On inequalities between convex, concave, and multilinear continuations of Boolean functions

The preprint is devoted to the study of inequalities between convex, concave, and multilinear continuations of Boolean functions. As a result of the study, it was proved that for a given arbitrary Boolean function of n variables, firstly, any of its convex continuation does not exceed its multilinear continuation, and secondly, its multilinear continuation does not exceed any of its concave continuations. It is also proved that equality in this sequence of inequalities can be achieved if and only if the number of essential variables of a given Boolean function is no more than one. The obtained result in a number of cases can be used when transforming systems of Boolean equations into numerical optimization problems and subsequent searches for their solutions.