Document Type : Research Paper

Authors

• Ishfaq Ahmad Dar
• Nisar Ahmad Rather
• Mohd Shafi Wani

Department of Mathematics, University of Kashmir, Srinagar-190006, India

Abstract

Let $r(z)=f(z)/w(z)$ where $f(z)$ be a polynomial of degree at most $n$ and $w(z)=\prod _{j=1}^n(z-a_j)$, $|a_j|>1$ for $1\le j\le n.$ If the rational function $r(z)\ne 0$ in $|z|<k$, then for $k=1$, it is known that $$|r(Rz)|\le \left(\frac{|B(Rz)|+1}{2}\right)\underset{|z|=1}{sup}|r(z)|for|z|=1$$ where $B(z)=\prod _{j=1}^n\{(1-\overline{a_j}z)/(z-a_j)\}$. In this paper, we consider the case $k\ge 1$ and obtain certain results concerning the growth of the maximum modulus of the rational functions with prescribed poles and restricted zeros in the Chebyshev norm on the unit circle in the complex plane.

Keywords

• Rational functions
• Polynomial Inequalities
• Zeros