Inequalities Concerning Rational Functions With Prescribed Poles

Let ${\mathrm{\Re }}_n$ be the set of all rational functions of the type $r(z)=p(z)\mathrm{/}w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z)={\prod }_{j=1}^n(z-a_j)$, $\mathrm{\mid }{a}_j\mathrm{\mid }>1$ for $1\le j\le n$. In this paper, we set up some results for rational functions with fixed poles and restricted zeros. The obtained results bring forth generalizations and refinements of some known inequalities for rational functions and in turn produce generalizations and refinements of some polynomial inequalities as well.