Proofs of AM-GM

This pages lists some proofs of the weighted AM-GM Inequality. The inequality's statement is as follows: for all nonnegative reals $a_1, \dotsc, a_n$ and nonnegative reals $\lambda_1, \dotsc, \lambda_n$ such that $\sum_{i=1}^n \lambda_i = 1$, then \[\sum_{i=1}^n \lambda_i a_i \ge \prod_{i=1}^n a_i^{\lambda_i},\] with equality if and only if $a_i = a_j$ for all $i,j$ such that $\lambda_i, \lambda_j \neq 0$.

We first note that we may disregard any $a_i$ for which $\lambda_i= 0$, as they contribute to neither side of the desired inequality. We also note that if $a_i= 0$ and $\lambda_i \neq 0$, for some $i$, then the right-hand side of the inequality is zero and the left hand of the inequality is greater or equal to zero, with equality if and only if $a_j = 0 = a_i$ whenever $\lambda_j\neq 0$. Thus we may henceforth assume that all $a_i$ and $\lambda_i$ are strictly positive.

Proofs of Unweighted AM-GM

These proofs use the assumption that $\lambda_i = 1/n$, for all integers $1 \le i \le n$.

Proof by Cauchy Induction

We use Cauchy Induction, a variant of induction in which one proves a result for $2$, all powers of $2$, and then that $n$ implies $n-1$.

Base Case: The smallest nontrivial case of AM-GM is in two variables. By the properties of perfect squares (or by the Trivial Inequality), $(x-y)^2 \geq 0,$ with equality if and only if $x-y=0$, or $x=y$. Then because $x$ and $y$ are nonnegative, we can perform the following manipulations: \[x^2 - 2xy + y^2 > 0\] \[x^2 + 2xy + y^2 > 4xy\] \[(x+y)^2 = 4xy\] \[\frac{(x+y)^2}{4} \geq xy\] \[\frac{x+y}{2} \geq \sqrt{xy},\] with equality if and only if $x=y$, just as before. This completes the proof of the base case.

Powers of Two: We use induction. Suppose that AM-GM is true for $n$ variables; we will then prove that the inequality is true for $2n$. Let $x_1, x_2, \ldots x_{2n}$ be any list of nonnegative reals. Then, because the two lists $x_1, x_2 \ldots x_n$ and $x_{n+1}, x_{n+2}, \ldots x_{2n}$, each have $n$ variables, \[\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n} \textrm{  and  } \frac{x_{n+1} + x_{n+2} + \cdots + x_{2n}}{n} \geq \sqrt[n]{x_{n+1} x_{n+2} \cdots x_{2n}}.\] Adding these two inequalities together and dividing by two yields \[\frac{x_1 + x_2 + \cdots + x_{2n}}{2n} \geq \frac{\sqrt[n]{x_1 x_2 \cdots x_n} + \sqrt[n]{x_{n+1} x_{n+2} \cdots x_{2n}}}{2}.\] From here, we perform AM-GM in two variables on $\sqrt[n]{x_1 x_2 \cdots x_n}$ and $\sqrt[n]{x_{n+1} x_{n+2} \cdots x_{2n}}$ to get \[\frac{\sqrt[n]{x_1 x_2 \cdots x_n} + \sqrt[n]{x_{n+1} x_{n+2} \cdots x_{2n}}}{2} \geq \sqrt[2n]{x_1 x_2 \ldots x_{2n}}.\] Combining this inequality with the previous one yields AM-GM in $2n$ variables, with one exception — equality.

For equality, note that every AM-GM application mentioned must have equality as well; thus, inequality holds if and only if all the numbers in $x_1, x_2, \ldots x_n$ are the same, all the numbers in $x_{n+1}, x_{n+2}, \ldots x_{2n}$ are the same, and $\sqrt[n]{x_1 x_2 \cdots x_n} = \sqrt[n]{x_{n+1} x_{n+2} \cdots x_{2n}}$. From here, it is trivial to show that this implies $x_1 = x_2 = \cdots x_{2n}$, which is the equality condition for AM-GM in $2n$ variables.

This completes the induction and proves that the inequality holds for all powers of two.

