# MacLaurin’s inequality

Let $a_{1},a_{2},\ldots,a_{n}$ be positive real numbers , and define the sums $S_{k}$ as follows :

 $S_{k}=\frac{\displaystyle\sum_{1\leq i_{1}

Then the following chain of inequalities is true :

 $S_{1}\geq\sqrt{S_{2}}\geq\sqrt[3]{S_{3}}\geq\cdots\geq\sqrt[n]{S_{n}}$

Note : $S_{k}$ are called the averages of the elementary symmetric sums
This inequality is in fact important because it shows that the arithmetic-geometric mean inequality is nothing but a consequence of a chain of stronger inequalities

Title MacLaurin’s inequality MacLaurinsInequality 2013-03-22 13:19:28 2013-03-22 13:19:28 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Definition msc 26D15