AM–GM inequality

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File:AM GM inequality animation.gif

Visual proof that Template:Math. Taking square roots and dividing by two gives the AM–GM inequality.<ref>Template:Citation</ref>

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 (in which case they are both that number).

The simplest non-trivial case is for two non-negative numbers Template:Mvar and Template:Mvar, that is,

<math>\frac{x+y}2 \ge \sqrt{xy}</math>

with equality if and only if Template:Math. This follows from the fact that the square of a real number is always non-negative (greater than or equal to zero) and from the identity Template:Math:

<math>\begin{align}

0 & \le (x-y)^2 \\ & = x^2-2xy+y^2 \\ & = x^2+2xy+y^2 - 4xy \\ & = (x+y)^2 - 4xy. \end{align}</math> Hence Template:Math, with equality when Template:Math, i.e. Template:Math. The AM–GM inequality then follows from taking the positive square root of both sides and then dividing both sides by 2.

For a geometrical interpretation, consider a rectangle with sides of length Template:Mvar and Template:Mvar; it has perimeter Template:Math and area Template:Mvar. Similarly, a square with all sides of length Template:Math has the perimeter Template:Math and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that Template:Math and that only the square has the smallest perimeter amongst all rectangles of equal area.

The simplest case is implicit in Euclid's Elements, Book 5, Proposition 25.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Extensions of the AM–GM inequality treat weighted means and generalized means.

Background

The arithmetic mean, or less precisely the average, of a list of Template:Mvar numbers Template:Math is the sum of the numbers divided by Template:Mvar:

<math>\frac{x_1 + x_2 + \cdots + x_n}{n}.</math>

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:

<math>\sqrt[n]{x_1 \cdot x_2 \cdots x_n}.</math>

If Template:Math, this is equal to the exponential of the arithmetic mean of the natural logarithms of the numbers:

<math>\exp \left( \frac{\ln {x_1} + \ln {x_2} + \cdots + \ln {x_n}}{n} \right).</math>

The inequality

Restating the inequality using mathematical notation, we have that for any list of Template:Mvar nonnegative real numbers Template:Math,

<math>\frac{x_1 + x_2 + \cdots + x_n}{n} \ge \sqrt[n]{x_1 \cdot x_2 \cdots x_n}\,,</math>

and that equality holds if and only if Template:Math.

Geometric interpretation

In two dimensions, Template:Math is the perimeter of a rectangle with sides of length Template:Math and Template:Math. Similarly, Template:Math is the perimeter of a square with the same area, Template:Math, as that rectangle. Thus for Template:Math the AM–GM inequality states that a rectangle of a given area has the smallest perimeter if that rectangle is also a square.

The full inequality is an extension of this idea to Template:Mvar dimensions. Consider an Template:Mvar-dimensional box with edge lengths Template:Math. Every vertex of the box is connected to Template:Mvar edges of different directions, so the average length of edges incident to the vertex is Template:Math. On the other hand, <math>\sqrt[n]{x_1 x_2 \cdots x_n}</math> is the edge length of an Template:Mvar-dimensional cube of equal volume, which therefore is also the average length of edges incident to a vertex of the cube.

Thus the AM–GM inequality states that only the [[Hypercube|Template:Mvar-cube]] has the smallest average length of edges connected to each vertex amongst all Template:Mvar-dimensional boxes with the same volume.<ref>Template:Cite book</ref>

Examples

Example 1

If <math>a,b,c>0</math>, then the AM-GM inequality tells us that

<math>(1+a)(1+b)(1+c)\ge 2\sqrt{1\cdot{a}} \cdot 2\sqrt{1\cdot{b}} \cdot 2\sqrt{1\cdot{c}} = 8\sqrt{abc}</math>

Example 2

A simple upper bound for <math>n!</math> can be found. AM-GM tells us

<math>1+2+\dots+n \ge n\sqrt[n]{n!}</math>

<math>\frac{n(n+1)}{2} \ge n\sqrt[n]{n!}</math>

and so

<math>\left(\frac{n+1}{2}\right)^n \ge n!</math>

with equality at <math>n=1</math>.

