On the Radii of univalence of Gel'fond-Leont'ev derivatives
Let $0 < R < +\infty,$ let $A(R)$ bethe class of functions $$f(z) = \sum_{k=0}^{\infty}f_kz^k,$$ analytic in $\{ z: |z| < R \}$, and let $$l(z) = \sum_{k=0}^{\infty}l_kz^k,\; l_k > 0$$ be a formal power series. We prove that if $l^2_k/l_{k+1}l_{k-1}$ is a nonincreasing s...
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| Datum: | 1995 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
1995
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5428 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511660314198016 |
|---|---|
| author | Sheremeta, M. M. Шеремета, М. М. |
| author_facet | Sheremeta, M. M. Шеремета, М. М. |
| author_sort | Sheremeta, M. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T08:59:05Z |
| description | Let $0 < R < +\infty,$ let $A(R)$ bethe class of functions
$$f(z) = \sum_{k=0}^{\infty}f_kz^k,$$
analytic in $\{ z: |z| < R \}$, and let
$$l(z) = \sum_{k=0}^{\infty}l_kz^k,\; l_k > 0$$
be a formal power series.
We prove that if $l^2_k/l_{k+1}l_{k-1}$ is a nonincreasing sequence, $f \in A(R)$, and
$|f_k/f_{k+1} \nearrow R,\; k \rightarrow \infty,\; 0 < R < +\infty$, then the sequence $(\rho_n)$ of radii of univalence of the Gel'fondLeont'ev derivatives
satisfies the relation
$$D^n_lf(z) = \sum_{k=0}^{\infty}\frac{l_kf_{k+n}}{l_{k+n}}z_k$$
The case where the condition $|f_k/f_{k+1}|\nearrow R,\quad k \rightarrow \infty$, is not satisfied is also considered. |
| first_indexed | 2026-03-24T03:16:25Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-5428 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:16:25Z |
| publishDate | 1995 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/90/6d95916e3f7dbde76b4ca266fe14eb90.pdf |
| spelling | umjimathkievua-article-54282020-03-19T08:59:05Z On the Radii of univalence of Gel'fond-Leont'ev derivatives Про радіуси однолистості похідних Гельфонда-Леонтьева Sheremeta, M. M. Шеремета, М. М. Let $0 < R < +\infty,$ let $A(R)$ bethe class of functions $$f(z) = \sum_{k=0}^{\infty}f_kz^k,$$ analytic in $\{ z: |z| < R \}$, and let $$l(z) = \sum_{k=0}^{\infty}l_kz^k,\; l_k > 0$$ be a formal power series. We prove that if $l^2_k/l_{k+1}l_{k-1}$ is a nonincreasing sequence, $f \in A(R)$, and $|f_k/f_{k+1} \nearrow R,\; k \rightarrow \infty,\; 0 < R < +\infty$, then the sequence $(\rho_n)$ of radii of univalence of the Gel'fondLeont'ev derivatives satisfies the relation $$D^n_lf(z) = \sum_{k=0}^{\infty}\frac{l_kf_{k+n}}{l_{k+n}}z_k$$ The case where the condition $|f_k/f_{k+1}|\nearrow R,\quad k \rightarrow \infty$, is not satisfied is also considered. Нехай $0 < R < +\infty, A(R)$ — клас аналітичних в $\{ z: |z| < R \}$ функцій $$f(z) = \sum_{k=0}^{\infty}f_kz^k,\;\; l(z) = \sum_{k=0}^{\infty}l_kz^k,\; l_k > 0$$ — формальный степеневий ряд. Доведено, що коли $l^2_k/l_{k+1}l_{k-1}$ — незростаюча послідовність, $f \in A(R)$ і $|f_k/f_{k+1} \nearrow R,\; k \rightarrow \infty,\; 0 < R < +\infty$, то послідовність $(\rho_n)$ радіусів однолистості похідних Гельфонда-Леонтьева $$D^n_lf(z) = \sum_{k=0}^{\infty}\frac{l_kf_{k+n}}{l_{k+n}}z_k$$ задовольняє співвідношення $$\rho_n \asymp \frac{l_{n+2}}{l_{n+1}}\left|\frac{f_{n+1}}{f_{n+2}}\right|$$ Вивчається також випадок, коли умова $|f_k/f_{k+1}|\nearrow R,\quad k \rightarrow \infty$ не виконується. Institute of Mathematics, NAS of Ukraine 1995-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5428 Ukrains’kyi Matematychnyi Zhurnal; Vol. 47 No. 3 (1995); 390–399 Український математичний журнал; Том 47 № 3 (1995); 390–399 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/5428/7545 https://umj.imath.kiev.ua/index.php/umj/article/view/5428/7546 Copyright (c) 1995 Sheremeta M. M. |
| spellingShingle | Sheremeta, M. M. Шеремета, М. М. On the Radii of univalence of Gel'fond-Leont'ev derivatives |
| title | On the Radii of univalence of Gel'fond-Leont'ev derivatives |
| title_alt | Про радіуси однолистості похідних Гельфонда-Леонтьева |
| title_full | On the Radii of univalence of Gel'fond-Leont'ev derivatives |
| title_fullStr | On the Radii of univalence of Gel'fond-Leont'ev derivatives |
| title_full_unstemmed | On the Radii of univalence of Gel'fond-Leont'ev derivatives |
| title_short | On the Radii of univalence of Gel'fond-Leont'ev derivatives |
| title_sort | on the radii of univalence of gel'fond-leont'ev derivatives |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5428 |
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