Let $F$ be a distribution function of a random variable $X$ and for a non-decreasing function $g$, $$g^\leftarrow (t)=\inf \{y:g(y)\geq t\}$$ the left-continuous inverse. $U$ is the left-continuous inverse of $1/(1-F)$. I am trying to figure the left-continuous inverse of
$$t^{-1}U\left(\frac{x}{1-F(t)}\right)$$ out, where $x\in \mathbb{R}$.
Thanks for any answers!