基本关系式
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取整恒等式
$x = ⌊x⌋ + [x]$
$⌈x⌉ = ⌊x⌋ + \mathrm{sgn} [x]$
典例：$2.71828 = ⌊2.71828⌋ + [2.71828] = 2 + 0.71828$，$-2.71828 = ⌊-2.71828⌋ + [-2.71828] = -2 + -0.71828$
典例：$3 = ⌈2.71828⌉ = ⌊2.71828⌋ + \mathrm{sgn} [2.71828] = 2 + 1$，$-3 = ⌈-2.71828⌉ = ⌊-2.71828⌋ + \mathrm{sgn} [-2.71828] = -2 + -1$
最值恒等式
| $\max\limits_{x∈X}\lbrace f (x), g (x) \rbrace = \dfrac{f (x) + g (x) + | f (x) - g (x) | }{2} = \dfrac{f (x) + g (x) + \sqrt{[ f (x) - g (x) ]^2} }{2}$ |
| $\min\limits_{x∈X}\lbrace f (x), g (x) \rbrace = \dfrac{f (x) + g (x) - | f (x) - g (x) | }{2} = \dfrac{f (x) + g (x) - \sqrt{[ f (x) - g (x) ]^2} }{2}$ |
$\min\limits_{x∈X} \lbrace f (x), g (x) \rbrace = -\max\limits_{x∈X} \lbrace -f (x), -g (x) \rbrace$
组合恒等式
$\mathrm{C} (n, i) = \dfrac{n!}{i! · (n - i)!} = \dfrac{n!}{(n - i)! · i!} = \mathrm{C} (n, n - i)$
$\mathrm{C} (n + 1, i) = \dfrac{(n + 1)!}{i! · (n + 1 - i)!} = \dfrac{n!}{i! · (n - i)!} + \dfrac{n!}{(i - 1)! · (n - i + 1)!} = \mathrm{C} (n, i) + \mathrm{C} (n, i - 1)$
$(y + x)^{n} = \sum\limits_{i=0}^{n} \dfrac{n!}{i! · (n - i)!} · y^{n-i} · x^{i}$
| $(y + x)^{0}$ | $1$ |
|---|---|
| $(y + x)^{1}$ | $y^{1} · x^{0} + y^{0} · x^{1}$ |
| $(y + x)^{n}$ | $\sum\limits_{i=0}^{n} \dfrac{n!}{i! · (n - i)!} · y^{n - i} · x^{i}$ |
| $(y + x)^{1} · (y + x)^{n}$ | $\sum\limits_{i=0}^{n} \dfrac{n!}{i! · (n - i)!} · \left[ y^{n-i+1} · x^{i} + y^{n-i} · x^{i+1} \right]$ |
| $y^{n+1} · x^{0} + \left[ \sum\limits_{i=1}^{n} \dfrac{n!}{i! · (n - i)!} y^{n-i+1} · x^{i} + \sum\limits_{i=1}^{n} \dfrac{n!}{(i - 1)! · (n - i + 1)!} · y^{n-(i-1)} · x^{i} \right] + y^{0} · x^{n+1}$ | |
| $y^{n+1} · x^{0} + \left[ \sum\limits_{i=1}^{n} \dfrac{(n + 1)!}{i! · (n + 1 - i)!} · y^{n+1-i} · x^{i} \right] + y^{0} · x^{n+1}$ | |
| $(y + x)^{n+1}$ | $\sum\limits_{i=0}^{n+1} \dfrac{(n + 1)!}{i! · (n + 1 - i)!} · y^{n-i+1} · x^{i}$ |
等比恒等式
$y^{n+1} - x^{n+1} = (y - x) · \sum\limits_{i=0}^{n} y^{n-i} · x^{i}$
$\dfrac{1 - x^{n+1} }{1 - x} = \sum\limits_{i=0}^{n} x^{i}$
| $y^{n+1} - x^{n+1}$ | $(y - x) · \sum\limits_{i=0}^{n} y^{n-i} · x^{i} = \sum\limits_{i=0}^{n} y^{n-i+1} · x^{i} - \sum\limits_{i=0}^{n} y^{n-i} · x^{i+1} = y^{n+1} + \left[ \sum\limits_{i=1}^{n} y^{n-i+1} · x^{i} - \sum\limits_{i=1}^{n} y^{n-(i-1)} · x^{i} \right] - x^{n+1} = y^{n+1} - x^{n+1}$ |
|---|---|
| $1 - x^{n+1}$ | $(1 - x) · \sum\limits_{i=0}^{n} x^{i}$ |
| $\dfrac{1 - x^{n+1} }{1 - x}$ | $\sum\limits_{i=0}^{n} x^{i}$ |
等幂恒等式
$I(m, n) = \dfrac{(m + 1)^{n+1} }{n + 1} - \sum\limits_{i=0}^{n-1} \dfrac{n!}{i! · (n + 1 - i)!} · I(m,i)$
$\sum\limits_{j=0}^{m} j^{0} = \dfrac{(m + 1)^{1} }{1}$
$\sum\limits_{j=0}^{m} j^{1} = \dfrac{(m + 1)^{2} }{2} - \dfrac{(m + 1)^{1} }{2}$
$\sum\limits_{j=0}^{m} j^{2} = \dfrac{(m + 1)^3}{3} - \dfrac{(m + 1)^{2} }{2} + \dfrac{(m + 1)^{1} }{6}$
$\sum\limits_{j=0}^{m} j^{3} = \dfrac{(m + 1)^{4} }{4} - \dfrac{(m + 1)^3}{2} + \dfrac{(m + 1)^{2} }{4}$
| $⇓$ | $(x + 1)^{n+1}$ | $= \sum\limits_{i=0}^{n+1} \dfrac{(n + 1)!}{i! · (n + 1 - i)!} · x^{i} = \sum\limits_{i=0}^{n} \dfrac{(n + 1)!}{i! · (n + 1 - i)!} · x^{i} + x^{n+1}$ | |
|---|---|---|---|
| $⇓$ | $(m + 1)^{n+1} - 0^{n+1} = \sum\limits_{j=0}^{m} [(j + 1)^{n+1} - j^{n+1}]$ | $= \sum\limits_{j=0}^{m} \sum\limits_{i=0}^{n} \dfrac{(n + 1)!}{i! · (n + 1 - i)!} · j^{i} = \sum\limits_{i=0}^{n} \dfrac{(n + 1)!}{i! · (n + 1 - i)!} · \sum\limits_{j=0}^{m} j^{i} = (n + 1) · \sum\limits_{j=0}^{m} j^{n} + \sum\limits_{i=0}^{n-1} \dfrac{(n + 1)!}{i! · (n + 1 - i)!} · \sum\limits_{j=0}^{m} j^{i}$ | |
| $⇓$ | $\sum\limits_{j=0}^{m} j^{n}$ | $= \dfrac{(m + 1)^{n + 1} }{n + 1} - \sum\limits_{i=0}^{n-1} \dfrac{n!}{i! · (n + 1 - i)!} · \sum\limits_{j=0}^{m} j^{i}$ | |
