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