Equivalently,

<math>(n+1)^n \ge 2^nn!</math>

Example 3

Consider the function

<math>f(x,y,z) = \frac{x}{y} + \sqrt{\frac{y}{z}} + \sqrt[3]{\frac{z}{x}}</math>

for all positive real numbers Template:Mvar, Template:Mvar and Template:Mvar. Suppose we wish to find the minimal value of this function. It can be rewritten as:

<math>

\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}</math> with

<math> 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}}.</math>

Applying the AM–GM inequality for Template:Math, we get

<math>

\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}</math>

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

<math>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}}.</math>

All the points Template:Math satisfying these conditions lie on a half-line starting at the origin and are given by

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

Applications

Cauchy-Schwarz inequality

The AM-GM equality can be used to prove the Cauchy–Schwarz inequality.Template:Cn

Annualized returns

In financial mathematics, the AM-GM inequality shows that the annualized return, the geometric mean, is less than the average annual return, the arithmetic mean.

Nonnegative polynomials Template:Anchor

The Motzkin polynomial <math display="block">x^4y^2+x^2y^4-3x^2y^2+1</math> is a nonnegative polynomial which is not a sum of square polynomials. It can be proven nonnegative using the AM-GM inequality with <math>x_1 = x^4 y^2</math>, <math>x_2 = x^2 y^4</math>, and <math>x_3 = 1</math>,<ref>Template:Cite book</ref> that is, <math display="block">{\sqrt[3]{(x^4 y^2) \cdot (x^2 y^4) \cdot (1)}} \le {{(x^4 y^2)+(x^2 y^4)+(1)}\over{3}}.</math> Simplifying and multiplying both sides by 3 gives <math display="block">{3 x^2 y^2} \le {x^4 y^2 + x^2 y^4 + 1},</math> so<ref> Aaron Potechin, Sum of Squares seminar, University of Chicago, "Lecture 5: SOS Proofs and the Motzkin Polynomial", slide 25</ref> <math display="block">{0 \le x^4 y^2 + x^2 y^4 - 3 x^2 y^2 + 1}.</math>

Proofs of the AM–GM inequality

The AM–GM inequality can be proven in many ways.

Proof using Jensen's inequality

Jensen's inequality states that the value of a concave function of an arithmetic mean is greater than or equal to the arithmetic mean of the function's values. Since the logarithm function is concave, we have

<math>\log \left(\frac { \sum x_i}{n} \right) \geq \frac{1}{n} \sum \log x_i = \frac{1}{n} \log \left( \prod x_i\right) = \log \left( \left( \prod x_i \right) ^{1/n} \right). </math>

Taking antilogs of the far left and far right sides, we have the AM–GM inequality.

Proof by successive replacement of elements

We have to show that

<math>\alpha = \frac{x_1+x_2+\cdots+x_n}{n} \ge \sqrt[n]{x_1x_2 \cdots x_n}=\beta</math>

with equality only when all numbers are equal.

If not all numbers are equal, then there exist <math>x_i,x_j</math> such that <math>x_i<\alpha<x_j</math>. Replacing Template:Mvar by <math>\alpha</math> and Template:Mvar by <math>(x_i+x_j-\alpha)</math> will leave the arithmetic mean of the numbers unchanged, but will increase the geometric mean because

<math>\alpha(x_j+x_i-\alpha)-x_ix_j=(\alpha-x_i)(x_j-\alpha)>0</math>

If the numbers are still not equal, we continue replacing numbers as above. After at most <math>(n-1)</math> such replacement steps all the numbers will have been replaced with <math>\alpha</math> while the geometric mean strictly increases at each step. After the last step, the geometric mean will be <math>\sqrt[n]{\alpha\alpha \cdots \alpha}=\alpha</math>, proving the inequality.