| $⇓$ | $I(m, n)$ | $= \dfrac{(m + 1)^{n+1} }{n + 1} - \sum\limits_{i=0}^{n-1} \dfrac{n!}{i! · (n + 1 - i)!} · I(m,i)$ | |
| $⇓$ | $0^{0} ≡ 1$ | $\sum\limits_{j=0}^{m} j^{0} = I(m, 0)$ | $= \dfrac{(m + 1)^{1} }{1} - \dfrac{1}{1} · \mathop{I(m, -1)}\limits_{≡0}$ |
| $⇓$ | $\sum\limits_{j=0}^{m} j^{1} = I (m, 1)$ | $= \dfrac{(m + 1)^{2} }{2} - \dfrac{1}{2} · I(m, 0) = \dfrac{(m + 1)^{2} }{2} - \dfrac{(m + 1)^{1} }{2}$ | |
| $⇓$ | $\sum\limits_{j=0}^{m} j^{2} = I(m, 2)$ | $= \dfrac{(m + 1)^{3} }{3} - \dfrac{1}{3} · I(m, 0) - \dfrac{1}{1} · I(m, 1) = \dfrac{(m + 1)^3}{3} - \dfrac{(m + 1)^{2} }{2} + \dfrac{(m + 1)^{1} }{6}$ | |
| $⇓$ | $\sum\limits_{j=0}^{m} j^{3} = I(m, 3)$ | $= \dfrac{(m + 1)^{4} }{4} - \dfrac{1}{4} · I(m, 0) - \dfrac{1}{1} · I(m, 1) - \dfrac{3}{2} · I(m, 2) = \dfrac{(m + 1)^{4} }{4} - \dfrac{(m + 1)^3}{2} + \dfrac{(m + 1)^{2} }{4}$ |
复数恒等式
$ә^{x} = \cosh x + \sinh x$
$ә^{τ·x} = \cos x + τ · \sin x$
$ә^{+τ·k·π} = (-1)^k$
$ә^{+τ·(k·π+\frac{π}{2})} = (-1)^{k} · τ$
| $\cosh x = \sum\limits_{i=0}^{∞^{+} } \dfrac{1}{(2 · i)!} · x^{2 · i}$ | $\sinh x = \sum\limits_{i=0}^{∞^{+} } \dfrac{1}{(2 · i + 1)!} · x^{2 · i + 1}$ |
|---|---|
| $ә^{x} = \cosh x + \sinh x$ | $ә^{x} = \sum\limits_{i=0}^{∞^{+} } \dfrac{1}{i!} · x^i = \sum\limits_{i=0}^{∞^{+} } \dfrac{1}{(2 · i)!} · x^{2 · i} + \sum\limits_{i=0}^{∞^{+} } \dfrac{1}{(2 · i + 1)!} · x^{2 · i + 1}$ |
| $\cos x = \sum\limits_{i=0}^{∞^{+} } \dfrac{(-1)^i}{(2 · i)!} · x^{2 · i}$ | $\sin x = \sum\limits_{i=0}^{∞^{+} } \dfrac{(-1)^{i} }{(2 · i + 1)!} · x^{2 · i + 1}$ |
| $ә^{τ·x} = \cos x + τ · \sin x$ | $ә^{τ·x} = \sum\limits_{i=0}^{∞^{+} } \dfrac{τ^{i} }{i!} · x^i = \sum\limits_{i=0}^{∞^{+} } \dfrac{(-1)^i}{(2 · i)!} · x^{2 · i} + τ · \sum\limits_{i=0}^{∞^{+} } \dfrac{(-1)^{i} }{(2 · i + 1)!} · x^{2 · i + 1}$ |
| $ә^{+τ·k·π} = (-1)^k$ | $ә^{+τ·(k·π+\frac{π}{2})} = (-1)^{k} · τ$ |
阶乘方不等式
$n! \mathop{≡≡}\limits^{0!≡1} n · (n - 1)!$
$[n!]^2 ≥ n^{n}$
$\left( \dfrac{n + 1}{ә} \right)^{n} < n! < (n + 1) · \left( \dfrac{n + 1}{ә} \right)^{n}$
| $n ≥ i+ 1$ | $n ≥ i + 1$ | $n = 1,2$ |
|---|---|---|
| $(n - i) · (i + 1) - n = i · [n - (i + 1)] ≥ 0$ | $[n!]^{2} = [(n - 0) · 1] · [(n - 1) · 2] ··· [2 · (n - 1)] · [1 · (n - 0)]$ | |
| $(n - i) · (i + 1) ≥ n$ | $[n!]^2 ≥ n^{n}$ | $[n!]^2 = n^n$ |
| $\dfrac{(n + 1)^{n} }{n^{n} } = \left( 1 + \dfrac{1}{n} \right)^{n} < ә < \left( 1 + \dfrac{1}{n} \right)^{n+1} = \dfrac{(n + 1)^{n+1} }{n^{n+1} }$ | $\dfrac{(n + 1)^{n} }{n!} = \prod\limits_{i=1}^{n} \dfrac{(n + 1)^{n} }{n^{n} } < ә^{n} < \prod\limits_{i=1}^{n} \dfrac{(n + 1)^{n+1} }{n^{n+1} } = \dfrac{(n + 1)^{n+1} }{n!}$ | $\left( \dfrac{n + 1}{ә} \right)^{n} < n! < (n + 1) · \left( \dfrac{n + 1}{ә} \right)^{n}$ |
凹函数不等式
函数$f(x)$在点$(x_0,f(x_0))$与点$(x_1,f(x_1))$此两点处的切割线函数。
$g(x) = \dfrac{x_1 - x}{x_1 - x_0} · f (x_0) + \dfrac{x - x_0}{x_1 - x_0} · f (x_1)$
$g[(1 - t) · x_0 + t · x_1] = (1 - t) · f(x_0) + t · f(x_1)$
| $⇓$ | $0 = T(t) ≡ \left | \begin{matrix} 1 & 1 & 1 \ 0 & t & 1 \ x_0 & x & x_1 \end{matrix}\right | = \left | \begin{matrix} 1 & 0 & 0 \ 0 & t & 1 \ x_0 & x - x_0 & x_1 - x_0 \end{matrix}\right | $ | $⇒$ | $0 = T(t) ≡ t · (x_1 - x_0) - (x - x_0)$ |
|---|---|---|---|---|---|---|---|
| $⇓$ | $x = (1 - t) · x_0 + t · x_1$ | $⇒$ | $(t, x(t)) = (0,x_0);(1,x_1)$ | ||||
| $⇓$ | $t = \dfrac{x - x_0}{x_1 - x_0}$ | $⇒$ | $1 - t = \dfrac{x_1 - x}{x_1 - x_0}$ | ||||
| $⇓$ | $0 = F(x) ≡ \left | \begin{matrix} 1 & 1 & 1 \ x_0 & x & x_1 \ f(x_0) & g(x) & f(x_1) \end{matrix}\right | = \left | \begin{matrix} 1 & 0 & 0 \ x_0 & x - x_0 & x_1 - x_0 \ f(x_0) & g(x) - f(x_0) & f(x_1) - f (x_0) \end{matrix}\right | $ | $⇒$ | $0 = F(x) ≡ (x - x_0) · [f(x_1) - f(x_0)] - (x_1 - x_0)·[g(x) - f(x_0)]$ |
