---
tag: 498VVT12YJ
kind: definition
name: Functional inequality
msc: [39B62, 39B72]
defines: YQB2B0EXY8
uses: [0EPAT51DXQ, SRAN4EKWS0]
sources:
  - url: https://en.wikipedia.org/wiki/Functional_equation
---

## Definition

Let $D$ be a set, let $\mathcal{F}$ be a set of [[functions|0EPAT51DXQ]] $f \colon D \to \R$, 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.

## Remarks

Example: the subadditivity inequality $f(x + y) \le f(x) + f(y)$ on $D = [0, \infty)$, with $S = D^2$.
The function $f(x) = \sqrt{x}$ is a solution: both sides are nonnegative, and squaring gives $x + y \le x + y + 2\sqrt{xy}$.

Non-example: $f(x) = x^2$ is not a solution of the same inequality, since $f(1 + 1) = 4 > 2 = f(1) + f(1)$.

As with a [[functional equation|SRAN4EKWS0]], the class $\mathcal{F}$ matters: one may ask for all solutions, or only for continuous, measurable or bounded ones.
A system that mixes equations and inequalities is also common, for example a function that is additive and bounded above.