Existence of a root $\alpha$ so that $|\alpha+i| <1$

Question

For some monic polynomial $P(z) = \displaystyle \sum_{k=0}^n a_k z^k, 0 < |P(i)| < 1, a_k \in \mathbb{R}, k=0,1,...,n$, how does one show that a complex root $\alpha$ exists such that $|\alpha + i|<1$? I have so far deduced that if there are $p$ real roots then their modulus is always greater than or equal to one, so the product of the moduli of all the $q=k-p$ complex roots is less than one. How does this lead to the required conclusion?

Answer 1

If you have a complex root $c_k$ such that $|c_k-i|<1, |=|\overline{c_k-i}|<1$ (modulus of complex number is equal to modulus of its conjugate).

So $|\bar{c_k}+i|<1$. But as all coefficients are real, $\bar{c_k}$ is also a root of P(z), by the complex conjugate root theorem. And so, there is at least one root $\alpha$, such that $|\alpha + i|<1$