Young's Inequality

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Young's Inequality

Let f be a real-valued, continuous, and strictly increasing function on [0,c] with c > 0. If f(0)=0, a in [0,c], and b in [0,f(c)], then

$\int_0^af(x)dx+\int_0^bf^{-1}(y)dy\ge ab,$

where f^-1 is the inverse function of f. Equality holds iff b=f(a). To prove this, draw the graph of f(x), and treat each integral as the area bounded by the x and y axes and the function. Clearly, all of the rectangle bounded by the axes, a, and b is included in the sum of these areas.

Taking the particular function f(x)=x^p-1 gives the special case

(a^p)/p+((p-1)/p)b^p/(p-1) ≥ ab,

which is often written in the symmetric form

(a^p)/p+(b^q)/q ≥ ab,

where a,b ≥ 0, p>1, and 1/p+1/q=1.

Internet References

Mathworld: Young's Inequality

Wikipedia: Young's Inequality, which treats only the particular case (a^p)/p+(b^q)/q ≥ ab, and notes that Young's inequality is used in the proof of the Hölder inequality.

Related pages in this website

The AM-GM Inequality: the Arithmetic Mean of positive numbers is always greater than the Geometric Mean. This is proved using Jensen's Inequality.
The Cauchy-Schwarz inequality

The Triangle Inequality
Puzzles
The Quadratic Formula

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