Inequality of arithmetic and geometric means

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.

The simplest non-trivial case — i.e., with more than one variable — for two non-negative numbers x and y, is the statement that

$\frac{x+y}2 \ge \sqrt{xy}$

with equality if and only if x = y. This case can be seen from the fact that the square of a real number is always non-negative (greater than or equal to zero) and from the elementary case (a ± b)² = a² ± 2ab + b² of the binomial formula:

$\begin{align} 0 & \le (x-y)^2 \\ & = x^2-2xy+y^2 \\ & = x^2+2xy+y^2 - 4xy \\ & = (x+y)^2 - 4xy. \end{align}$

In other words (x + y)² ≥ 4xy, with equality precisely when (x − y)² = 0, i.e. x = y. For a geometrical interpretation, consider a rectangle with sides of length x and y, hence it has perimeter 2x + 2y and area xy. Similarly, a square with all sides of length √xy has the perimeter 4√xy and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that 2x + 2y ≥ 4√xy and that only the square has the smallest perimeter amongst all rectangles of equal area.

The general AM–GM inequality corresponds to the fact that the natural logarithm, which converts multiplication to addition, is a strictly concave function; using Jensen's inequality the general proof of the inequality follows.

$\frac{\ln x + \ln y}{2} \leq \ln \left( \frac{x+y}{2}\right)$

Extensions of the AM–GM inequality are available to include weights or generalized means.

Background

The arithmetic mean, or less precisely the average, of a list of n numbers x₁, x₂, . . . , x_n is the sum of the numbers divided by n:

$\frac{x_1 + x_2 + \cdots + x_n}{n}.$

The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:

$\sqrt[n]{x_1 \cdot x_2 \cdots x_n}.$

If x₁, x₂, . . . , x_n > 0, this is equal to the exponential of the arithmetic mean of the natural logarithms of the numbers:

$\exp \left( \frac{\ln {x_1} + \ln {x_2} + \cdots + \ln {x_n}}{n} \right).$

The inequality

Restating the inequality using mathematical notation, we have that for any list of n nonnegative real numbers x₁, x₂, . . . , x_n,

$\frac{x_1 + x_2 + \cdots + x_n}{n} \ge \sqrt[n]{x_1 \cdot x_2 \cdots x_n}\,,$

and that equality holds if and only if x₁ = x₂ = · · · = x_n.

Geometric interpretation

In two dimensions, 2x₁ + 2x₂ is the perimeter of a rectangle with sides of length x₁ and x₂. Similarly, 4√x₁x₂ is the perimeter of a square with the same area as the rectangle previously discussed. Thus for n = 2 the AM–GM inequality states that only the square has the smallest perimeter amongst all rectangles of equal area.

The full inequality is an extension of this idea to n dimensions. Every vertex of an n-dimensional box is connected to n edges. If these edges' lengths are x₁, x₂, . . . , x_n, then x₁ + x₂ + · · · + x_n is the total length of edges incident to the vertex. There are 2ⁿ vertices, so we multiply this by 2ⁿ; since each edge, however, meets two vertices, every edge is counted twice. Therefore we divide by 2 and conclude that there are 2ⁿ⁻¹n edges. There are equally many edges of each length and n lengths; hence there are 2ⁿ⁻¹ edges of each length and the total edge-length is 2ⁿ⁻¹(x₁ + x₂ + · · · + x_n). On the other hand,

$2^{n-1} n \sqrt[n]{x_1 x_2 \cdots x_n}$

is the total length of edges connected to a vertex on an n-dimensional cube of equal volume. Since the inequality says

${x_1 + x_2 +\cdots + x_n \over n} \ge \sqrt[n]{x_1 x_2\cdots x_n},$

we get

$2^{n-1}(x_1 + x_2 + \cdots + x_n) \ge 2^{n-1} n \sqrt[n]{x_1 x_2\cdots x_n}\,$

with equality if and only if x₁ = x₂ = · · · = x_n.

