I consider a smooth function $f$ over the real numbers, the derivatives are $f^{(k)}$. I suppose that:

$$0< f^{(n+1)}(x) \leq f^{(n)}(x)$$

for all $n \in {\bf N}$ and all $x \in {\bf R}$. Then I suppose that for $x \cong -\infty$, $f (x)\sim e^x$, ($f$ is equivalent to $e^x$ in $-\infty$); have we in these conditions:

$$f(x)=e^x$$

This problem may have a meaning in physics because if $t$ is the time, $x=\ln(t)$, we have near the Big Bang $t=0$, $x\cong -\infty$, so that $f$ is in fact the time.