If $f(z) = z^2 + 1$ we want to find the maximum value of $|f(z)|$ in $D^-(i;1)$
What I have tried is to apply the maximum module to find out we can find maximum values on the edge of the disc. After doing that I tried $|z^2+1| = |(z+i)(z-i)| \leq (|z|+1)^2 \leq (|2i|+1)^2 = 9$ but I do not know if this is correct.
Solution 1:
By MMP the maximum is atttained on the boundary. Boundary points are of the type $z=i+e^{it}$ with $t$ real. For this $z$ we have $f(z)=-1+e^{2it}+2ie^{it}+1=e^{it}(e^{it}+2i)$. So we have $|f(z)| \leq (1)(1+2)=3$ and the value $3$ is attained when $z=2i$.