We present a computer program solving inequalities of the form \( a_1f(\alpha _1x+(1-\alpha _1)y)+\cdots +a_nf(\alpha _nx+(1-\alpha _n)y)\leqslant \frac{1}{y-x}\int _x^yf(t)\, dt, \) where the unknown function \(f:\mathbb R\rightarrow \mathbb R\) is assumed to be continuous. This inequality includes, in particular cases, many well-known inequalities such as the classical Hermite–Hadamard inequality, the Hermite–Hadamard inequalities of higher orders, the Bullen inequality, and others. In the construction of our program, we use three theoretical results. The first one is used to show that every continuous solution of the inequality is a convex function of some order. The second one is a simple sufficient condition under which every convex function of this order satisfies the inequality we are considering. If this condition fails, the program checks a more complicated condition which is necessary and sufficient for the inequality to be satisfied by every convex function of that order. If this third condition is satisfied, then our inequality is solved completely; in the opposite case, the exact form of the solutions of the inequality in question remains unknown. However, if this is the case, we know that the quadrature in question is not definite. In the paper, we provide many examples that would be hard to calculate by hand. In the list of references, we listed many papers where computer programs were used to obtain the solution of functional equations; however, we do not know any papers of this kind dealing with inequalities.
