Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.