  Let $(R, \mathfrak m)$ be a Noetherian local ring and $I$ a $\mathfrak
m$-primary ideal. In this paper, we study an inequality involving the number of
generators, the Loewy length and the multiplicity of $I$. There is strong
evidence that the inequality holds for all integrally closed ideals of finite
colength if and only if $R$ has sufficiently nice singularities. We verify the
inequality for regular local rings in all dimensions, for rational singularity
in dimension $2$, and cDV singularities in dimension $3$. In addition, we can
classify when the inequality always hold for a Cohen-Macaulay $R$ of dimension
at most two. We also discuss relations to various topics: classical results on
rings with minimal multiplicity and rational singularities, the recent work on
$p_g$ ideals by Okuma-Watanabe-Yoshida, multiplicity of the fiber cone, and the
$h$-vector of the associated graded ring.