It may be noted that the replacement strategy works just as well from the right hand side. If any of the numbers is 0 then so will the geometric mean thus proving the inequality trivially. Therefore we may suppose that all the numbers are positive. If they are not all equal, then there exist <math>x_i,x_j</math> such that <math>0<x_i<\beta<x_j</math>. Replacing <math>x_i</math> by <math>\beta</math> and <math>x_j</math> by <math>\frac{x_ix_j}{\beta}</math>leaves the geometric mean unchanged but strictly decreases the arithmetic mean since

<math>x_i + x_j - \beta - \frac{x_i x_j}\beta = \frac{(\beta - x_i)(x_j - \beta)}\beta > 0</math>. The proof then follows along similar lines as in the earlier replacement.

Induction proofs

Proof by induction #1

Of the non-negative real numbers Template:Math, the AM–GM statement is equivalent to

<math>\alpha^n\ge x_1 x_2 \cdots x_n</math>

with equality if and only if Template:Math for all Template:Math.

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

Induction basis: For Template:Math the statement is true with equality.

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

Induction step: Consider Template:Math non-negative real numbers Template:Math, . Their arithmetic mean Template:Mvar satisfies

<math> (n+1)\alpha=\ x_1 + \cdots + x_n + x_{n+1}.</math>

If all the Template:Mvar are equal to Template:Mvar, then we have equality in the AM–GM statement and we are done. In the case where some are not equal to Template:Mvar, there must exist one number that is greater than the arithmetic mean Template:Mvar, and one that is smaller than Template:Mvar. Without loss of generality, we can reorder our Template:Mvar in order to place these two particular elements at the end: Template:Math and Template:Math. Then

<math>x_n - \alpha > 0\qquad \alpha-x_{n+1}>0</math>

<math>\implies (x_n-\alpha)(\alpha-x_{n+1})>0\,.\qquad(*)</math>

Now define Template:Math with

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

and consider the Template:Mvar numbers Template:Math which are all non-negative. Since

<math>(n+1)\alpha=x_1 + \cdots + x_{n-1} + x_n + x_{n+1}</math>

<math>n\alpha=x_1 + \cdots + x_{n-1} + \underbrace{x_n+x_{n+1}-\alpha}_{=\,y},</math>

Thus, Template:Mvar is also the arithmetic mean of Template:Mvar numbers Template:Math and the induction hypothesis implies

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

Due to (*) we know that

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

hence

<math>y\alpha>x_nx_{n+1}\,,\qquad({*}{*}{*})</math>

in particular Template:Math. Therefore, if at least one of the numbers Template:Math 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 can substitute (***) into (**) to get

<math>\alpha^{n+1}>x_1x_2 \cdots x_{n-1} x_nx_{n+1}\,,</math>

which completes the proof.

Proof by induction #2

First of all we shall prove that for real numbers Template:Math and Template:Math there follows

<math> x_1 + x_2 > x_1x_2+1.</math>

Indeed, multiplying both sides of the inequality Template:Math by Template:Math, gives

<math> x_2 - x_1x_2 > 1 - x_1,</math>

whence the required inequality is obtained immediately.

Now, we are going to prove that for positive real numbers Template:Math satisfying Template:Math, there holds

<math>x_1 + \cdots + x_n \ge n.</math>

The equality holds only if Template:Math.

Induction basis: For Template:Math the statement is true because of the above property.

Induction hypothesis: Suppose that the statement is true for all natural numbers up to Template:Math.

Induction step: Consider natural number Template:Math, i.e. for positive real numbers Template:Math, there holds Template:Math. There exists at least one Template:Math, so there must be at least one Template:Math. Without loss of generality, we let Template:Math and Template:Math.

Further, the equality Template:Math we shall write in the form of Template:Math. Then, the induction hypothesis implies

<math>(x_1 + \cdots + x_{n-2}) + (x_{n-1} x_n ) > n - 1.</math>

However, taking into account the induction basis, we have

<math>\begin{align}

x_1 + \cdots + x_{n-2} + x_{n-1} + x_n & = (x_1 + \cdots + x_{n-2}) + (x_{n-1} + x_n ) \\ &> (x_1 + \cdots + x_{n-2}) + x_{n-1} x_n + 1 \\ & > n, \end{align}</math>

which completes the proof.

