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 QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the

harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean ...

(HM),

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 ...

(GM),

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 ...

(AM), and

quadratic mean In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denoted M ...

(QM; also known as root mean square). Suppose that

x_1, x_2, \ldots, x_n

are positive

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. Then :

leq\frac \leq\sqrt.

In other words, QM≥AM≥GM≥HM. These inequalities often appear in mathematical competitions and have applications in many fields of science.

Proof

There are three inequalities between means to prove. There are various methods to prove the inequalities, including

mathematical induction Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a ...

, the

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...

Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function (mathematics), function subject to constraint (mathematics), equation constraints (i.e., subject to the conditio ...

s, and

Jensen's inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier p ...

. For several proofs that GM ≤ AM, see

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 ...

AM-QM inequality

From the Cauchy–Schwarz inequality on real numbers, setting one vector to : :

\left( \sum_^n x_i \cdot 1 \right)^ \leq \left( \sum_^n x_i^2 \right) \left( \sum_^n 1^2 \right) = n \,\sum_^n x_i^2,

hence

\left( \frac \right)^ \leq \frac

. For positive

x_i

the square root of this gives the inequality.

AM–GM inequality

HM-GM inequality

The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals

1/x_1  , \dots, 1/x_n

, and it exceeds

1/\sqrt /math> by the AM-GM inequality. x_i > 0 implies the inequality:

: \frac \leq \sqrt

The ''n'' = 2 case

When ''n'' = 2, the inequalities become :

\frac  \leq \sqrt \leq \frac\leq\sqrt

for all

x_1, x_2 > 0,

which can be visualized in a semi-circle whose diameter is 'AB''and center ''D''. Suppose ''AC'' = ''x''₁ and ''BC'' = ''x''₂. Construct perpendiculars to 'AB''at ''D'' and ''C'' respectively. Join 'CE''and 'DF''and further construct a perpendicular 'CG''to 'DF''at ''G''. Then the length of ''GF'' can be calculated to be the harmonic mean, ''CF'' to be the geometric mean, ''DE'' to be the arithmetic mean, and ''CE'' to be the quadratic mean. The inequalities then follow easily by the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

References

{{reflist

External links

The HM-GM-AM-QM InequalitiesUseful inequalities cheat sheet
entry "means" on the right of page 1 Inequalities (mathematics) Means Articles containing proofs