## 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

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.

. . . . . . a picture will be added to this web page in due course.

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
H�lder inequality.

### Related pages in this website

The
Nondecreasing Sequence
Two puzzle whose most elegant solution relies on Young's Inequality.

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

The Quadratic Formula

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