Stability of roots of polynomials under linear combinations of derivatives

Let $T=\alpha_0 I + \alpha_1 D + ...+\alpha_n D^n$, where $D$ is the differentiation operator and $\alpha_0\not= 0$, and let $f$ be a square-free polynomial with large minimum root separation. We prove that the roots of $Tf$ are close to the roots of $f$ translated by $-\alpha_1/\alpha_0$.