A sharp bound for the Frobenius test exponents in generalized Cohen-Macaulay local rings

Let $(R,\mathfrak{m})$ be a generalized Cohen-Macaulay local ring of prime characteristic $p$. In this paper we give a sharp bound for the Frobenius test exponent of parameter ideals. Namely, we prove that $$\mathrm{F}\mathrm{t}\mathrm{e}(R)\le \lceil {\log }_p(2n_0)\rceil +\mathrm{H}\mathrm{S}\mathrm{L}(R),$$ where $n_0$ is the integer such that ${\mathfrak{m}}^{n_0}{H}_{\mathfrak{m}}^i(R)=0$ for all $i<\mathrm{d}\mathrm{i}\mathrm{m}(R)$, and $\lceil x\rceil$ is the smallest integer that is greater than or equal to $x$.