Let $Z \sim \pi N(\mu, \sigma^2) + (1-\pi)\delta_0$ and $z_i \sim Z$ are iid for $i=1,\ldots,n$. I would like to obtain a result of the form $ P[n^{-1}\sum_i z_i - \pi\mu > \epsilon]\leq\exp(-c n \epsilon^2) $ with an explicit constant c.
I computed ${\mathbb E}[\exp(t(Z - \pi\mu)] = \exp(-t\pi\mu)[1-\pi + \pi\exp(\mu t + \frac{\sigma^2}{2}t^2)]$ and tried upper bounding RHS of
$ P[n^{-1}\sum_i z_i - \pi\mu > \epsilon]\leq \min_{t > 0} \exp(-nt\epsilon)\exp(-nt\pi\mu)[1-\pi + \pi\exp(\mu t + \frac{\sigma^2}{2}t^2)]^n, $ however, I have problems finding explicit $t$ that minimizes the RHS objective.
Are there alternative approaches to finding desired upper bound?