For positive real numbers Template:Math, let's denote

<math>x_1 = \frac{a_1}{\sqrt[n]{a_1\cdots a_n}}, . . ., x_n = \frac{a_n}{\sqrt[n]{a_1\cdots a_n}}. </math>

The numbers Template:Math satisfy the condition Template:Math. So we have

<math>\frac{a_1}{\sqrt[n]{a_1\cdots a_n}} + \cdots + \frac{a_n}{\sqrt[n]{a_1\cdots a_n}} \ge n, </math>

whence we obtain

<math>\frac{a_1 + \cdots + a_n}n \ge \sqrt[n]{a_1\cdots a_n}, </math>

with the equality holding only for Template:Math.

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.<ref>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.</ref>

The case where all the terms are equal

If all the terms are equal:

<math>x_1 = x_2 = \cdots = x_n,</math>

then their sum is Template:Math, so their arithmetic mean is Template:Math; and their product is Template:Math, so their geometric mean is Template:Math; 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 Template:Math.

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

The subcase where n = 2

If Template:Math, then we have two terms, Template:Math and Template:Math, and since (by our assumption) not all terms are equal, we have:

<math>\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} </math>

hence

<math>

\frac{x_1 + x_2}{2} > \sqrt{x_1 x_2}</math>

as desired.

The subcase where n = 2^k

Consider the case where Template:Math, where Template:Mvar is a positive integer. We proceed by mathematical induction.

In the base case, Template:Math, so Template:Math. We have already shown that the inequality holds when Template:Math, so we are done.

Now, suppose that for a given Template:Math, we have already shown that the inequality holds for Template:Math, and we wish to show that it holds for Template:Math. To do so, we apply the inequality twice for Template:Math numbers and once for Template:Math numbers to obtain:

<math>

\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} </math>

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

<math>x_1 = x_2 = \cdots = x_{2^{k-1}}</math>

and

<math>x_{2^{k-1}+1} = x_{2^{k-1}+2} = \cdots = x_{2^k}</math>

(in which case the first arithmetic mean and first geometric mean are both equal to Template:Math, 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 Template:Math numbers are equal, it is not possible for both inequalities to be equalities, so we know that:

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

as desired.

The subcase where n < 2^k

If Template:Mvar is not a natural power of Template:Math, then it is certainly less than some natural power of 2, since the sequence Template:Math is unbounded above. Therefore, without loss of generality, let Template:Mvar be some natural power of Template:Math that is greater than Template:Mvar.

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

<math>x_{n+1} = x_{n+2} = \cdots = x_m = \alpha.</math>

We then have:

<math>\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}) (\because x_1 + x_2 + \cdots + x_n = \frac{{n} (x_1 + x_2 + \cdots + x_n)}Template:N) \\[3pt] & = \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] & \ge \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}</math>

so

<math>\alpha^m \ge x_1 x_2 \cdots x_n \alpha^{m-n}</math>

and

<math>\alpha \ge \sqrt[n]{x_1 x_2 \cdots x_n}</math>

as desired.

Proof by induction using basic calculus

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

Induction basis: For Template:Math the statement is true with equality.

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

Induction step: In order to prove the statement for Template:Math non-negative real numbers Template:Math, we need to prove that

<math>\frac{x_1 + \cdots + x_n + x_{n+1}}{n+1} - ({x_1 \cdots x_n x_{n+1}})^{\frac{1}{n+1}}\ge0</math>

with equality only if all the Template:Math 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 Template:Math numbers are positive.

We consider the last number Template:Math as a variable and define the function

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

Proving the induction step is equivalent to showing that Template:Math for all Template:Math, with Template:Math only if Template:Math and Template:Mvar are all equal. This can be done by analyzing the critical points of Template:Mvar using some basic calculus.

