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.