Difference between revisions of "Jensen's Inequality"
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<math>F(a_1x_1+\dots+a_n x_n)\le a_1F(x_1)+\dots+a_n F(x_n)</math> | <math>F(a_1x_1+\dots+a_n x_n)\le a_1F(x_1)+\dots+a_n F(x_n)</math> | ||
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| − | 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>\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 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>. | 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>. | ||
Revision as of 12:01, 21 June 2006
Let 
be a convex function of one real variable. Let 
and let 
satisfy 
. Then
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 with the linear function 
, whose graph is tangent to the graph of at the point 
. Then the left hand side of the inequality is the same for and , while the right hand side is smaller for . But the inequality for is an identity!
The simplest example of the use of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Take 
and 
. You'll get 
. Similarly, arithmetic mean-geometric mean inequality can be obtained from Jensen's inequality by considering 
.
