Difference between revisions of "Jensen's Inequality"

Revision as of 21:07, 7 October 2007

Let ${F}$ be a convex function of one real variable. Let $x_1,\dots,x_n\in\mathbb R$ and let $a_1,\dots, a_n\ge 0$ satisfy $a_1+\dots+a_n=1$ . Then

$F(a_1x_1+\dots+a_n x_n)\le a_1F(x_1)+\dots+a_n F(x_n)$

The proof of Jensen's inequality is very simple. Since the graph of every convex function lies above its tangent line at every point, we can compare the function ${F}$ with the linear function ${L}$ , whose graph is tangent to the graph of ${F}$ at the point $a_1x_1+\dots+a_n x_n$ . Then the left hand side of the inequality is the same for ${F}$ and ${L}$ , while the right hand side is smaller for ${L}$ . But the inequality for ${L}$ is an identity!

The simplest example of the use of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Take $F(x)=x^2$ and $a_1=\dots=a_n=\frac 1n$ . You'll get $\left(\frac{x_1+\dots+x_n}{n}\right)^2\le \frac{x_1^2+\dots+ x_n^2}{n}$ . Similarly, arithmetic mean-geometric mean inequality can be obtained from Jensen's inequality by considering $F(x)=-\log x$ . This article is a stub. Help us out by expanding it. Template:Wikify

@@ Line 1: / Line 1: @@
-Let <math>\displaystyle{F}</math> be a [[convex function]] of one real variable. Let <math>\displaystyle x_1,\dots,x_n\in\mathbb R</math> and let <math>a_1,\dots, a_n\ge 0</math> satisfy <math>a_1+\dots+a_n=1</math>. Then
+Let <math>{F}</math> be a [[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb R</math> and let <math>a_1,\dots, a_n\ge 0</math> satisfy <math>a_1+\dots+a_n=1</math>. Then
 <br><center>
 <math>F(a_1x_1+\dots+a_n x_n)\le a_1F(x_1)+\dots+a_n F(x_n)</math>
 </center><br>
-The proof of Jensen's inequality is very simple. Since the graph of every convex function lies above its tangent line at every point, we can compare the function <math>\displaystyle{F}</math> with the linear function <math>\displaystyle{L}</math>, whose graph is tangent to the graph of <math>\displaystyle{F}</math> at the point <math>a_1x_1+\dots+a_n x_n</math>. Then the left hand side of the inequality is the same for <math>\displaystyle{F}</math> and <math>\displaystyle{L}</math>, while the right hand side is smaller for <math>\displaystyle{L}</math>. But the inequality for <math>\displaystyle{L}</math> is an identity!
+The proof of Jensen's inequality is very simple. Since the graph of every convex function lies above its tangent line at every point, we can compare the function <math>{F}</math> with the linear function <math>{L}</math>, whose graph is tangent to the graph of <math>{F}</math> at the point <math>a_1x_1+\dots+a_n x_n</math>. Then the left hand side of the inequality is the same for <math>{F}</math> and <math>{L}</math>, while the right hand side is smaller for <math>{L}</math>. But the inequality for <math>{L}</math> is an identity!
 The simplest example of the use of Jensen's inequality is the [[quadratic mean]] - [[arithmetic mean]] inequality. Take <math>F(x)=x^2</math> and <math>a_1=\dots=a_n=\frac 1n</math>. You'll get <math>\left(\frac{x_1+\dots+x_n}{n}\right)^2\le \frac{x_1^2+\dots+ x_n^2}{n} </math>. Similarly, [[arithmetic mean]]-[[geometric mean]] inequality can be obtained from Jensen's inequality by considering <math>F(x)=-\log x</math>.
+{{stub}}
+{{wikify}}
+[[Category:Number theory]]

Page

Toolbox

Search

Difference between revisions of "Jensen's Inequality"

Revision as of 21:07, 7 October 2007