The first derivative of Template:Mvar is given by

<math>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.</math>

A critical point Template:Math has to satisfy Template:Math, which means

<math>({x_1 \cdots x_n})^{\frac{1}{n+1}}t_0^{-\frac{n}{n+1}}=1.</math>

After a small rearrangement we get

<math>t_0^{\frac{n}{n+1}}=({x_1 \cdots x_n})^{\frac{1}{n+1}},</math>

and finally

<math>t_0=({x_1 \cdots x_n})^{\frac{1}n},</math>

which is the geometric mean of Template:Math. This is the only critical point of Template:Mvar. Since Template:Math for all Template:Math, the function Template:Mvar is strictly convex and has a strict global minimum at Template:Math. Next we compute the value of the function at this global minimum:

<math>

\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}</math>

where the final inequality holds due to the induction hypothesis. The hypothesis also says that we can have equality only when Template:Math are all equal. In this case, their geometric mean  Template:Math has the same value, Hence, unless Template:Math are all equal, we have Template:Math. 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 Template:Math.

Proof by Pólya using the exponential function

George Pólya provided a proof similar to what follows. Let Template:Math for all real Template:Mvar, with first derivative Template:Math and second derivative Template:Math. Observe that Template:Math, Template:Math and Template:Math for all real Template:Mvar, hence Template:Mvar is strictly convex with the absolute minimum at Template:Math. Hence Template:Math for all real Template:Mvar with equality only for Template:Math.

Consider a list of non-negative real numbers Template:Math. If they are all zero, then the AM–GM inequality holds with equality. Hence we may assume in the following for their arithmetic mean Template:Math. By Template:Mvar-fold application of the above inequality, we obtain that

<math>\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}</math>

with equality if and only if Template:Math for every Template:Math. The argument of the exponential function can be simplified:

<math>\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 \\ & = \frac{n \alpha}{\alpha} - n \\ & = 0. \end{align}</math>

Returning to Template:Math,

<math>\frac{x_1 x_2 \cdots x_n}{\alpha^n} \le e^0 = 1,</math>

which produces Template:Math, hence the result<ref>Template:Cite book</ref>

<math>\sqrt[n]{x_1 x_2 \cdots x_n} \le \alpha.</math>

Proof by Lagrangian multipliers

If any of the <math>x_i</math> are <math>0</math>, then there is nothing to prove. So we may assume all the <math>x_i</math> are strictly positive.

Because the arithmetic and geometric means are homogeneous of degree 1, without loss of generality assume that <math>\prod_{i=1}^n x_i = 1</math>. Set <math>G(x_1,x_2,\ldots,x_n)=\prod_{i=1}^n x_i</math>, and <math>F(x_1,x_2,\ldots,x_n) = \frac{1}{n}\sum_{i=1}^n x_i</math>. The inequality will be proved (together with the equality case) if we can show that the minimum of <math>F(x_1,x_2,...,x_n),</math> subject to the constraint <math>G(x_1,x_2,\ldots,x_n) = 1,</math> is equal to <math>1</math>, and the minimum is only achieved when <math>x_1 = x_2 = \cdots = x_n = 1</math>. Let us first show that the constrained minimization problem has a global minimum.

Set <math>K = \{(x_1,x_2,\ldots,x_n) \colon 0 \leq x_1,x_2,\ldots,x_n \leq n\}</math>. Since the intersection <math>K \cap \{G = 1\}</math> is compact, the extreme value theorem guarantees that the minimum of <math>F(x_1,x_2,...,x_n)</math> subject to the constraints <math>G(x_1,x_2,\ldots,x_n) = 1</math> and <math> (x_1,x_2,\ldots,x_n) \in K </math> is attained at some point inside <math>K</math>. On the other hand, observe that if any of the <math>x_i > n</math>, then <math>F(x_1,x_2,\ldots,x_n) > 1 </math>, while <math>F(1,1,\ldots,1) = 1</math>, and <math>(1,1,\ldots,1) \in K \cap \{G = 1\} </math>. This means that the minimum inside <math>K \cap \{G = 1\}</math> is in fact a global minimum, since the value of <math>F</math> at any point inside <math>K \cap \{G = 1\}</math> is certainly no smaller than the minimum, and the value of <math>F</math> at any point <math>(y_1,y_2,\ldots, y_n)</math> not inside <math>K</math> is strictly bigger than the value at <math>(1,1,\ldots,1)</math>, which is no smaller than the minimum.

