Czechoslovak Mathematical Journal, Vol. 72, No. 2, pp. 331-348, 2022
Inequalities for Taylor series involving the divisor function
Horst Alzer, Man Kam Kwong
Received October 28, 2020. Published online July 15, 2021.
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 $\alpha\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)} < T(q)< \beta\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)}$, $0<q<1,$ 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 $\alpha=\frac12$ and $\beta=1$.
Affiliations: Horst Alzer (corresponding author), Morsbacher Strasse 10, 51545 Waldbröl, Germany, e-mail: h.alzer@gmx.de; Man Kam Kwong, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, e-mail: mankwong@connect.polyu.hk
