Ky-Fan Inequality

In mathematics , the Ky-Fan inequality is an inequality that was discovered by Ky Fan and published for the first time by ( Lit .: Beckenbach and Bellman, 1983). Its importance lies primarily in the fact that, due to its similarity to the inequality of the arithmetic and geometric mean, it is the starting point for further generalizations.

formulation

In its simplest form, the Ky-Fan inequality is as follows:

If for numbers are with , then applies ${\displaystyle x_{i}}$ ${\displaystyle i=1,\dots ,n\;}$ ${\displaystyle 0<x_{i}<{\frac {1}{2}}}$

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

The equal sign applies if and only if . ${\displaystyle x_{1}=\cdots =x_{n}\;}$

If one designates with the arithmetic mean and with the geometric mean of the numbers as well as with the arithmetic mean and with the geometric mean of the numbers , then the Ky-Fan inequality takes the form ${\displaystyle A_{n}:={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$ ${\displaystyle G_{n}:=\left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}}$ ${\displaystyle x_{i}\;}$ ${\displaystyle A_{n}^{\prime }:={\frac {1}{n}}\sum _{i=1}^{n}(1-x_{i})}$ ${\displaystyle G_{n}^{\prime }:=\left(\prod _{i=1}^{n}(1-x_{i})\right)^{\frac {1}{n}}}$ ${\displaystyle 1-x_{i}\;}$

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

on; the similarity to the inequality of the arithmetic and geometric mean becomes clear. ${\displaystyle G_{n}\leq A_{n}}$

proof

A simple proof of the Ky-Fan inequality is obtained by applying Jensen's inequality to the function that is for concave . This proof immediately provides a generalization of the Ky-Fan inequality with weighted averages: ${\displaystyle f(x)=\ln x-\ln(1-x)\;}$ ${\displaystyle 0<x\leq {\frac {1}{2}}}$

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

where for the weights and must apply. ${\displaystyle \gamma _{i}\geq 0}$ ${\displaystyle \sum _{i=1}^{n}\gamma _{i}=1}$

Related inequalities

( Ref : Wang and Wang, 1984), the Ky Fan inequality have on the harmonic mean values and extended: ${\displaystyle H_{n}:={\frac {1}{{\frac {1}{n}}\sum _{i=1}^{n}1/x_{i}}}}$ ${\displaystyle H_{n}^{\prime }:={\frac {1}{{\frac {1}{n}}\sum _{i=1}^{n}1/(1-x_{i})}}}$

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

literature

• Horst Alzer: Exacerbation of an inequality by Ky Fan . Aequationes Mathematicae 36 (1988) 246-250.
• EF Beckenbach and R. Bellman: Inequalities. Springer-Verlag, Berlin 1983, quoted from ( Lit .: Alzer, 1988).
• Edward Neuman and Jósef Sándor: On the Ky Fan inequality and related inequalities I . Mathematical Inequalities & Applications. Volume 5, Number 1 (2002), 49-56.
• Edward Neuman and Jósef Sándor: On the Ky Fan inequality and related inequalities II (PDF; 575 kB). Bulletin of the Australian Mathematical Society. Volume 72, Number 1 (2005), 87-107.
• Jósef Sándor and Tiberiu Trif: A new refinement of the Ky Fan inequality . Mathematical Inequalities & Applications. Volume 2, Number 4 (1999), 529-533.
• W. Wang and P. Wang: A class of inequalities for the symmetric functions (Chinese) Acta Math Sinica 27 (1984) 485-497, quoted in. ( Ref : Alzer, 1988).