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# Prove the product of sums of n numbers and their reciprocals is greater than n^2

Let and let for . Prove

Proof. First,

for all . We then apply the Cauchy-Schwarz inequality to the numbers and . So, Cauchy-Schwarz gives us,

# Prove the pth power mean is less than the (2p)th power mean

We recall the definition of the th power mean .

For , and with , we define the th power-mean as:

Now, for , prove for not all equal.

Proof. From the Cauchy-Schwarz inequality we know that for real numbers and , we have

with equality if and only if there is some such that for all . Letting and we have

This inequality is strict since if equality held there would exist some such that for all , but this would imply for all , contradicting our assumption that the are not all equal. Since (see here), this implies

# Prove an if and only if condition for equality in the Cauchy-Schwarz inequality

Recall the Cauchy-Schwarz inequality,

For arbitrary real numbers and we have

The claim is then that the equality sign holds if and only if there is a real number such that for each .

Proof. () If for all , then equality clearly holds. Assume then that for at least one .

Then, considering the equation and defining,

We have,

But, since we know (by assumption), we have which is in (since since for at least one and each term in nonnegative, so the sum is strictly positive).
() Assume there exists such that for each . Then, . So,