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The Hurwitz zeta function: monotonicity, convexity and inequalities

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Abstract

We present monotonicity and convexity properties of functions defined in terms of the Hurwitz zeta function

$$\zeta(s, a)=\sum_{k=0}^\infty \frac{1}{(k+a)^s} \quad{(s > 1; \, a > 0)}$$

and apply these results to obtain new inequalities involving \({\zeta(s, a)}\). Among others, we prove that the double-inequality

$$\frac{1}{2} < a^s \zeta(s, a)+b^s \zeta(s, b)-(a+b)^s \zeta(s, a+b) < 1$$

holds for all s > 1 and a, b >  0. Both bounds are sharp.

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  1. Morsbacher Str. 10, 51545, Waldbröl, Germany

    Horst Alzer

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  1. Horst Alzer
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Correspondence to Horst Alzer.

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Dedicated to Professor Janos Aczél on the occasion of his 90th birthday

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Alzer, H. The Hurwitz zeta function: monotonicity, convexity and inequalities. Aequat. Math. 89, 1401–1414 (2015). https://doi.org/10.1007/s00010-015-0371-1

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  • DOI: https://doi.org/10.1007/s00010-015-0371-1

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