## Inequalities for Taylor series involving the divisor function.(English)Zbl 1524.11008

Summary: 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 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 $\alpha\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)}<T(q)<\beta\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)},\quad 0<q<1,$ holds with the best possible constant factors $$\alpha =\gamma$$ and $$\beta =1$$. Here, $$\gamma$$ denotes Euler’s constant. This refines a result of A. Salem [Math. Inequal. Appl. 23, No. 3, 855–872 (2020; Zbl 1453.33011)], who proved the inequalities with $$\alpha =\frac{1}{2}$$ and $$\beta =1$$.

### MSC:

 11A25 Arithmetic functions; related numbers; inversion formulas 26A48 Monotonic functions, generalizations 26D15 Inequalities for sums, series and integrals 33D05 $$q$$-gamma functions, $$q$$-beta functions and integrals

Zbl 1453.33011

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### References:

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