let-p-be-a-non-constant-polynomial-with-degree-n-such-that-p-z-a-n-z-n-a-0-show-that-if-z-C-with-z-max-1-2-a-n-i-1-n-1-a-i-then-1-2-a-n-z-n-p-z-3-2-a-n-z-n-i

Question Number 171145 by floor(10²Eta[1]) last updated on 08/Jun/22

let p be a non constant polynomial with

degree n such that : p (z) = a_{n} z^{n} + \dots + a_{0}

show that if z \in C with ∣ z ∣⩾ \max {1, \frac{2}{∣ a_{n} ∣} \underset{i = 1}{\sum^{n - 1}} ∣ a_{i} ∣}

then \frac{1}{2} ∣ a_{n} ∣∣ z ∣^{n} ⩽∣ p (z) ∣⩽ \frac{3}{2} ∣ a_{n} ∣∣ z ∣^{n}

(i already did the right side of the inequality

so try to show ∣ p (z) ∣⩾ \frac{1}{2} ∣ a_{n} ∣∣ z ∣^{n})