Thus the AM–GM inequality states that only the n-cube has the smallest sum of lengths of edges connected to each vertex amongst all n-dimensional boxes with the same volume.^[1]

Example application

Consider the function

$f(x,y,z) = \frac{x}{y} + \sqrt{\frac{y}{z}} + \sqrt[3]{\frac{z}{x}}$

for all positive real numbers x, y and z. Suppose we wish to find the minimal value of this function. First we rewrite it a bit:

$\begin{align} f(x,y,z) &= 6 \cdot \frac{ \frac{x}{y} + \frac{1}{2} \sqrt{\frac{y}{z}} + \frac{1}{2} \sqrt{\frac{y}{z}} + \frac{1}{3} \sqrt[3]{\frac{z}{x}} + \frac{1}{3} \sqrt[3]{\frac{z}{x}} + \frac{1}{3} \sqrt[3]{\frac{z}{x}} }{6}\\ &=6\cdot\frac{x_1+x_2+x_3+x_4+x_5+x_6}{6} \end{align}$

with

$x_1=\frac{x}{y},\qquad x_2=x_3=\frac{1}{2} \sqrt{\frac{y}{z}},\qquad x_4=x_5=x_6=\frac{1}{3} \sqrt[3]{\frac{z}{x}}.$

Applying the AM–GM inequality for n = 6, we get

$\begin{align} f(x,y,z) &\ge 6 \cdot \sqrt[6]{ \frac{x}{y} \cdot \frac{1}{2} \sqrt{\frac{y}{z}} \cdot \frac{1}{2} \sqrt{\frac{y}{z}} \cdot \frac{1}{3} \sqrt[3]{\frac{z}{x}} \cdot \frac{1}{3} \sqrt[3]{\frac{z}{x}} \cdot \frac{1}{3} \sqrt[3]{\frac{z}{x}} }\\ &= 6 \cdot \sqrt[6]{ \frac{1}{2 \cdot 2 \cdot 3 \cdot 3 \cdot 3} \frac{x}{y} \frac{y}{z} \frac{z}{x} }\\ &= 2^{2/3} \cdot 3^{1/2}. \end{align}$

Further, we know that the two sides are equal exactly when all the terms of the mean are equal:

$f(x,y,z) = 2^{2/3} \cdot 3^{1/2} \quad \mbox{when} \quad \frac{x}{y} = \frac{1}{2} \sqrt{\frac{y}{z}} = \frac{1}{3} \sqrt[3]{\frac{z}{x}}.$

All the points (x, y, z) satisfying these conditions lie on a half-line starting at the origin and are given by

$(x,y,z)=\biggr(t,\sqrt[3]{2}\sqrt{3}\,t,\frac{3\sqrt{3}}{2}\,t\biggr)\quad\mbox{with}\quad t>0.$

Practical applications

An important practical application in financial mathematics is to computing the rate of return: the annualized return, computed via the geometric mean, is less than the average annual return, computed by the arithmetic mean (or equal if all returns are equal). This is important in analyzing investments, as the average return overstates the cumulative effect.

Proofs of the AM–GM inequality

There are several ways to prove the AM–GM inequality; for example, it can be inferred from Jensen's inequality, using the concave function ln(x). It can also be proven using the rearrangement inequality. Considering length and required prerequisites, the elementary proof by induction given below is probably the best recommendation for first reading.

Idea of the first two proofs

We have to show that

$\frac{x_1+x_2+\cdots+x_n}{n} \ge \sqrt[n]{x_1x_2 \cdots x_n}$

with equality only when all numbers are equal. If x_i ≠ x_j, then replacing both x_i and x_j by (x_i + x_j)/2 will leave the arithmetic mean on the left-hand side unchanged, but will increase the geometric mean on the right-hand side because

$\Bigl(\frac{x_i+x_j}{2}\Bigr)^2-x_ix_j=\Bigl(\frac{x_i-x_j}{2}\Bigr)^2>0 .$

Thus the right-hand side will be largest when all x_is are equal to the arithmetic mean

$\alpha=\frac{x_1+x_2+\ldots+x_n}{n},$

thus as this is then the largest value of right-hand side of the expression, we have

$\frac{x_1+x_2+\ldots+x_n}{n}=\alpha=\sqrt[n]{\alpha\alpha \ldots \alpha}\ge\sqrt[n]{x_1x_2 \ldots x_n}.$

This is a valid proof for the case n = 2, but the procedure of taking iteratively pairwise averages may fail to produce n equal numbers in the case n ≥ 3. An example of this case is x₁ = x₂ ≠ x₃: Averaging two different numbers produces two equal numbers, but the third one is still different. Therefore, we never actually get an inequality involving the geometric mean of three equal numbers.

