Difference between revisions of "Proofs of AM-GM"

Latest revision as of 13:45, 11 October 2024

This page 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 $\sum_{i=1}^n \lambda_i a_i \geq \prod_{i=1}^n a_i^{\lambda_i},$ 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 $2$ 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 $2$ .

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 backward 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. $\blacksquare$

@@ Line 1: / Line 1: @@
-This pages lists some proofs of the weighted [[AM-GM]] Inequality.  The inequality's statement is as follows: for all nonnegative reals <math>a_1, \dotsc, a_n</math> and nonnegative reals <math>\lambda_1, \dotsc, \lambda_n</math> such that <math>\sum_{i=1}^n \lambda_i = 1</math>, then
+This page lists some proofs of the weighted [[AM-GM Inequality]]. The inequality's statement is as follows: for all nonnegative reals <math>a_1, \dotsc, a_n</math> and nonnegative reals <math>\lambda_1, \dotsc, \lambda_n</math> such that <math>\sum_{i=1}^n \lambda_i = 1</math>, then
-<cmath> \sum_{i=1}^n \lambda_i a_i \ge \prod_{i=1}^n a_i^{\lambda_i}, </cmath>
+<cmath> \sum_{i=1}^n \lambda_i a_i \geq \prod_{i=1}^n a_i^{\lambda_i}, </cmath>
 with equality if and only if <math>a_i = a_j</math> for all <math>i,j</math> such that <math>\lambda_i, \lambda_j \neq 0</math>.
-We first note that we may disregard any <math>a_i</math> for which <math>\lambda_i= 0</math>, as they contribute to neither side of the desired inequality.  We also note that if <math>a_i= 0</math> and <math>\lambda_i \neq 0</math>, for some <math>i</math>, 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 <math>a_j = 0 = a_i</math> whenever <math>\lambda_j\neq 0</math>.  Thus we may henceforth assume that all <math>a_i</math> and <math>\lambda_i</math> are ''strictly positive.''
+We first note that we may disregard any <math>a_i</math> for which <math>\lambda_i= 0</math>, as they contribute to neither side of the desired inequality. We also note that if <math>a_i= 0</math> and <math>\lambda_i \neq 0</math>, for some <math>i</math>, 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 <math>a_j = 0 = a_i</math> whenever <math>\lambda_j\neq 0</math>. Thus we may henceforth assume that all <math>a_i</math> and <math>\lambda_i</math> are ''strictly positive.''
-== Complete Proofs ==
+== Proofs of Unweighted AM-GM ==
+These proofs use the assumption that <math>\lambda_i = 1/n</math>, for all integers <math>1 \le i \le n</math>.
+=== Proof by Cauchy Induction ===
+We use [[Cauchy Induction]], a variant of induction in which one proves a result for <math>2</math>, all powers of <math>2</math>, and then that <math>n</math> implies <math>n-1</math>.
+'''Base Case''': The smallest nontrivial case of AM-GM is in two variables. By the properties of perfect squares (or by the [[Trivial Inequality]]), <math>(x-y)^2 \geq 0,</math> with equality if and only if <math>x-y=0</math>, or <math>x=y</math>. Then because <math>x</math> and <math>y</math> are nonnegative, we can perform the following manipulations: <cmath>x^2 - 2xy + y^2 > 0</cmath> <cmath>x^2 + 2xy + y^2 > 4xy</cmath> <cmath> (x+y)^2 = 4xy</cmath> <cmath>\frac{(x+y)^2}{4} \geq xy</cmath> <cmath>\frac{x+y}{2} \geq \sqrt{xy},</cmath> with equality if and only if <math>x=y</math>, just as before. This completes the proof of the base case.
+'''Powers of Two''': We use induction. Suppose that AM-GM is true for <math>n</math> variables; we will then prove that the inequality is true for <math>2n</math>. Let <math>x_1, x_2, \ldots, x_{2n}</math> be any list of nonnegative reals. Then, because the two lists <math>x_1, x_2 \ldots, x_n</math> and <math>x_{n+1}, x_{n+2}, \ldots, x_{2n}</math>, each have <math>n</math> variables, <cmath>\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}}.</cmath> Adding these two inequalities together and dividing by <math>2</math> yields <cmath>\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}.</cmath> From here, we perform AM-GM in two variables on <math>\sqrt[n]{x_1 x_2 \cdots x_n}</math> and <math>\sqrt[n]{x_{n+1} x_{n+2} \cdots x_{2n}}</math> to get <cmath>\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}}.</cmath> Combining this inequality with the previous one yields AM-GM in <math>2n</math> 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 <math>x_1, x_2, \ldots x_n</math> are the same, all the numbers in <math>x_{n+1}, x_{n+2}, \ldots x_{2n}</math> are the same, and <math>\sqrt[n]{x_1 x_2 \cdots x_n} = \sqrt[n]{x_{n+1} x_{n+2} \cdots x_{2n}}</math>. From here, it is trivial to show that this implies <math>x_1 = x_2 = \cdots x_{2n}</math>, which is the equality condition for AM-GM in <math>2n</math> variables.
