For complex number z, how to show that max_{|z|<=1} |az^{n}+b|=|a|+|b| where a, b are real number?
Use the triangle inequality: $\displaystyle \displaystyle \begin{align*} |x + y| \leq |x| + |y| \end{align*}$. So
$\displaystyle \displaystyle \begin{align*} \left|a\,z^n + b\right| &\leq \left|a\,z^n\right| + |b| \\ &= |a|\left|z^n\right| + |b| \\ &= |a||z|^n + |b| \\ &\leq |a| \cdot 1^n + |b| \textrm{ since } |z| \leq 1 \\ &= |a| + |b| \end{align*}$
Therefore $\displaystyle \displaystyle \begin{align*} \max_{|z| \leq 1} \left|a\,z^n + b\right| = |a| + |b| \end{align*}$.