 Solution

Put $\mu =\mathbb{E}Z$ and $h(u)=(1+u)\log (1+u)-u$. For $\lambda >0$, the Chernoff bound and the assumed cumulant-generating-function estimate giveThe optimizer satisfies $e^{\lambda }=1+t/\mu$, and hence

For the left tail, apply the same argument at a negative parameter. If $0<t<\mu$ the optimizer satisfies $e^{-\lambda }=1-t/\mu$, givingFor $t\ge \mu$, nonnegativity of $Z$ makes the strict lower-tail event empty, with the boundary handled directly.