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 +...+a_0 show that if z∈C with∣z∣≥max{1,(2/(∣a_n ∣))Σ_(i=1) ^(n−1) ∣a_i ∣} then (1/2)∣a_n ∣∣z∣^n ≤∣p(z)∣≤(3/2)∣a_n ∣∣z∣^n (i already did the right side of the inequality so try to show ∣p(z)∣≥(1/2)∣a_n ∣∣z∣^n )

$$\mathrm{let}\:\mathrm{p}\:\mathrm{be}\:\mathrm{a}\:\mathrm{non}\:\mathrm{constant}\:\mathrm{polynomial}\:\mathrm{with}\: \\ $$$$\mathrm{degree}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}:\:\mathrm{p}\left(\mathrm{z}\right)=\mathrm{a}_{\mathrm{n}} \mathrm{z}^{\mathrm{n}} +…+\mathrm{a}_{\mathrm{0}} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:\mathrm{z}\in\mathbb{C}\:\mathrm{with}\mid\mathrm{z}\mid\geqslant\mathrm{max}\left\{\mathrm{1},\frac{\mathrm{2}}{\mid\mathrm{a}_{\mathrm{n}} \mid}\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\mid\mathrm{a}_{\mathrm{i}} \mid\right\} \\ $$$$\mathrm{then}\:\frac{\mathrm{1}}{\mathrm{2}}\mid\mathrm{a}_{\mathrm{n}} \mid\mid\mathrm{z}\mid^{\mathrm{n}} \leqslant\mid\mathrm{p}\left(\mathrm{z}\right)\mid\leqslant\frac{\mathrm{3}}{\mathrm{2}}\mid\mathrm{a}_{\mathrm{n}} \mid\mid\mathrm{z}\mid^{\mathrm{n}} \\ $$$$\left(\mathrm{i}\:\mathrm{already}\:\mathrm{did}\:\mathrm{the}\:\mathrm{right}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\:\mathrm{inequality}\right. \\ $$$$\left.\mathrm{so}\:\mathrm{try}\:\mathrm{to}\:\mathrm{show}\:\mid\mathrm{p}\left(\mathrm{z}\right)\mid\geqslant\frac{\mathrm{1}}{\mathrm{2}}\mid\mathrm{a}_{\mathrm{n}} \mid\mid\mathrm{z}\mid^{\mathrm{n}} \right) \\ $$

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