| Abstrakt: |
Abstract: Let $T(q)=\sum_{k=1}^\infty d(k) q^k$, $|q|<1,$ 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 $\alpha\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)} < T(q)< \beta\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)}$, $0
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