Is this inequality true? Clearly if $a,b >0$ and $p \in \mathbb{N}$
$$
a^{p} + b^{p} \le (a+b)^{p}
$$
Is there a constante $C = C(p)$ such that if  $a,b >0$ and $p \in \mathbb{N}$ then
\begin{equation}
a^{p} - b^{p} \le C(p)(a-b)^{p} ?
\end{equation}

Hint: Suppose that $p>1$. Let $b$ be huge and let $a=b+1$. 
Added: One might argue that choosing $a-b=1$ is unfair. However, since the proposed inequality is homogeneous, it really makes no difference. For let $b=kx$ and $b=k(x+1)$, where perhaps $k$ is very large.  Then $a^p-b^p=k^p\left((x+1)^p-x^p\right)$, and $(a-b)^p=k^p$, so our inequality becomes
$$(x+1)^p -x^p \le C(p).$$
If $x$ is large enough, this cannot hold.
