- Prove that if and , then
.- Similarly to the previous problem:
Prove that if and , then
.- Let's generalize the problems above:
Prove that if , , , and then
.Find the maximum possible value of .
I post the remaining separately because a small trick there, wont work any longer
I will directly attack the generalization,
Claim:
Proof:
Denote LHS of your inequality by I,
AM-HM says:
-------------(*)
Applying CS:
But
So .
But by (*),
Thus
Equality for all s equal.
K cant get any better than that because we showed equality achieving