Some Inequalities for the Maximum Modulus of Rational Functions

For a polynomial $p\left(z\right)$ of degree $n$ , it follows from the maximum modulus theorem that ${\max }_{\left|z\right|=R\ge 1}\left|p\left(z\right)\right|\le R^n{\max }_{\left|z\right|=1}\left|p\left(z\right)\right|$ . It was shown by Ankeny and Rivlin that if $p\left(z\right)\ne 0$ for $\left|z\right|<1$ , then ${\max }_{\left|z\right|=R\ge 1}\left|p\left(z\right)\right|\le \left(\left(R^n+1\right)/2\right){\max }_{\left|z\right|=1}\left|p\left(z\right)\right|$ . 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.