# 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.

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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