樊𰋀不等式

樊𰋀不等式，用最簡單的形式可以表述為：

如果實數${\displaystyle x_{1},\ldots ,x_{n}\ }$都符合${\displaystyle 0<x_{i}\leq {1 \over 2}}$，那麼

${\displaystyle {\frac {\left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}}{\left[\prod _{i=1}^{n}(1-x_{i})\right]^{\frac {1}{n}}}}\leq {\frac {{1 \over n}\sum _{i=1}^{n}x_{i}}{{1 \over n}\sum _{i=1}^{n}(1-x_{i})}}}$，

等式成立當且僅當${\displaystyle x_{1}=\cdots =x_{n}}$。

若分別記${\displaystyle x_{i}\;}$的算術平均和幾何平均為${\displaystyle A_{n}\equiv {\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$，${\displaystyle G_{n}\equiv \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}}$；又記${\displaystyle 1-x_{i}\;}$的這兩種平均為${\displaystyle A_{n}^{\prime }\equiv {\frac {1}{n}}\sum _{i=1}^{n}(1-x_{i})}$，${\displaystyle G_{n}^{\prime }\equiv \left[\prod _{i=1}^{n}(1-x_{i})\right]^{\frac {1}{n}}}$，那麼不等式可寫作

${\displaystyle {\frac {G_{n}}{G_{n}^{\prime }}}\leq {\frac {A_{n}}{A_{n}^{\prime }}}}$。

如此可以看出它和算幾不等式${\displaystyle G_{n}\leq A_{n}}$的相似處。

利用函數${\displaystyle f(x)=\ln x-\ln(1-x)\;}$在${\displaystyle 0<x\leq {\frac {1}{2}}}$的凹性，套用延森不等式，這樣得到一個簡單的證明。這證明可以直接推廣至不等式的加權形式：

${\displaystyle {\frac {\prod _{i=1}^{n}x_{i}^{\gamma _{i}}}{\prod _{i=1}^{n}\left(1-x_{i}\right)^{\gamma _{i}}}}\leq {\frac {\sum _{i=1}^{n}\gamma _{i}x_{i}}{\sum _{i=1}^{n}\gamma _{i}\left(1-x_{i}\right)}}}$，

其中${\displaystyle \gamma _{i}\geq 0}$，${\displaystyle \sum _{i=1}^{n}\gamma _{i}=1}$。

如果記調和平均為${\displaystyle H_{n}:={\frac {1}{{\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{x_{i}}}}}}$，${\displaystyle H_{n}^{\prime }:={\frac {1}{{\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{1-x_{i}}}}}}$，在W. Wang, P. Wang (1984)有如下推廣：

${\displaystyle {\frac {H_{n}}{H_{n}^{\prime }}}\leq {\frac {G_{n}}{G_{n}^{\prime }}}\leq {\frac {A_{n}}{A_{n}^{\prime }}}}$。

• Horst Alzer: Verschärfung einer Ungleichung von Ky Fan. Aequationes Mathematicae 36 (1988) 246-250.
• E. F. Beckenbach, R. Bellman: Inequalities. Springer-Verlag, Berlin 1983, 引用在 Alzer (1988).
• Edward Neuman, Jósef Sándor: On the Ky Fan inequality and related inequalities I. Mathematical Inequalities & Applications. Volume 5, Number 1 (2002), 49–56.
• Edward Neuman, Jósef Sándor: On the Ky Fan inequality and related inequalities II （页面存档备份，存于）. Bulletin of the Australian Mathematical Society. Volume 72, Number 1 (2005), 87-107.
• Jósef Sándor, Tiberiu Trif: A new refinement of the Ky Fan inequality. Mathematical Inequalities & Applications. Volume 2, Number 4 (1999), 529–533.
• W. Wang, P. Wang: A class of inequalities for the symmetric functions (中文) Acta Math. Sinica 27 (1984), 485-497, 引用在 Alzer (1988).