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Claim: If $x, y, z >0$ and $x+y+z = 3\pi, $ then $x^x + y^y + z^z > 81.$

My attempt: Let $f(w) = w^w$, so $f$ is convex on $(0, \infty).$ By Jensen's inequality, $f(x\frac{x}{3\pi}+ y\frac{y}{3\pi} + z\frac{z}{3\pi}) \leq \frac{x}{3\pi}f(x) + \frac{y}{3\pi}f(y) + \frac{z}{3\pi}f(z) < x^x + y^y + z^z. $

Could anyone advise me how to continue from here?

wythagoras
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Alexy Vincenzo
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1 Answers1

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You rather just want to use that $$\frac{1}{3}f(x)+\frac{1}{3}f(y)+\frac{1}{3}f(z) \geq f\left(\frac{1}{3}x+\frac{1}{3}y+\frac{1}{3}z\right)=f(\pi)=\pi^\pi > 3^3 =27$$

Multiply by 3 to get the desired inequality.

wythagoras
  • 25,026