+This completes the induction and proves that the inequality holds for all powers of <math>2</math>.
+'''Backward Step''': Assume that AM-GM holds for <math>n</math> variables. We will then use a substitution to derive AM-GM for <math>n-1</math> variables. Letting <math>x_n = \frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}</math>, we have that <cmath>\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)}.</cmath> Because we assumed AM-GM in <math>n</math> variables, equality holds if and only if <math>x_1 = x_2 = \cdots = x_{n-1} = \frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}</math>. However, note that the last equality is implied if all the numbers of <math>x_1, x_2, \ldots, x_{n-1}</math> are the same; thus, equality holds if and only if <math>x_1 = x_2 = \cdots = x_{n-1}</math>.
+We first simplify the lefthand side. Multiplying both sides of the fraction by <math>n-1</math> and combining like terms, we get that <cmath>\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}.</cmath> Plugging this into the earlier inequality yields <cmath>\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)}.</cmath> Raising both sides to the <math>n</math>th power yields <cmath>\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).</cmath> From here, we divide by <math>\frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}</math> and take the <math>(n-1)^{\textrm{th}}</math> root to get that <cmath>\frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1} \geq \sqrt[n-1]{x_1 x_2 \cdots x_{n-1}}.</cmath> This is the inequality in <math>n-1</math> variables. Note that every step taken also preserves equality, which completes the backward step. Then by Cauchy Induction, the AM-GM inequality holds. <math>\square</math>
+=== Proof by Rearrangement ===
+Define the <math>n</math> sequence <math>\{ r_{i,j}\}_{i=1}^{n}</math> as <math>r_{i,j} = \sqrt[n]{a_i}</math>, for all integers <math>1 \le i,j \le n</math>. Evidently, these sequences are similarly sorted. Then by the [[Rearrangement Inequality]],
+<cmath> \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} , </cmath>
+where we take our indices modulo <math>n</math>, with equality exactly when all the <math>r_{i,j}</math>, and therefore all the <math>a_i</math>, are equal. Dividing both sides by <math>n</math> gives the desired inequality. <math>\blacksquare</math>
+=== Proof by Calculus ===
+We will start the proof by considering the function <math>f(x) = \ln x - x</math>. We will now find the maximum of this function. We can do this simply using calculus. We need to find the critical points of <math>f(x)</math>, we can do that by finding <math>f'(x)</math> and setting it equal to <math>0</math>. Using the linearity of the derivative <math>f'(x) = \frac{1}{x} - 1</math>. We need <math>f'(x) = 0</math>
+<cmath>
+f'(x) = \frac{1}{x} - 1 = 0
+</cmath>
+<cmath>
+\frac{1}{x} = 1
+</cmath>
+<cmath>
+x = 1
+</cmath>
+Note that this is the only critical point of <math>f(x)</math>. We can confirm it is the maximum by finding it's second derivative and making sure it is negative. <math>f''(x) = -\frac{1}{x^2}</math> letting x = 1 we get <math>f''(1) = -\frac{1}{1} < 0</math>. Since the second derivative <math>f''(x) < 0</math>, <math>f(1)</math> is a maximum. <math>f(1) = \ln 1 - 1 = -1</math>. Now that we have that <math>f(1) = -1</math> is a maximum of <math>f(x)</math>, we can safely say that <math>\ln x - x \leq -1</math> or in other words <math>\ln x \leq x - 1</math>. We will now define a few more things and do some manipulations with them. Let <math>A = \frac{\sum_{i=1}^{n} a_i}{n}</math>, with this notice that <math>nA = \sum_{i=1}^{n} a_i</math>. This fact will come into play later. now we can do the following, let <math>x = \frac{a_i}{A}</math> and plug this into <math>\ln x < x - 1</math>, we get
