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In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
Newton inequalities are named after
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
. Suppose
a_{1},
a_{2}, ...,
a_{n} are real numbers and let
denote the
kth elementary symmetric function in
a_{1},
a_{2}, ...,
a_{n}. Then the
elementary symmetric means, given by
satisfy the inequality
with equality if and only if all the numbers
a_{i} are equal. Note that
S_{1} is simply the
arithmetic meanIn mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
, and
S_{n} is of course the
geometric meanThe geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
.
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