Hence, an additional trick or a modified argument is necessary to turn the above idea into a valid proof for the case n ≥ 3.

Proof by induction

With the arithmetic mean

$\alpha=\frac{\ x_1 + \cdots + x_n}n$

of the non-negative real numbers x₁, . . . , x_n, the AM–GM statement is equivalent to

$\alpha^n\ge x_1 x_2 \cdots x_n\,$

with equality if and only if α = x_i for all i ∈ {1, . . . , n}.

For the following proof we apply mathematical induction and only well-known rules of arithmetic.

Induction basis: For n = 1 the statement is true with equality.

Induction hypothesis: Suppose that the AM–GM statement holds for all choices of n non-negative real numbers.

Induction step: Consider n + 1 non-negative real numbers x₁, . . . , x_n+1, . Their arithmetic mean α satisfies

$(n+1)\alpha=\ x_1 + \cdots + x_n + x_{n+1}.\,$

If all numbers are equal to α, then we have equality in the AM–GM statement and we are done. Otherwise we may find one number that is greater than α and one that is smaller than α, say x_n > α and x_n+1 < α. Then

$(x_n-\alpha)(\alpha-x_{n+1})>0\,.\qquad(*)$

Now consider the n numbers x₁, . . . , x_n–1, y with

$y:=x_n+x_{n+1}-\alpha\ge x_n-\alpha>0\,,$

which are also non-negative. Since

$n\alpha=x_1 + \cdots + x_{n-1} + \underbrace{x_n+x_{n+1}-\alpha}_{=\,y},$

Thus,α is also the arithmetic mean of n numbers x₁, . . . , x_n–1, y and the induction hypothesis implies

$\alpha^{n+1}=\alpha^n\cdot\alpha\ge x_1x_2 \cdots x_{n-1} y\cdot\alpha.\qquad(**)$

Due to (*) we know that

$(\underbrace{x_n+x_{n+1}-\alpha}_{=\,y})\alpha-x_nx_{n+1}=(x_n-\alpha)(\alpha-x_{n+1})>0,$

hence

$y\alpha>x_nx_{n+1}\,,\qquad({*}{*}{*})$

in particular α > 0. Therefore, if at least one of the numbers x₁, . . . , x_n–1 is zero, then we already have strict inequality in (**). Otherwise the right-hand side of (**) is positive and strict inequality is obtained by using the estimate (***) to get a lower bound of the right-hand side of (**). Thus, in both cases we get

$\alpha^{n+1}>x_1x_2 \cdots x_{n-1} x_nx_{n+1}\,,$

which completes the proof.

Proof by Cauchy using forward–backward induction

The following proof by cases relies directly on well-known rules of arithmetic but employs the rarely used technique of forward-backward-induction. It is essentially from Augustin Louis Cauchy and can be found in his Cours d'analyse.^[2]

The case where all the terms are equal

If all the terms are equal:

$x_1 = x_2 = \cdots = x_n,$

then their sum is nx₁, so their arithmetic mean is x₁; and their product is x₁ⁿ, so their geometric mean is x₁; therefore, the arithmetic mean and geometric mean are equal, as desired.

The case where not all the terms are equal

It remains to show that if not all the terms are equal, then the arithmetic mean is greater than the geometric mean. Clearly, this is only possible when n > 1.

This case is significantly more complex, and we divide it into subcases.

The subcase where n = 2

If n = 2, then we have two terms, x₁ and x₂, and since (by our assumption) not all terms are equal, we have:

$\begin{align} \Bigl(\frac{x_1+x_2}{2}\Bigr)^2-x_1x_2 &=\frac14(x_1^2+2x_1x_2+x_2^2)-x_1x_2\\ &=\frac14(x_1^2-2x_1x_2+x_2^2)\\ &=\Bigl(\frac{x_1-x_2}{2}\Bigr)^2>0, \end{align}$

hence

$\frac{x_1 + x_2}{2} > \sqrt{x_1 x_2}$

as desired.

