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The Turán-type inequality in the space L0 on the unit interval

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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″0n(e/4)‖P0, 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 · nPr 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 Lp [−1, 1]. Our theorems complement the well-known results of P. Turán, A. K. Varma, S. P. Zhou.

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The author is grateful to the referee for careful reading and useful comments.

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Correspondence to M. A. Komarov.

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Komarov, M.A. The Turán-type inequality in the space L0 on the unit interval. Anal Math 47, 843–852 (2021).

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