Our main result states that for all real numbers s>1 we have
\gamma < s (\frac{\zeta'(s)}{\zeta(s)}+\frac{1}{s-1}). \eqno (\ast)
The constant lower bound \gamma is sharp.
This refines an inequality published by Delange in 1987.
Applications of (\ast) lead to a monotonicity theorem, namely, that
\frac{(s-1)\zeta(s)}{s^{\alpha}}
is strictly increasing on (1,\infty) if and only if \alpha \leq \gamma,
and to additional inequalities for the zeta function.
