Proofs of AM-GM inequality


Solution 1

This is a fairly old answer of mine with a proof that was not very motivated, so I thought I'd update it with some geometric intuition about convexity.

Consider for simplicity the two-variable case $(a+b)/2 \ge \sqrt{ab}$ and fix, say $a = 1$. The plot of $(1+b)/2$ and $\sqrt{b}$ show intuitively how the concave nature of the geometric mean implies it will always lie below the arithmetic mean. Also observe the equality at one point. In fact, this concavity will extend to any number of variables, but obviously a plot is not a proof.

plot example

The proof for more than two variables presented requires elementary properties of logarithms which transforms dealing with multiplication to dealing with addition. The inequality is rewritten in terms of logarithms:

$$ \frac{x_1 + \dots + x_n}{n}\ge (x_1 \dots x_n)^{1/n} $$

Taking logs preserves the inequality since $\log$ is an increasing function:

$$\iff \log \left(\frac{x_1 + \dots + x_n}{n}\right) \ge \frac 1 n \log (x_1 \dots x_n) = \frac{\log x_1 + \dots + \log x_n}{n}$$

$\DeclareMathOperator{\E}{E}$ If we write $\E[X]$ as the mean of $x_i$'s and $\E[\log(X)]$ as the mean of $\log x_i$'s, we can also understand this in the language of expectation:

$$\log(\E[X]) \ge \E[\log (X)]$$

Using the concavity of $\log$, by Jensen's Inequality (proved inductively starting from the definition of convexity), the inequality holds.

Original post of Pólya's Proof, using similar ideas of convexity of $e^x$:

Let $f(x) = e^{x-1}-x$. The first derivative is $f'(x)=e^{x-1}-1$ and the second derivative is $f''(x) = e^{x-1}$.

$f$ is convex everywhere because $f''(x) > 0$, and has a minimum at $x=1$. Therefore $x \le e^{x-1}$ for all $x$, and the equation is only equal when $x=1$.

Using this inequality we get

$$\frac{x_1}{a} \frac{x_2}{a} \cdots \frac{x_n}{a} \le e^{\frac{x_1}{a}-1} e^{\frac{x_2}{a}-1} \cdots e^{\frac{x_n}{a}-1}$$

with $a$ being the arithmetic mean. The right side simplifies

$$\exp \left(\frac{x_1}{a} -1 \ +\frac{x_1}{a} -1 \ + \cdots + \frac{x_n}{a} -1 \right)$$

$$=\exp \left(\frac{x_1 + x_2 + \cdots + x_n}{a} - n \right) = \exp(n - n) = e^0 = 1$$

Going back to the first inequality

$$\frac{x_1x_2\cdots x_n}{a^n} \le 1$$

So we end with

$$\sqrt[n]{x_1x_2\cdots x_n} \le a$$

Solution 2

I shall provide a simple geometric proof of the inequality in the case of two variables (which I have not been able to find anywhere else - a proof involving a triangle in a circle seems to be popular).

Consider the square of side $a + b$ in the figure below.

The area of the square is $(a + b)^2$. But as it completely contains the four blue rectangles, each of area $ab$, it follows that

$(a + b)^2 \ge 4ab \Rightarrow\\ \dfrac{a + b}{2} \ge \sqrt{ab} $

Further, note that there is a square in middle, of side $(b - a)$, and hence area $(b - a)^2$. Therefore the inequality is strict except when $a = b$.

This proves the two-variable case. The same can be extended to the $n$-variable case. I have tried extending it to three variables, but it is difficult to argue why exactly $27$ rectangular parallelepipeds (of sides $a, b, c$) fit in the cube (of side $a + b + c$), though I can see it is so. Any suggestions?

Solution 3

LEMMA. If $a_1,\dots,a_n$ are positive numbers whose product is equal to $1,$ then $a_1+\dots+a_n\ge n,$ with equality only when $a_1=\cdots=a_n=1.$

Proof by induction on $n.$ The case $n=1$ being trivial, we suppose $n\ge2.$ Let $a_1,\ a_2,\ \dots,\ a_n$ be positive numbers with $a_1a_2\dots a_n=1.$ Without loss of generality, we can assume that $a_1=\max\{a_1,\dots,a_n\}\ge1$ and $a_2=\min\{a_1,\dots,a_n\}\le1.$ Thus we have $$a_1+a_2-a_1a_2-1=(a_1-1)(1-a_2)\ge0,$$ i.e., $$a_1+a_2-a_1a_2\ge1.\tag1$$ Since $a_1a_2,\ a_3,\ a_4,\ \dots,\ a_n$ are $n-1$ positive numbers with product equal to $1,$ by the inductive hypothesis we have $$a_1a_2+a_3+\cdots+a_n\ge n-1.\tag2$$

Adding the inequalities (1) and (2), we get $$a_1+a_2+a_3+\cdots+a_n\ge1+(n-1)=n.\tag3$$ If equality holds in (3) then we must have equality in (1), that is, $a_1=1$ or $a_2=1,$ whence $a_1=\dots=a_n=1.$

