Inequality problem

Question

Inequality problem

anonymous · Answer

what

anonymous · Answer

If x, y, z are positive numbers such that x + y + z = 1, then prove $$\frac{ x ^{k+2} }{ x^{k+1} + y^k + z^k} + \frac{ y ^{k+2} }{ y^{k+1} + x^k + z^k} + \frac{ z ^{k+2} }{ z^{k+1} + x^k + y^k} >= \frac{ 1 }{ 7 }$$

anonymous · Answer

k >= 0

anonymous · Answer

So, I have an idea on how to go about showing this. It has to do with the denominators of each term. Basically, in order to maximize the denominator while minimizing the numerator, the denominator of each term should be the same. Is this correct reasoning?

anonymous · Answer

What I have noticed, is that when x = y = z = 1/3, then equality holds for all k.

amistre64 · Answer

one thought also, is that as one of them, say, x grows to 1, the others diminish to zero

amistre64 · Answer

does it work for x=y=1/2, z=0  .... or rather as the limit of these go there.

amistre64 · Answer

not sure if my idea is proof enough ...

essentially:

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