On removing restrictions in the Bernstein theorem and its modifications

In 1930, S. N. Bernstein proved the following theorem: Let f and F be complex polynomials such that 1)deg f ≤ deg F = n; 2)F has all its zeros in closure of the disc Δ = {z∈ℂ:|z| < 1}; 3)|f(z)| ≤ |F(z)| for |z| = 1. Then 4)Then |f'(z)| ≤ |F'(z)| in ℂ\Δ. 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 re move these restrictions and find out how this will affect the final inequality |f'(z)| ≤ |F'(z)| of the Bernstein theorem and many of its modifications, generalizations, and consequences.