$p(x) \geq 0$ is convex, and $q(x) > 0$ is concave.
How to prove $f(x) = \frac{p(x)}{q(x)}$ is quasiconvex?
My proof is using t-sublevel set:
$\{x | \frac{p(x)}{q(x)} \leq t\}$ is equivalent to $\{x | p(x)-tq(x) \leq 0\}$
So, if $t \geq 0$, then we can know that $p(x)-tq(x)$ is a convex function with its domain a convex set.
So we can prove $f(x)$ is quasi-convex.
My question is:
- t is restricted to $\geq0$, which is not for all t, how could we say $f(x)$ is quasi-convex?
- The reason we cannot say $f(x)$ is convex is we cannot prove epigraph of $f(x)$ is convex?
