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.

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Great title, philosophically speaking. Now if I could only figure out what the quetsion has to do with it... –  Marc van Leeuwen Sep 12 '12 at 13:06
The title comes from my (completely unfounded) belief that the proof of this inequality will come from examining some properties of the power means. –  ivan Sep 12 '12 at 13:44