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Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $M$ be a finitelygenerated $R$-module of dimension $d$. We prove that the set$\left\{\frac{l(M/IM)}{e(I, M)} \right\}_{\sqrt{I}=\mathfrak{m}}$ is boundedbelow by ${1}/{d!e(\overline{R})}$ where $\overline{R}=R/Ann(M)$. Moreover,when $\widehat{M}$ is equidimensional, this set is bounded above by a finiteconstant depending only on $M$. The lower bound extends a classical inequalityof Lech, and the upper bound answers a question of St\"{u}ckrad--Vogel in theaffirmative. As an application, we obtain results on uniform behavior of thelengths of Koszul homology modules.

Lech's inequality, the St\"{u}ckrad--Vogel conjecture, and uniform behavior of Koszul homology