Math Challenges

Solution:
\( \displaylines{z=x+iy\\ z^2=x^2-y^2+i2xy\\ \left|\exp\left(z^2\right)\right|=\exp\left(x^2-y^2\right)\text{, while }\left.\exp(\right|z\left|^2)=\right.\exp\left(x^2+y^2\right)\\ \text{Factoring gives}\\ \exp\left(x^2\right)\exp\left(-y^2\right)\leq\exp\left(x^2\right)\exp\left(y^2\right)\\ \text{which is clearly true, as }y^2\geq0\implies1\leq\exp\left(2y^2\right)} \)