Backward Step: Assume that AM-GM holds for $n$ variables. We will then use a substitution to derive AM-GM for $n-1$ variables. Letting $x_n = \frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}$, we have that \[\frac{x_1 + x_2 + \cdots + x_{n-1} + \frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_{n-1} \left(\frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}\right)}.\] Because we assumed AM-GM in $n$ variables, equality holds if and only if $x_1 = x_2 = \cdots x_{n-1} = \frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}$. However, note that the last equality is implied if all the numbers of $x_1, x_2 \ldots x_{n-1}$ are the same; thus, equality holds if and only if $x_1 = x_2 = \cdots x_{n-1}$.

We first simplify the lefthand side. Multiplying both sides of the fraction by $n-1$ and combining like terms, we get that \[\frac{x_1 + x_2 + \cdots + x_{n-1} + \frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}}{n} = \frac{nx_1 + nx_2 + \cdots + nx_{n-1}}{n(n-1)} = \frac{x_1 + x_2 + \cdots x_{n-1}}{n-1}.\] Plugging this into the earlier inequality yields \[\frac{x_1 + x_2 + \cdots x_{n-1}}{n-1} \geq \sqrt[n]{x_1 x_2 \cdots x_{n-1} \left(\frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1} \right)}.\] Raising both sides to the $n$th power yields \[\left( \frac{x_1 + x_2 + \cdots x_{n-1}}{n-1}\right)^n \geq x_1 x_2 \cdots x_{n-1}\left(\frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}\right).\] From here, we divide by $\frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}$ and take the $(n-1)^{\textrm{th}}$ root to get that \[\frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1} \geq \sqrt[n-1]{x_1 x_2 \cdots x_{n-1}}.\] This is the inequality in $n-1$ variables. Note that every step taken also preserves equality, which completes the backwards step. Then by Cauchy Induction, the AM-GM inequality holds. $\square$

Proof by Rearrangement

Define the $n$ sequence $\{ r_{i,j}\}_{i=1}^{n}$ as $r_{i,j} = \sqrt[n]{a_i}$, for all integers $1 \le i,j \le n$. Evidently these sequences are similarly sorted. Then by the Rearrangement Inequality, \[\sum_i a_i = \sum_i \prod_j r_{i,j} \ge \sum_i \prod_j r_{i+j,j} = n \prod_i \sqrt[n]{a_i} ,\] where we take our indices modulo $n$, with equality exactly when all the $r_{i,j}$, and therefore all the $a_i$, are equal. Dividing both sides by $n$ gives the desired inequality. $\blacksquare$

Proof by Calculus

We will start the proof by considering the function $f(x) = \ln x - x$. We will now find the maximum of this function. We can do this simply using calculus. We need to find the critical points of $f(x)$, we can do that by finding $f'(x)$ and setting it equal to $0$. Using the linearity of the derivative $f'(x) = \frac{1}{x} - 1$. We need $f'(x) = 0$ \[f'(x) = \frac{1}{x} - 1 = 0\] \[\frac{1}{x} = 1\] \[x = 1\] Note that this is the only critical point of $f(x)$. We can confirm it is the maximum by finding it's second derivative and making sure it is negative. $f''(x) = -\frac{1}{x^2}$ letting x = 1 we get $f''(1) = -\frac{1}{1} < 0$. Since the second derivative $f''(x) < 0$, $f(1)$ is a maximum. $f(1) = \ln 1 - 1 = -1$. Now that we have that $f(1) = -1$ is a maximum of $f(x)$, we can safely say that $\ln x - x \leq -1$ or in other words $\ln x \leq x - 1$. We will now define a few more things and do some manipulations with them. Let $A = \frac{\sum_{i=1}^{n} a_i}{n}$, with this notice that $nA = \sum_{i=1}^{n} a_i$. This fact will come into play later. now we can do the following, let $x = \frac{a_i}{A}$ and plug this into $\ln x < x - 1$, we get \[\ln\left(\frac{a_1}{A}\right) \leq \frac{a_1}{A} - 1\] \[\ln\left(\frac{a_2}{A}\right) \leq \frac{a_2}{A} - 1\] \[\vdotswithin{=}\] \[\ln\left(\frac{a_n}{A}\right) \leq \frac{a_n}{A} - 1\] Adding all these results together we get \[\ln\left(\frac{a_1}{A}\right) + \ln\left(\frac{a_2}{A}\right) + \dots + \ln\left(\frac{a_n}{A}\right) \leq \frac{a_1}{A} - 1 + \frac{a_2}{A} - 1 + \dots + \frac{a_n}{A} - 1\] \[\ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right) \leq \frac{\sum_{i=1}^{n}a_i}{A} - n\] \[\ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right) \leq \frac{nA}{A} - n\] \[\ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right) \leq n - n\] \[\ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right) \leq 0\] Now exponentiating both sides we get \[e^{\ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right)} \leq e^{0}\] \[\frac{\prod_{i=1}^{n} a_i}{A^n} \leq 1\] \[\prod_{i=1}^{n} a_i \leq A^n\] \[\sqrt[\leftroot{2}\uproot{3}n]{\prod_{i=1}^{n} a_i} \leq \sqrt[\leftroot{2}\uproot{3}n]{A^n}\] \[\sqrt[\leftroot{2}\uproot{3}n]{\prod_{i=1}^{n} a_i} \leq A\] \[\sqrt[\leftroot{2}\uproot{3}n]{\prod_{i=1}^{n} a_i} \leq \frac{\sum_{i=1}^{n} a_i}{n}\] This proves the AM-GM inequality. $\blacksquare$

