How can I prove that $_{max}|Az^{n}+b|$ =$ |A|$ + $|B|$ when |Z| $\leq$ 1?

Question

How can I prove that $_{max}$$|Az^{n}+B|$ = $|A| + |B|$ when $|Z|$ $\leq$ 1?

Remember that Z is a complex number which is why I had to include the magnitude.

Answer 1

first, if $|z|\leq1$, then $|Az^{n}+B|\leq |A|+|B|$

Let $A=r_1e^{i\alpha}, B=r_2e^{i\beta}$. Take

$$z=e^{i\frac{\beta-\alpha}n}$$

then $Az^{n}+B=r_1e^{i\beta}+r_2e^{i\beta}$, so $|Az^{n}+B|=r_1+r_2=|A|+|B|$