TITLE: complex polynomial has zeroes only in the upper half plane
QUESTION [7 upvotes]: Let $f(z)=z^{n}+a_{1}z^{n-1}+...+a_{n}$ be a polynomial with complex coefficients and suppose it has $n$ zeros in the upper half plane, that is $\operatorname{Im} z>0$, and let $\alpha_ {k}$ be the real part of $a_{k}$. Show that $\alpha(x)=x^{n}+\alpha_{1}x^{n-1}+...+\alpha_{n}$ has $n$ real distinct roots.

REPLY [2 votes]: the distinct comes from the following :
Since $$\alpha(x)=\prod_{k=1}^{n}(x-\alpha_k)$$
$$\alpha'(x)=\alpha(x)\sum_{k=1}^{n}\frac{1}{x-\alpha_k}$$