Inequalities for Taylor series involving the divisor function

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 $\beta=1$.