Inequality over the entropy of an integer-valued random variable
Man, that's a nasty inequality.

Anyways, given a certain expectation $E[Z]=\mu$, it is known that the distribution that maximizes $H(Z)$ while keeping $E[Z]=\mu$ 
 is the the Geometric Distribution with parameter $p=1/\mu$. 

Then, it is easy to check that the inequality is satisfied (in fact the equality holds).
Indeed, if $Z\sim Geo(p)$ then $E[Z]=1/p$ and $H(Z)=-\log p -\frac{1-p}{p}\log(1-p)$.
Since this distribution maximizes $H(Z)$, any other distribution with the same expectation will have a lower left-hand side, while the right-hand side won't change (since $E[Z]=\mu$ didn't change). This proves the claim.