The method of Lagrange multipliers says that the global minimum is attained at a point <math>(x_1,x_2,\ldots,x_n)</math> where the gradient of <math>F(x_1,x_2,\ldots,x_n)</math> is <math>\lambda</math> times the gradient of <math>G(x_1,x_2,\ldots,x_n)</math>, for some <math>\lambda</math>. We will show that the only point at which this happens is when <math>x_1 = x_2 = \cdots = x_n = 1</math> and <math>F(x_1,x_2,...,x_n) = 1.</math>

Compute <math>\frac{\partial F}{\partial x_i} = \frac{1}{n}</math> and

<math>\frac{\partial G}{\partial x_i} = \prod_{j \neq i}x_j = \frac{G(x_1,x_2,\ldots,x_n)}{x_i} = \frac{1}{x_i}</math>

along the constraint. Setting the gradients proportional to one another therefore gives for each <math>i</math> that <math>\frac{1}{n} = \frac{\lambda}{x_i},</math> and so <math>n\lambda= x_i.</math> Since the left-hand side does not depend on <math>i</math>, it follows that <math>x_1 = x_2 = \cdots = x_n</math>, and since <math>G(x_1,x_2,\ldots, x_n) = 1</math>, it follows that <math> x_1 = x_2 = \cdots = x_n = 1</math> and <math>F(x_1,x_2,\ldots,x_n) = 1</math>, as desired.

Generalizations

Weighted AM–GM inequality

There is a similar inequality for the weighted arithmetic mean and weighted geometric mean. Specifically, let the nonnegative numbers Template:Math and the nonnegative weights Template:Math be given. Set Template:Math. If Template:Math, then the inequality

<math>\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}}</math>

holds with equality if and only if all the Template:Mvar with Template:Math are equal. Here the convention Template:Math is used.

If all Template:Math, this reduces to the above inequality of arithmetic and geometric means.

One stronger version of this, which also gives strengthened version of the unweighted version, is due to Aldaz. Specifically, let the nonnegative numbers Template:Math and the nonnegative weights Template:Math be given. Assume further that the sum of the weights is 1. Then

<math>\sum_{i=1}^n w_ix_i \geq \prod_{i=1}^n x_i^{w_i} + \sum_{i=1}^n w_i\left(x_i^{\frac{1}{2}} -\sum_{i=1}^n w_ix_i^{\frac{1}{2}} \right)^2 </math>.<ref>Template:Cite journal</ref>

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 Template:Mvar with weight Template:Math has no influence on the inequality, we may assume in the following that all weights are positive. If all Template:Mvar 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 Template:Mvar 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 Template:Mvar 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

<math>\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}</math>

Since the natural logarithm is strictly increasing,

<math>

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

Matrix arithmetic–geometric mean inequality

Most matrix generalizations of the arithmetic geometric mean inequality apply on the level of unitarily invariant norms, since, even if the matrices <math>A</math> and <math>B</math> are positive semi-definite, the matrix <math>A B</math> may not be positive semi-definite and hence may not have a canonical square root. In <ref>Template:Cite journal</ref> Bhatia and Kittaneh proved that for any unitarily invariant norm <math>|||\cdot|||</math> and positive semi-definite matrices <math>A</math> and <math>B</math> it is the case that

<math>

|||AB|||\leq \frac{1}{2}|||A^2 + B^2||| </math> Later, in <ref>Template:Cite journal</ref> the same authors proved the stronger inequality that

<math>

|||AB||| \leq \frac{1}{4}|||(A+B)^2||| </math> Finally, it is known for dimension <math>n=2</math> that the following strongest possible matrix generalization of the arithmetic-geometric mean inequality holds, and it is conjectured to hold for all <math>n</math>

