The log canonical threshold of products of ideals and mixed Łojasiewicz exponents

Given two ideals $I$ and $J$ of the ring ${\mathcal{O}}_n$ of analytic function germs $f:({\mathbb{C}}^n,0)\to \mathbb{C}$, we show a sharp lower bound for the log canonical threshold of $IJ$ in terms of the sequences of mixed Łojasiewicz exponents of them. In particular, in the case where $J$ is the maximal ideal, the corresponding equality holds if and only if the integral closure of $I$ equals some power of the maximal ideal.