The subcase where n = 2^k

Consider the case where n = 2^k, where k is a positive integer. We proceed by mathematical induction.

In the base case, k = 1, so n = 2. We have already shown that the inequality holds when n = 2, so we are done.

Now, suppose that for a given k > 1, we have already shown that the inequality holds for n = 2^k−1, and we wish to show that it holds for n = 2^k. To do so, we apply the inequality twice for 2^k-1 numbers and once for 2 numbers to obtain:

$\begin{align} \frac{x_1 + x_2 + \cdots + x_{2^k}}{2^k} & {} =\frac{\frac{x_1 + x_2 + \cdots + x_{2^{k-1}}}{2^{k-1}} + \frac{x_{2^{k-1} + 1} + x_{2^{k-1} + 2} + \cdots + x_{2^k}}{2^{k-1}}}{2} \\[7pt] & \ge \frac{\sqrt[2^{k-1}]{x_1 x_2 \cdots x_{2^{k-1}}} + \sqrt[2^{k-1}]{x_{2^{k-1} + 1} x_{2^{k-1} + 2} \cdots x_{2^k}}}{2} \\[7pt] & \ge \sqrt{\sqrt[2^{k-1}]{x_1 x_2 \cdots x_{2^{k-1}}} \sqrt[2^{k-1}]{x_{2^{k-1} + 1} x_{2^{k-1} + 2} \cdots x_{2^k}}} \\[7pt] & = \sqrt[2^k]{x_1 x_2 \cdots x_{2^k}} \end{align}$

where in the first inequality, the two sides are equal only if

$x_1 = x_2 = \cdots = x_{2^{k-1}}$

and

$x_{2^{k-1}+1} = x_{2^{k-1}+2} = \cdots = x_{2^k}$

(in which case the first arithmetic mean and first geometric mean are both equal to x₁, and similarly with the second arithmetic mean and second geometric mean); and in the second inequality, the two sides are only equal if the two geometric means are equal. Since not all 2^k numbers are equal, it is not possible for both inequalities to be equalities, so we know that:

$\frac{x_1 + x_2 + \cdots + x_{2^k}}{2^k} > \sqrt[2^k]{x_1 x_2 \cdots x_{2^k}}$

as desired.

The subcase where n < 2^k

If n is not a natural power of 2, then it is certainly less than some natural power of 2, since the sequence 2, 4, 8, . . . , 2^k, . . . is unbounded above. Therefore, without loss of generality, let m be some natural power of 2 that is greater than n.

So, if we have n terms, then let us denote their arithmetic mean by α, and expand our list of terms thus:

$x_{n+1} = x_{n+2} = \cdots = x_m = \alpha.$

We then have:

$\begin{align} \alpha & = \frac{x_1 + x_2 + \cdots + x_n}{n} \\[6pt] & = \frac{\frac{m}{n} \left( x_1 + x_2 + \cdots + x_n \right)}{m} \\[6pt] & = \frac{x_1 + x_2 + \cdots + x_n + \frac{m-n}{n} \left( x_1 + x_2 + \cdots + x_n \right)}{m} \\[6pt] & = \frac{x_1 + x_2 + \cdots + x_n + \left( m-n \right) \alpha}{m} \\[6pt] & = \frac{x_1 + x_2 + \cdots + x_n + x_{n+1} + \cdots + x_m}{m} \\[6pt] & > \sqrt[m]{x_1 x_2 \cdots x_n x_{n+1} \cdots x_m} \\[6pt] & = \sqrt[m]{x_1 x_2 \cdots x_n \alpha^{m-n}}\,, \end{align}$

so

$\alpha^m > x_1 x_2 \cdots x_n \alpha^{m-n}$

and

$\alpha > \sqrt[n]{x_1 x_2 \cdots x_n}$

as desired.

Proof by induction using basic calculus

The following proof uses mathematical induction and some basic differential calculus.

