  In this paper we study upper and lower bounds on the Bregman divergence
$\Delta_{\mathcal{F}}^{\xi}(y,x):=\mathcal{F}(y)-\mathcal{F}(x)-\langle \xi,
y-x\rangle $ for some convex functional $\mathcal{F}$ on a normed space
$\mathcal{X}$, with subgradient $\xi\in\partial\mathcal{F}(x)$. We give a
considerably simpler new proof of the inequalities by Xu and Roach for the
special case $\mathcal{F}(x)=\left\| x\right\|^p, p>1$. The results can be
transfered to more general functions as well.
