On the Smirnov-Type Inequality for Polynomials

The presented article is devoted to differential inequalities for polynomials. The theme goes back to the problem posed by the famous chemist D. I. Mendeleev. This problem was repeatedly modificated and extended by many mathematicians. In these studies, it was usually assumed that all the zeros of a majorizing polynomial belong to the closed unit disk. We remove this requirement, replacing it with a weaker one and obtain a generalization of the Smirnov type inequality for polynomials having one zero in the exterior of the unit disk. This allow us to obtain a refinement of the Bernstein inequality, proving it not only outside the unit disk, but also in a part of the this disk.