| $⇓$ | $g(x) = \dfrac{x_1 - x}{x_1 - x_0} · f (x_0) + \dfrac{x - x_0}{x_1 - x_0} · f (x_1)$ | $⇒$ | $(x, f (x)) = (x_0, f(x_0));(x_1,f(x_1))$ | ||||
| $⇓$ | $g[(1 - t) · x_0 + t · x_1] = (1 - t) · f(x_0) + t · f(x_1)$ |
凹函数的运算性质。凸函数的运算性质完全类似。
$f(x) \mathop{≤}\limits^{x_0≤x≤x_1} \dfrac{x_1 - x}{x_1 - x_0} · f(x_0) + \dfrac{x - x_0}{x_1 - x_0} · f(x_1)$
$\dfrac{f(x) - f (x_0)}{x - x_0} ≤ \dfrac{f(x_1) - f(x_0)}{x_1 - x_0} ≤ \dfrac{f(x_1) - f(x)}{x_1 - x}$
| $\left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | _{x_0^{+} } ≤ \dfrac{f(x_1) - f(x_0)}{x_1 - x_0} ≤ \left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | _{x_1^{-} }$ |
$\dfrac{\mathrm{d} }{\mathrm{d} x} \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} = \dfrac{\mathrm{d^2 f(x)} }{\mathrm{d}^2 x} ≥ 0$
| $f(x) ≥ f(x_0) + \left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | _{x_0} · (x - x_0)$ |
| $⇓$ | $f(x) ≤ g(x)$ | $⇒$ | $f(x) ≤ \dfrac{x_1 - x}{x_1 - x_0} · f(x_0) + \dfrac{x - x_0}{x_1 - x_0} · f(x_1)$ | |||||
|---|---|---|---|---|---|---|---|---|
| $⇓$ | $\dfrac{f(x) - f (x_0)}{x - x_0} ≤ \dfrac{f(x_1) - f(x_0)}{x_1 - x_0}$ | $⇐$ | $f(x) - f(x_0) ≤ \left[\dfrac{x_1 - x}{x_1 - x_0} · f(x_0) - f(x_0)\right] + \dfrac{x - x_0}{x_1 - x_0} · f(x_1) = \dfrac{x - x_0}{x_1 - x_0} · [f(x_1) - f(x_0)]$ | |||||
| $⇓$ | $\dfrac{f(x_1) - f(x_0)}{x_1 - x_0} ≤ \dfrac{f(x_1) - f(x)}{x_1 - x}$ | $⇐$ | $f(x_1) - f(x) ≥ -\dfrac{x_1 - x}{x_1 - x_0} · f(x_0) + \left[ f(x_1) - \dfrac{x - x_0}{x_1 - x_0} · f(x_1) \right] = \dfrac{x_1 - x}{x_1 - x_0} · [f(x_1) - f(x_0)]$ | |||||
| $⇓$ | $\lim\limits_{x⇝x_0^+} \dfrac{f (x) - f (x_0)}{x - x_0} = \left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | _{x_0^{+} } ≤ \dfrac{f(x_1) - f(x_0)}{x_1 - x_0} ≤ \left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | {x_1^{-} } = \lim\limits{x⇝x_0^-} \dfrac{f (x_1) - f (x)}{x_1 - x}$ | $⇔$ | $\dfrac{\mathrm{d^2 f(x)} }{\mathrm{d}^2 x} ≥ 0$ | |||
| $⇓$ | $f(x) ≥ f(x_0) + \left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | _{x_0} · (x - x_0)$ | $⇐$ | $\left[ f(x) - f(x_0) ≥ \left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | _{x_0^+} · (x - x_0) \right] ∧ \left[ f(x_1) - f(x) ≤ \left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | _{x_1^-} · (x_1 - x) \right]$ | ||
| $⇓$ | $f(x) ≤ \dfrac{x_1 - x}{x_1 - x_0} · f(x_0) + \dfrac{x - x_0}{x_1 - x_0} · f(x_1)$ | $⇐$ | $f(x) = \dfrac{x_1 - x}{x_1 - x_0} · \left[ f(x) + \left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | _{x} · (x_0 - x) \right] + \dfrac{x - x_0}{x_1 - x_0} · \left[ f(x) + \left.\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}\right | _{x} · (x_1 - x) \right]$ |
凹区间满足$\dfrac{\mathrm{d}^2 f(x)}{\mathrm{d}^2 x} ≥ 0$，凸区间满足$\dfrac{\mathrm{d}^2 f(x)}{\mathrm{d}^2 x} ≤ 0$。凹区间与凸区间的分界点称为拐点，此点处满足$\dfrac{\mathrm{d}^2 f(x)}{\mathrm{d}^2 x} = 0$，反之不对。
典例：函数$f(x) = x^{3}$的二阶导函数为$\dfrac{\mathrm{d}^{2} f(x)}{\mathrm{d}^{2} x} = 6 · x$，凹区间为$[0⁺, ∞⁺)$，凸区间为$(∞⁻, 0⁻]$，拐点为$x = 0$。
| 典例：函数$f(x) = x^{4}$的二阶导函数为$\dfrac{\mathrm{d}^{2} f(x)}{\mathrm{d}^{2} x} = 12 · x^{2}$，凹区间为$(∞⁻, ∞⁺)$，无凸区间，无拐点，但$\left.\dfrac{\mathrm{d}^2 f(x)}{\mathrm{d}^2 x}\right | _{x=0} = 0$。 |
凹函数不等式。凸函数不等式完全类似。
$f\left[ \sum\limits_{i=0}^{n} t_i · x_i \right] \mathop{≤}\limits_{\sum\limits_{i=0}^{n} t_i =1}^{0≤t_i≤1} \sum\limits_{i=0}^{n} t_i · f(x_i)$
$f\left[ \sum\limits_{i=0}^{n} \dfrac{t_i · x_i}{\sum\limits_{i=0}^{n} t_i} \right] \mathop{≤}\limits^{0≤t_i} \sum\limits_{i=0}^{n} \dfrac{t_i · f(x_i)}{\sum\limits_{i=0}^{n} t_i}$
| $⇓$ | $t_0 = 1$ | $f (t_0 · x_0) ≤ t_0 · f(x_0)$ |
|---|---|---|
