Is there another method besides the minimum modulus theorem than can help me solve this problem?

Question

The question is, find the minimum of $|exp(z^2)|$ for those z with $|z|\leq 1$. Im aware the modulus theorem would solve this, but we haven't gotten there yet. The section of the book this question is on is differentiation of elementary complex functions. Don't solve for me, just hints please

Answer 1

I would do it like this:

Write $z = x+iy$ where $|z| = \sqrt{x^2 + y^2} \leq 1$.

1) How do we simplify $|e^{z^2}|$ in terms of $x$ and $y$?

2) How can we use the bounds on $x$ and $y$ implied by the inequality?

Answer 2

The function $g(z):=z^2$ assumes the same values in $\bar D$ as does $f(z)=z$. Now $\bigl|e^z\bigr|=e^x\geq{1\over e}$ in $\bar D$, and the minimal value is taken at $z=-1$. In the original problem we therefore should choose $z$ such that $z^2=-1$, which is $z=\pm i$.