For positive real numbers $x,y,z$, Prove that

Question

For positive real numbers $x,y,z$, Prove that,

$$\bigg(\frac{x^2+y^2+z^2}{x+y+z}\bigg)^{(x+y+z)}\geq x^xy^yz^z \geq
\bigg(\frac{x+y+z}{3}\bigg)^{(x+y+z)}$$

I have no idea how to begin this? I am getting a hint for A.M. – G.M. equality but numbers don't seem to fit.

Thanks for you time.

Answer 1

Since $\ln$ is a concave function we obtain: $$\sum_{cyc}\frac{x}{x+y+z}\ln{x}\leq\ln\left(\sum_{cyc}\frac{x}{x+y+z}\cdot x\right)=\ln\frac{x^2+y^2+z^2}{x+y+z},$$ which gives a left inequality.

The right inequality it's just Jensen for the convex function $f(x)=x\ln{x}:$ $$\frac{\sum\limits_{cyc}x\ln{x}}{3}\geq\frac{x+y+z}{3}\ln\frac{x+y+z}{3}.$$