On the finiteness of the $l$-$\mathfrak{M}$-index of entire functions represented by series in а system of functions
UDC517.5 Let $f$ be an entire transcendental function and let $(\lambda_n)$ be a sequence of positive numbers increasing to $+\infty.$ Suppose that the series $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ is regularly convergent in ${\mathbb C},$ i.e., $\mathfrak{M}(r,A):=\sum_{n=1}^{\inf...
Збережено в:
| Дата: | 2024 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2024
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7866 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512794292518912 |
|---|---|
| author | Sheremeta, M. Шеремета, Мирослав |
| author_facet | Sheremeta, M. Шеремета, Мирослав |
| author_sort | Sheremeta, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-19T00:35:29Z |
| description | UDC517.5
Let $f$ be an entire transcendental function and let $(\lambda_n)$ be a sequence of positive numbers increasing to $+\infty.$ Suppose that the series $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ is regularly convergent in ${\mathbb C},$ i.e., $\mathfrak{M}(r,A):=\sum_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ for all $r\in [0,+\infty).$  For a positive function $l$ continuous on $[0,\,+\infty),$ the function $A$ is called a function of bounded $l$-$\mathfrak{M}$-index if there exists $N\in{\Bbb Z}_+$ such that $\dfrac{\mathfrak{M}(r,A^{(n)})}{n!l^n(r)}\le\max\left\{\dfrac{\mathfrak{M}(r,A^{(k)})}{k!l^k(r)}\colon 0\le k\le N\right\}$ for all $n\in{\Bbb Z}_+$ and all $r\in [0,+\infty).$  We study the properties of growth of the functions of bounded $l$-$\mathfrak{M}$-index  and formulate some unsolved problems. |
| doi_str_mv | 10.3842/umzh.v74i4.7866 |
| first_indexed | 2026-03-24T03:34:27Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-7866 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:34:27Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1a/a103c467650cc65463b8835bd3cdc81a |
| spelling | umjimathkievua-article-78662024-06-19T00:35:29Z On the finiteness of the $l$-$\mathfrak{M}$-index of entire functions represented by series in а system of functions Про обмеженість $l$-$\mathfrak{M}$-індексу цілих функцій, зображених рядами за системою функцій Sheremeta, M. Шеремета, Мирослав Ціла фукнція Ряд за системою функцій Обмежений індекс UDC517.5 Let $f$ be an entire transcendental function and let $(\lambda_n)$ be a sequence of positive numbers increasing to $+\infty.$ Suppose that the series $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ is regularly convergent in ${\mathbb C},$ i.e., $\mathfrak{M}(r,A):=\sum_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ for all $r\in [0,+\infty).$  For a positive function $l$ continuous on $[0,\,+\infty),$ the function $A$ is called a function of bounded $l$-$\mathfrak{M}$-index if there exists $N\in{\Bbb Z}_+$ such that $\dfrac{\mathfrak{M}(r,A^{(n)})}{n!l^n(r)}\le\max\left\{\dfrac{\mathfrak{M}(r,A^{(k)})}{k!l^k(r)}\colon 0\le k\le N\right\}$ for all $n\in{\Bbb Z}_+$ and all $r\in [0,+\infty).$  We study the properties of growth of the functions of bounded $l$-$\mathfrak{M}$-index  and formulate some unsolved problems. УДК 517.5 Нехай $f$ –ціла трансцендентна функція і $(\lambda_n)$ –зростаюча до $+\infty$ послідовність додатних чисел. Припустимо, що ряд $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ регулярно збіжний в ${\mathbb C},$ тобто $\mathfrak{M}(r,A):=\sum_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ для всіх $r\in [0,+\infty).$ Для додатної неперервної на $[0,\,+\infty)$ функції $l$ функція $A$ називається функцією обмеженого $l$-$\mathfrak{M}$-індексу, якщо існує таке $N\in{\Bbb Z}_+,$ що $\dfrac{\mathfrak{M}(r,A^{(n)})}{n!l^n(r)}\le\max\left\{\dfrac{\mathfrak{M}(r,A^{(k)})}{k!l^k(r)}\colon 0\le k\le N\right\}$ для всіх $n\in{\Bbb Z}_+$ і $r\in [0,+\infty).$ Вивчено зростання функцій обмеженого $l$-$\mathfrak{M}$-індексу і сформульовано нерозв'язані задачі.  Institute of Mathematics, NAS of Ukraine 2024-04-26 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7866 10.3842/umzh.v74i4.7866 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 4 (2024); 599 - 606 Український математичний журнал; Том 76 № 4 (2024); 599 - 606 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/7866/9919 Copyright (c) 2024 Мирослав Шеремета |
| spellingShingle | Sheremeta, M. Шеремета, Мирослав On the finiteness of the $l$-$\mathfrak{M}$-index of entire functions represented by series in а system of functions |
| title | On the finiteness of the $l$-$\mathfrak{M}$-index of entire functions represented by series in а system of functions |
| title_alt | Про обмеженість $l$-$\mathfrak{M}$-індексу цілих функцій, зображених рядами за системою функцій |
| title_full | On the finiteness of the $l$-$\mathfrak{M}$-index of entire functions represented by series in а system of functions |
| title_fullStr | On the finiteness of the $l$-$\mathfrak{M}$-index of entire functions represented by series in а system of functions |
| title_full_unstemmed | On the finiteness of the $l$-$\mathfrak{M}$-index of entire functions represented by series in а system of functions |
| title_short | On the finiteness of the $l$-$\mathfrak{M}$-index of entire functions represented by series in а system of functions |
| title_sort | on the finiteness of the $l$-$\mathfrak{m}$-index of entire functions represented by series in а system of functions |
| topic_facet | Ціла фукнція Ряд за системою функцій Обмежений індекс |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7866 |
| work_keys_str_mv | AT sheremetam onthefinitenessofthelmathfrakmindexofentirefunctionsrepresentedbyseriesinasystemoffunctions AT šeremetamiroslav onthefinitenessofthelmathfrakmindexofentirefunctionsrepresentedbyseriesinasystemoffunctions AT sheremetam proobmeženístʹlmathfrakmíndeksucílihfunkcíjzobraženihrâdamizasistemoûfunkcíj AT šeremetamiroslav proobmeženístʹlmathfrakmíndeksucílihfunkcíjzobraženihrâdamizasistemoûfunkcíj |