Smirnov and Bernstein-type inequalities, taking into account higher-order coefficients and free terms of polynomials

The starting point in the theory of differential inequalities for polynomials is the book "Investigation of aqueous solutions by specific gravity" by D. I. Mendeleev. In this work, he dealt not only with chemical, but also mathematical problems. The question raised in this book led to appearance of a large number of works on various types of differential inequalities for polynomials. In our paper, we obtain Smirnov and Bernstein-type inequalities that use higher-order coefficients and free terms of polynomials.