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.



a inequality of $L^p$
Suppose $y>x\ge 0$. Then we have
$$(y-x)^p\le y^p-x^p=\int\_x^y pz^{p-1}dz.$$
For any $p\ge 1$, the right hand side is bounded by 
$$p(y-x)y^{p-1}.$$ 
For $p\ge 2$, the function $z\mapsto z^{p-1}$ is convex, and so the integral is bounded by its trapezoidal approximation. This leads to the bound
$$p(y-x){x^{p-1}+y^{p-1}\over 2}.$$


