Is my logic true?

Question

$\sup\space{|a_1(z_1-z_2)+a_2(z_1^2-z_2^2)+\space...|}\space\ge\space \sup\space{|a_1(z_1-z_2)|}$

If it is, then how can I prove it?

Here $z_1,z_2\space\in\space\mathbb C\space$ such that $|z_1|,|z_2|\le1$

Answer 1

Take $a_1=1000, a_2=-1,$ all others $0$. The right is $2000,$ achieved when $z_1,z_2$ are opposite on the unit circle. In those cases the left will be $1998$