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  "body": "\n## Statement\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]] on $(0, \\infty)$.\nThen $\\Gamma$ is [[logarithmically convex|E97P413DKM]]: the function $x \\mapsto \\ln \\Gamma(x)$ is convex on $(0, \\infty)$.\nEquivalently, for all $x, y \u003e 0$ and $\\lambda \\in [0, 1]$,\n$$\\Gamma(\\lambda x + (1 - \\lambda) y) \\le \\Gamma(x)^{\\lambda} \\, \\Gamma(y)^{1 - \\lambda}.$$\n\n## Remarks\n\nWrite $t^{\\lambda x + (1-\\lambda) y - 1} e^{-t} = \\big(t^{x-1}e^{-t}\\big)^{\\lambda}\\big(t^{y-1}e^{-t}\\big)^{1-\\lambda}$ and apply [[Hölder's inequality|JQ9WXC729D]] with exponents $1/\\lambda$ and $1/(1-\\lambda)$ to Euler's integral; this gives the inequality.\nA second proof: the second derivative is $(\\ln\\Gamma)''(x) = \\sum_{n \\ge 0} \\frac{1}{(x + n)^2} \u003e 0$, so $\\ln \\Gamma$ is even strictly convex.\nExample: with $x = 1$, $y = 3$ and $\\lambda = \\tfrac12$ the inequality says $\\Gamma(2) = 1 \\le \\sqrt{\\Gamma(1)\\Gamma(3)} = \\sqrt{2}$.\nLog-convexity implies ordinary convexity of $\\Gamma$, so $\\Gamma$ has a single minimum on $(0, \\infty)$, near $x \\approx 1.4616$.\nIn Mathlib the statement is `Real.convexOn_log_Gamma`.\n\n## Explanations\n\n### L1 human\n\nThe graph of $\\ln \\Gamma$ on the positive axis bends upward everywhere.\nThis is a strong form of convexity for $\\Gamma$ itself.\n\n### L2 human\n\nLog-convexity is the property that singles out $\\Gamma$ among all functions with $f(1) = 1$ and $f(x + 1) = x f(x)$: this is the [[Bohr–Mollerup theorem|QFF87AXWN2]].\nIt gives inequalities such as Gautschi's bounds for ratios $\\Gamma(x + s)/\\Gamma(x + 1)$.\nThe positivity of $(\\ln\\Gamma)''$ says that the [[digamma function|HWRRXB0YXS]] is strictly increasing on $(0, \\infty)$.\n",
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