estimate on a function with compact support

Question

let $U\subseteq\mathbb{R}^n$ be open and $f\in C_c^2(U)$ satisfy $f\geq0$ in $U$. I want to get an estimate of the function which is $$\sup_U\frac{|Df|^2}{f}\leq2\sup_U|D^2f|$$

Answer 1

Chances are your estimation doesn't even exist.
For example, let $n=1$, $U=(1,2)$, $f(x)=x^p$ for some $p>1$, then $(f')^2/f=\frac{p}{p-1}f''$ so

$$\sup_U\frac{|Df|^2}{f}= \frac{p}{p-1}\sup_U|D^2f|$$ Now, take $p$ converge to $1$.

Answer 2

define that $$\varphi=|D\sqrt{\eta+\epsilon}|^{2}=\frac{|D\eta|^2}{4(\eta+\epsilon)}$$ where $\epsilon>0$ fixed
since $\eta\in C_c^2(U)$ then $\varphi\in C_c^1(U)$
so $\varphi$ must attain a maximum inside $U$, say at point $x_0$
then $D_i\varphi(x_0)=0$ for any $1\leq i\leq n$ i.e. $$2D^2\eta\cdot D\eta \cdot(\eta+\epsilon)-|D\eta|^2D\eta=0$$ then \begin{align*} D\eta^T|D\eta|^2D\eta&=2(\eta+\epsilon)D\eta^TD^2\eta D\eta\\ &\leq2(\eta+\epsilon)|D\eta|\cdot|D^2\eta|\cdot|D\eta|\mbox{ (by cauthy-schwarz inequality)}\\ &=2(\eta+\epsilon)|D\eta|^2\cdot|D^2\eta| \end{align*} therefore $$\frac{|D\eta|^2}{(\eta+\epsilon)}\leq2|D^2\eta|$$ let $\epsilon\to0^+$ and take $\sup$ by both side, we get $$\sup\limits_{U}\frac{|D\eta|^2}{\eta}\leq2\sup\limits_{U}|D^2\eta|$$