  A comparison theorem is proved for a pair of solutions that satisfy in a weak
sense opposite differential inequalities with nonlinearity of the form $f (u)$
with $f$ belonging to the class $L^p_{loc}$. The solutions are assumed to have
non-vanishing gradients in the domain, where the inequalities are considered.
The comparison theorem is applied to the problem describing steady, periodic
water waves with vorticity in the case of arbitrary free-surface profiles
including overhanging ones. Bounds for these profiles as well as
streamfunctions and admissible values of the total head are obtained.
