If the real part of a polynomial P(is), where s is real, is always positive, then is it possible for the real part of the polynomial to have negative value as s goes to infinity?
This means that the polynomial never crosses the imaginary axis except possibly at infinity, when the domain is the imaginary axis.
For example, Re P(z)= z^4 +8z^3 + 3z^2 + 8z + 3, has no zero, when z=is, and s is real. Also P(iR) and P(-IR) both are in the right half plane when R goes to infinity. Will P(iR) and P(-iR) always be in the right half plane for all polynomials, when P(is) is always positive?
If a polynomial $p$ is not constant, then the equation $p(z)=-1$ has a solution, so the real part of any non-constant polynomial cannot be "always positive".