## Abstract

Let *P*(*x*) be an arbitrary algebraic polynomial of degree *n* with all zeros in the unit interval −1 ≤ *x* ≤ 1. We establish the Turán-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 Turán-type inequality in different metrics: ‖*P″*‖_{s} >*C · n*‖*P*‖_{r} for 0 < *r* < 1, *r*/(1 − *r*) < *s* ≤ ∞, where *C* > 0 is a constant only depending on *s, r* and ‖·‖_{p} is the standard norm in *L*_{p} [−1, 1]. Our theorems complement the well-known results of P. Turán, A. K. Varma, S. P. Zhou.

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Komarov, M.A. The Turán-type inequality in the space *L*_{0} on the unit interval.
*Anal Math* **47**, 843–852 (2021). https://doi.org/10.1007/s10476-021-0097-3

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DOI: https://doi.org/10.1007/s10476-021-0097-3