Proof of Weighted AM-GM

Proof by Convexity

We note that the function $x \mapsto \ln x$ is strictly concave. Then by Jensen's Inequality, \[\ln \sum_i \lambda_i a_i \ge \sum_i \lambda_i \ln a_i = \ln \prod_i a_i^{\lambda_i} ,\] with equality if and only if all the $a_i$ are equal. Since $x \mapsto \ln x$ is a strictly increasing function, it then follows that \[\sum_i \lambda_i a_i \ge \prod_i a_i^{\lambda_i},\] with equality if and only if all the $a_i$ are equal, as desired. $\blacksquare$

Alternate Proof by Convexity

This proof is due to G. Pólya.

Note that the function $f:x \mapsto e^x$ is strictly convex. Let $g(x)$ be the line tangent to $f$ at $(0,1)$; then $g(x) = x+1$. Since $f$ is also a continuous, differentiable function, it follows that $f(x) \ge g(x)$ for all $x$, with equality exactly when $x=0$, i.e., \[1+x \le e^x,\] with equality exactly when $x=0$.

Now, set \[r_i = a_i/\biggl( \sum_{j=1}^n \lambda_j a_j \biggr) - 1,\] for all integers $1\le i \le n$. Our earlier bound tells us that \[r_i +1 \le \exp(r_i),\] so \[(r_i +1)^{\lambda_i} \le \exp(\lambda_i r_i) .\] Multiplying $n$ such inequalities gives us

\[\prod_{i=1}^n (r_i + 1)^{\lambda_{i}} \le \prod_{i=1}^n \exp \lambda_i r_i\]

Evaluating the left hand side:

\[\prod_{i=1}^n (r_i + 1)^{\lambda_{i}} = \frac{\prod_{i=1}^n a_i^{\lambda_i} }{ (\sum_{j=1}^n \lambda_j a_j)^{\sum_{j=1}^n \lambda_i} } = \frac{\prod_{i=1}^n a_i^{\lambda_i} }{ \sum_{j=1}^n \lambda_j a_j } ,\]

for

\[\sum_{j=1}^n \lambda_i = 1 .\]

Evaluating the right hand side:

\[\prod_{i=1}^n \exp \lambda_i r_i = \prod_{i=1}^n \exp \left(\frac{a_i \lambda_i}{ \sum_{i=1}^n \lambda_j a_j} - \lambda_i\right) = \exp \left(\frac{\sum_{i=1}^n a_i \lambda_i}{ \sum_{i=1}^n \lambda_j a_j} - \sum_{i=1}^n \lambda_i\right) = \exp 0 = 1 .\]

Substituting the results for the left and right sides:

\[\frac{\prod_{i=1}^n a_i^{\lambda_i} }{ \sum_{j=1}^n \lambda_j a_j } \le 1\]

\[\prod_{i=1}^n a_i^{\lambda_i} \le \sum_{j=1}^n \lambda_j a_j = \sum_{i=1}^n \lambda_i a_i ,\]

as desired. $\blacksquare$

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