Correction to: On Bernstein’s inequality for polynomials

1 Correction to: Analysis and Mathematical Physics https://doi.org/10.1007/s13324-019-00294-x

In this addendum to the paper On Bernstein’s inequality for polynomials [Anal. Math. Phys. online 20 March 2019], we rectify the beginning of Section 5 where we mentioned a proof of Mahler’s result, i.e. the case $p=0$ in Bernstein’s inequality, using subharmonicity. In particular, we take into account a reference that we previously missed, and that Paul Nevai, whom we thank, has very recently brought to our attention.

2 Case $p=0$, Mahler’s result

This section (as well as Section 6) owes much to old conversations with Nazarov [3] during the fall of 1994, when the first-named author was finishing the joint work [4] with B. Saffari. F. Nazarov then emphasized the importance of subharmonicity.

The possible use of this subharmonicity was first alluded to by the referee (M. Marden) of Mahler’s 1961 paper [2]. But then it was also intensively used in the 1989 paper [1], which fully reproved Bernstein’s inequality in $L^0$ and then in $L^p, 0<p\le \infty$. The authors of the implied survey paper were not aware of [1] when their work was accepted in “Analysis and Mathematical Physics” and were informed of this important paper by P. Nevai , whom they thank, very shortly after their work appeared in arXiv. They apologize for this overview.