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function_integral_on_ℝⁿ [2016/02/06 15:02] nikolaj |
function_integral_on_ℝⁿ [2016/02/06 15:07] nikolaj |
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| ^ $ {\mathrm d}\left(\int_{v(y)}^{w(y)}\,f(x)\,{\mathrm d}x\right) = f(v(y))\,{\mathrm d}v(y)-f(w(y))\,{\mathrm d}w(y) $ ^ | ^ $ {\mathrm d}\left(\int_{v(y)}^{w(y)}\,f(x)\,{\mathrm d}x\right) = f(v(y))\,{\mathrm d}v(y)-f(w(y))\,{\mathrm d}w(y) $ ^ | ||
| - | For $f$ convex: | + | For $f$ convex and |
| - | ^ $\dfrac{1}{b - a}\int_a^b f(x)\,dx \ge f \left(\dfrac{a+b}{2}\right) $ ^ | + | $\langle f\rangle_{[a,b]}:=\dfrac{1}{b - a}\int_a^b f(x)\,{\mathrm d}x$ |
| - | ^ $\dfrac{1}{b - a}\int_a^b f(x)\,dx \le \dfrac{f(a) + f(b)}{2}$ ^ | + | |
| - | ==== Parents ==== | + | ^ $\dfrac{f(a) + f(b)}{2} \ge \langle f\rangle_{[a,b]} \ge f \left(\dfrac{a+b}{2}\right) $ ^ |
| + | |||
| + | See references. | ||
| + | |||
| + | === References === | ||
| + | Wikipedia: | ||
| + | [[https://en.wikipedia.org/wiki/Hermite%E2%80%93Hadamard_inequality|Hermite–Hadamard inequality]] | ||
| + | |||
| + | ----- | ||
| === Subset of === | === Subset of === | ||
| [[Function integral]] | [[Function integral]] | ||
