Inequality for strictly increasing convex function

Question

Suppose I have a strictly increasing convex function $f:[0,\infty)\mapsto [0,\infty)$. How do I show that $$f^{-1}(x^2) \leq C_f f^{-1}(x)$$ for any $x\geq 0$, and $C_f$ is a constant that is dependent only on $f$.

Answer 1

The function $f$ defined via $f(x) = x$ is a counterexample.