\text { A bit on the zeros of } D_\alpha f(z) \text { of a polynomial } f(z)

Abstract

In this paper, we prove a variant of enestrom and kakeya theorem. Indeed, for a given a polynomial $f(z)$ with real coefficients, we are providing a bounded region such that any zero of $D_\alpha f(z)$ lie in this region must be a simple zero if coefficients of $D_\alpha^{\prime} f(z)$ are monotonic, and any zero of $D_\alpha f(z)$ which does not lie in this region must be a simple zero if coefficients of $D_\alpha^{\prime} f(z)$ are alternative.