  The $F$-threshold $c^J(\a)$ of an ideal $\a$ with respect to an ideal $J$ is
a positive characteristic invariant obtained by comparing the powers of $\a$
with the Frobenius powers of $J$. We study a conjecture formulated in an
earlier paper \cite{HMTW} by the same authors together with M. Musta\c{t}\u{a},
which bounds $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when
$\a$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of
parameters. We prove the conjecture when $\a$ and $J$ are generated by
homogeneous systems of parameters in a Noetherian graded $k$-algebra. We also
prove a similar inequality involving, instead of the $F$-threshold, the jumping
number for the generalized parameter test submodules introduced in \cite{ST}.
