An inequality using AM – GM

Let x, y , z be positive numbers. Prove that

\displaystyle \frac{3xy}{xy+x+y}+ \frac{3yz}{yz+y+z}+ \frac{3zx}{zx+z+x}\leq 2+ \frac{x^2+y^2+z^2}{3}

Solution

We are invoking the AM – GM inequality

    \begin{align*} 2+ \frac{x^2+y^2+z^2}{3} &=\sum \frac{x^2+y^2+4}{6} \\ &\geq \sum \sqrt[6]{x^2y^2} \\ &=\sum \frac{xy}{\sqrt[3]{(xy)xy}} \\ &\geq \sum \frac{xy}{\frac{xy+x+y}{3}}\\ &=\sum \frac{3xy}{xy+x+y} \end{align*}

and we conclude the result.

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