Multiplicities and log canonical threshold

Given an n n -dimensional local ring R R of a smooth variety, and a zero-dimensional ideal I ⊂ R I⊂ R , we prove the following inequality involving the Samuel multiplicity and the log canonical threshold: e ( I ) ≥ n n / lc ⁡ ( I ) n e(I)≥ n^n/lc(I)^n . Moreover, equality holds if and only if the integral closure of I I is a power of the maximal ideal in R R . When n = 2 n=2 , we give a similar inequality for an arbitrary ideal I I .