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 $G(1/2)$?


1 Answer 1


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 $G_n(\frac 12) = 2^{-n}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.