Zeros of the Hurwitz zeta function in the interval (0,1)

We first give a condition on the parameters $s,w$ under which the Hurwitz zeta function $\zeta(s,w)$ has no zeros and is actually negative. As a corollary we derive that it is nonzero for $w\geq 1$ and $s\in(0,1)$ and, as a particular instance, the known result that the classical zeta function has no zeros in $(0,1)$.