Power means inequality Rummaging through old notebooks from high school I found the following inequality. Let $x, y, z$ and $t$ be positive numbers such that $x+y+z+t=1$. Then the following inequality holds:
$$
1 < \frac{\sqrt[3]{x^4 + y^4 + z^4 + t^4} - (x^2 + y^2 + z^2 + t^2)}{\sqrt[3]{x^4 + y^4 + z^4 + t^4} - \sqrt{x^3 + y^3 + z^3 + t^3}}<4
$$
While the left part is almost obvious from the power means monotonicity, the right part escapes me. I verified it with Mathematica and it seems correct. Any suggestions?
EDIT: Also asked at Math Overflow now.

You can find the proof at https://mathoverflow.net/questions/107551/an-inequality-involving-sums-of-powers/107594
