Inequalities for a rational function with restricted zeros

Recently, several authors have investigated the extremal problems for rational functions in the supremum norm on the unit disk in the plane. These investigations primarily focus on Bernstein and Tur?n-type norm estimates, which provide non-trivial generalizations and refinements of various polynomial inequalities. In this paper, we focus on obtaining bounds for the derivative of rational functions in both directions, which leads to more precise versions of certain polynomial inequalities for derivatives and polar derivatives. Furthermore, our findings generalize several well-known results from the classical literature on rational inequalities.