Article
| Title: | Inequalities for Taylor series involving the divisor function (English) |
| Author: | Alzer, Horst |
| Author: | Kwong, Man Kam |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 2 |
| Year: | 2022 |
| Pages: | 331-348 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $$ T(q)=\sum _{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we prove that $$ H(q) = T(q)- \frac {\log (1-q)}{\log (q)} $$ is strictly increasing on $ (0,1)$, while $$ F(q) = \frac {1-q}{q} H(q) $$ is strictly decreasing on $(0,1)$. These results are then applied to obtain various inequalities, one of which states that the double inequality $$ \alpha \frac {q}{1-q}+\frac {\log (1-q)}{\log (q)} < T(q)< \beta \frac {q}{1-q}+\frac {\log (1-q)}{\log (q)}, \quad 0<q<1, $$ holds with the best possible constant factors $\alpha =\gamma $ and $\beta =1$. Here, $\gamma $ denotes Euler's constant. This refines a result of Salem, who proved the inequalities with $\alpha =\frac 12$ and $\beta =1$. (English) |
| Keyword: | divisor function |
| Keyword: | infinite series |
| Keyword: | inequality |
| Keyword: | monotonicity |
| Keyword: | $q$-digamma function |
| Keyword: | Euler's constant |
| MSC: | 11A25 |
| MSC: | 26D15 |
| MSC: | 33D05 |
| idZBL: | Zbl 07547207 |
| idMR: | MR4412762 |
| DOI: | 10.21136/CMJ.2021.0464-20 |
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| Date available: | 2022-04-21T18:58:41Z |
| Last updated: | 2022-09-08 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/150404 |
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