Remark on a double-inequality for the Riemann zeta function

Let ζ be the Riemann zeta function and δ(x)=1/(2x-1). For all x>0 we have(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),with the best possible constant factorsα=log2-12=0.1931…andβ=12.This improves a recently published result of Cerone et al., J. Inequalities Pure Appl. Math. 5(2) (43) (2004), who showed that the double-inequality holds with α=18 and β=12.