# variance-based upper bound for entropy: proof?

I found the inequality in wikipedia https://en.wikipedia.org/wiki/Entropic_uncertainty

$$H(\phi )\leq \log {\sqrt {2\pi eV(\phi )}},$$ with $$\phi$$ as "any probability density function on the real line".

Can anyone point to the proof of this statement? What about discrete case?

• The proof is standard and available in wikipedia. In the discrete case, the entropy is upper bounded by $\log M$, where $M$ is the number of possible values of the random variable. – Stelios Mar 27 at 12:47