Mechanical decision for a class of integral inequalities

Abstract

A class of integral inequalities is transformed into homogeneous symmetric polynomial inequalities beyond Tarski model, where the number of elements of the polynomial, say n, is also a variable and the coefficients are functions of n. This is closely associated with some open problems formulated recently by Yang et al. Using Timofte's dimension-decreasing method for symmetric polynomial inequalities, combined with the inequality-proving package BOTTEMA and a program of implementing the method known as successive difference substitution, we provide a procedure for deciding the nonnegativity of the corresponding polynomial inequality such that the original integral inequality is mechanically decidable; otherwise, a counterexample will be given. The effectiveness of the algorithm is illustrated by some more examples.