Inequalities for Taylor series involving the divisor function

Alzer, Horst, and Kwong, Man Kam. "Inequalities for Taylor series involving the divisor function." Czechoslovak Mathematical Journal 72.2 (2022): 331-348. <http://eudml.org/doc/298318>.

@article{Alzer2022,
abstract = {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 $\alpha =\frac\{1\}\{2\}$ and $\beta =1$.},
author = {Alzer, Horst, Kwong, Man Kam},
journal = {Czechoslovak Mathematical Journal},
keywords = {divisor function; infinite series; inequality; monotonicity; $q$-digamma function; Euler’s constant},
language = {eng},
number = {2},
pages = {331-348},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Inequalities for Taylor series involving the divisor function},
url = {http://eudml.org/doc/298318},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Alzer, Horst
AU - Kwong, Man Kam
TI - Inequalities for Taylor series involving the divisor function
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 2
SP - 331
EP - 348
AB - 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 $\alpha =\frac{1}{2}$ and $\beta =1$.
LA - eng
KW - divisor function; infinite series; inequality; monotonicity; $q$-digamma function; Euler’s constant
UR - http://eudml.org/doc/298318
ER -