Induction basis: For n = 1 the statement is true with equality.

Induction hypothesis: Suppose that the AM–GM statement holds for all choices of n non-negative real numbers.

Induction step: In order to prove the statement for n + 1 non-negative real numbers x₁, . . . , x_n, x_n+1, we need to prove that

$\frac{x_1 + \cdots + x_n + x_{n+1}}{n+1} - ({x_1 \cdots x_n x_{n+1}})^{\frac{1}{n+1}}\ge0$

with equality only if all the n + 1 numbers are equal.

If all numbers are zero, the inequality holds with equality. If some but not all numbers are zero, we have strict inequality. Therefore, we may assume in the following, that all n + 1 numbers are positive.

We consider the last number x_n+1 as a variable and define the function

$f(t)=\frac{x_1 + \cdots + x_n + t}{n+1} - ({x_1 \cdots x_n t})^{\frac{1}{n+1}},\qquad t>0.$

Proving the induction step is equivalent to showing that f(t) ≥ 0 for all t > 0, with f(t) = 0 only if x₁, . . . , x_n and t are all equal. This can be done by analyzing the critical points of f using some basic calculus.

The first derivative of f is given by

$f'(t)=\frac{1}{n+1}-\frac{1}{n+1}({x_1 \cdots x_n})^{\frac{1}{n+1}}t^{-\frac{n}{n+1}},\qquad t>0.$

A critical point t₀ has to satisfy f′(t₀) = 0, which means

$({x_1 \cdots x_n})^{\frac{1}{n+1}}t_0^{-\frac{n}{n+1}}=1.$

After a small rearrangement we get

$t_0^{\frac{n}{n+1}}=({x_1 \cdots x_n})^{\frac{1}{n+1}},$

and finally

$t_0=({x_1 \cdots x_n})^{\frac{1}n},$

which is the geometric mean of x₁, . . . , x_n. This is the only critical point of f. Since f′′(t) > 0 for all t > 0, the function f is strictly convex and has a strict global minimum at t₀. Next we compute the value of the function at this global minimum:

$\begin{align} f(t_0) &= \frac{x_1 + \cdots + x_n + ({x_1 \cdots x_n})^{1/n}}{n+1} - ({x_1 \cdots x_n})^{\frac{1}{n+1}}({x_1 \cdots x_n})^{\frac{1}{n(n+1)}}\\ &= \frac{x_1 + \cdots + x_n}{n+1} + \frac{1}{n+1}({x_1 \cdots x_n})^{\frac{1}n} - ({x_1 \cdots x_n})^{\frac{1}n}\\ &= \frac{x_1 + \cdots + x_n}{n+1} - \frac{n}{n+1}({x_1 \cdots x_n})^{\frac{1}n}\\ &= \frac{n}{n+1}\Bigl(\frac{x_1 + \cdots + x_n}n - ({x_1 \cdots x_n})^{\frac{1}n}\Bigr)\ge0, \end{align}.$

where the final inequality holds due to the induction hypothesis. The hypothesis also says that we can have equality only when x₁, . . . , x_n are all equal. In this case, their geometric mean  t₀ has the same value, Hence, unless x₁, . . . , x_n, x_n+1 are all equal, we have f(x_n+1) > 0. This completes the proof.

This technique can be used in the same manner to prove the generalized AM–GM inequality and Cauchy–Schwarz inequality in Euclidean space Rⁿ.

Proof by Pólya using the exponential function

George Pólya provided a proof similar to what follows. Let f(x) = e^x–1 – x for all real x, with first derivative f′(x) = e^x–1 – 1 and second derivative f′′(x) = e^x–1. Observe that f(1) = 0, f′(1) = 0 and f′′(x) > 0 for all real x, hence f is strictly convex with the absolute minimum at x = 1. Hence x ≤ e^x–1 for all real x with equality only for x = 1.

