Application of Jensen's inequality to $x^x+y^y+z^z$ 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?

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.
