Remark on a double-inequality for the Riemann zeta function (Q813949)

The author gives an elementary proof that, for \(x>0\), \[ (1-\delta(x))\zeta(x)+\alpha\delta(x)<\zeta(x+1)<(1-\delta(x))\zeta(x) +\beta\delta(x), \] where \(\delta(x) = 1/(2^x-1)\) and \(\alpha = \log 2 - {1\over2}\), \(\beta = {1\over2}\) are best possible constants. The lower bound improves a recent result of \textit{P. Cerone} et al. [JIPAM, J. Inequal. Pure Appl. Math. 5, No. 2, Paper No. 43, 17 p., electronic only (2004; Zbl 1070.11034)], who had the lower bound with the constant 1/8.