On removing restrictions in the Bernstein theorem and its modifications

E. G. Kompaneets, V. V. Starkov, E. S. Shmidt

Petrozavodsk State University, 33 Lenina pr., Petrozavodsk 185910, Russia

Abstract: In 1930, S. N. Bernstein proved the following theorem: Let $f$ and $F$ be complex polynomials such that 1) $\mathrm{deg}\, f \le \mathrm{deg}\, F=n$; 2) $F$ has all its zeros in closure of the disc $\Delta=\{z\in \mathbb{C}\colon |z|<1\}$; 3) $|f(z)| \le |F(z)|$ for $|z|=1$. Then $|f'(z)| \le |F'(z)|$ in $\mathbb{C}\setminus\Delta$. In a huge number of papers that appeared after 1930 and related to this theorem, the restrictions on the geometry of domains and conditions 1) and 2) of the theorem usually remained unchanged. In this article, we consistently remove these restrictions and find out how this will affect the final inequality $|f'(z)| \le |F'(z)|$ of the Bernstein theorem and many of its modifications, generalizations, and consequences.

Keywords: differential inequalities for polynomials, differential operator, $L^p$ inequalities, convex sets.

UDC: 517.53

MSC: 30C10, 30A10

Received: 20.09.2025
Revised: 03.11.2025
Accepted: 03.11.2025

Language: English

DOI: 10.15393/j3.art.2025.19212