Extremal problems of Tur´an-type involving the location of all zeros of a class of rational functions

Abstract

In this paper, we prove a Tur´an-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 Tur´antype inequalities for rational functions, but also improve as well as generalize some known polynomial inequalities.