樊<span class="inline-unihan" style="border-bottom: 1px dotted; font-variant: normal;cursor: help; font-family: sans-serif, &#039;Unicode内码天珩输入法配套字体&#039;, &#039;方正宋体S-超大字符集&#039;, &#039;方正宋体S-超大字符集(SIP)&#039;, &#039;文泉驿等宽正黑&#039;, &#039;BabelStone Han&#039;, &#039;HanaMinB&#039;, &#039;FZSong-Extended&#039;, &#039;Arial Unicode MS&#039;, Code2002, DFSongStd, &#039;STHeiti SC&#039;, unifont;" title="字符描述：⿰土畿 &#10;※如果您看到空白、方块或问号，代表您的系统无法显示该字符。">𰋀</span>不等式

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).