On the concavity and convexity of 1/ζ

Let [Formula: see text] be the Riemann zeta function. In 2020, Alkan proved that [Formula: see text] is strictly concave on [Formula: see text]. We extend this result and show that [Formula: see text] is strictly concave on the maximal interval [Formula: see text]. Moreover, we study concavity and convexity properties of [Formula: see text] on [Formula: see text]. Applications lead to functional inequalities involving the zeta function.