+<cmath>
+\ln\left(\frac{a_1}{A}\right) \leq \frac{a_1}{A} - 1
+</cmath>
+<cmath>
+\ln\left(\frac{a_2}{A}\right) \leq \frac{a_2}{A} - 1
+</cmath>
+<cmath>
+\vdotswithin{=}
+</cmath>
+<cmath>
+\ln\left(\frac{a_n}{A}\right) \leq \frac{a_n}{A} - 1
+</cmath>
+Adding all these results together we get
+<cmath>
+    \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
+</cmath>
+<cmath>
+    \ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right) \leq \frac{\sum_{i=1}^{n}a_i}{A} - n
+</cmath>
+<cmath>
+    \ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right) \leq \frac{nA}{A} - n
+</cmath>
+<cmath>
+    \ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right) \leq n - n
+</cmath>
+<cmath>
+    \ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right) \leq 0
+</cmath>
+Now exponentiating both sides we get
+<cmath>
+    e^{\ln\left(\frac{\prod_{i=1}^{n} a_i}{A^n}\right)} \leq e^{0}
+</cmath>
+<cmath>
+    \frac{\prod_{i=1}^{n} a_i}{A^n} \leq 1
+</cmath>
+<cmath>
+    \prod_{i=1}^{n} a_i \leq A^n
+</cmath>
+<cmath>
+    \sqrt[\leftroot{2}\uproot{3}n]{\prod_{i=1}^{n} a_i} \leq \sqrt[\leftroot{2}\uproot{3}n]{A^n}
+</cmath>
+<cmath>
+    \sqrt[\leftroot{2}\uproot{3}n]{\prod_{i=1}^{n} a_i} \leq A
+</cmath>
+<cmath>
+    \sqrt[\leftroot{2}\uproot{3}n]{\prod_{i=1}^{n} a_i} \leq \frac{\sum_{i=1}^{n} a_i}{n}
+</cmath>
+This proves the AM-GM inequality. <math>\blacksquare</math>
+== Proof of Weighted AM-GM ==
 === Proof by Convexity ===
-We note that the [[function]] <math>x \mapsto \ln x</math> is strictly [[concave]].  Then by [[Jensen's Inequality]],
+We note that the [[function]] <math>x \mapsto \ln x</math> is strictly [[concave]]. Then by [[Jensen's Inequality]],
-<cmath> \ln \sum_i \lambda_i a_i \ge \lambda_i \sum_i \ln a_i = ln \prod_i a_i^{\lambda_i} , </cmath>
+<cmath> \ln \sum_i \lambda_i a_i \ge \sum_i \lambda_i \ln a_i = \ln \prod_i a_i^{\lambda_i} , </cmath>
 with equality if and only if all the <math>a_i</math> are equal.
 Since <math>x \mapsto \ln x</math> is a strictly increasing function, it then follows that
 <cmath> \sum_i \lambda_i a_i \ge \prod_i a_i^{\lambda_i}, </cmath>
-with equality if and only if all the <math>a_i</math> are equal, as desired.  <math>\blacksquare</math>
+with equality if and only if all the <math>a_i</math> are equal, as desired. <math>\blacksquare</math>
 === Alternate Proof by Convexity ===
@@ Line 20: / Line 107: @@
 ''This proof is due to [[G. Pólya]].''
-Note that the function <math>f:x \mapsto e^x</math> is strictly convex.  Let <math>g(x)</math> be the line tangent to <math>f</math> at <math>(0,1)</math>; then <math>g(x) = x+1</math>.  Since <math>f</math> is also a [[continuous]], [[differentiable]] function, it follows that <math>f(x) \ge g(x)</math> for all <math>x</math>, with equality exactly when <math>x=0</math>, i.e.,
+Note that the function <math>f:x \mapsto e^x</math> is strictly convex. Let <math>g(x)</math> be the line tangent to <math>f</math> at <math>(0,1)</math>; then <math>g(x) = x+1</math>. Since <math>f</math> is also a [[continuous]], [[differentiable]] function, it follows that <math>f(x) \ge g(x)</math> for all <math>x</math>, with equality exactly when <math>x=0</math>, i.e.,
 <cmath> 1+x \le e^x, </cmath>
 with equality exactly when <math>x=0</math>.
@@ Line 26: / Line 113: @@
 Now, set
 <cmath> r_i = a_i/\biggl( \sum_{j=1}^n \lambda_j a_j \biggr) - 1, </cmath>
-for all integers <math>1\le i \le n</math>.  Our earlier bound tells us that
+for all integers <math>1\le i \le n</math>. Our earlier bound tells us that
 <cmath> r_i +1 \le \exp(r_i), </cmath>
 so
 <cmath> (r_i +1)^{\lambda_i} \le \exp(\lambda_i r_i) .</cmath>
 Multiplying <math>n</math> such inequalities gives us
-<cmath> \frac{\prod_{i=1}^n a_i^{\lambda_i}}{ \sum_{i=1}^n \lambda_i r_i} = \prod_{i=1}^n (r_i + 1) \le \prod_{i=1}^n \exp \lambda_i r_i = \exp \sum_{i=1}^n \lambda_i a_i / \sum_{j=1}^n \lambda_j a_j - \lambda_i = \exp 0 = 1 ,</cmath>
