{\mathbb D},$ where $D^jf$ is the Sălăgean derivative. For Dirichlet series $F(s) = e^{s}-\sum _{k = 1 }^{\infty}f_k\exp\{s\lambda_k\}$ with $f_k\ge0$ absolutely convergent in the half plane $\Pi_0 = \{s\colon \operatorname{Re} s<0\},$ an analog of the Sălăgean class is the class$D_{j}(\alpha)$ defined by the condition $\operatorname{Re} \frac{F^{(j + 1)}(s)}{F^{(j)}(s)} > \alpha$ for each $s\in \Pi_0.$ By analogy with the neighborhood of an analytic function in ${\mathbb D}$ defined by A. V. Goodman, for $F\in D_{j}(\alpha),$ we introduce the concept of a neighborhood $O_{j,\delta}(F)$ and establish the conditions under which all functions from $O_{j,\delta}(F)$ belong to $D_{j}(\alpha_1),$ $0\le \alpha_1<\alpha<1,$ and vice versa. The problem of belonging of solutions of the differential equation  $\frac{d^2w}{ds^2} + (\gamma_0e^{2s} + \gamma_1 e^s + \gamma_2)w = 0$ with real parameters to the class $D_{j}(\alpha)$ is investigated. |
| doi_str_mv | 10.3842/umzh.v76i9.8555 |
| first_indexed | 2026-03-24T03:40:16Z |
| format | Article |
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| id | umjimathkievua-article-8555 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:40:16Z |
| publishDate | 2025 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1f/7bf376da959aa70c2df1ffaf3b89241f |
| spelling | umjimathkievua-article-85552025-04-16T12:01:20Z On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients Про аналог класу Салагеана для рядів Діріхле і розв'язки одного лінійного диференціального рівняння з екпоненціальними коефіцієнтами Sheremeta, M. Mulyava, O. Medvedev, M. Шеремета, Мирослав Мулява, Оксана Медвєдєв, Микола - UDC 517.537 In his study of the geometric properties of functions analytic in a disk ${\mathbb D} = \{z\colon |z|<1\},$ G. S. Sălăgean introduced a class $S_j(\alpha)$ of functions $f(z) = z + \sum _{k = 2}^{\infty}f_kz^k$ such that $\operatorname{Re} \frac{D^{j + 1}f(z)}{D^{j}f(z)} > \alpha\in [0, 1)$ for each $ z\in{\mathbb D},$ where $D^jf$ is the Sălăgean derivative. For Dirichlet series $F(s) = e^{s}-\sum _{k = 1 }^{\infty}f_k\exp\{s\lambda_k\}$ with $f_k\ge0$ absolutely convergent in the half plane $\Pi_0 = \{s\colon \operatorname{Re} s<0\},$ an analog of the Sălăgean class is the class$D_{j}(\alpha)$ defined by the condition $\operatorname{Re} \frac{F^{(j + 1)}(s)}{F^{(j)}(s)} > \alpha$ for each $s\in \Pi_0.$ By analogy with the neighborhood of an analytic function in ${\mathbb D}$ defined by A. V. Goodman, for $F\in D_{j}(\alpha),$ we introduce the concept of a neighborhood $O_{j,\delta}(F)$ and establish the conditions under which all functions from $O_{j,\delta}(F)$ belong to $D_{j}(\alpha_1),$ $0\le \alpha_1<\alpha<1,$ and vice versa. The problem of belonging of solutions of the differential equation  $\frac{d^2w}{ds^2} + (\gamma_0e^{2s} + \gamma_1 e^s + \gamma_2)w = 0$ with real parameters to the class $D_{j}(\alpha)$ is investigated. УДК 517.537 Вивчаючи геометричні властивості функцій, аналітичних у крузі ${\mathbb D} = \{z\colon |z|<1\},$ Г. С. Салагеан увів клас $S_j(\alpha)$ функцій $f(z) = z + \sum _{k = 2}^{\infty}f_kz^k,$ для яких $\operatorname{Re} \frac{D^{j + 1}f(z)}{D^{j}f(z)} > \alpha\in [0, 1)$ для кожного $z\in{\mathbb D},$ де $D^jf$ – похідна Салагеана. Для абсолютно збіжних у півплощині $\Pi_0 = \{s\colon \operatorname{Re} s<0\}$ рядів Діріхле $F(s) = e^{s}-\sum _{k = 1}^{\infty}f_k\exp\{s\lambda_k\}$ з $f_k\ge0$ аналогом класу Салагеaна є клас $D_{j}(\alpha),$ означений умовою $\operatorname{Re} \frac{F^{(j + 1)}(s)}{F^{(j)}(s)} > \alpha$ для кожного $s\in \Pi_0.$  Подібно до означеного А. В. Гудманом околу аналітичної в ${\mathbb D}$ функції для $F\in D_{j}(\alpha)$ введено поняття околу $O_{j,\delta}(F)$ і знайдено умови, за яких усі функції з $O_{j,\delta}(F)$ належать до $D_{j}(\alpha_1),$ $0\le \alpha_1<\alpha<1,$ і навпаки. Досліджено належність розв'язків диференціального рівняння $\frac{d^2w}{ds^2} + (\gamma_0e^{2s} + \gamma_1 e^s + \gamma_2)w = 0$ з дійсними параметрами до класу $D_{j}(\alpha).$  Institute of Mathematics, NAS of Ukraine 2025-04-16 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/8555 10.3842/umzh.v76i9.8555 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 9 (2024); 1412 - 1418 Український математичний журнал; Том 76 № 9 (2024); 1412 - 1418 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/8555/10170 Copyright (c) 2024 Мирослав Шеремета |
| spellingShingle | Sheremeta, M. Mulyava, O. Medvedev, M. Шеремета, Мирослав Мулява, Оксана Медвєдєв, Микола On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients |
| title | On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients |
| title_alt | Про аналог класу Салагеана для рядів Діріхле і розв'язки одного лінійного диференціального рівняння з екпоненціальними коефіцієнтами |
| title_full | On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients |
| title_fullStr | On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients |
| title_full_unstemmed | On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients |
| title_short | On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients |
| title_sort | on the analog of the sălăgean class for dirichlet series and the solutions of one linear differential equation with exponential coefficients |
| topic_facet | - |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8555 |
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