Real functions Classical inequalities Theorem

Young's inequality for products

Builds on Inequality of arithmetic and geometric means

Statement

Let $p, q > 1$ be real numbers with $\frac{1}{p} + \frac{1}{q} = 1$ . Then for all real $a, b \ge 0$ , $a b \le \frac{a^p}{p} + \frac{b^q}{q}.$ Equality holds if and only if $a^p = b^q$ .

Remarks

It is the weighted AM–GM inequality for the numbers $a^p$ and $b^q$ , with weights $1/p$ and $1/q$ . For $p = q = 2$ it is $a b \le (a^2 + b^2)/2$ . It is the main step in the proof of Hölder's inequality. Mathlib states it as Real.young_inequality.

Reading path

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Inequality of arithmetic and geometric means statement
Young's inequality for products statement

Where this leads

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Tag	`07VHQ31EYJ`
URI	`m2://07VHQ31EYJ`
Kind	statement, theorem
Writing	W1 stated
MSC	26D15
Status	This copy of the site has no build data. Run `m2 status 07VHQ31EYJ`.
Uses	Inequality of arithmetic and geometric means
Same as	Wikidata Q910563 Wikipedia Young's inequality for products

Young's inequality for products

Statement

Remarks

Reading path

Where this leads

Further reading

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