Some Inequalities for the Maximum Modulus of Rational Functions

For a polynomial p z of degree n , it follows from the maximum modulus theorem that max z = R ≥ 1 p z ≤ R n max z = 1 p z . It was shown by Ankeny and Rivlin that if p z ≠ 0 for z < 1 , then max z = R ≥ 1 p z ≤ R n + 1 / 2 max z = 1 p z . In 1998, Govil and Mohapatra extended the above two inequalities to rational functions, and in this paper, we study the refinements of these results of Govil and Mohapatra.