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  "body": "\n## Summary\n\nA functional inequality is an inequality between expressions in an unknown function.\nIt is required to hold for all values of the variables.\nIt is the inequality analogue of a [[functional equation|SRAN4EKWS0]].\nStandard examples are subadditivity $f(x + y) \\le f(x) + f(y)$ and midpoint convexity $f\\big(\\tfrac{x + y}{2}\\big) \\le \\tfrac{f(x) + f(y)}{2}$.\nTypical questions are which functions satisfy a given inequality, and when mild regularity forces its solutions to be nice.\n\n## Explanations\n\n### L1 human\n\nA functional equation asks for functions that make two expressions equal everywhere; a functional inequality only asks that one expression is never larger than the other.\nFor example, the inequality $f(x + y) \\le f(x) + f(y)$ says that the value at a sum is at most the sum of the values.\nThe square root on $[0, \\infty)$ satisfies it, while the squaring map does not.\nConvex functions, norms and many quantities in probability and analysis are described by such inequalities.\n",
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