| $⇓$ | $t_0 + t_1 = 1$ | $f(t_0 · x_0 + t_1 · x_1) ≤ t_0 · f(x_0) + t_1 · f(x_1) = g(t_0 · x_0 + t_1 · x_1)$ |
| $⇓$ | $\sum\limits_{i=0}^{n} t_i = 1$ | $f\left[ \sum\limits_{i=0}^{n} t_i · x_i \right] ≤ \sum\limits_{i=0}^{n} t_i · f(x_i)$ |
| $⇓$ | $\sum\limits_{i=0}^{n+1} t_i = 1$ | $f\left[ \sum\limits_{i=0}^{n+1} t_i · x_i \right] = f\left[ (1 - t_{n+1}) · \sum\limits_{i=0}^{n} \dfrac{t_i}{1 - t_{n+1} } · x_i + t_{n+1} · x_{n+1} \right]$ |
| $⇓$ | $\dfrac{1 - t_{n+1} }{1 - t_{n+1} } = \sum\limits_{i=0}^{n} \dfrac{t_i}{1 - t_{n+1} } = 1$ | $f\left[ \sum\limits_{i=0}^{n+1} t_i · x_i \right] ≤ (1 - t_{n+1}) · f\left[ \sum\limits_{i=0}^{n} \dfrac{t_i}{1 - t_{n+1} } · x_i \right] + t_{n+1} · f(x_{n+1}) ≤ (1 - t_{n+1}) · \sum\limits_{i=0}^{n} \dfrac{t_i}{1 - t_{n+1} } · f(x_i) + t_{n+1} · f(x_{n+1}) = \sum\limits_{i=0}^{n+1} t_i · f(x_i)$ |
| $⇓$ | $\sum\limits_{i=0}^{n} \dfrac{t_i}{\sum\limits_{i=0}^{n} t_i} = 1$ | $f\left[ \sum\limits_{i=0}^{n} \dfrac{t_i · x_i}{\sum\limits_{i=0}^{n} t_i} \right] \mathop{≤}\limits^{0≤t_i} \sum\limits_{i=0}^{n} \dfrac{t_i · f(x_i)}{\sum\limits_{i=0}^{n} t_i}$ |
| $\dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \mathop{=====}\limits^{f(x_i)≡g(x_i)} \mathrm{Con}$ | $f\left[ \sum\limits_{i=0}^{n} \dfrac{t_i · x_i}{\sum\limits_{i=0}^{n} t_i} \right] \mathop{==}\limits^{0≤t_i} \sum\limits_{i=0}^{n} \dfrac{t_i · f(x_i)}{\sum\limits_{i=0}^{n} t_i}$ |
权均值不等式
$\prod\limits_{i=0}^{n} x_i^{t_i} \mathop{≤}\limits_{0<x_i}^{0<p} \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$
$\prod\limits_{i=0}^{n} x_i^{t_i} \mathop{≥}\limits_{0<x_i}^{p<0} \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$
| $\sum\limits_{i=0}^{n} t_i · x_i \mathop{≤}\limits_{0≤x_i}^{\left | p-\frac{1}{2}\right | <\frac{1}{2} } \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$ |
| $\sum\limits_{i=0}^{n} t_i · x_i \mathop{≥}\limits_{0≤x_i}^{\left | p-\frac{1}{2}\right | >\frac{1}{2} } \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$ |
$x_n^{↘} = \inf\limits_{i} x_i \mathop{≤}\limits_{0<x_i} \mathop{\left[ \dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} x_i^{-1} \right]^{-1} }\limits^{\mathrm{Harmonic\underline{}Mean} } \mathop{≤}\limits_{0<x_i} \mathop{\left[ \prod\limits_{i=0}^{n} x_i \right]^{\frac{1}{n+1} } }\limits^{\mathrm{Geometric\underline{}Mean} } \mathop{≤}\limits_{0<x_i} \mathop{\left[ \dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} x_i^{+1} \right]^{+1} }\limits^{\mathrm{Arithmetic\underline{}Mean} } \mathop{≤}\limits_{0≤x_i} \mathop{\left[ \dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} x_i^{2} \right]^{\frac{1}{2} } }\limits^{\mathrm{Quadratic\underline{}Mean} } \mathop{≤}\limits_{0≤x_i} \sup\limits_{i} x_i = x_n^{↗}$
$\dfrac{s_n^{↘} }{\mathop{\mathrm{ave} }\limits_{i} t_i} = \dfrac{\inf\limits_{i} s_i}{\mathop{\mathrm{ave} }\limits_{i} t_i} ≤ \dfrac{\mathop{\mathrm{ave} }\limits_{i} s_i}{\mathop{\mathrm{ave} }\limits_{i} t_i} ≤ \dfrac{\mathop{\mathrm{ave} }\limits_{i} s_i}{\inf\limits_{i} t_i} = \dfrac{\mathop{\mathrm{ave} }\limits_{i} s_i}{t_n^{↘} }$
$\dfrac{\mathop{\mathrm{ave} }\limits_{i} s_i}{t_n^{↗} } = \dfrac{\mathop{\mathrm{ave} }\limits_{i} s_i}{\sup\limits_{i} t_i} ≤ \dfrac{\mathop{\mathrm{ave} }\limits_{i} s_i}{\mathop{\mathrm{ave} }\limits_{i} t_i} ≤ \dfrac{\sup\limits_{i} s_i}{\mathop{\mathrm{ave} }\limits_{i} t_i} = \dfrac{s_n^{↗} }{\mathop{\mathrm{ave} }\limits_{i} t_i}$
