| Title:
|
Some lower bounds for the quotients of normalized error function and their partial sums (English) |
| Author:
|
Frasin, Basem Aref |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
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1212-5059 (online) |
| Volume:
|
61 |
| Issue:
|
2 |
| Year:
|
2025 |
| Pages:
|
73-83 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The purpose of the present paper is to determine lower bounds for $\mathfrak{R}\left\rbrace \frac{\mathcal{E}_{k}f(z)}{(\mathcal{E}_{k}f)_{m}(z)}\right\lbrace $, $\mathfrak{R}\left\rbrace \frac{(\mathcal{E}_{k}f)_{m}(z)}{\mathcal{E}_{k}f(z)}\right\lbrace , \mathfrak{R}\left\rbrace \frac{\mathcal{E}_{k}^{\prime }f(z)}{(\mathcal{E}_{k}f)_{m}^{\prime }(z)}\right\lbrace $ and $\mathfrak{R}\left\rbrace \frac{(\mathcal{E}_{k}f)_{m}^{\prime }(z)}{\mathcal{E}_{k}^{\prime }f(z)}\right\lbrace $, where $\mathcal{E}_{k}f$ is the generalized normalized error function of the form $\mathcal{E}_{k}f\left( z\right) =z+\sum _{n=2}^{\infty }\frac{\left( -1\right) ^{n-1}}{(\left( n-1\right) k+1)\left( n-1\right) !}z^{n}$ and $(\mathcal{E}_{k}f)_{m}$ its partial sum. Furthermore, we give lower bounds for $\mathfrak{R}\left\rbrace \frac{\mathbb{I}\left[ \mathcal{E}_{k}f\right] (z)}{(\mathbb{I}\left[ \mathcal{E}_{k}f\right] )_{m}(z)}\right\lbrace $ and $\mathfrak{R}\left\rbrace \frac{(\mathbb{I}\left[ \mathcal{E}_{k}f\right] )_{m}(z)}{\mathbb{I}\left[ \mathcal{E}_{k}f\right] (z)}\right\lbrace $, where $\mathbb{I}\left[ \mathcal{E}_{k}f\right] $ is the Alexander transform of $\mathcal{E}_{k}f$. Several examples of the main results are also considered. (English) |
| Keyword:
|
partial sums |
| Keyword:
|
analytic functions |
| Keyword:
|
generalized error function |
| MSC:
|
30C45 |
| DOI:
|
10.5817/AM2025-2-73 |
| . |
| Date available:
|
2025-07-01T07:31:13Z |
| Last updated:
|
2025-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153019 |
| . |
| Reference:
|
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