Question

Inequality problem

Answer 1

what

Answer 2

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

Answer 3

k >= 0

Answer 4

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?

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

not sure if my idea is proof enough ...

essentially: