Maximizing a ratio of convex functions by minimizing a difference?

Maximizing a ratio of convex functions by minimizing a difference?

Given that $g(.),h(.)$ are convex real functions whose domain is the set
of all real matrices while the range is the set of real numbers:
Is maximizing the term $g(X)/h(X)$ w.r.t X the same as minimizing
$h(X)-\nu g(X)$ for some scalar $\nu$? Is this always valid- why or why
not? Otherwise, is the problem ill-posed, and if so why?
I was thinking there is a different trade-off between how much, $g(X)$ is
maximized while $h(X)$ is minimized in both the formulations.