The $F$ -threshold $c^{J}({\mathfrak{a}})$ of an ideal $\mathfrak{a}$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\mathfrak{a}$ with the Frobenius powers of $J$ . We study a conjecture formulated in an earlier article that we authored with M. Mustaţă, which bounds $c^{J}({\mathfrak{a}})$ in terms of the multiplicities $e({\mathfrak{a}})$ and $e(J)$ when $\mathfrak{a}$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\mathfrak{a}$ 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.