In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Newton inequalities are named after
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
. Suppose ''a''
1, ''a''
2, ..., ''a''
''n'' are
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s and let
denote the ''k''th
elementary symmetric polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sym ...
in ''a''
1, ''a''
2, ..., ''a''
''n''. Then the elementary symmetric means, given by
:
satisfy the
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
:
If all the numbers ''a''
''i'' are non-zero, then equality holds
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
all the numbers ''a''
''i'' are equal.
It can be seen that ''S''
1 is the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
, and ''S''
''n'' is the ''n''-th power of the
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
.
See also
*
Maclaurin's inequality 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'' be positive real numbers, and for ''k'' = 1,&nbs ...
References
*
*
*D.S. Bernstein ''Matrix Mathematics: Theory, Facts, and Formulas'' (2009 Princeton) p. 55
*
*
*
{{Isaac Newton
Isaac Newton
Inequalities
Symmetric functions