Inequalities for Taylor series involving the divisor function

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 proved that $$ H(q) := T(q)- \frac{\log(1-q)}{\log(q)} $$ is strictly increasing in $ (0,1) $ while $$ F(q) := \frac{1-q}{q} \,H(q) $$ is strictly decreasing in $ (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 $\alpha=1/2$ and $\beta=1$.