mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Newton inequalities are named after

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...

. Suppose ''a''₁, ''a''₂, ..., ''a''_''n'' are non-negative

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s and let

e_k

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 sy ...

in ''a''₁, ''a''₂, ..., ''a''_''n''. Then the elementary symmetric means, given by :

S_k = \frac,

satisfy the inequality :

S_S_ \le S_k^2.

Equality holds

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

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 ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

, 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 finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...

References

* * *D.S. Bernstein ''Matrix Mathematics: Theory, Facts, and Formulas'' (2009 Princeton) p. 55 * * * {{Isaac Newton Isaac Newton Inequalities (mathematics) Symmetric functions

See also

References