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 ...

, Mahler's inequality, named after

Kurt Mahler Kurt Mahler FRS (26 July 1903 – 25 February 1988) was a German mathematician who worked in the fields of transcendental number theory, diophantine approximation, ''p''-adic analysis, and the geometry of numbers.

, states that 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 ...

of the term-by-term sum of two finite sequences of positive numbers is greater than or equal to the sum of their two separate geometric means: :

\prod_^n (x_k + y_k)^ \ge \prod_^n x_k^ + \prod_^n y_k^

when

x_k,\ y_k>0

for all

k

Proof

By the

inequality of arithmetic and geometric means Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of in ...

, we have: :

\prod_^n \left(\right)^ \le  \sum_^n ,

and :

\prod_^n \left(\right)^ \le  \sum_^n .

Hence, :

\prod_^n \left(\right)^  + \prod_^n \left(\right)^ \le  n = 1.

Clearing denominators In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions. Example C ...

then gives the desired result.

References

Minkowski inequality
in the ''

Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...

'' Inequalities (mathematics) Articles containing proofs {{mathanalysis-stub

Proof

See also

References