Some remarks on Turán's inequality (Q1183164)

The main result in this paper is as follows: Theorem. If \(f\in H_ n\), then for \(1\leq p\leq q\leq\infty\), \[ \| f'\|_{L^ p[-1,1]}\geq Cn^{{1\over2}-{1\over2p}+{1\over2q}}\| f\|_{L^ q[-1,1]}, \] where \(H_ n\) denotes the class of real algebraic polynomials of degree \(n\) whose zeros all lie in \([-1,1]\).