Inequalities for Riemann's zeta function

Let zeta and Lambda the the Riemann zeta function and the von Mangoldt function, respectively. Further, let c > 0. We prove that the double-inequality exp (-c Sigma(infinity)(n=1) Lambda(n)/n(s+alpha)) < zeta(s + c)/zeta(s) < exp (-c Sigma(infinity)(n=1) Lambda(n)/n(s+beta)) holds for all s > 1 with the best possible constants alpha = 0 and beta = 1/log 2 log (c log 2/1 - 2(-c)). This extends and refines a recent result of Cerone and Dragomir.