Young's inequality for three variables

The function $\ln$ is concave, so if $\sum_n \lambda_n =1$, with $\lambda_n \ge 0$, then $\ln ( \sum_n \lambda_n x_n ) \ge \sum_n \lambda_n \ln x_n$ (with $x_n >0$, of course).

Hence $\ln ( \frac{x^p}{p} + \frac{y^q}{q} + \frac{z^r}{r} ) \ge \frac{1}{p} \ln x^p + \frac{1}{q} \ln y^q + \frac{1}{r} \ln x^r = \ln (x y z)$.

Taking exponents yields the desired result.

Since $\ln$ is strictly concave, we have equality iff $x_i = x_j$ in the first inequality, which corresponds to $x^p = y^q = z^r$.

Here's a solution by applying Young's inequality twice.

First apply the inequality to $x$ and $yz$ with $p$ and $\frac{p}{p-1}$ to get $$xyz \le \frac{x^p}{p} + \frac{(yz)^{\frac{p}{p-1}}}{\frac{p}{p-1}}.$$ Then apply it to $y^{\frac{p}{p-1}}$ and $z^{\frac{p}{p-1}}$ with $\frac{p-1}{p} q$ and $\frac{\frac{p-1}{p}q}{\frac{p-1}{p}q - 1} = \frac{(p-1)q}{pq-p-q}$ to get $$xyz \le \frac{x^p}{p} + \frac{p-1}{p} \left(\frac{y^q}{\frac{p-1}{p}q} + \frac{z^{\frac{pq}{pq-p-q}}}{\frac{(p-1)q}{pq-p-q}}\right)$$ Notice that $\frac{pq}{pq-p-q} = r$, so you get $$xyz \le \frac{x^p}{p} + \frac{y^q}{q} + \frac{z^r}{r}$$ as wanted.