In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a 1 , a 2 , … , a n {\displaystyle a_{1},a_{2},\ldots ,a_{n}} be non-negative real numbers, and for k = 1 , 2 , … , n {\displaystyle k=1,2,\ldots ,n} , define the averages S k {\displaystyle S_{k}} as follows:
S k = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n a i 1 a i 2 ⋯ a i k ( n k ) . {\displaystyle S_{k}={\frac {\displaystyle \sum _{1\leq i_{1}<\cdots <i_{k}\leq n}a_{i_{1}}a_{i_{2}}\cdots a_{i_{k}}}{\displaystyle {n \choose k}}}.}
The numerator of this fraction is the elementary symmetric polynomial of degree k {\displaystyle k} in the n {\displaystyle n} variables a 1 , a 2 , … , a n {\displaystyle a_{1},a_{2},\ldots ,a_{n}} , that is, the sum of all products of k {\displaystyle k} of the numbers a 1 , a 2 , … , a n {\displaystyle a_{1},a_{2},\ldots ,a_{n}} with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient ( n k ) . {\displaystyle {\tbinom {n}{k}}.} Maclaurin's inequality is the following chain of inequalities:
S 1 ≥ S 2 ≥ S 3 3 ≥ ⋯ ≥ S n n {\textstyle S_{1}\geq {\sqrt {S_{2}}}\geq {\sqrt[{3}]{S_{3}}}\geq \cdots \geq {\sqrt[{n}]{S_{n}}}} , with equality if and only if all the a i {\displaystyle a_{i}} are equal. Maclaurin's inequality can be proved using Newton's inequalities or a generalised version of Bernoulli's inequality.
Examples For n = 2 {\displaystyle n=2} , Maclaurin's inequality gives the arithmetic mean-geometric mean inequality for two non-negative numbers. For n = 4 {\displaystyle n=4} , Maclaurin's inequality states:
a 1 + a 2 + a 3 + a 4 4 ≥ a 1 a 2 + a 1 a 3 + a 1 a 4 + a 2 a 3 + a 2 a 4 + a 3 a 4 6 ≥ a 1 a 2 a 3 + a 1 a 2 a 4 + a 1 a 3 a 4 + a 2 a 3 a 4 4 3 ≥ a 1 a 2 a 3 a 4 4 . {\begin{aligned}&\quad {\frac {a_{1}+a_{2}+a_{3}+a_{4}}{4}}\\[8pt]&\geq {\sqrt {\frac {a_{1}a_{2}+a_{1}a_{3}+a_{1}a_{4}+a_{2}a_{3}+a_{2}a_{4}+a_{3}a_{4}}{6}}}\\[8pt]&\geq {\sqrt[{3}]{\frac {a_{1}a_{2}a_{3}+a_{1}a_{2}a_{4}+a_{1}a_{3}a_{4}+a_{2}a_{3}a_{4}}{4}}}\\[8pt]&\geq {\sqrt[{4}]{a_{1}a_{2}a_{3}a_{4}}}.\end{aligned}}
See also Bernoulli's inequality Bonferroni inequality Generalized mean inequality Muirhead's inequality Newton's inequalities
References Biler, Piotr; Witkowski, Alfred (1990). Problems in mathematical analysis. New York, N.Y.: M. Dekker. ISBN 0-8247-8312-3. This article incorporates material from MacLaurin's Inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. letter| 30 Apr 1729 IV. A second letter from Mr. Colin McLaurin, Professor of Mathematicks in the University of Edinburgh and F. R. S. to Martin Folkes, Esq; concerning the roots of equations, with the demonstration of other rules in algebra; being the continuation of the letter published in the Philosophical Transactions, N° 394 Free Colin MacLaurin Author & article information Phil. Trans. R. Soc. (1729) 36 (408): 59–96. https://doi.org/10.1098/rstl.1729.0011 https://royalsocietypublishing.org/rstl/article/36/408/59/110768/IV-A-second-letter-from-Mr-Colin-McLaurin