Consider two functions $f: [0,1) \rightarrow \mathbb{R}$ and $g: [0,1) \rightarrow \mathbb{R}$. Suppose $f(x) > g(x)$ for all $x \in [0,1)$. Suppose further that $f$ and $g$ are infinitely differentiable and all derivatives of both $f$ and $g$ are strictly positive on the interior $(0,1)$ of the domain and non-negative at $x=0$: $f^{(n)}(x) > 0$ and $g^{(n)}(x) > 0$ for all $x \in (0,1)$, $f^{(n)}(0) \geq 0$, $g^{(n)}(0) \geq 0$, for all $n$. (For lack of a better term, I call $f$ and $g$ 'fully monotonic', since 'totally' or 'completely' monotonic requires the signs to alternative. See here.) Let $x_0 \in (0,1)$ be a fixed interior point.
Is it true that $ \frac{f(x)}{f(x_0)} = \frac{g(x)}{g(x_0)} $ only holds at $x = x_0$? That is, do the functions $f(x)/f(x_0)$ and $g(x)/g(x_0)$ cross only once?
A standard sufficient condition for uniqueness is $f' > g'$, or the slightly weaker $f'(x)/f(x_0) > g'(x)/g(x_0)$, but this is not assumed here. Intuitively it seems that full monotonicity and $f > g$ imply any rescaled functions $f(x)/C_1$ and $g(x)/C_2$ where $C_1 > 1$ and $0 < C_2 < 1$ can cross at most once, but I have not been able to prove this, nor can I construct a counterexample.
Also, in case these added properties matter: I know that $f(0) = 1$, $g(0) = 0$ and both $f(x)$ and $g(x)$ diverge to infinity as $x \rightarrow 1$. $f$ and $g$ are both power series.