Table of Contents
- The Geometric Mean as a Limit
- Power Mean Bound for the Geometric Mean
- Power Mean Inequality
- The Integral Analogs
The Geometric Mean as a Limit
对非负实数 $ x_ {k}, k = 1, 2, \ldots, n $，和非负重量值 $ p_ {k}, k = 1, 2, \ldots, n $ 且 $ p_ {1} + p_ {2} + \cdots + p_ {n} = 1 $，其有极限
$ \lim_ {t \to 0} \{ \sum^{n}_ {k=1}p_ {k}x^{t}_ {k} \}^{\frac{1}{t}} = \prod^{n}_ {k=1}x^{p_ {k}}_ {k} $
Power Mean Bound for the Geometric Mean
对任意非负重量值 $ p_ {k}, k = 1, 2, \ldots, n $ 且 $ p_ {1} + p_ {2} + \cdots + p_ {n} = 1 $ 且对任意非负实数 $ x_ {k}, k = 1, 2, \ldots, n $，有边界
$ \prod^{n}_ {k=1}x^{p_ {k}}_ {k} \le \{ \sum^{n}_ {k=1}p_ {k}x^{t}_ {k} \}^{\frac{1}{t}} \quad \forall t > 0 $
Power Mean Inequality
正重量值 $ p_ {k}, k = 1, 2, \ldots, n $ 有 $ p_ {1} + p_ {2} + \cdots + p_ {n} = 1 $ 且对非负实数 $ x_ {k}, k = 1, 2, \ldots, n $，映射 $ t \mapsto M_ {t}, M_ {t} = \{ \sum^{n}_ {k=1}p_ {k}x^{t}_ {k} \}^{\frac{1}{t}} $ 在 $ \mathbb{R} $ 上是非减函数。则对 $ - \infty < s < t < \infty $ 有
$ \{ \sum^{n}_ {k=1}p_ {k}x^{s}_ {k} \}^{\frac{1}{s}} \le \{ \sum^{n}_ {k=1}p_ {k}x^{t}_ {k} \}^{\frac{1}{t}} $
当且仅当 $ x_ {1} = x_ {2} = \cdots = x_ {n} $ 时等式才成立
Some Special Means
当 t = -1，均值 $ M_ {-1} $ 被称为调和均值
$ M_ {-1} = M_ {-1}[x; p] = \frac{1}{\frac{p_ {1}}{x_ {1}} + \frac{p_ {2}}{x_ {2}} + \cdots + \frac{p_ {n}}{x_ {n}}} $
特别的，我们有调和平均几何平均不等式
$ \frac{1}{\frac{p_ {1}}{x_ {1}} + \frac{p_ {2}}{x_ {2}} + \cdots + \frac{p_ {n}}{x_ {n}}} \le x^{p_ {1}}_ {1} x^{p_ {2}}_ {2} \cdots x^{p_ {n}}_ {n} $
同时我们有调和平均算术平均不等式
$ \frac{1}{\frac{p_ {1}}{x_ {1}} + \frac{p_ {2}}{x_ {2}} + \cdots + \frac{p_ {n}}{x_ {n}}} \le p_ {1}x_ {1} + p_ {2}x_ {2} + \cdots + p_ {n}x_ {n} $
The Integral Analogs
设 $ D \subset \mathbb{R} $ 且我们考虑一个重量值函数 $ w : D \to [0, \infty) $ 满足
$ \int_ {D} w(x)dx = 1 $ 且 $ w(x) > 0 \quad \quad \forall x \in D $
则对 $ f : D \to [0, \infty) $ 且 $ t \in (- \infty, 0) \cup (0, \infty) $ 我们用如下公式定义 f 的第 t 个中值
$ M_ {t} = M_ {t}[f; w] \equiv \{ \int_ {D} f^{t}(x) w(x) dx\}^{\frac{1}{t}} $
同时对 $ M_ {0} $ 的特殊情况，我们定义其为
$ M_ {0}[f; w] \equiv exp \left( \int_ {D} \{ \log{f(x)} \} w(x)dx \right) $
则我们有
$ M_ {s}[f; w] \le M_ {t}[f; w] \quad \quad \forall - \infty < s < t < \infty $