Multiplicity bounds in graded rings

The $F$-threshold $c^J(\text{a})$ of an ideal $\text{a}$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\text{a}$ 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 $c^J(\text{a})$ in terms of the multiplicities $e(\text{a})$ and $e(J)$, when $\text{a}$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\text{a}$ 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}.