A Turán type inequality for rational functions with prescribed poles (Q5921650)

Let \(\Pi_n\) be the set of all real algebraic polynomials of degree at most \(n\). Put \[ P_n (a_1,\dots,a_n) := \left\{ P(x)\left/\prod^n_{k=1}\right. (x-a_k); \;P\in \Pi_n\right\}, \] where \(\{a_k\}^{\infty}_{k=1} \subset \mathbb C \setminus [-1,1]\) is a fixed set of poles such that \(\prod^n_{k=1} (x-a_k)\in \Pi_n\) (that is, the non-real poles are paired by complex conjugation). \textit{G.\,Min} [Acta Math. Hung. 82, No. 1--2, 11-20 (1999; Zbl 0929.26010)] proved the following results (Turán-type inequalities in the \(L^{\infty}\) norm and in the \(L^2\) norm): Let the non-real elements in \(\{a_k\}^n_{k=1} \subset \mathbb C \setminus [-1,1]\) be paired by complex conjugation. (i) If \(|a_k|-1 > \rho\) with \(\rho >2\) for all \(k=1,\dots,n\), then \[ \|P'\|_{L^{\infty}(-1,1)} > (\sqrt{\rho^2-4}/6\rho) \sqrt{n} \|P\|_{L^{\infty} (-1,1)} \] for all \(P \in P_n(a_1,\dots,a_n)\) with all its zeros in \([-1,1]\) provided that \(n\geq \max\{(\rho +2)/9(\rho-2)\), \(4\rho^2/(\rho^2-4)\}\). (ii) For all \(P \in P_n(a_1,\dots,a_n)\) with all its zeros in \([-1,1]\), \[ \|(1-x^2)^{1/2} P'(x)\|_{L^2(-1,1)} \geq 2^{-1/2} \|(B_n(x))^{1/2} P(x) \|_{L^2(-1,1)}, \] where \[ B_n (x) = \sum^n_{k=1} (a^2_k - 1)/(x-a_k)^2 > 0, \;x \in [-1,1]. \] The aim of the paper under review is to establish the following result (a Turán-type inequality in the \(L^p\) norm, \(1\leq p \leq \infty\)): Let the non-real elements in \(\{a_k\}^n_{k=1} \subset \mathbb C \setminus [-1,1]\) be paired by complex conjugation and let \(|a_k|-1>\rho\) for some \(\rho >0\) and all \(k=1,\dots,n\). Then, for every \(f\in P_n (a_1,\dots,a_n)\) with all its zeros in \([-1,1]\), \[ \|f'\|_{L^p(-1,1)} \geq C(\rho) \sqrt{n} \|f\|_{L^p(-1,1)}, \;1\leq p \leq \infty. \] There is a lot of shortcomings (concerning printing) in the paper which makes problems to the reader.