<math>

|||(AB)^{\frac{1}{2}}|||\leq \frac{1}{2}|||A+B||| </math>

This conjectured inequality was shown by Stephen Drury in 2012. Indeed, he proved<ref>S.W. Drury, On a question of Bhatia and Kittaneh, Linear Algebra Appl. 437 (2012) 1955–1960.</ref>

<math>\sqrt{\sigma_j(AB)}\leq \frac{1}{2}\lambda_j(A+B), \ j=1, \ldots, n.</math>

Finance: Link to geometric asset returns

In finance much research is concerned with accurately estimating the rate of return of an asset over multiple periods in the future. In the case of lognormal asset returns, there is an exact formula to compute the arithmetic asset return from the geometric asset return.

For simplicity, assume we are looking at yearly geometric returns Template:Math over a time horizon of Template:Mvar years, i.e.

<math>r_n=\frac{V_n - V_{n-1}}{V_{n-1}},</math>

where:

<math>V_n</math> = value of the asset at time <math>n</math>,

<math>V_{n-1}</math> = value of the asset at time <math>n-1</math>.

The geometric and arithmetic returns are respectively defined as

<math>g_N=\left(\prod_{n = 1}^N(1+r_n)\right)^{1/N},</math>

<math>a_N=\frac1N \sum_{n = 1}^Nr_n.</math>

When the yearly geometric asset returns are lognormally distributed, then the following formula can be used to convert the geometric average return to the arithemtic average return:<ref>Template:Cite journal</ref>

<math>1+g_N=\frac{1+a_N}{\sqrt{1+\frac{\sigma^2}{(1+a_N)^2}}},</math>

where <math>\sigma^2</math> is the variance of the observed asset returns This implicit equation for Template:Mvar can be solved exactly as follows. First, notice that by setting

<math>z=(1+a_N)^2,</math>

we obtain a polynomial equation of degree 2:

<math>z^2 - (1+g)^2 - (1+g)^2\sigma^2 = 0.</math>

Solving this equation for Template:Mvar and using the definition of Template:Mvar, we obtain 4 possible solutions for Template:Mvar:

<math>a_N = \pm \frac{1+g_N}{\sqrt{2}}\sqrt{1 \pm \sqrt{1+\frac{4\sigma^2}{(1+g_N)^2}}}-1.</math>

However, notice that

<math> \sqrt{1+\frac{4\sigma^2}{(1+g_N)^2}} \geq 1. </math>

This implies that the only 2 possible solutions are (as asset returns are real numbers):

<math>a_N = \pm \frac{1+g_N}{\sqrt{2}}\sqrt{1 + \sqrt{1+\frac{4\sigma^2}{(1+g_N)^2}}}-1.</math>

Finally, we expect the derivative of Template:Mvar with respect to Template:Mvar to be non-negative as an increase in the geometric return should never cause a decrease in the arithmetic return. Indeed, both measure the average growth of an asset's value and therefore should move in similar directions. This leaves us with one solution to the implicit equation for Template:Mvar, namely

<math>a_N = \frac{1+g_N}{\sqrt{2}}\sqrt{1 + \sqrt{1+\frac{4\sigma^2}{(1+g_N)^2}}}-1.</math>

Therefore, under the assumption of lognormally distributed asset returns, the arithmetic asset return is fully determined by the geometric asset return.

Other generalizations

Template:QM AM GM HM inequality visual proof.svg Other generalizations of the inequality of arithmetic and geometric means include:

• Muirhead's inequality,
• Maclaurin's inequality,
• QM-AM-GM-HM inequalities,
• Generalized mean inequality,
• Means of complex numbers.<ref>cf. Iordanescu, R.; Nichita, F.F.; Pasarescu, O. Unification Theories: Means and Generalized Euler Formulas. Axioms 2020, 9, 144.</ref>

See also

• Hoffman's packing puzzle
• Ky Fan inequality
• Young's inequality for products

External links

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