How do we evaluate this limit -- a product of a series, which is entire, and a decaying exponential function,

Question

$$lim_{s-> \infty}\phi(sz)e^{-s}$$

where $\phi$ is the series $$\sum_0^\infty \frac{a_n}{n!}(sz)^n$$ and is entire.

Also, assume that we have this upper bound: $|\phi(sz)|$ $\le$M$e^{|sz|}$.

Then, we can look at this limit instead:

$$lim_{s-> \infty}\ Me^{s(|z|-1)}$$

which is just zero for |z| < 1 easily -- but it is not zero for |z| =1 or |z|>1.

Note: s is real, while z is complex.

Thanks,