Functional inequality

Definition

Let $D$ be a set, let $\mathcal{F}$ be a set of functions $f \colon D \to \R$, and let $n \ge 1$. Let $S \subseteq D^n$ be a set. Let $E_1$ and $E_2$ be rules that assign a real number to each $f \in \mathcal{F}$ and each $(x_1, \ldots, x_n) \in S$. The rules are built from $f$, the variables and fixed given functions by composition and arithmetic operations. The functional inequality given by $E_1$ and $E_2$ is the condition $$E_1(f; x_1, \ldots, x_n) \le E_2(f; x_1, \ldots, x_n) \quad \text{for all } (x_1, \ldots, x_n) \in S,$$ and a solution is a function $f \in \mathcal{F}$ that satisfies it.