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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

Citations:

Zbl 1453.33011

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References:

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