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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: 2024-07-01
Stable URL: http://hdl.handle.net/10338.dmlcz/150404
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