Stability of roots of polynomials under linear combinations of derivatives

Let $T={\alpha }_{0}I+{\alpha }_{1}D+\mathrm{.}\mathrm{.}\mathrm{.}+{\alpha }_n{D}^n$, where $D$ is the differentiation operator and ${\alpha }_0≠0$, 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 $-{\alpha }_1\mathrm{/}{\alpha }_0$.