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  "body": "\n## Statement\n\nLet $p, q \u003e 1$ be real numbers with $\\frac{1}{p} + \\frac{1}{q} = 1$.\nThen for all real $a, b \\ge 0$,\n$$a b \\le \\frac{a^p}{p} + \\frac{b^q}{q}.$$\nEquality holds if and only if $a^p = b^q$.\n\n## Remarks\n\nIt is the weighted [[AM–GM inequality|0DRF0F3XC9]] for the numbers $a^p$ and $b^q$, with weights $1/p$ and $1/q$.\nFor $p = q = 2$ it is $a b \\le (a^2 + b^2)/2$.\nIt is the main step in the proof of [[Hölder's inequality|0BKDAGMGX9]]. Mathlib states it as `Real.young_inequality`.\n\n## Explanations\n\n### L1 human\n\nYoung's inequality bounds a product by a sum of powers.\nIt splits the product in a way that depends on the pair of exponents $p$ and $q$.\n",
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