Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq 2{G'(x)}^2.$$ Is there a constant positive bound from below on $G(1/2)$?



A differential inequality and a special value
Consider $G\_n(x) := (1-x)^n$. We have $G\_n(0) = 1$, $G\_n(1) = 0$ and $G\_n$ is monotonic (strictly) decreasing on $[0,1]$. Furthermore,


\begin{equation}
G\_n(x) G\_n''(x) = n(n-1) (1-x)^{2(n-1)} \leq 2n^2 (1-x)^{2(n-1)} = 2 G\_n'(x)^2.
\end{equation}


And witness that $G\_n(\frac 12) = 2^{-n}$.


