Calculate $\sup_{(a,b)\in B_2(0,1)}\left\{\max(|a|,|a+b|)\right\}$ Let
$$B_2(0,1):=\{(a,b)\in \mathbb{C}^2;\;|a|^2+|b|^2<1\}.$$

I want to calculate
  $$M:=\sup_{(a,b)\in B_2(0,1)}\left\{\max(|a|,|a+b|)\right\}.$$

I hope to show that $M\neq 1$.

Since for all $(a,b)\in B_2(0,1)$ we have
$$\max\{|a|,|a+b|\}\leq|a|+|b|\leq\sqrt{2}\sqrt{|a|^2+|b|^2}<\sqrt{2}$$
it follows that  $M\leq\sqrt{2}$.
On the other hand, if $0\leq a<1$ and $0\leq b<1$ then $\max\{|a|,|a+b|\}=a+b$. It follows that for all $r<1$ we have
$$M\geq \max\bigl\{a+b\bigm| a^2+b^2=r^2\bigr\}=\sqrt{2} r\ .$$
Since this is true for all $r<1$ we conclude that $M\geq\sqrt{2}$, so that in fact $M=\sqrt{2}$. 
