Hi,

I have the following problem that came up. It is not a homework problem or something similar. I did my simulations and it seems to hold but i was unable to prove it.## Heading ##

Let $P$ and $Q$ be two discrete probability distributions on the alphabet $\{1,2,\dots n\}$. Prove that:

$H(P)-H(Q) \leq \sum\limits_{i=1}^n \big [ (P_i-Q_i)\log(\frac{1}{\frac{P_i}{e}+(1-\frac{1}{e})Q_i}) \big ]$, where $e$ is the base of the natural log. All entropies are measured in nats.

Thank you very much for your help! Any ideas would be very helpful.