Let $a_k$, $b_k$, $c_k$ for $k \in \{1,\dots,n\}$ be nonnegative real numbers such that $a_k+b_k > 0$ for all $k$. Let $$ p(z) = \prod_{k=1}^n (a_k z^2 + b_k z + c_k) - c_1c_2\dots c_n, \qquad z \in \mathbb{C}. $$ Clearly $p(x) > 0$ for all $x > 0$, that is $p$ has no roots on the positive half-line. Is it true for the right half-plane?
Is it true that $p(z) \neq 0$ for all $z$ such that $\mathrm{Re}(z) > 0$?
Is it possible to somehow bound $|p(z)|$ from below on the right half-plane? Or maybe I should use Rouche's theorem?