-so
-<cmath> \prod_{i=1}^n a_i^{\lambda_i} \le \sum_{i=1}^n \lambda_i r_i , </cmath>
-as desired.  Equality holds when <math>r_i</math> is always equal to zero, i.e., when <math>a_i = a_j</math> for all <math>i,j</math> such that <math>\lambda_i , \lambda_j \neq 0</math>.  <math>\blacksquare</math>
-== Proofs of Unweighted AM-GM ==
+<cmath>\prod_{i=1}^n (r_i + 1)^{\lambda_{i}} \le \prod_{i=1}^n \exp \lambda_i r_i </cmath>
+Evaluating the left hand side:
-These proofs use the assumption that <math>\lambda_i = 1/n</math>, for all integers <math>1 \le i \le n</math>.
+<cmath>\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 } ,</cmath>
-=== Proof by Rearrangement ===
+for
-Define the <math>n</math> sequence <math>\{ r_{ij}\}_{i=1}^{n}</math> as <math>r_{ij} = \sqrt[n]{a_i}</math>, for all integers <math>1 \le i,j \le n</math>.  Evidently these sequences are similarly sorted.  Then by the [[Rearrangement Inequality]],
+<cmath> \sum_{j=1}^n \lambda_i = 1 .</cmath>
-<cmath> \sum_i a_i = \sum_i \prod_j r_{ij} \ge \sum_i \prod_j r_{i,i+j} = n \prod_i \sqrt[n]{a_i} , </cmath>
-where we take our indices modulo <math>n</math>, with equality exactly when all the <math>r_{ij}</math>, and therefore all the <math>a_i</math>, are equal.  Dividing both sides by <math>n</math> gives the desired inequality.  <math>\blacksquare</math>
-=== Proof by [[Cauchy Induction]] ===
+Evaluating the right hand side:
-We first prove that the inequality holds for two variables.  Note that <math>(\sqrt{a}, \sqrt{b}), (\sqrt{a}, \sqrt{b})</math> are similarly sorted sequences.  Then by the Rearrangement Inequality,
+<cmath> \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 .</cmath>
-<cmath> \sqrt{ab} = \frac{ \sqrt{a} \sqrt{b} + \sqrt{b} \sqrt{a}}{2} \le \frac{ \sqrt{a} \sqrt{a} + \sqrt{b} \sqrt{b}}{2} = \frac{a+b}{2} , </cmath>
-with equality exactly when <math>a=b</math>.
-We next prove that if the inequality holds for <math>n</math> variables (with equality when all are equal to zero), than it holds for <math>2n</math> variables (with equality when all are equal to zero).  Indeed, suppose the inequality holds for <math>n</math> variables.  Let <math>A_n, A'_n, A_{2n}</math> denote the arithmetic means of <math>(a_1, \dotsc, a_n)</math>, <math>(a_{n+1}, \dotsc, a_{2n})</math>, <math>(a_1, \dotsc, a_{2n})</math>, respectively; let <math>G_n, G'_n, G_{2n}</math> denote their respective geometric means.  Then
+Substituting the results for the left and right sides:
-<cmath> G_{2n} = \sqrt{G_n G'_n} \le \frac{G_n + G'_n}{2} \le \frac{A_n + A'_n}{2} = A_2n, </cmath>
-with equality when all the numbers <math>a_1, \dotsc, a_{2n}</math> are equal, as desired.
-These two results show by induction that the theorem holds for <math>n=2^k</math>, for all integers <math>k</math>.  In particular, for every integer <math>n</math>, there is an integer <math>N >n</math> such that the theorem holds for <math>N</math> variables.
+<cmath> \frac{\prod_{i=1}^n a_i^{\lambda_i} }{ \sum_{j=1}^n \lambda_j a_j } \le 1 </cmath>
-Finally, we show that if <math>N>n</math> and the theorem holds for <math>N</math> variables, then it holds for <math>n</math> variables <math>a_1, \dotsc, a_n</math> with arithmetic mean <math>A_n</math> and geometric mean <math>G_n</math>.  Indeed, set <math>b_k = a_k</math> for <math>1\le k \le n</math>, and let <math>b_k = A_k</math> for <math>n < k \le N</math>.  Then
+<cmath> \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 , </cmath>
-<cmath> G_k^{n/N} \cdot A_k^{(N-n)/N} = \prod_{i=1}^N b_i^{1/N} \le \sum_{i=1}^N b_i/N = A_n . </cmath>
-It follows that <math>G_k^{n/N} \le A_k^{n/N}</math>, or <math>G_n \le A_n</math>, with equality exactly when all the <math>a_i</math> are equal to <math>A_k</math>.
-The last two results show that for all positive integers <math>n</math>, the theorem holds for <math>n</math> variables.  Therefore the theorem is true.  <math>\blacksquare</math>
+as desired. <math>\blacksquare</math>
-[[Category:Inequality]]
+[[Category:Algebra]]
+[[Category:Inequalities]]

Page

Toolbox

Search