If , then show that there exists a real such thatfor

Question

If $p(z)=a_{0}+a_{1}z+....+a_{n-1}z^{n-1}+z^{n}(n\geq 1)$ , then show that there exists a real $R> 0$ such that $2^{^{-1}}\left | z \right |^{n}\leq p(z)\left | \leq 2 \right |z|^n$ for $\left | z \right |\geq R$ .

21 Mar 2023

Answer :

Word Count : 281

To show the existence of a real R > 0 such that the inequality holds, we can use the triangle inequality for complex numbers. First, let's rewrite the polynomial p(z) in the following form:

p(z) = z^n + a_{n-1}z^{n-1} + ... + a_1z + a_0

Now, let's consider the absolute value of p(z):

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