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  "body": "\n## Definition\n\nLet $I$ be an [[interval|0TK8Y3J5RW]] of $\\R$ and let $f \\colon I \\to (0, \\infty)$ be a function.\nThe function $f$ is *logarithmically convex*, or *log-convex*, if the function $\\ln \\circ f \\colon I \\to \\R$ is [[convex|0VV0HSW1ZS]], where $\\ln$ is the [[natural logarithm|0044CPHJBV]].\n\n## Remarks\n\nEquivalently, $f(t x + (1 - t) y) \\le f(x)^{t} f(y)^{1 - t}$ for all $x, y \\in I$ and $t \\in [0, 1]$.\nBy the [[AM–GM inequality|0DRF0F3XC9]], $f(x)^t f(y)^{1-t} \\le t f(x) + (1 - t) f(y)$, so a log-convex function is convex.\n\nExample: $f(x) = e^{x^2}$ on $\\R$ is log-convex, because $\\ln f(x) = x^2$ and $x^2$ is convex.\nThe [[gamma function|0V3FZMVB4K]] on $(0, \\infty)$ is also log-convex.\n\nNon-example: $f(x) = x^2$ on $(0, \\infty)$ is convex, but $\\ln f(x) = 2 \\ln x$ is strictly concave, so $f$ is not log-convex.\n\nSome authors allow $f$ to take the value $0$, with the convention $\\ln 0 = -\\infty$.\n",
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