Inequalities for the derivatives of rational functions with real zeros (Q1307447)

One of the main results of the paper is the following extension of Turán's inequality: Let the nonreal elements \(\{a_k\}_{k=1}^n\subset\mathbb{C}\setminus[-1,1]\) be paired by complex conjugation with \(| a_k| >\rho+1>3\). Then, for all rational functions \(P\) of the form \(P(x)=Q(x)/[(x-a_1)\cdots(x-a_n)]\), where \(Q\) is a polynomial of degree \(n\) with all its zeros in \([-1,1]\), the following inequality holds: \[ \| P'\| _{[-1,1]}>\frac{\sqrt{\rho^2-1}}{6\rho}\sqrt{n}\| P\| _{[-1,1]} \] for \(n\geq\max\{\frac{\rho+2}{9(\rho-2)}, \frac{4\rho^2}{\rho^2-4}\}\). The limiting case when \(a_1,\dots,a_k,\rho\to\infty\) reduces this inequality to a well-known result of Turán. The paper also offers \(L^2\)-norm generalizations of Turán's inequality and also of an inequality due to Erdős.