樊畿不等式（英语：Ky Fan inequality）是华裔数学家樊畿发现的一个不等式。这个不等式的意义，在于因其与算几不等式相似，从中可以推广出很多结果。

樊𰋀不等式，用最简单的形式可以表述为：

如果实数${\displaystyle x_{1},\ldots ,x_{n}\ }$都符合${\displaystyle 0<x_{i}\leq {1 \over 2))$，那么

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

等式成立当且仅当${\displaystyle x_{1}=\cdots =x_{n))$。

若分别记${\displaystyle x_{i}\;}$的算术平均和几何平均为${\displaystyle A_{n}\equiv {\frac {1}{n))\sum _{i=1}^{n}x_{i))$，${\displaystyle G_{n}\equiv \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n))}$；又记${\displaystyle 1-x_{i}\;}$的这两种平均为${\displaystyle A_{n}^{\prime }\equiv {\frac {1}{n))\sum _{i=1}^{n}(1-x_{i})}$，${\displaystyle G_{n}^{\prime }\equiv \left[\prod _{i=1}^{n}(1-x_{i})\right]^{\frac {1}{n))}$，那么不等式可写作

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

如此可以看出它和算几不等式${\displaystyle G_{n}\leq A_{n))$的相似处。

利用函数${\displaystyle f(x)=\ln x-\ln(1-x)\;}$在${\displaystyle 0<x\leq {\frac {1}{2))}$的凹性，套用延森不等式，这样得到一个简单的证明。这证明可以直接推广至不等式的加权形式：

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

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

如果记调和平均为${\displaystyle H_{n}:={\frac {1}((\frac {1}{n))\sum _{i=1}^{n}{\frac {1}{x_{i))))))$，${\displaystyle H_{n}^{\prime }:={\frac {1}((\frac {1}{n))\sum _{i=1}^{n}{\frac {1}{1-x_{i))))))$，在W. Wang, P. Wang (1984)有如下推广：

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

• Horst Alzer: Verschärfung einer Ungleichung von Ky Fan^{[永久失效链接]}. 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，存于互联网档案馆）. 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).