I'm investigating about some relations between random variables through stochastic orders.
In particular, I have to prove the following thing:
"Let $F$ and $G$ be respectively the cdfs of the random variables $U$ e $V$. Both the relations $U \leq_{*} V$ and $V \leq_{*} U$ hold simultaneously, and then we write $U=_{*}V$, if the random variables $U$ and $V$ have proportional quantile functions."
The quantile functions are $F^{-1}$ and $G^{-1}$.
If they're proportional, then $$ F^{-1}=aG^{-1}, $$
where $a$ is a constant and $a \neq 0$.
I know also that $U$ is smaller than $V$ in the star order, with $U$ and $V$ nonegative, and write $U \leq_{*} V$, if and only if $G^{-1}$ is starshaped in x, or equivalently $$ \frac{G^{-1}(F(x))}{x} \qquad \text{is increasing in $x$ on the support of $F(x)$}. $$
MY IDEA
Because $U$ and $V$ have proportional quantile functions, then I could write:
$$ F^{-1}=aG^{-1}, \quad a \neq 0. $$
When $U \leq_{*} V$, then $$ \frac{G^{-1}(F(x))}{x}=\frac{\frac{1}{a}F^{-1}(F(x))}{x}=\frac{1}{a} $$
than is constant in $x$.
When $U \geq_{*} V$, then $$ \frac{G^{-1}(F(x))}{x}=\frac{aF^{-1}(F(x))}{x}=a. $$
Does this prove the result I have to prove?