Consider the sequence of polynomials given as, $p^{a}_k (x) = (1 - a \frac{d}{dx})^k x^n $ for some parameter $a>0$ and $k$ being a positive integer. For any positive integer $d$ it seems to be known that the ratio of the largest to the smallest root of $p_{k=dn}^a (x)$ is at most $\frac{d+1+2\sqrt{d} }{d+1-2\sqrt{d} } $
Can someone kindly give me a reference for a proof of this result?
Is this bound saturated by the polynomial?
