I want to prove the inequality 
$$
|x-y|^p \le \frac{p}{2}\big|x-y\big|\;\big(x^{p-1}+y^{p-1}\big)
$$
if $p \ge 1$.  For the case $p$ is an integer, it's easy to do, but I have no idea when $p$ is not an integer!  Thank you for the answer.