| $f(x) \mathop{≡≡}\limits_{0<x} \ln x$ | $\dfrac{\mathrm{d}^{2} f(x)}{\mathrm{d}^{2} x} = -\dfrac{1}{x^{2} } < 0$ | $\dfrac{\mathrm{d}^{1} f(x)}{\mathrm{d}^{1} x} = +\dfrac{1}{x^{1} } > 0$ | $x_i ≡ x$ | $\sum\limits_{i=0}^{n} t_i \mathop{===}\limits_{0≤t_i≤1} 1$ | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $0 < p$ | $\ln \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p} = p · \ln \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right] ≥ p · \sum\limits_{i=0}^{n} t_i · \ln x_i^{\frac{1}{p} } = \ln \prod\limits_{i=0}^{n} x_i^{t_i}$ | $\prod\limits_{i=0}^{n} x_i^{t_i} \mathop{≤}\limits_{0<x_i}^{0<p} \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$ | $\prod\limits_{i=0}^{n} x_i^{t_i} \mathop{=}\limits_{0<x}^{0<p} \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$ | $\left[ \prod\limits_{i=0}^{n} x_i \right]^{\frac{1}{n + 1} } \mathop{≤}\limits_{0<x_i}^{0<p} \left[ \dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p}$ | ||||||||
| $p < 0$ | $\ln \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p} = p · \ln \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right] ≤ p · \sum\limits_{i=0}^{n} t_i · \ln x_i^{\frac{1}{p} } = \ln \prod\limits_{i=0}^{n} x_i^{t_i}$ | $\prod\limits_{i=0}^{n} x_i^{t_i} \mathop{≥}\limits_{0<x_i}^{p<0} \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$ | $\left[ \prod\limits_{i=0}^{n} x_i \right]^{\frac{1}{n + 1} } \mathop{≥}\limits_{0<x_i}^{p<0} \left[ \dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p}$ | |||||||||
| $f(x) \mathop{≡≡}\limits_{0≤x} x^{p}$ | $\dfrac{\mathrm{d}^{2} f(x)}{\mathrm{d}^{2} x} = p · (p - 1) · x^{p-2} = \mathop{-0}\limits_{\left | p-\frac{1}{2}\right | <\frac{1}{2} };\mathop{+0}\limits_{\left | p-\frac{1}{2}\right | >\frac{1}{2} }$ | $x_i ≡ u_i^{p}$ | $x_i ≡ x$ | $\sum\limits_{i=0}^{n} t_i \mathop{===}\limits_{0≤t_i≤1} 1$ | ||||
| $\left | p - \dfrac{1}{2} \right | < \dfrac{1}{2}$ | $\left[ \sum\limits_{i=0}^{n} t_i · u_i \right]^{p} ≥ \sum\limits_{i=0}^{n} t_i · u_i^{p}$ | $\sum\limits_{i=0}^{n} t_i · x_i \mathop{≤}\limits_{0≤x_i}^{\left | p-\frac{1}{2}\right | <\frac{1}{2} } \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$ | $\sum\limits_{i=0}^{n} t_i · x_i \mathop{=}\limits_{0≤x}^{\left | p-\frac{1}{2}\right | <\frac{1}{2} } \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$ | $\dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} x_i \mathop{≤}\limits_{0≤x_i}^{\left | p-\frac{1}{2}\right | <\frac{1}{2} } \left[ \dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p}$ |
| $\left | p - \dfrac{1}{2} \right | > \dfrac{1}{2}$ | $\left[ \sum\limits_{i=0}^{n} t_i · u_i \right]^{p} ≤ \sum\limits_{i=0}^{n} t_i · u_i^{p}$ | $\sum\limits_{i=0}^{n} t_i · x_i \mathop{≥}\limits_{0≤x_i}^{\left | p-\frac{1}{2}\right | >\frac{1}{2} } \left[ \sum\limits_{i=0}^{n} t_i · x_i^{\frac{1}{p} } \right]^{p}$ | $[p < 0] ⇒ [ 0 < x_i ]$ | $\dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} x_i \mathop{≥}\limits_{0≤x_i}^{\left | p-\frac{1}{2}\right | >\frac{1}{2} } \left[ \dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p}$ |
对函数不等式
$\dfrac{x}{1 + x} \mathop{≤}\limits_{x>-1} \ln (1 + x) \mathop{≤}\limits_{x>-1} x$
$\dfrac{α}{x + α} \mathop{≤}\limits_{x·(x+α)>0} \ln\left( 1 + \dfrac{α}{x} \right) \mathop{≤}\limits_{x·(x+α)>0} \dfrac{α}{x}$
$\dfrac{α}{1 + \dfrac{α}{x} } \mathop{≤}\limits_{x>0}^{x>-α} \ln\left( 1 + \dfrac{α}{x} \right)^{x} \mathop{≤}\limits_{x>0}^{x>-α} α$
$\dfrac{α}{1 + \dfrac{α}{x} } \mathop{≥}\limits_{x<0}^{x<-α} \ln\left( 1 + \dfrac{α}{x} \right)^{x} \mathop{≥}\limits_{x<0}^{x<-α} α$
$\dfrac{α}{x} \mathop{≤}\limits_{x·(x+α)>0} \ln\left( 1 + \dfrac{α}{x} \right)^{1+\frac{α}{x} } \mathop{≤}\limits_{x·(x+α)>0} \dfrac{α}{x} · \left( 1 + \dfrac{α}{x} \right)$
| $x∈(-1,+∞)$ | $x∈(-1,+∞)$ | $x=0$ | ||
|---|---|---|---|---|
| $f (x) \mathop{≡≡≡}\limits_{-1<x} \ln (1 + x)$ | $1+θ∈(1,1+x)⊆(0, +∞)$ | $\dfrac{x}{1+θ}∈\left(\dfrac{x}{1+x},\dfrac{x}{1}\right)$ | ||
