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

We prove the strict concavity of Dirichlet's eta function eta(s) = Sigma(infinity)(j=1) (-1)(j-1)/j(n) on (0,infinity). This extends a result of Wang, who proved in 1998 that eta is strictly logarithmically concave on (0, infinity). Several new inequalities satisfied by eta are also presented. Among them is the double-inequality log 2 < eta(x)(1/root x)eta(y)(1/root y)/eta(xy)(1/root xy) < 1, for all x, y is an element of (1, infinity). Both bounds are sharp. (C) 2015 Elsevier Inc. All rights reserved.