What are the functions $f$ on $[0,1]$ (with continuous first and second derivatives) that satisfy the following conditions:

* Monotonically increasing in $[0,1]$, with $f(0)\geq 0$ and $f(1)=1$;
* for all $x\in[0,1]$:

$$
{f(x) f''(x) \over f'(x)^2} < 2
$$

?

Some simple examples are:

* $f(x) = x^k$ for some $k> 0$. Then: $f(x)f''(x)=k(k-1)x^{2k-2}$ and $f'(x)^2 = k^2 x^{2k-2}$ so the quotient is $(k-1)/k < 1 < 2$.
* $f(x) = e^{k (x-1)}$ for some $k\geq 0$. Then: $f(x)f''(x) = k^2 f(x) = f'(x)$ so the quotient is $1 < 2$.

Is there a general form of functions that satisfy these conditions?