if $z \in \mathbb{C}$ Then Find the value of $$\min_{|z| \le 1} \left(max\left\{|1+z|,|1+z^2|\right\}\right)$$
i assumed $z=r(\cos\theta+i\sin \theta)$
Then we have
$$|1+z|=\sqrt{(r\cos\theta+1)^2+(r\sin\theta)^2}=\sqrt{r^2+2r\cos\theta+1}$$ and similarly
$$|1+z^2|=\sqrt{r^4+2r^2\cos2\theta+1}=\sqrt{(r^2+1)^2-4r^2\sin^2\theta}\le r^2+1$$
also $$|1+z| \le r+1$$
but how to decide maximum of these two