On zero-sector reducing operators

We prove a Jensen-disc type theorem for polynomials p∈R[z] having all their zeros in a sector of the complex plane. This result is then used to prove the existence of a collection of linear operators T:R[z]→R[z] which map polynomials with their zeros in a closed convex sector |arg⁡z|≤θ<π/2 to polynomials with zeros in a smaller sector |arg⁡z|≤γ<θ. We, therefore, provide the first example of a zero-sector reducing operator.