Is the quotient between convex and positive concave function a convex function?

Question

Given a strictly convex function $f(x)$ and a positive strictly concave function $g(x)$. Is $$h(x) = \frac{f(x)}{g(x)}$$ a (strictly) convex function?

I can only prove that $h(x)$ would be quasiconvex. Is there any counterexample or proof to this problem?

Answer 1

For example, take $f(x) = x^{\frac{4}{3}}$ and $g(x) = x^{\frac{2}{3}}$, both over the interval $(0, \infty)$.