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 give a sharp lower bound for the log canonical threshold of $IJ$ in terms of the sequences of mixed Łojasiewicz exponents of the two ideals. In particular, when $J$ is the maximal ideal, the corresponding equality holds if and only if the integral closure of $I$ equals some power of the maximal ideal.