Some remarks on Turán's inequality. III: The completion (Q1908572)

[For part II see the author in J. Math. Anal. Appl. 180, No. 1, 138-143 (1993; Zbl 0816.41010).] Let \(H_n\) be the class of real algebraic polynomials of degree \(n\) with all zeros lying in the interval \([- 1,1]\). In a series of research papers the author has examined Turán's inequality. His final version is as follows: Theorem. If \(f \in H_n\), then for \(0 < p \leq v \leq \infty\), \(1 - {1 \over p} + {1 \over q} \geq 0\) \[ |f' |_{L^p} \geq C^* (\sqrt n)^{1 - {1 \over p} + {1 \over q}} |f |_{L^p}. \]