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# Real Analysis : Young's Inequality

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Note: * = infinite

Suppose that the function f:[0,*)->R is continuous and strictly increasing, with f(0) = 0 and f([0,*)) = [0,*). Then define

F(x) = the integral from 0 to x of f and
G(x) = the integral from 0 to x of f^-1 for all x>=0

(a) Prove Young's Inequality:
ab <= F(a) + G(b) for all a >= 0 and b >= 0

(b) Now use Young's Inequality with f(x) = x^(p-1) for all x>=0, and p>1 fixed, to prove that if the number q is chosen to have the property that 1/p + 1/q = 1, then

ab <= a^p/p + b^q/q for a >= 0 and b >= 0.

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#### Solution Summary

Young's inequality is proven and a functional property is proven using Young's inequality.

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