From Wikipedia, the free encyclopedia
In
mathematics , the Newton inequalities are named after
Isaac Newton . Suppose a 1 , a 2 , ..., a n are non-negative
real numbers and let
e
k
{\displaystyle e_{k}}
denote the k th
elementary symmetric polynomial in a 1 , a 2 , ..., a n . Then the elementary symmetric means , given by
S
k
=
e
k
(
n
k
)
,
{\displaystyle S_{k}={\frac {e_{k}}{\binom {n}{k}}},}
satisfy the
inequality
S
k
−
1
S
k
+
1
≤
S
k
2
.
{\displaystyle S_{k-1}S_{k+1}\leq S_{k}^{2}.}
Equality holds
if and only if all the numbers a i are equal.
It can be seen that S 1 is the
arithmetic mean , and S n is the n -th power of the
geometric mean .
See also
References
Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities . Cambridge University Press.
ISBN
978-0521358804 .
Newton, Isaac (1707). Arithmetica universalis: sive de compositione et resolutione arithmetica liber .
D.S. Bernstein Matrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55
Maclaurin, C. (1729).
"A second letter to Martin Folks, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra" . Philosophical Transactions . 36 (407–416): 59–96.
doi :
10.1098/rstl.1729.0011 .
Whiteley, J.N. (1969). "On Newton's Inequality for Real Polynomials". The American Mathematical Monthly . 76 (8). The American Mathematical Monthly, Vol. 76, No. 8: 905–909.
doi :
10.2307/2317943 .
JSTOR
2317943 .
Niculescu, Constantin (2000).
"A New Look at Newton's Inequalities" . Journal of Inequalities in Pure and Applied Mathematics . 1 (2). Article 17.