Solving inequalities using radical and adjacent functions

Within the framework of the nonlinear method of angular boundary functions, the existence of solutions to nonlinear boundary value problems is proven through the construction of barrier functions. Barrier functions are constructed through specially designated support barriers. The support barriers themselves can also act as barrier functions. In this case, it is necessary to prove the fulfillment of certain inequalities that are of independent functional interest. The study of these inequalities leads to cumbersome calculations. This paper proposes a method that significantly simplifies obtaining results. Possible solutions to inequalities are constructed in the form of polynomials. The initial stage involves identifying the polynomial of the highest degree of interest. Such a polynomial is called radical. Next, polynomials of lower degrees, called adjacent polynomials, are successively added to the radical polynomial.