Difference between revisions of "Jensen's Inequality"
Durianaops (talk | contribs) (→Proof) |
Durianaops (talk | contribs) (→Proof) |
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By the fundamental theorem of calculus, we have | By the fundamental theorem of calculus, we have | ||
<cmath>\int_{x_i}^{\bar{x}} F'(t) \, dt = F(\bar{x}) - F(x_i) .</cmath> | <cmath>\int_{x_i}^{\bar{x}} F'(t) \, dt = F(\bar{x}) - F(x_i) .</cmath> | ||
− | Evaluating the integrals, the last two inequalities | + | Evaluating the integrals, each of the last two inequalities implies the same result: |
+ | <cmath>F(\bar{x})-F(x_i) \ge F'(\bar{x})(\bar{x}-x_i)</cmath> | ||
+ | so this is true for all <math>x_i</math>. Then we have | ||
<cmath> | <cmath> | ||
\begin{align*} | \begin{align*} |
Revision as of 08:28, 31 July 2020
Jensen's Inequality is an inequality discovered by Danish mathematician Johan Jensen in 1906.
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, then 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.
One of the simplest examples of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Take (verify that and ) and . You'll get . Similarly, arithmetic mean-geometric mean inequality can be obtained from Jensen's inequality by considering .
Problems
Introductory
Prove AM-GM using Jensen's Inequality
Intermediate
- Prove that for any , we have .
- Show that in any triangle we have
Olympiad
- Let be positive real numbers. Prove that
(Source)