Generalization of certain well-known inequalities for rational functions

Abstract

Let be a class of all polynomials of degree at most m and let where and denote the class of rational functions. It is proved that if the rational function having all its zeros in , then for . The main purpose of this paper is to improve the above inequality for rational functions having all its zeros in with -fold zeros at the origin and some other related inequalities. The obtained results sharpen some well-known estimates for the derivative and polar derivative of polynomials.