Difference between revisions of "Jensen's Inequality"

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(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>
 
</center><br>
 
</center><br>
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If <math>{F}</math> is a [[concave function]]:
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<br><center>
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<math>F(a_1x_1+\dots+a_n x_n)< a_1F(x_1)+\dots+a_n F(x_n)</math>
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</center><br>
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==Proof==
 
==Proof==
 
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 equality case holds for all linear functions!  (check it yourself)
 
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 equality case holds for all linear functions!  (check it yourself)

Revision as of 18:46, 8 May 2013

Jensen's Inequality is an inequality discovered by a mathematician of that name in 1906.

Inequality

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


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


If ${F}$ is a concave function:


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


Proof

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 equality case holds for all linear functions! (check it yourself)

One of the simplest examples of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Take $F(x)=x^2$ (verify that $F'(x)=2x$ and $F''(x)=2>0$) 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 $F(x)=-\log x$.

Problems

Introductory

Seeing as this is quite a complicated theorem, there are no introductory problems.

Intermediate

  • Prove that for any $\triangle ABC$, we have $\sin{A}+\sin{B}+\sin{C}\leq \frac{3\sqrt{3}}{2}$.

Olympiad

  • Let $a,b,c$ be positive real numbers. Prove that

$\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$ (Source)