  Let $f:\mathbb{R}^n_{> 0} \rightarrow \mathbb{R}^n_{>0}$ be an
order-preserving and homogeneous function. We show that the set of eigenvectors
of $f$ in $\mathbb{R}^n_{>0}$ is nonempty and bounded in Hilbert's projective
metric if and only if $f$ satisfies a condition involving upper and lower
Collatz-Wielandt numbers of readily computed auxiliary functions. This
condition generalizes a test for the existence of eigenvectors using
hypergraphs that was proved by Akian, Gaubert, and Hochart. We include several
examples to show how the new condition can be combined with the hypergraph test
to give a systematic approach to determine when homogeneous and
order-preserving functions have eigenvectors in $\mathbb{R}^n_{>0}$. We also
observe that if the entries of $f$ are real analytic functions on
$\mathbb{R}^n_{>0}$, then the set of eigenvectors of $f$ in $\mathbb{R}^n_{>0}$
is nonempty and bounded in Hilbert's projective metric if and only if the
eigenvector is unique, up to scaling.
