Lower bound for negative part of a real function

Question

Let $u:\mathbb{R}\to\mathbb{R}$ be a function and let $u_-$ its negative part.

My question is: what is an lower bound ($u_-\ge \dots$) for $u_-$? I mean, it is possible to estimate $u_-$ by using $\inf(u_-), \sup(u_-)$ or somethinf similar?

Thank you in advance!