Inequalities for the derivative of rational functions with prescribed poles and restricted zeros

In this paper, instead of assuming that a rational function $r(z)$ with prescribed poles has a zero of order $s$ at origin, we suppose that it has a zero of multiplicity s at any point inside the unit circle, whereas the remaining zeros are within or outside a circle of radius $k$ and prove some results which besides generalizing some inequalities for rational functions include refinements of some polynomial inequalities as special cases.