Young’s inequality

Let $\varphi :ℝ\to ℝ$ be a continuous , strictly increasing function such that $\varphi (0)=0$ . Then the following inequality holds:

$$ab\le {\int }_0^a\varphi (x)𝑑x+{\int }_0^b{\varphi }^{-1}(y)𝑑y$$

Equality only holds when $b=\varphi (a)$. This inequality can be demonstrated by drawing the graph of $\varphi (x)$ and by observing that the sum of the two areas represented by the integrals above is greater than the area of a rectangle of sides $a$ and $b$, as is illustrated in http://planetmath.org/node/5575an attachment.

Title	Young’s inequality
Canonical name	YoungsInequality
Date of creation	2013-03-22 13:19:25
Last modified on	2013-03-22 13:19:25
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	8
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 26D15
Related topic	YoungInequality