  We define upper bound and lower bounds for order-preserving homogeneous of
degree one maps on a proper closed cone in $\R^n$ in terms of the cone spectral
radius. We also define weak upper and lower bounds for these maps. For a proper
closed cone $C \subset \R^n$, we prove that any order-preserving homogeneous of
degree one map $f: \inter C \rightarrow \inter C$ has a lower bound. If $C$ is
polyhedral, we prove that the map $f$ has a weak upper bound. We give examples
of weak upper bounds for certain order-preserving homogeneous of degree one
maps defined on the interior of $\R^n_+$.