| $\dfrac{\mathrm{d} f (x)}{\mathrm{d} x} = \dfrac{1}{1 + x} > 0$ | $\dfrac{f (x) - f (0)}{x - 0} = \dfrac{\ln (1 + x)}{x} = \left. \dfrac{\mathrm{d} f (x)}{\mathrm{d} x} \right | _{∃θ∈(0,x)} = \dfrac{1}{1 + θ} > 0$ | $\dfrac{1}{1 + \dfrac{1}{x} } = \dfrac{x}{1 + x} \mathop{≤}\limits_{x>-1} \ln (1 + x) \mathop{≤}\limits_{x>-1} x$ | $\dfrac{1}{1 + \dfrac{1}{x} } = \dfrac{x}{1 + x} \mathop{=}\limits_{x=0} \ln (1 + x) \mathop{=}\limits_{x=0} x$ |
| $t \mathop{≡≡≡≡≡}\limits_{x·(x+α)>0}^{\frac{α}{x}>-1} \dfrac{α}{x}$ | $\dfrac{α}{x} · \dfrac{1}{1 + \dfrac{α}{x} } = \dfrac{α}{x + α} \mathop{≤}\limits_{x·(x+α)>0} \ln\left( 1 + \dfrac{α}{x} \right) \mathop{≤}\limits_{x·(x+α)>0} \dfrac{α}{x}$ | $\dfrac{α}{x} \mathop{≤}\limits_{x·(x+α)>0} \ln\left( 1 + \dfrac{α}{x} \right)^{1+\frac{α}{x} } \mathop{≤}\limits_{x·(x+α)>0} \dfrac{α}{x} · \left( 1 + \dfrac{α}{x} \right)$ | $α = 0$ | |
| $x > 0$ | $\dfrac{α}{1 + \dfrac{α}{x} } \mathop{≤}\limits_{x>0}^{x>-α} \ln\left( 1 + \dfrac{α}{x} \right)^{x} \mathop{≤}\limits_{x>0}^{x>-α} α$ | $\dfrac{α}{1 + \dfrac{α}{x} } \mathop{=}\limits_{x>0}^{α=0} \ln\left( 1 + \dfrac{α}{x} \right)^{x} \mathop{=}\limits_{x>0}^{α=0} α$ | ||
| $x < 0$ | $\dfrac{α}{1 + \dfrac{α}{x} } \mathop{≥}\limits_{x<0}^{x<-α} \ln\left( 1 + \dfrac{α}{x} \right)^{x} \mathop{≥}\limits_{x<0}^{x<-α} α$ | $\dfrac{α}{1 + \dfrac{α}{x} } \mathop{=}\limits_{x<0}^{α=0} \ln\left( 1 + \dfrac{α}{x} \right)^{x} \mathop{=}\limits_{x<0}^{α=0} α$ |
幂函数不等式
| $x^p \mathop{≤}\limits_{0≤x}^{\left | p-\frac{1}{2}\right | ≤\frac{1}{2} } p · x + q$ |
| $x^p \mathop{≥}\limits_{0≤x}^{\left | p-\frac{1}{2}\right | ≥\frac{1}{2} } p · x + q$ |
| $(1 + x)^{p} \mathop{≤}\limits_{-1≤x}^{\left | p-\frac{1}{2}\right | ≤\frac{1}{2} } 1 + p · x$ |
| $(1 + x)^{p} \mathop{≥}\limits_{-1≤x}^{\left | p-\frac{1}{2}\right | ≥\frac{1}{2} } 1 + p · x$ |
| $p + q = 1$ | $\dfrac{\mathrm{d} f (x)}{\mathrm{d} x} = p · (x^{p-1} - 1)$ | $f (x) = x^{p} - p · x + p - 1$ | $0 ≤ x$ | $x=1$ | $t \mathop{≡≡≡}\limits_{-1≤x}^{0≤t} x + 1$ | |||
|---|---|---|---|---|---|---|---|---|
| $\left | p - \dfrac{1}{2} \right | ≤ \dfrac{1}{2}$ | $\dfrac{\mathrm{d} f (x)}{\mathrm{d} x} = \mathop{+0}\limits_{0<x<1};\mathop{0}\limits_{x=1};\mathop{-0}\limits_{1<x}$ | $f (x) = \mathop{-0}\limits_{0<x<1};\mathop{0}\limits_{x=1};\mathop{-0}\limits_{1<x}$ | $x^p ≤ p · x + 1 - p$ | $x^{p} ≤ p · x + q$ | $x^{p} = p · x + q$ | $(1 + x)^{p} ≤ 1 + p · x$ |
| $\left | p - \dfrac{1}{2} \right | ≥ \dfrac{1}{2}$ | $\dfrac{\mathrm{d} f (x)}{\mathrm{d} x} = \mathop{-0}\limits_{0<x<1};\mathop{0}\limits_{x=1};\mathop{+0}\limits_{1<x}$ | $f (x) = \mathop{+0}\limits_{0<x<1};\mathop{0}\limits_{x=1};\mathop{+0}\limits_{1<x}$ | $x^p ≥ p · x + 1 - p$ | $x^{p} ≥ p · x + q$ | $[p < 0] ⇒ [ 0 < x ]$ | $(1 + x)^{p} ≥ 1 + p · x$ |
连乘积不等式
$\prod\limits_{i=0}^{n} (1 + x_i)^{p} \mathop{≥}\limits_{0≤x_i}^{1≤n,p} 1 + p · \sum\limits_{i=0}^{n} x_i + p^{n+1} · \prod\limits_{i=0}^{n} x_i$
| $(1 + x_i)^{p} \mathop{≥}\limits^{1≤p}$ | $1 + p · x_i$ | |
|---|---|---|
| $\prod\limits_{i=0}^{0} (1 + x_i)^{p} ≥$ | $1 + p · x_0$ | $≥ 1 + p · \sum\limits_{i=0}^{0} x_i$ |
| $\prod\limits_{i=0}^{1} (1 + x_i)^{p} ≥$ | $\left[ 1 + p · x_0 \right] · (1 + p · x_1)$ | $≥ 1 + p · \sum\limits_{i=0}^{1} x_i + p^{2} · \prod\limits_{i=0}^{1} x_i$ |
| $\prod\limits_{i=0}^{2} (1 + x_i)^{p} ≥$ | $\left[ 1 + p · \sum\limits_{i=0}^{1} x_i + p^{2} · \prod\limits_{i=0}^{1} x_i \right] · (1 + p · x_2)$ | $≥ 1 + p · \sum\limits_{i=0}^{2} x_i + p^{3} · \prod\limits_{i=0}^{2} x_i$ |
| $\prod\limits_{i=0}^{n} (1 + x_i)^{p} ≥$ | $≥ 1 + p · \sum\limits_{i=0}^{n} x_i + p^{n+1} · \prod\limits_{i=0}^{n} x_i$ | |
