Suppose we have that $$f(z)=(z^3+1)e^{z^2}$$ on the unit disk.
My textbook says,
"obviously"
$$|f(z)| \le 2 |e^{z^2}|=e^{x^2-y^2} \le e$$ for all $z \in D(0,1)$
So I fully understand where the 2 comes from but I am not so sure about the $|e^{z^2}|$
$$e^{z^2}=e^{x^2-y^2+2xyi}=e^{x^2-y^2}e^{2xyi}$$
So my thoughts are maybe that the norm of $e^{2xyi}=1$ because it can be written using euler formula and then it will follow from Pythagoras.
But if that is the case, where does the 2 go in the first estimate?
So in the comments I explain why I am still confused on what is correct and not