Let $z \in \mathbb{C}$. For large $c>0$, how do we find some $c'$ such that $$e^{2 \pi c|z|^2 + \log(2) c|z|} \le e^{c' |z|^2}$$ for all $z$?
This is part of a hint in Exercise 5.4(a) of Stein and Shakarchi's Complex Analysis. We have $F_1(z) = \Pi_{n=1}^N (1-e^{-2\pi nt} e^{2\pi iz})$ for a fixed $t>0$. By choosing $N \approx c|z|$, the hint says we get, from $$|1-e^{-2\pi nt} e^{2\pi iz}| \le 1+e^{2\pi |z|} \le 2e^{2 \pi |z|},$$ $|F_1(z) | \le 2^N e^{2 \pi N |z|} \le e^{c'|z|^2}$.
Thus, if we take $N=c|z|$, we get $|F_1(z)| \le (2 e^{2\pi |z|})^N = 2^N e^{2\pi N|z|} \approx e^{2 \pi c|z|^2 + \log (2) c|z|}$. So it seems like we need some $c'$ such that $$e^{2 \pi c|z|^2 + \log(2) c|z|} \le e^{c' |z|^2}.$$
I am struggling with the case $|z| <1$. How can we bound all such $z$ by some multiple of $|z|^2$? I would greatly appreciate any help.