Inequalities for rational functions in the complex plane

This paper investigates inequalities for rational functions with prescribed poles and zeros located inside or outside a disk of radius k. We extend classical polynomial inequalities due to Govil (1973, 1980) to the setting of rational functions. In particular, we establish a relationship between the maximum modulus of a rational function on the circle of radius k (where k ? 1) and on the unit circle. Our findings refine inequalities for rational functions previously obtained by Li, Mohapatra, and Rodriguez (1993), providing sharper bounds under certain conditions.