Maximum modulus of $f(z)=z^2$ on the disc $D=\{z \in \mathbb C~:~|z-(1+i)|=1\}$ How to find the maximum modulus of $f(z)=z^2$ on the disc $D=\{z \in \mathbb C~:~|z-(1+i)|=1\}$.
Clearly, by maximum modulus principle $\max_{z \in D}|f|$ attains on points of the form $z=(1+i)+e^{i \theta},~\theta \in \mathbb R$, Or how to find maximum of $g(x,y)=(x^2-y^2)^2+4x^2y^2,$ subject to $(x-1)^2+(y-1)^2=1$.

Geometrically, what is the point on $D$ that is the furthest from the origin? \
Analytically, why do you want $x$ and $y$?
Maximize, as you suggested, $g(\theta)=\vert 1+i+e^{i\theta }\vert^2=(\cos \theta+1)^2+(\sin \theta+1)^2=3+2\cos \theta+2 \sin \theta$, for $\theta \in [0,2\pi]$.
