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 $G(1/2)$?
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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 $G_n(\frac 12) = 2^{-n}$.
