Inequalities for Taylor series involving the divisor function

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 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 $$\alpha \frac{q}{1-q}+\frac{\log (1-q)}{\log (q)}<T(q)<\beta \frac{q}{1-q}+\frac{\log (1-q)}{\log (q)},0<q<1,$$ 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 $\beta =1$.

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