LAUSR

Let P(x) be an arbitrary algebraic polynomial of degree n with all zeros in the unit interval −1 ≤ x ≤ 1. We establish the Turan-type inequality ‖P″‖0 ≥ n(e/4)‖P‖0, where $${\left\| f \right\|_0} = \exp \left( {{1 \over 2}\int_{ - 1}^1 {\ln \left| {f(x)} \right|\,dx} } \right)$$ is the geometric mean of a function. This estimate is extremal for any even n. We also obtain the following Turan-type inequality in different metrics: ‖P″‖s >C · n‖P‖r for 0 0 is a constant only depending on s, r and ‖·‖p is the standard norm in Lp [−1, 1]. Our theorems complement the well-known results of P. Turan, A. K. Varma, S. P. Zhou.