A Turán type inequality for rational functions with prescribed poles (Q5919864)

Let \(\Pi_n\) be the class of real algebraic polynomials of degree \(n\). \textit{P. Turán} [Compos. Math. 7, 89--95 (1939; Zbl 0021.39504)] proved that every polynomial \(f\in\Pi_n\) whose zeros all lie in the interval \([-1,1]\) satisfies the estimate \(\| f'\|\geq C\sqrt{n}\| f\|\) with a constant \(C>0\) independent of \(f\). \textit{G. Min} [Acta Math. Hung. 82, No. 1--2, 11--20 (1999; Zbl 0929.26010)] extended this result for the class of rational functions \(f\in\mathcal P(a_1,a_2,\dots,a_n):=\{P(x)\prod_{k=1}^n(x-a_k)^{-1}: P\in\Pi_n\}\) with all zeros in \([-1,1]\), where \(a_k\in\mathbb C\setminus[-1,1]\), \(k=1,\dots,n\), are nonreal numbers paired by complex conjugation and such that \(| a_k|-1>\rho\) for some \(\rho>2\). He proved also a version of this estimate for the \(L^2\) norms. In the present paper, Min's result has been extended for \(L^p\)-norms, \(1\leq p\leq\infty\), and, moreover, the assumption on \(\rho\) has been weakened in the form \(\rho>0\).