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> | ||
</center><br> | </center><br> | ||
− | If <math>{F}</math> is a | + | If <math>{F}</math> is a concave function, we have: |
<br><center> | <br><center> | ||
<math>F(a_1x_1+\dots+a_n x_n)\ge a_1F(x_1)+\dots+a_n F(x_n)</math> | <math>F(a_1x_1+\dots+a_n x_n)\ge a_1F(x_1)+\dots+a_n F(x_n)</math> | ||
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==Example== | ==Example== | ||
− | One of the simplest examples of Jensen's inequality is the [[quadratic mean]] - [[arithmetic mean]] inequality. | + | One of the simplest examples of Jensen's inequality is the [[quadratic mean]] - [[arithmetic mean]] inequality. Taking <math>F(x)=x^2</math>, which is convex (because <math>F'(x)=2x</math> and <math>F''(x)=2>0</math>), and <math>a_1=\dots=a_n=\frac 1n</math>, we obtain |
+ | <cmath>\left(\frac{x_1+\dots+x_n}{n}\right)^2\le \frac{x_1^2+\dots+ x_n^2}{n} .</cmath> | ||
+ | |||
+ | Similarly, [[arithmetic mean]]-[[geometric mean]] inequality ([[AM-GM]]) can be obtained from Jensen's inequality by considering <math>F(x)=-\log x</math>. | ||
+ | |||
+ | In fact, the [[power mean inequality]], a generalization of AM-GM, follows from Jensen's inequality. | ||
==Problems== | ==Problems== | ||
+ | |||
===Introductory=== | ===Introductory=== | ||
+ | |||
+ | ====Problem 1==== | ||
Prove AM-GM using Jensen's Inequality | Prove AM-GM using Jensen's Inequality | ||
+ | |||
+ | ====Problem 2==== | ||
+ | Prove the weighted [[AM-GM Inequality|AM-GM inequality]]. (It states that <math>x_1^{\lambda_1}x_2^{\lambda_2} \cdots x_n^{\lambda_n} \leq \lambda_1 x_1 + \lambda_2 x_2 +\cdots+ \lambda_n x_n</math> when <math>\lambda_1 + \cdots + \lambda_n = 1</math>) | ||
===Intermediate=== | ===Intermediate=== | ||
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<math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1</math> ([[2001 IMO Problems/Problem 2|Source]]) | <math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1</math> ([[2001 IMO Problems/Problem 2|Source]]) | ||
− | [[Category: | + | [[Category:Algebra]] |
− | [[Category: | + | [[Category:Inequalities]] |
Latest revision as of 12:10, 20 February 2024
Jensen's Inequality is an inequality discovered by Danish mathematician Johan Jensen in 1906.
Contents
Inequality
Let be a convex function of one real variable. Let and let satisfy . Then
If is a concave function, we have:
Proof
We only prove the case where is concave. The proof for the other case is similar.
Let . As is concave, its derivative is monotonically decreasing. We consider two cases.
If , then If , then By the fundamental theorem of calculus, we have Evaluating the integrals, each of the last two inequalities implies the same result: so this is true for all . Then we have as desired.
Example
One of the simplest examples of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Taking , which is convex (because and ), and , we obtain
Similarly, arithmetic mean-geometric mean inequality (AM-GM) can be obtained from Jensen's inequality by considering .
In fact, the power mean inequality, a generalization of AM-GM, follows from Jensen's inequality.
Problems
Introductory
Problem 1
Prove AM-GM using Jensen's Inequality
Problem 2
Prove the weighted AM-GM inequality. (It states that when )
Intermediate
- Prove that for any , we have .
- Show that in any triangle we have
Olympiad
- Let be positive real numbers. Prove that
(Source)