# An inequality for Holder norms

Let $f$ be a non-negative $C^2$ function on a compact domain $\Omega$ in $\mathbb R^n$. I am trying to prove the inequality

$$\|\sqrt f\|_{C^{0,1}(\Omega)}\leq C(1 + \|f\|_{C^{1,1}(\Omega)})$$

where $C^{k,\alpha}(\Omega)$ denotes the Holder space. It seems like this should be a consequence of the mean value theorem and/or the fundamental theorem of calculus, but I am not seeing an elementary proof of this nature. Any suggestions?

• Jensen's Inequality? – treble Dec 21 '12 at 21:20

It is incorrect. For example, let $n=1$, $\Omega=[0,1]$ and $f(x)=x$. Then $\|\sqrt{f}\|_{C^{0,1}(\Omega)}=\infty$.
• Does the inequality hold if $M$ is instead assumed to be a compact manifold? – user15464 Dec 21 '12 at 21:42
• @user15464: If you mean $M$ is compact without boundary, I think so. The problem here is that the zero of $f$ is located at the boundary of $\Omega$. If $f(x)=0$ for some interior point $x$, then since $f\ge 0$, $f'(x)=0$, so $\sqrt{f}$ is well behaved around $x$. – 23rd Dec 21 '12 at 22:00
• @user15464: Sorry, my last comment was misleading. I think in your inequality, $C$ is supposed to be dependent only on $\Omega$ or $M$. Then the inequality fails even when $M$ is a compact manifold without boundary. – 23rd Dec 21 '12 at 23:11