Maclaurin's inequality

The Maclaurin inequality (after Colin Maclaurin ) is a statement from analysis , a branch of mathematics . It aggravates the inequality of the arithmetic and geometric mean , which says that the arithmetic mean of a finite number of positive real numbers is always at least as large as their geometric mean , in formulas

${\displaystyle {\frac {a_{1}+a_{2}+\ldots +a_{n}}{n}}\geq {\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}}}}$

for a natural number and . In the tightening, further mean values ​​are specified that lie between the arithmetic and geometric mean, for example says the inequality for three numbers ${\displaystyle n}$ ${\displaystyle a_{1},a_{2},\ldots ,a_{n}>0}$ ${\displaystyle x,y,z}$

${\displaystyle {\frac {x+y+z}{3}}\geq {\sqrt {\frac {xy+yz+zx}{3}}}\geq {\sqrt[{3}]{xyz}}.}$

statement

Are positive real numbers, and is ${\displaystyle a_{1},...,a_{n}}$

${\displaystyle S_{k}={\frac {\displaystyle \sum _{1\leq i_{1}<...<i_{k}\leq n}a_{i_{1}}\cdots a_{i_{k}}}{n \choose k}},}$

then applies

${\displaystyle S_{1}\geq {\sqrt {S_{2}}}\geq ...\geq {\sqrt[{n}]{S_{n}}}.}$

Note: is the arithmetic mean of the numbers, the geometric mean. The numerator of is the elementary symmetric polynomial of degree  in . ${\displaystyle S_{1}}$ ${\displaystyle {\sqrt[{n}]{S_{n}}}}$ ${\displaystyle S_{k}}$ ${\displaystyle k}$ ${\displaystyle a_{1},\ldots ,a_{n}}$

proof

${\displaystyle f(x)=(x+a_{1})\cdots (x+a_{n})}$can be written according to Vieta's theorem as ${\displaystyle \sum _{k=0}^{n}{n \choose k}S_{k}\,x^{n-k}}$

${\displaystyle f^{(n-m)}(x)={\frac {n!}{m!}}\,(x+b_{1})\cdots (x+b_{m})={\frac {n!}{m!}}\,\sum _{k=0}^{m}{\frac {m!}{k!}}\,S_{k}\,{\frac {x^{m-k}}{(m-k)!}}}$

According to the principle of Rolle , all are also positive. ${\displaystyle b_{1},...,b_{m}\,}$

Again after Vieta's theorem is and ${\displaystyle b_{1}\cdots b_{m}=S_{m}}$ ${\displaystyle {\hat {b_{1}}}\,b_{2}\cdots b_{m}+b_{1}\,{\hat {b_{2}}}\cdots b_{m}+...+b_{1}\,b_{2}\cdots {\hat {b_{m}}}=m\,S_{m-1}}$

According to the AM-GM inequality is ${\displaystyle {\frac {m\,S_{m-1}}{m}}\geq {\sqrt[{m}]{S_{m}^{m-1}}}\,\Longrightarrow \,{\sqrt[{m-1}]{S_{m-1}}}\geq {\sqrt[{m}]{S_{m}}}}$