On the concavity of Dirichlet's eta function and related functional inequalities

Abstract

We prove the strict concavity of Dirichlet's eta functionη(s)=∑j=1∞(−1)j−1js on (0,∞). This extends a result of Wang, who proved in 1998 that η is strictly logarithmically concave on (0,∞).Several new inequalities satisfied by η are also presented. Among them is the double-inequalitylog⁡2<η(x)1/xη(y)1/yη(xy)1/xy<1, for all x,y∈(1,∞). Both bounds are sharp.