Smirnov's inequality for polynomials having zeros outside the unit disc

In 1887, the famous chemist D. I. Mendeleev posed the following problem: to estimate $|f'(x)|$ for a real polynomial $f(x)$, satisfying the condition $|f(x)|\leq M$ on $[a, b]$. This question arose when Mendeleev was studying aqueous solutions. The problem was solved by the famous mathematician A. A. Markov, and over the following 100 years was repeatedly modified and extended. For complex polynomials, important inequalities were obtained by S. N. Bernstein and V. I. Smirnov. Many other well-known mathematicians, such as Ch. Pommerenke, G. Szegö, Q. I. Rahman, G. Schmeisser, worked in this subject. Almost all results in this direction significantly use the following condition: all zeros of a majorizing polynomial belong to the closed unit disc. In this paper, we remove this condition. Here a majorizing polynomial may have zeros outside the unit disc. This allows to extend the inequalities of Bernstein and Smirnov.