  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 $\beta=1$.
