The Interplay Between Linear Functional Equations and Inequalities

We study the connections between the equation \(\begin{aligned} \sum _{i=1}^na_if(\alpha _ix+(1-\alpha _i)y)=0 \end{aligned}\) $$\begin{array}{c}\hfill \sum _{i=1}^n{a}_{i}f({\alpha }_{i}x+(1-{\alpha }_i)y)=0\end{array}$$ and the corresponding inequality. \(\begin{aligned} \sum _{i=1}^na_if(\alpha _ix+(1-\alpha _i)y)\ge 0. \end{aligned}\) $$\begin{array}{c}\hfill \sum _{i=1}^n{a}_{i}f({\alpha }_{i}x+(1-{\alpha }_i)y)\ge 0.\end{array}$$ At present, it is clear that linear functional equations should be solved with the use of the results due to L. Székelyhidi. We show that the simplest and most efficient way of dealing with the (continuous) solutions of linear inequalities of the above form is connected with the use of stochastic orderings tools. It will be shown that, in order to solve the inequality, we need to know the continuous solutions of the corresponding equation. In the last part of the paper, we obtain some simple but unexpected connections in the other direction.