Inequalities for Rational Functions with Prescribed Poles

For rational functions $R(z)=P(z)/W(z)$, where $P$ is a polynomial of degree at the most $n$ and $W(z)=\prod_{j=1}^{n}(z-a_j)$, with $|a_j|>1,$ $j\in \{1,2,\dots,n\},$ we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdős-Lax and Turán to rational functions $R$. In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdős-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we will demonstrate how some recently obtained results could have been proved without invoking the results of Dubinin [Distortion theorems for polynomials on the circle, Sb. Math. 191(12) (2000) 1797–1807].