Upper Bound for indicator function

Question

For a given $t \in \mathbb{R}$, I want to know if there is a tighter bound on the function $u(x) =\mathbb{1}_{(x \geq t)}$ than $\bar{u}(s,x) = e^{2s(x-t)}$, $s > 0$.

Answer 1

Certainly $\min(\bar u(s_1,x),\bar u(s_2,x),\ldots,\bar u(s_n,x))$, for a selection of positive numbers $s_1,\ldots,s_n$, will give a tighter bound than any one of them by itself. It might be helpful for us to know what your goal is.