On the derivative with respect to a point

Abstract:

The derivative of a polynomial $p(z)$ with respect to a point $\varsigma$ is defined by the formula ${A_\varsigma }p(z) = (\varsigma - z)p’(z) + np(z)$, where $n$ is the degree of the polynomial. Let $p(z)$ have all its zeros in the unit disk and one zero at $z = 1$. We determine a minimal region that must contain at least one zero of ${A_\varsigma }p(z)$.