THEOREM. If $x_1,\ x_2,\ \dots,\ x_n$ are positive numbers, then $$\frac{x_1+x_2+\cdots+x_n}n\ge(x_1x_2\cdots x_n)^{\frac1n},$$ with equality only when $x_1=x_2=\cdots=x_n.$

Proof. Let $g=(x_1x_2\cdots x_n)^{\frac1n}.$ According to the lemma we have $$\frac{x_1}g+\frac{x_2}g+\cdots+\frac{x_n}g\ge n,$$ i.e., $$\frac{x_1+x_2+\cdots+x_n}n\ge g=(x_1x_2\cdots x_n)^{\frac1n}.$$ Equality holds only if $\frac{x_1}g=\frac{x_2}g=\dots=\frac{x_n}g=1,$ that is, $x_1=x_2=\cdots=x_n.$

Solution 4

As requested by dani_s, I will give the thermodynamic proof of the AM-GM inequality. This is certainly an example of an original proof, although you might argue about whether or not it's rigorous.

Let's start with a list of numbers $x_i$ for which we want to prove the inequality. Take $n$ identical heat reservoirs with the same heat capacity $c$. Reservoir $i$ had initial temperature $x_i$. Bring those reservoirs in contact with each other such that this system evolves to an equilibrium temperature A.

The first law of thermodynamics (conservation of energy) implies that A equals the arithmetic mean of the $x_i$, AM.

The second law of thermodynamics states that the entropy increases until the equilibrium is reached, where the entropy has a maximum. The corresponding formula of change in entropy is $$\Delta S=c \ln{\frac{T}{T_0}}$$ where $c$ is the heat capacity, $T_0$ the initial temperature and $T$ the end temperature.

In our case $T_i=A$ for all $i$ and $T_{0,i}=x_i$. The total entropy didn't decrease and therefore, $$\sum_{i=1}^n c \ln\frac{A}{x_i} \geq 0$$

By writing the sum of logarithms as a logarithm of a product, we recognize the geometric mean. Therefore (since $A=AM$): $$\frac{AM^n}{GM^n} \geq 1$$ This proves the AM-GM inequality.

Solution 5

As $(\sqrt{x_1}-\sqrt{x_2})^2 \geq 0$ we have $$\sqrt{x_1 \cdot x_2} \leq \frac{x_1+x_2}{2}.$$ Applying this inequality twice, we get $$(x_1 x_2 x_3 x_4)^{\frac{1}{4}} \leq \frac{\sqrt{x_1 x_2}+\sqrt{x_3 x_4}}{2} \leq \frac{x_1+x_2+x_3+x_4}{4}.$$ By induction, it is not difficult to see that $$(x_1 \cdots x_{2^k})^{\frac{1}{2^k}} \leq \frac{x_1+\ldots+x_{2^k}}{2^k} \tag{1}$$ for all $k \geq 1$.

It remains to fill the gaps between the powers of two. So let $x_1,\ldots,x_n$ be arbitrary positive numbers and choose $k$ such that $n\leq 2^k$. We set

$$\alpha_i := \begin{cases} x_i & i \leq n \\ A & n< i \leq 2^k \end{cases}$$

where $A:= \frac{x_1+\ldots+x_n}{n}$. Applying $(1)$ to the $(\alpha_1,\ldots,\alpha_{2^k})$ yields

$$\bigg( x_1 \ldots x_n A^{2^k-n} \bigg)^{\frac{1}{2^k}} \leq \frac{x_1+\ldots+x_n+(2^k-n) A}{2^k} = A.$$


$$(x_1 \ldots x_n)^{1/n} \leq A = \frac{x_1+\ldots+x_n}{n}.$$


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Michiel Van Couwenberghe
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Michiel Van Couwenberghe

I am a PhD student in mathematics at Ghent University (Belgium). My main research interest is non associative algebras and their connections to finite group theory, linear algebraic groups and incidence geometry.

Updated on October 30, 2021


  • Michiel Van Couwenberghe
    Michiel Van Couwenberghe over 1 year

    The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone knew a proof that is unexpected in some way. e.g. can you link the theorem to some famous theorem, can you find a non-trivial geometric proof (I can find some of those), proofs that use theory that doesn't link to this inequality at first sight (e.g. differential equations …)?

    Induction, backward induction, use of Jensen inequality, swapping terms, use of Lagrange multiplier, proof using thermodynamics (yeah, I know, it's rather some physical argument that this theorem might be true, not really a proof), convexity, … are some of the proofs I know.

