I want to show $[(f^k)^{(n)}]^2 \geq (f^{k-1})^{(n)} (f^{k+1})^{(n)}$, where $f$ satisfies $f^{(n)} \geq 0$ for all integer $n$ and $f^k$ denotes the $k$-th power of $f$.
I believe it's right but I can't see it clearly.
Thanks.
I want to show $[(f^k)^{(n)}]^2 \geq (f^{k-1})^{(n)} (f^{k+1})^{(n)}$, where $f$ satisfies $f^{(n)} \geq 0$ for all integer $n$ and $f^k$ denotes the $k$-th power of $f$.
I believe it's right but I can't see it clearly.
Thanks.
Take $n=3, k=2$, then your inequality will be equivalent to show the following:
$36{f'}^2{f''}^2+24ff'f''f'''+4f^2{f'''}^2 \ge 6{f'}^3f'''+18ff'f''f'''+3f^2{f'''}^2$,
so equivalent to $36{f'}^2{f''}^2+6ff'f''f'''+f^2{f'''}^2 \ge 6{f'}^3f'''$.
Now pick some function $f(x)$, such that $f'$ is big enough compared to $f,f''$ and $f'''$ at least at some point and then your inequality will not hold at that point.