Consider a list of non-negative real numbers x₁, x₂, . . . , x_n. If they are all zero, then the AM–GM inequality holds with equality. Hence we may assume in the following for their arithmetic mean α > 0. By n-fold application of the above inequality, we obtain that

$\begin{align}{ \frac{x_1}{\alpha} \frac{x_2}{\alpha} \cdots \frac{x_n}{\alpha} } &\le { e^{\frac{x_1}{\alpha} - 1} e^{\frac{x_2}{\alpha} - 1} \cdots e^{\frac{x_n}{\alpha} - 1} }\\ & = \exp \Bigl( \frac{x_1}{\alpha} - 1 + \frac{x_2}{\alpha} - 1 + \cdots + \frac{x_n}{\alpha} - 1 \Bigr), \qquad (*) \end{align}$

with equality if and only if x_i = α for every i ∈ {1, . . . , n}. The argument of the exponential function can be simplified:

$\begin{align} \frac{x_1}{\alpha} - 1 + \frac{x_2}{\alpha} - 1 + \cdots + \frac{x_n}{\alpha} - 1 & = \frac{x_1 + x_2 + \cdots + x_n}{\alpha} - n \\ & = n - n \\ & = 0. \end{align}$

Returning to (*),

$\frac{x_1 x_2 \cdots x_n}{\alpha^n} \le e^0 = 1,$

which produces x₁ x₂ · · · x_n ≤ αⁿ, hence the result^[3]

$\sqrt[n]{x_1 x_2 \cdots x_n} \le \alpha.$

Generalizations

Weighted AM–GM inequality

There is a similar inequality for the weighted arithmetic mean and weighted geometric mean. Specifically, let the nonnegative numbers x₁, x₂, . . . , x_n and the nonnegative weights w₁, w₂, . . . , w_n be given. Set w = w₁ + w₂ + · · · + w_n. If w > 0, then the inequality

$\frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w} \ge \sqrt[w]{x_1^{w_1} x_2^{w_2} \cdots x_n^{w_n}}$

holds with equality if and only if all the x_k with w_k > 0 are equal. Here the convention 0⁰ = 1 is used.

If all w_k = 1, this reduces to the above inequality of arithmetic and geometric means.

Proof using Jensen's inequality

Using the finite form of Jensen's inequality for the natural logarithm, we can prove the inequality between the weighted arithmetic mean and the weighted geometric mean stated above.

Since an x_k with weight w_k = 0 has no influence on the inequality, we may assume in the following that all weights are positive. If all x_k are equal, then equality holds. Therefore, it remains to prove strict inequality if they are not all equal, which we will assume in the following, too. If at least one x_k is zero (but not all), then the weighted geometric mean is zero, while the weighted arithmetic mean is positive, hence strict inequality holds. Therefore, we may assume also that all x_k are positive.

Since the natural logarithm is strictly concave, the finite form of Jensen's inequality and the functional equations of the natural logarithm imply

$\begin{align} \ln\Bigl(\frac{w_1x_1+\cdots+w_nx_n}w\Bigr) & >\frac{w_1}w\ln x_1+\cdots+\frac{w_n}w\ln x_n \\ & =\ln \sqrt[w]{x_1^{w_1} x_2^{w_2} \cdots x_n^{w_n}}. \end{align}$

Since the natural logarithm is strictly increasing,

$\frac{w_1x_1+\cdots+w_nx_n}w >\sqrt[w]{x_1^{w_1} x_2^{w_2} \cdots x_n^{w_n}}.$

Other generalizations

Other generalizations of the inequality of arithmetic and geometric means include:

• Muirhead's inequality,
• Maclaurin's inequality,
• Generalized mean inequality.

See also

• Ky Fan inequality
• Young's inequality

Notes

References

1. Steele, J. Michael (2004). The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. MAA Problem Books Series. Cambridge University Press. ISBN 978-0-521-54677-5. OCLC 54079548.
2. Cauchy, Augustin-Louis (1821). Cours d'analyse de l'École Royale Polytechnique, première partie, Analyse algébrique, Paris. The proof of the inequality of arithmetic and geometric means can be found on pages 457ff.
3. Arnold, Denise; Arnold, Graham (1993). Four unit mathematics. Hodder Arnold H&S. p. 242. ISBN 978-0-340-54335-1. OCLC 38328013.

External links

• Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.