Criteria for the constant sign property for real polynomials on a segment

It is well known that classic theorems of Markov and Lukach for real polynomials which have a constant sign on a segment are ineffective. In this paper we obtain criteria for the constant sign property on a segment for real polynomials of the fourth degree and formulate certain generalizations of them. The mentioned criteria are stated in terms of the coefficients of the polynomials under consideration.