  We prove a Jensen-disc type theorem for polynomials $p\in\mathbb{R}[z]$
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
$T\colon\mathbb{R}[z]\to\mathbb{R}[z]$ which map polynomials with their zeros
in a closed convex sector $|\arg z| \leq \theta<\pi/2$ to polynomials with
zeros in a smaller sector $|\arg z| \leq \gamma<\theta$. We, therefore, provide
the first example of a zero-sector reducing operator.