| $\prod\limits_{i=0}^{n+1} (1 + x_i)^{p} ≥$ | $\left[ 1 + p · \sum\limits_{i=0}^{n} x_i + p^{n+1} · \prod\limits_{i=0}^{n} x_i \right] · (1 + p · x_{n+1})$ | $≥ 1 + p · \sum\limits_{i=0}^{n+1} x_i + p^{n+2} · \prod\limits_{i=0}^{n+1} x_i$ |
互补项不等式
| $x · y \mathop{≤}\limits_{0≤x,0≤y}^{\left | p-\frac{1}{2}\right | <\frac{1}{2} } p · x^{\frac{1}{p} } + q · y^{\frac{1}{q} }$ |
| $x · y \mathop{≥}\limits_{0≤x,0≤y}^{\left | p-\frac{1}{2}\right | >\frac{1}{2} } p · x^{\frac{1}{p} } + q · y^{\frac{1}{q} }$ |
| $p + q = 1$ | $t ≡ \dfrac{u}{v}$ | $0 ≤ u, 0 ≤ v$ | $x ≡ u^{p}, y ≡v^q$ | $u = x^{\frac{1}{p} } ≡ y^{\frac{1}{q} } = v$ | ||
|---|---|---|---|---|---|---|
| $\left | p - \dfrac{1}{2} \right | < \dfrac{1}{2}$ | $\left[ \dfrac{u}{v} \right]^{p} ≤ p · \dfrac{u}{v} + q$ | $u^{p} · v^{q} ≤ p · u + q · v$ | $x · y ≤ p · x^{\frac{1}{p} } + q · y^{\frac{1}{q} }$ | $x · y = p · x^{\frac{1}{p} } + q · y^{\frac{1}{q} }$ |
| $\left | p - \dfrac{1}{2} \right | > \dfrac{1}{2}$ | $\left[ \dfrac{u}{v} \right]^p ≥ p · \dfrac{u}{v} + q$ | $u^{p} · v^{q} ≥ p · u + q · v$ | $x · y ≥ p · x^{\frac{1}{p} } + q · y^{\frac{1}{q} }$ | $[p < 0] ⇒ [ 0 < x · y ]$ |
向量积不等式
| $\sum\limits_{i=0}^{n} x_i · y_i \mathop{≤}\limits_{0≤x_i,0≤y_i}^{\left | p-\frac{1}{2}\right | <\frac{1}{2} } \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} · \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } \right]^{q}$ |
| $\sum\limits_{i=0}^{n} x_i · y_i \mathop{≥}\limits_{0≤x_i,0≤y_i}^{\left | p-\frac{1}{2}\right | >\frac{1}{2} } \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} · \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } \right]^{q}$ |
| $p + q = 1$ | $u_i ≡ \dfrac{x_i}{\left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} }, v_i ≡ \dfrac{y_i}{\left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } \right]^{q} }$ | $0 ≤ x_i, 0 ≤ y_i$ | $x_i^{\frac{1}{p} } ≡ λ · y_i^{\frac{1}{q} }$ | ||
|---|---|---|---|---|---|
| $\left | p - \dfrac{1}{2} \right | < \dfrac{1}{2}$ | $\sum\limits_{i=0}^{n} \dfrac{x_i}{\left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} } · \dfrac{y_i}{\left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } \right]^{q} } ≤ p · \sum\limits_{i=0}^{n} \dfrac{x_i^{\frac{1}{p} } }{\sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } } + q · \sum\limits_{i=0}^{n} \dfrac{y_i^{\frac{1}{q} } }{\sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } } = 1$ | $\sum\limits_{i=0}^{n} x_i · y_i ≤ \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} · \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } \right]^{q}$ | $\sum\limits_{i=0}^{n} x_i · y_i = \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} · \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } \right]^{q}$ |
| $\left | p - \dfrac{1}{2} \right | > \dfrac{1}{2}$ | $\sum\limits_{i=0}^{n} \dfrac{x_i}{\left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} } · \dfrac{y_i}{\left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } \right]^{q} } ≥ p · \sum\limits_{i=0}^{n} \dfrac{x_i^{\frac{1}{p} } }{\sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } } + q · \sum\limits_{i=0}^{n} \dfrac{y_i^{\frac{1}{q} } }{\sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } } = 1$ | $\sum\limits_{i=0}^{n} x_i · y_i ≥ \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} · \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{q} } \right]^{q}$ | $[p < 0] ⇒ [ 0 < x_i · y_i ]$ |
拓扑距不等式
| $\left[ \sum\limits_{i=0}^{n} (x_i + y_i)^{\frac{1}{p} } \right]^{p} \mathop{≤}\limits_{0≤x_i,0≤y_i}^{\left | p- \frac{1}{2}\right | <\frac{1}{2} } \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} + \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{p} } \right]^{p}$ |
