Let $p(z)=z^{n}+a_{n-1}z^{n-1}+\dots+a_{1}z+a_{0}$ (where $a_{0},a_{1},\dots,a_{n-1}$ are complex numbers). If $|p(z)|\leq 1$ for all $z\in\mathbb{C}$ with $|z|\leq 1$ then prove that $a_{0}=a_{1}=\dots=a_{n-1}=0$. I have tried to get an estimate for each $|a_{j}|$ by using the fact that $|p(z)|\leq 1$ but I didn't succeed. Is there a way to prove the result without using Maximum Modulus principle, Cauchy estimate, Cauchy Integral Formula etc and by just using simple and basic inequalities?
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@MartinR given OP's subsequent edit, I think a better duplicate is this one: https://math.stackexchange.com/questions/2278087/maximum-value-of-a-complex-polynomial-on-the-unit-disk e.g. it follows by inspection for degree 1 polynomials [base case] and for degree $n\geq 2$ the linked manipulation of roots of unity & triangle inequality proves that $a_0=0$, so $p(z)=z\cdot q(z)$ and the result follows by induction hypothesis. – user8675309 Aug 05 '23 at 17:15
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1@user8675309: I have added that to the list of duplicate targets. – Martin R Aug 05 '23 at 17:22