I'm tasked with the following problem:
Find $c\in\mathbb{R}$ such that, for arbitrary complex constants: $a_n,a_{n-1},...,a_0$, the polynomial: $$P(z)=a_n(2\pi iz)^n+a_{n-1}(2\pi iz)^{n-1}+...+a_0$$ does not vanish on $R$, where $R$ is defined as: $$R=\{z:z=x+ic, x\in\mathbb{R}\}$$
I figured that I probably had to find a $|z|$ large enough so that $z^n$ outweighed the rest of the terms so that it could never be zero, but I was not certain how to determine what $c$ would be sufficiently large to produce this result (or, conversely, what the lower bound of this $c$ would be). Any suggestions would be greatly appreciated.
Let $a=2\pi iz$. Then we just want $|a_na^n|$ to “outweigh” all the other terms. This is done by setting it to be larger than $n\max(|a_i|)|a^{n-1}|$; by the triangle inequality $|a+b|\le |a|+|b|$ so this sum is at least as large as the rest of the terms for $|a|>1$. Thus all it remains to do is choose $$|a|>\frac{n\max(|a_i|)}{ |a_n|}, 1$$and from here it should be easy to solve for $z$.