Inequalities for a class of meromorphic functions whose zeros are within or outside a given disk

In this paper, we consider a class of meromorphic functions $r(z)$ having an $s$-fold zero at the origin and establish some inequalities of Bernstein and Turán type for the modulus of the derivative of rational functions in the sup-norm on the disk in the complex plane. These results produce some sharper inequalities while taking into account the placement of zeros of the underlying rational function. Moreover, many inequalities for polynomials and polar derivatives follow as special cases. In particular, our results generalize as well as refine a result due Dewan et al. [6].