Extremal problems of Turán-type involving the location of all zeros of a class of rational functions

In this paper, we prove a Turán-type inequality for rational functions and thereby extend it to a more general class of rational functions $r(s(z))$ of degree $mn$ with prescribed poles, where $s(z)$ is a polynomial of degree $m$. These results not only generalize some Turán-type inequalities for rational functions, but also improve as well as generalize some known polynomial inequalities.