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  "body": "\n## Definition\n\nLet $D$ be a set, let $\\mathcal{F}$ be a set of [[functions|0EPAT51DXQ]] $f \\colon D \\to \\R$, and let $n \\ge 1$.\nLet $S \\subseteq D^n$ be a set.\nLet $E_1$ and $E_2$ be rules that assign a real number to each $f \\in \\mathcal{F}$ and each $(x_1, \\ldots, x_n) \\in S$.\nThe rules are built from $f$, the variables and fixed given functions by composition and arithmetic operations.\nThe *functional inequality* given by $E_1$ and $E_2$ is the condition\n$$E_1(f; x_1, \\ldots, x_n) \\le E_2(f; x_1, \\ldots, x_n) \\quad \\text{for all } (x_1, \\ldots, x_n) \\in S,$$\nand a *solution* is a function $f \\in \\mathcal{F}$ that satisfies it.\n\n## Remarks\n\nExample: the subadditivity inequality $f(x + y) \\le f(x) + f(y)$ on $D = [0, \\infty)$, with $S = D^2$.\nThe function $f(x) = \\sqrt{x}$ is a solution: both sides are nonnegative, and squaring gives $x + y \\le x + y + 2\\sqrt{xy}$.\n\nNon-example: $f(x) = x^2$ is not a solution of the same inequality, since $f(1 + 1) = 4 \u003e 2 = f(1) + f(1)$.\n\nAs with a [[functional equation|SRAN4EKWS0]], the class $\\mathcal{F}$ matters: one may ask for all solutions, or only for continuous, measurable or bounded ones.\nA system that mixes equations and inequalities is also common, for example a function that is additive and bounded above.\n",
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