Bounding successive images of function that is a linear combination of two monotonly increasing functions.

Question

Let $f , g$ be differentiable with $f' >0$, $g'>0$. Define $h = (1-t) f + t g$.

We know that $min \{g(x), f(x) \} \leq h(x) \leq max\{g(x), f(x)\}$.

Can we find similar about $h^n(x) = h \circ \dots \circ h (x)$? That is can we find bounds in terms of $g^n$ and $f^n$ ?