| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2022-04-21T18:58:41Z |
| Last updated:
|
2024-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150404 |
| . |
| Reference:
|
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