    • chubakueno
      chubakueno about 9 years
      What have you seen, so that we don't repeat what you already know?
    • Hayden
      Hayden about 9 years
    • Michiel Van Couwenberghe
      Michiel Van Couwenberghe about 9 years
      Induction, backward induction, use of Jensen inequality, swapping terms, use of Lagrange multiplier, proof using thermodynamics are some of the proofs I know. Unfortunately, the proofs suggested by Hayden are also already on my list.
    • chubakueno
      chubakueno about 9 years
      Would you include that in your post? (And,thermodynamics? ._. )
    • dani_s
      dani_s about 9 years
      I'm curious about the thermodynamics proof $\ddot \smile$
    • abnry
      abnry about 9 years
      Community wiki?
    • vonbrand
      vonbrand about 9 years
      Please add the proofs you know as answers (just to get them all together).
    • Jose Antonio
      Jose Antonio about 9 years
    • ASB
      ASB about 8 years
      possible duplicate of Proving the AM:GM inequality
    • Admin
      Admin about 8 years
      @aruna This version is more comprehensive, better to close the other one as a duplicate of this. Age of questions doesn't matter.
  • Martín-Blas Pérez Pinilla
    Martín-Blas Pérez Pinilla about 9 years
    Almost all your inequalities are reversed.
  • qwr
    qwr about 9 years
    @Martín-BlasPérezPinilla Woops, fixed
  • Siméon
    Siméon almost 9 years
    I believe this one is due to Cauchy.
  • Ragib Zaman
    Ragib Zaman almost 9 years
    This reminds me of a trick I found: $e^x \geq 1+x$ by similar calculations as the start of this answer. Then $e^{1/n} > 1 + 1/n = (n+1)/n,$ so $e^{1 + 1/2 + 1/3 + \ldots + 1/n} > (2/1) (3/2) \ldots ( (n+1)/n ) = n+1,$ so $H_n:= 1 + 1/2 + \ldots + 1/n > \log (n+1),$ and in particular it diverges.
  • Admin
    Admin almost 8 years
    Can you please fix the formatting.
  • Mariano Suárez-Álvarez
    Mariano Suárez-Álvarez almost 8 years
    There is no need to argue whether this is or not a rigorous proof, as it isn't.
  • DanielWainfleet
    DanielWainfleet over 7 years
    The assumption is that entropy is minimum when all $T_i$ are equal, is known to be true only because of AGM is known to hold.
  • Saikat
    Saikat over 7 years
    I enjoyed this proof a lot. There were so many nice elements. I like how the product is always positive, and how nicely the a_1a_2 terms cancel each other out when you add the inequalities, and how elegantly this lemma is applied to prove the problem. Did you come up with this on your own?
  • ghosts_in_the_code
    ghosts_in_the_code over 6 years
    How did you get step (2)? Shouldn't it be $a_1+a_2+a_3+\cdots$?
  • ghosts_in_the_code
    ghosts_in_the_code over 6 years
    Oh ok, u've considered $a_1a_2$ as a single number, not as two distinct ones. Got it.
  • hukachaka
    hukachaka over 5 years
    @qwr : i know this thread is a few years old but i still have a question. How is exp(xi/a - n)=exp(n-n) ?
  • qwr
    qwr over 5 years
    @hukachaka $a$ is defined as the arithmetic mean.
  • Jack D'Aurizio
    Jack D'Aurizio over 5 years
    Please avoid posting duplicate answers: You have already been contacted by a moderator in such regard.
  • Charlie Tian
    Charlie Tian about 5 years
    interesting and fun concept, but nonetheless very circular
  • Charlie Tian
    Charlie Tian about 5 years
    my favorite proof. utilizing the fact that this is a corollary of the behavior of convex functions and tying it together is much better IMO than ad hoc algebraic manipulations
  • tortue
    tortue almost 5 years
    I think Hölder's inequality is way more general than AM-GM and proving the latter through the former is something artificial.
  • Bob
    Bob almost 5 years
    @pointguard0 If that is your problem, you can always use Cauchy Schwarz to get the result for $p=1/2,q=1$, then $p=1/4, q=1/2$, and so on...
  • Sandeep Silwal
    Sandeep Silwal over 3 years
    Very nice proof! :)
  • CopyPasteIt
    CopyPasteIt about 3 years
    I wanted an elementary algebraic proof (with the examination of rectangles/squares in 2-dim driving it) which leads to $\quad \sum_{i=1}^n x_i^n \ge n \prod_{i=1}^n x_i$. $\quad$ But this would inexorably lead to proving your lemma! Although dependent on 'proof aesthetics', can't understand why this is #1 answer. (+1)
  • CopyPasteIt
    CopyPasteIt about 3 years
    I wanted to extend the 2-dim proof to higher dimension and prove $\quad \sum_{i=1}^n x_i^n \ge n \prod_{i=1}^n x_i$. $\quad$ The techniques employed in bof's answer gets the job done.
  • Batominovski
    Batominovski almost 3 years
    Please copy your answer here to this link:…. In a week, I may copy your answer into the link so that the answer is preserved. The answer will be a Community Wiki post, and you can remove the part that belongs to you, and put it in your own separate answer. (I'm not sure if you were notified from my comment in the deleted post.)
  • Philipp
    Philipp about 1 year
    The fact that $a_1=a_2=1$ is clear but I don't see why you can conclude $a_1=\dots=a_n=1$? It would be possible to have, say $a_3<1$ and $a_4>1$ and still satisfy $a_1+a_2+a_3+\cdots+a_n=n$. Can you elaborate a bit?
  • bof
    bof about 1 year
    @Philipp I assumed that $a_1=\max\{a_1,\dots,a_n\}$ and $a_2=\min\{a_1,\dots,a_n\}$.