| $\left[ \sum\limits_{i=0}^{n} (x_i + y_i)^{\frac{1}{p} } \right]^{p} \mathop{≥}\limits_{0≤x_i,0≤y_i}^{\left | p-\frac{1}{2}\right | >\frac{1}{2} } \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} + \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{p} } \right]^{p}$ |
| $(x + y)^{p} \mathop{≤}\limits_{0≤x,0≤y}^{\left | p-\frac{1}{2}\right | <\frac{1}{2} } x^{p} +y^{p}$ |
| $(x + y)^{p} \mathop{≥}\limits_{0≤x,0≤y}^{\left | p-\frac{1}{2}\right | >\frac{1}{2} } x^{p} +y^{p}$ |
| $p + q = 1$ | $\sum\limits_{i=0}^{n} (x_{i} + y_{i})^{\frac{1}{p} } = \sum\limits_{i=0}^{n} x_{i} · (x_{i} + y_{i})^{\frac{1}{p} - 1} + \sum\limits_{i=0}^{n} y_{i} · (x_{i} + y_{i})^{\frac{1}{p} - 1}$ | $\left( \dfrac{1}{p} - 1 \right) · \dfrac{1}{q} = \dfrac{1}{p}$ | $x_i ≡ λ · y_i$ | ||||||
|---|---|---|---|---|---|---|---|---|---|
| $\left | p - \dfrac{1}{2} \right | < \dfrac{1}{2}$ | $\left[ \sum\limits_{i=0}^{n} (x_i + y_i)^{\frac{1}{p} } \right]^{p+q} ≤ \left[ \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} + \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{p} } \right]^{p} \right] · \left[ \sum\limits_{i=0}^{n} (x_i + y_i)^{\left( \frac{1}{p} - 1 \right) · \frac{1}{q} } \right]^{q}$ | $\left[ \sum\limits_{i=0}^{n} (x_i + y_i)^{\frac{1}{p} } \right]^{p} ≤ \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} + \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{p} } \right]^{p}$ | $\left[ \sum\limits_{i=0}^{n} (x_i + y_i)^{\frac{1}{p} } \right]^{p} = \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} + \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{p} } \right]^{p}$ | ||||
| $\left | p - \dfrac{1}{2} \right | > \dfrac{1}{2}$ | $\left[ \sum\limits_{i=0}^{n} (x_i + y_i)^{\frac{1}{p} } \right]^{p+q} ≥ \left[ \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} + \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{p} } \right]^{p} \right] · \left[ \sum\limits_{i=0}^{n} (x_i + y_i)^{\left( \frac{1}{p} - 1 \right) · \frac{1}{q} } \right]^{q}$ | $\left[ \sum\limits_{i=0}^{n} (x_i + y_i)^{\frac{1}{p} } \right]^{p} ≥ \left[ \sum\limits_{i=0}^{n} x_i^{\frac{1}{p} } \right]^{p} + \left[ \sum\limits_{i=0}^{n} y_i^{\frac{1}{p} } \right]^{p}$ | $[p < 0] ⇒ [ 0 < x_i · y_i ]$ | ||||
| $p + q = 1$ | $u_1 = 0, v_0 = 0$ | $x ≡ v_0^{\frac{1}{p} }, y ≡ u_1^{\frac{1}{p} }$ | $x ≡ 0, y ≡ 0$ | ||||||
| $\left | p - \dfrac{1}{2} \right | < \dfrac{1}{2}$ | $\left[ u_0^{\frac{1}{p} } + v_1^{\frac{1}{p} } \right]^{p} ≤ \left[ u_0^{\frac{1}{p} } \right]^{p} + \left[ v_1^{\frac{1}{p} } \right]^{p}$ | $(x + y)^{p} \mathop{≤}\limits_{0≤x,0≤y}^{\left | p-\frac{1}{2}\right | <\frac{1}{2} } x^{p} + y^{p}$ | $(x + y)^{p} \mathop{=}\limits_{0≤x,0≤y}^{\left | p-\frac{1}{2}\right | <\frac{1}{2} } x^{p} + y^{p}$ |
| $\left | p - \dfrac{1}{2} \right | > \dfrac{1}{2}$ | $\left[ u_0^{\frac{1}{p} } + v_1^{\frac{1}{p} } \right]^{p} ≥ \left[ u_0^{\frac{1}{p} } \right]^{p} + \left[ v_1^{\frac{1}{p} } \right]^{p}$ | $(x + y)^{p} \mathop{≥}\limits_{0≤x,0≤y}^{\left | p-\frac{1}{2}\right | >\frac{1}{2} } x^{p} + y^{p}$ | $[p < 0] ⇒ [ 0 < x_i · y_i ]$ |
全排序不等式
若数列$\lbrace a_i : a_0 ≤ a_1 ≤ ··· ≤ a_n \rbrace$，数列$\lbrace b_i : b_0 ≤ b_1 ≤ ··· ≤ b_n \rbrace$，数列$\lbrace b_i \rbrace$有乱序数列$\lbrace p_i : p_0,p_1,···,p_n \rbrace$，则数列的乘积满足：反序和 ≤ 乱序和 ≤ 顺序和。
$a_0 · b_n + a_1 · b_{n-1} + ··· + a_n · b_0 ≤ a_0 · p_0 + a_1 · p_1 + ··· + a_n · p_n ≤ a_0 · b_0 + a_1 · b_1 + ··· + a_n · b_n$
若数列$\lbrace a_i · p_i \rbrace$中$p_0 ≠ b_0, p_j = b_0$，则将$p_0$与$p_j$对换，记对换前和为$S$，记对换后和为$T$，则$T - S = a_0·p_j + a_j·p_0 - a_0·p_0-a_j·p_j = (a_j - a_0) · (p_0 - p_j) ≥ 0$。
依此类推逐项调整，因此有乱序和 ≤ 顺序和，同理有反序和 ≤ 乱序和。