estimate on a function with compact support 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 $$\sup_U\frac{|Df|^2}{f}\leq2\sup_U|D^2f|$$
 A: 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 $1$.
A: 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 $$\sup\limits_{U}\frac{|D\eta|^2}{\eta}\leq2\sup\limits_{U}|D^2\eta|$$
